Mcqs On Fourier Series - Rtmnu Winter-2020
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DkdjndMCQs on Fourier Series | RTMNU Winter-2020
MCQs on Fourier Series
Ans : (b)
( | |
Ans : (b)
3)
d)
Ans : (c)
b) an even
d) trigonometric
Ans : (a)
b) an odd
d) logarithmic
Ans : (c)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
(
( (
b) 0
( (
c) d) None of these
Ans : (b)
7) ( {
(
a) 0 b) 1 c) 2 d) 4
Ans : (a)
8) ( {
( (
a) 0, b) 0, c) 0,0 d) None of these
Ans : (c)
9)
a) First and second quadrant b) Second and third quadrant
c) First and third quadrant d) Third and fourth quadrant
Ans : (a)
10) ( {
( (
a)
Ans : (d )
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
11) Fourier series of a function f(x) converges to f(x) if x is a point of
a) Continuity b) Discontinuity c) Both continuity and discontinuity d) None of
these
Ans : (b)
12) The Fourier series for f(x)=| | (
a) Sine terms b) Cosine terms c) Both sine and cosine terms d) None of
these
Ans : (b)
13) Any waveform can be expressed in Fourier series if
a) Sampling conditions are satisfied b) Dirichlet conditions are satisfied
c) Maxwells` conditions are satisfied d) Leibnitzs` conditions satisfied
Ans : (b)
14) In a Fourier series for f(x) = | | in (-
( (
a) 1 b) c) d) 0
Ans : (d )
15) For a function f(x) having Fourier expansion
∑ ∫ ( ∑ is
called
a) Dirichlets` identity b) Eulers` identity c) Parsevals` identity d) None of
these
Ans : (a)
16) The constant term in the Fourier series for function f(x) = - 2 in (-2, 2) is
a)
Ans : (d )
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MCQs on Fourier Series | RTMNU Winter-2020
17) Find the value of
f(x)=1 in
a) b) c) d)
Ans : (a)
18) What is the value of , if f(x) = in the interval 0 to 5
a) 25/2 b) 125/2 c) 625/2 d) 625/5
Ans : (b)
19) Find the sum of using Fourier series expansion if
( {
a) b) c) d)
Ans : (a)
20 W D ’
a) f(x) is periodic b) f(x) is single valued
c) f(x) has infinite number of discontinuities d) f(x) is infinite
Ans : (d )
21 W D ’
a) (1/2) [f( ) – f( ] b) (1/2) [f( ) + f( ]
c) [f( ) + f( ] d) [f( ) – f( ]
Ans : (b)
22) Find the value of
f(x)=x in the interval
a) b) c) d)
Ans : (d )
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
23) The graph of even function is symmetric about
a) origin b) x-axis c) y-axis d) none of these
Ans : (c)
24) The Fourier series expansion of f(x) = x in ( , ) is
a) ∑ b) ∑ (
c) ∑ ( d) none of these
Ans : (c)
25) The function f(x) = { is
a) Even function b) Containing cosine terms only in Fourier series
expansion
c) Both A and B d) None of these
Ans : (c)
26) In the Half Range Fourier Cosine series of f(x) = xsin x in(0 is equal
to _______
a)
Ans : (b)
27) ( | |
( (
a) 0, b) 0, c) 0,0 d) None of these
Ans : (c)
28) In the Half Range Fourier Sine series of f(x) = lx- in(0 is equal to
_______
a)
Ans : (c)
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MCQs on Fourier Series | RTMNU Winter-2020
Q29) f(x)= { is
a) an even b) an odd c) neither even nor odd d) trigonometric
Ans : (b)
30) The Fourier series expansion of an even function contains
a) only cosine terms b) cosine terms and a constant
c) only sine terms d) sine terms and a constant
Ans : (b)
31) What is the Fourier series expansion of the function f(x) in the interval (c, c+2 ?
a) ∑ ∑ b) ∑ ∑
c) ∑ ∑ d) ∑ ∑
Ans : (a)
32 W D ’ Fourier series expansion
a) f(x) is periodic b) f(x) is single valued
c) f(x) has finite number of discontinuities d) f(x) is finite.
Ans : (d )
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MCQs on Fourier Transform | RTMNU Winter-2020
MCQs on Fourier Tranform
| |
Q.1) The Fourier sine Transform of
A)√ B) √ C) √ D) √
Ans : ( A)
Q.2) Fourier transform of dirac delta function 𝜹 (t) is given as .
A) 0 B) 1 C) 2π(W) D) π𝜹(W)
Ans : ( B)
Q.3) if { } = ̅ (s), then F{ }=
A) ̅ ( ) B) ̅
(s) C) ̅
(a) D) (̅ )
Ans : ( D)
Q.4) if f(x) = x2 then Fourier integral of f(x) is same as
A)Fourier sine transforms B) Fourier cosine transform C) Fourier sine integral D) Fourier
cosine integral
Ans : ( D)
Q.5) Fourier transform of
A) B) C) D) √
Ans : ( B)
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MCQs on Fourier Transform | RTMNU Winter-2020
Q.6) Fourier cosine transform of f(x) = 1,0≤x≤ 1 is
A) B) C) s D) √
√ √
Ans : ( D)
1, x 1
Q.7) The Fourier integral of
0 x 1
A) Fourier cosine integral B) fourier sin integral C) both a and b D) None
Ans : ( A)
Q.8) The Fourier sine transform of is
√ B) √ C) √ D) 1
Ans : (C )
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MCQs on Fourier Transform | RTMNU Winter-2020
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MCQs on Fourier Transform | RTMNU Winter-2020
Q.9) If F[ ] F[ ] then F[ ]=?
A) B) C) D)
Ans : ( B)
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MCQs on Fourier Transform | RTMNU Winter-2020
Q.10) Fourier sine transform of is
A)√ B) √ C) √ D) √
Ans : ( B)
1, | x | 1
Q.11) Fourier sine transform of f ( x)
0, | x | 1
A)√ B) C) D) √
Ans : ( D)
Q.12) If Fc ( f ( x)) Fc (s) then of f (x) is
2 2
A)
Fc (s) cos(sx)ds
0
B)
F (s) cos(sx)dx
0
c
2 2
C)
Fc (s) cos(sx)ds
D)
F (s) cos(sx)dx
c
Ans : ( A)
Q.13) If f (x) is even function then Fourier transform of f (x) is
A) Fourier sine transform B) Fourier cosine transform
C) Fourier sine integral D) Fourier cosine integral
Ans : ( B)
Q.14) The Fourier transform of unit step function is given by
1 j
A) F ( j ) B) F ( j ) j C) F ( j ) D) F ( j )
j j
Ans : ( A)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.15) If the Fourier transform of f (t ) is F ( j ) the what is the Fourier transform of f (t )
A) F ( j ) B) F ( j ) C) F ( j ) D) None of these
Ans : ( D)
Q.16) The Fourier integral is useful for
A) Periodic Function B) Non-periodic Function
C) Logarithmic Function D) Discontinuous Function
Ans : ( D)
x sin mx
|x|
Q.17) If he Fourier transform e is √ then dx ?
0
1 x2
A) e m B) e 2 m C) 0 D) None of these
2
Ans : ( A)
cos xd
Q.18) The Fourier cosine integral of function e x the value of
0
1 2
A) ex B) ex C) ex D) ex
2 3 4 6
Ans : ( A)
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
MCQs on Lapalce Transform (Winter 2020)
Q.1) Find ,∫ -
A) B) C) D)
Ans : ( A)
Q. 2) L[ ] is
n 1 n n 1 n
A) n
B) n 1 C) n 1
D) n
s s s s
Ans : (C )
Q.3) Find the Laplace transform of
A) ( ) B) - ( ) C) ( ) C) ( )
Ans : ( B)
Q.4) If f (t) is periodic function with period T then L{ } is
A) ∫ f(u)du B) ∫ f(u)du
C) ∫ f(u)du D) ∫ f(u)du
Ans : ( A)
Q.5) , -n
A) B) C) D)
Ans : ( D)
Q.6) * + in positive when n is
A) Zero B) Negative integers C) Negative Rational D) Positive Integer
Ans : ( D)
Q.7) Laplace transform of 1 is
A) A) A) A)
Ans : ( A)
Q.8) If y(t) is then solution of -2 y = 4, y(0) = 1,then L[ ]=
A) -2 B) -2 C) -3 D) -2
Ans : ( B)
Q.9) [ ] is
A) t B) C) D)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Ans : ( B)
Q.10) Laplace transform of 3t is
A) B) C) D) [ ]
Ans : ( D)
Q.11) * +
A) sin(3t/2) B) 3sin(3t/2) C) sin (3t/2) D) sin (3t/2)
Ans : (C )
Q.12) { }
A) B) t sint C) D) t sint
Ans : (C )
Q.13) Laplace transform of ( )
A) B) [ ]
C) [ ]
D) [ ]
Ans : ( B)
Q.14) *, - +
A)[1-cos2(t-1)]u(t-1) B) [1-cos2(t-1)] C) [1-cos2t]u(t-1) D) [1-cos2(t+1)]u(t+1)
Ans : ( A)
Q.15) Laplace Transform of sinh 3t is
A) B) C) D)
Ans : ( D)
Q.16) Laplace Transform of 3t is
A) B) C) 1 D)
Ans : ( B)
Q.17) , - is
A) sint u(t) B) sin (t+π) C) sin(t+π) u(t+π) D) sin (t-π) u(t-π)
Ans : ( D)
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MCQs on Laplace Transform | Winter-2020
Q.18) Laplace Transform of u(t-a) is
A) B) C) D)
Ans : (C )
Q.19) Inverse Laplace transform of * + is
A) * + B) * + C) * + D) * +
Ans : (C )
Q.20) If L(f(t)) = √ then L ( f(t))
A√ B) √ C) √ D) √
Ans : ( B)
Q.21) If f(t) = then L [f(t)] is
A) B) * + C) * + D) None of these
Ans : ( B)
Q.22) ∫ * + =
A) B) -4 C) D)
Ans : (C )
Q.23) * + is
A) Sin(t-4)u(t-4) B) cos4(t-2)u(t-2) C) cos (t-2)u(t-2 ) D) sin4(t-2)u(t-2)
Ans : ( B)
Q.24) If L[f(t)] =F(s)and L[g(t)]= G(s) then [ ] is
A) ∫ B) ∫
C) ∫ D) None of these
Ans : ( A)
Q.25) Given L[ √ ] = ⁄
then L * +=
√
A) B) C) D) None of these
√ √
Ans : ( A)
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.26) L ,∫ -=
A) B) C) D)
Ans : (C )
Q.26) ∫ sin 2t dt =
A) L { } B) C), - D) All are Correct
Ans : ( D)
Q.27) * +
A) B) C) D) none of these
Ans : (C )
Q.28) If f(t) is periodic function , where f(t+2)=f(t) then
Laplace transform of f(t) is
A) ∫ B) ∫
C) ∫ D) ∫
Ans : ( A)
Q.29) ∫
A)* + C)* + D)
Ans : (C )
Q.30) , can be written in term of unit step function as
A) e-tH (t)-e-t H(t+3) B)
C) D)
Ans : (C )
Q.31) L* +=
A) cot-1s B) tan-1s C) -1
s D) none of these
Ans : ( A)
Q.32) * +
A) * ( )+ B) * ( )+ C)* ( )+ D)* ( )+
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Ans : ( A)
Q.32) The inverse L.T. of is
A) [ ] B) A) [ ] C) [ ] D) [ ]
Ans : ( D)
Q.33) L[ sin2(t- ]=
A) B) C) D)
Ans : (C )
Q.34) Find the Laplace transform of
A) * + B) * + C) * + D) * +
Ans : ( A)
Q.35) Find the Laplace transform of t sinhat
A) B) C) D)
Ans : ( D)
Q.36) L *∫ +=
A) ( ) B) ( ) C) ( ) D) None of these
Ans : ( A)
Q.37) The Laplace transform of f(t)= is
A) B) C) D)
Ans : ( B)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.38) Laplace transform of function ,where a and b are constant are given by
A) B) C) D)
Ans : ( D)
Q.39) 2t with y(0)=,Y’(0)=0 is
A) B)
D) D)
Ans : (C )
{ } { }
–
A) B) C) D) * +
Ans : ( A)
{ }
B) C) D)
Ans : ( A)
{ } , then L{ }
√
B) C) D)
√ √ √ √
Ans : (C )
[ ]
A) B) C) D)
Ans : (C )
Q.44) Laplace transform of ∫
A) ( ) [ ] ( ) B) ( ) [ ] ( )
C) ( ) [ ] ( ) D) None of these
Ans : ( B)
Q.45) * +
√ √ √ √ √
A) ( ) B) ( ) C) ( ) D) ( )
√ √
Ans : (C )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
+y=0,
A) [ ] [ ] B) [ ] [ ]
C) [ ] [ ] D) None of these
Ans : ( B)
{ }
A) B)Zero C) D)1
Ans : ( A)
A) B) D)
Ans : ( A)
, -=
A) Cosh8t B) cos8t C) sinh8t D) sin8t
Ans : ( B)
[ ] { } , -
A) ∫ B) ∫
C) ∫ D) ∫
Ans : (C )
{ } ̅ { }
A) ̅ B) ∫ ̅ C) ̅ ̅
Ans : (C )
{ } ̅ { }
A) ̅ B) ∫ ̅ C) ̅ ̅
Ans : ( D)
{ } ∫
A) B) C) D) 0
Ans : (C )
is
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
A) B) C)
Ans : ( A)
{ }
A)sin u(t) B)sin (t+ C) D
Ans : ( D)
{∫ }
A) B) ⁄ D)
Ans : ( A)
∫ √
√
A)1/t B)1/2 C)1/3 D)1/u
Ans : ( B)
[ { }]
( ) ( ) ( ) ( )
A) B) C) D)
Ans : ( B)
{ }
A) B) C) D)
Ans : ( A)
{ }
A) B) C) D)
Ans : ( D)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
9 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
10 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
MCQs on Matrix (Winter 2020)
Q.1) If if the characteristic equation of a matrix A then Eigen values of A
are______
( ) ( )
( ) ( )
Ans : ( D)
Q.2) Using Sylvester theorem the value of is for A [ ]
( )[ ] ( )[ ]
( )[ ] ( )[ ]
Ans : (C )
Q.3) Using Caley theorem , for the matrix A [ ] is calculate from
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Ans : ( B)
Q.4) Model matrix B corresponding to matrix A [ ] is________
( )[ ] ( )[ ]
( )[ ] ( )
Ans : (C )
Q.5) Examine the following system of vectors for linearly dependent:
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
Ans : ( A)
Q.6) Which of the following is condition for quadratic to canonical form
( ) ( )
( ) ( )
Ans : ( A)
Q.7) Inverse of orthogonal matrix is equal to
( ) its transpose ( ) its diagonal
( ) orthogonal ( )none
Ans : (C )
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.8) If the characteristic equation for the matrix A is 3 then the Eigen
value of the matrix A are
( ) ( )
( ) ( )
Ans : (C )
Q.9) Using Sylvester for matrix A [ ]
( ) [ ] ( ) [ ]
( ) [ ] ( ) [ ]
Ans : ( A)
Q.10) The characteristic equation for square matrix A is
( )| | ( )| |
( )| | ( )
Ans : ( A)
Q.11) On what value of the matrix ,A [ ] is an orthogonal
( ) ( )
√ √ √ √ √
( ) ( )
√ √ √ √ √ √
Ans : ( A)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.12) Investigate the vectors ( ) ( ) ( ) are
( ) Linearly dependent ( ) Linearly independent
( ) Orthogonal ( ) None of these
Ans : ( A)
Q.13) If 2,2,3 are Eigan value of a matrix [ ] then the determinant of A is
( ) ( ) ( ) ( )
Ans : ( D)
Q.14) Statement of Caley Hamilton theorem is
( ) Every square matrix satisfies its own characteristics equation
( ) The some of the Eigan value of matrix is sum of element of principle diagonal
( )The Eigan value of matrix A and its transpose are same.
( )None of these
Ans : ( A)
Q.15) If [ ] is a non-singular square matrix and is the Eigen value of [ ] then the Eigen value
of is ____
( ) ( ) ( ) ( )
Ans : (C )
Q.16) If A [ ] then the matrix represented by is equal to ___
( ) ( )
( ) ( )
Ans : ( B)
Q.17) The characteristic equation of matrix A [ ]is ______
( ) ( )
( ) ( )
Ans : ( A)
Q.18) If A [ ] then s t n __________
( )I ( ) ( ) ( )1
Ans : ( A)
Q.19) Which of the following statement is true?
A) Every matrix satisfies its own characteristics equation.
B) Every matrix satisfies characteristics equation.
C) Every square matrix satisfies its own characteristic equation.
D) Every square matrix satisfies characteristic equation.
Ans : (C )
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.19) Inverse of the matrix [ ]
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( A)
Q.20) If [ ] then what is
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : (C )
Q.21) [ ] and [ ] then is
A) [ ]
B) [ ]
C) [ ]
D) None of these
Ans : (C )
Q.22) Eigen values of the matrix [ ] are
A)
B)
C)
D)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Ans : ( A)
Q.23) If [ ] then the characteristic equation is
A)
B)
C)
D)
Ans : ( A)
Q.24) If [ ] then [ ]
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( D)
Q.25) The characteristics equation of the matrix A of order 3x3 is Using
Caley Hamilton theorem simplified form of expression
is
A) 5A+3I
B) -5A+3I
C) 5A-3I
D) None of these
Ans : (C )
Q.26) The relation between the vector ; ( ) ( ) ( ) is
A)
B)
C)
D)
Ans : ( A)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.27) For an orthogonal matrix [ ] is
√
A) [ ]
√
B) [ ]
√
C) [ ]
√
D)
Ans : (C )
Q.28) The characteristics equation of the matrix [ ] is
A)
B)
C)
D)
Ans : ( A)
Q.30) The set of vectors are said to be linearly dependent then there exists
scalars not all zeros, such that
A) True
B) False
C) Could be either
D) None
Ans : ( A)
Q.31) If are eigen values of a matrix [A] then
A) Determinant of A
B) Zero
C) Sum of the elements of principal diagonal
D) None
Ans : (C )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.32) If A is the given matrix and be the eigen values of [ ] then the
Canonical form is _________
A)
B)
C)
D)
Ans : ( B)
Q.33) If 2,0,3 are Eigen values of a matrix [A] then the inverse of a A is exists
A) True
B) False
C) Could either
D) None
Ans : ( B)
Q.34) The linearly dependency of vector ( ) ( ) ( )
A)
B)
C)
D) None
Ans : (C )
Q.35) The functional relation of the vector ( ) ( ) ( )
A)
B)
C)
D) None
Ans : (C )
Q.36) The Eigen values of [ ] are
A) -3,-3,5
B) 1,2,3
C) 2,2,8
D) 0,3,15
Ans : ( A)
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.37) What are the eigen values of the diagonal matrix [ ]
A) 0,1,2
B) -1,0,2
C) -1,1,0
D) -1,1,2
Ans : ( D)
Q.38) The characteristic equation of a matrix [ ] is
A)
B)
C)
D)
Ans : ( B)
Q.39) By Sylv st r’s th or m ( ) ∑ ( ) ( ) where ( )
[ ]
A)
[ ]
[ ]
B)
[ ]
[ ]
C)
[ ]
D) None
Ans : (C )
Q.40) The eigen values of [ ] are
A) 3,3,5
B) 1,2,3
C) 2,-2,8
D) 0,3,15
Ans : ( D)
Q.41) s non z ro n v tor o th n s lso n n v tor wh r s n
non z ro onst nt
) ru B) ls ) oul th r D) None
Ans : ( A)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) r n v u o m tr x [ ] th n
A) Determine of A B) Zero
C) Sum of the element of principle diagonal D) None
Ans : ( A)
) th v n m tr x n th n v lu s o n
x
y
[ ] th n th non l From is ________
z
A) B) x y z ) D) x y z
Ans : ( B)
) th v n s y th n =?
A) A=[ ] B) A=[ ] C) A=[ ] D) A=[ ]
Ans : ( A)
)Solv n th qu t on x y z x y z
y r m r s rul th v lu o x y and z are respectively
A)1,2,3 B)1,2,-2 C)1,2,2 D)1,-2,2
Ans : (C )
) s squ r m tr x o or r h v n l n rly n p n nt n tor th n non s n ul r m tr x
B can be found such that
A)Canonical form B)Diagonal form C) Quadratic form D)None
Ans : ( B)
) Sp tr l r us o th m tr x [ ] is
A)2 B) 4 C) -6 D) 5
Ans : ( D)
) v tor o m tr x [ ]
A)[ ] B)[ ] C)[ ] D)[ ]
Ans : ( B)
) th or m n us to n
A) nv rs o m tr x B) Power of matrix
C) Any power of matrix D) All of the above
Ans : ( D)
)The largest eigen value for the matrix [ ] is
A)3 B)1 C)4 D)2
Ans : ( A)
9 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)Coefficient matrix obtained from the quadratic form is
A) Real symmetric matrix
B) Skew symmetric matrix
C) Orthogonal matrix
D) Modal matrix
Ans : ( A)
)Eigen vectors of a real symmetric matrix A are orthogonal if the eigen values are ______
A Repeated B Non-Repeated
C Complex D None of these
Ans : ( B)
)Which of the following is linear polynomial of matrix A?
A) A2 + 5A + I B) A2 + I C) A + 5I D) None of these
Ans : (C )
)If Q is an orthogonal matrix of the orthogonal eigen vectors then:
(a) (b) (c) (d) None
Ans : (a)
)If 1,2,3 are the eigen values of A, then the eigen values of are:
(a) 4,-12,20 (b) -4,12,-20 (c) -4,-12,20 (d) -4,-12,-20
Ans : (d )
)If [ ] , then is
(a) [ ]
(b) [ ]
(c) [ ]
(d) [ ]
Ans : (c)
)If [ ] , then =?
(a) [ ]
(b) [ ]
(c) [ ]
(d) [ ]
Ans : (a)
10 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)The matrix [ ] satisfy which of the following equations:
(a)
(b)
(c)
(d)
Ans : (b)
)The eigen vector of the matrix [ ] for eigen value is
(a) ( )
(b) ( )
(c) ( )
(d) ( )
Ans : (a)
)If the transformation of , , is regular
then
(a) [ ] (b) [ ] (c) [ ] (d) [ ]
Ans : (a)
)The eigen values of differential equation , are:
(a) 1,5 (b) 3,5 (c) 8,9 (d) -1,-5
Ans : (a)
)The sum of the Eigen values of the matrix A = [ ]
a) 2 b) 1 c) -2 d) -1
Ans : (c)
) The correct set of Eigen Values of [ ]
a)(-3,4,2) b) (0,7) c) (1,4,7) d) (-3,0,1)
Ans : (a)
) If A= [ ]
a)A+3I+2 b) (A+I)(A+2I) = 0 c) d) 0
Ans : (b)
11 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) If Y=AX is an orthogonal linear transformation then the matrix A is
a)non singular b) orthogonal c)singular d)none of these
Ans : (b)
)On what value of scalars the vectors: ( ) ( ) ( ) are
linearly dependent.
a)-1,-1,1 b)1,2,3 c)1,-1,2 d)-1,-1,3
Ans : (a)
)The system of equations AX=B are consistent if
a)Rank of A Rank of augmented matrix b)Rank of A=Rank of augmented matrix
c)Rank of A d) Rank of A
Ans : (b)
) If is the characteristics equation for the matrix A then the eigen
values of A are
A)3,3,5 B) C)3,-3,5 D)
Ans : ( D)
) If [ ] th n non l orm o
A) B)
) )
Ans : ( A)
) Eigen values of the matrix A=[ ]
a)1 and 6 b)-1 and -6 c)1 and -6 d)-1 and 6
Ans : (b)
) The sum of the Eigen values of the matrix A = [ ]
a) 5 b) 7 c) 4 d) none of these
Ans : (b)
)If A=[ ]
a) b) c) d)
Ans : (c)
12 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)If A=[ ] then Eigen values of are
a)-1,-9,-4 b)1,9,4 c)-1,-3,2 d)1,3,-2
Ans : (b)
) If A=[ ] ( sum of minor of diagonal element)
a)12 b)24 c)6 d)36
Ans : (d )
) If [ ] is an Eigen vector of matrix [ ] then Eigen value corresponding to
the given Eigen vector is
A) 1
B) -1
C) 3
D) -3
Ans : (C )
) Which of the following pair of vectors are linearly dependent?
A) [ ] ,[ ]
B) [ ] [ ]
C) [ ] ,[ ]
D) [ ] ,[ ]
Ans : ( A)
) The trace & determinant of a 2*2 matrix is -2 & -35 respectively eigen values are
A) -30 & -5
B) -37 & -1
C) -7 & 5
D) 17.5 & -2
Ans : (c)
) If A is an orthogonal matrix then
A) is an orthogonal
B) is an orthogonal
C) ( )
D) All are correct
Ans : ( D)
13 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) If eigen values of A are 1,2,3 the eigen values of matrix are
A)
B)
C)
D) 1,8,27
Ans : ( D)
) If [ ] then what will be the value of y Sylv st r’s th or m
A)
B)
C)
D) 0
Ans : ( A)
) The eigen value of the following matrix are [ ]
A)
B)
C)
D)
Ans : ( A)
) If the characteristics equation of a matrix A is its diagonal
form is
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( D)
) Atleast one Eigen value of a singular matrix is
A) Positive
B) Negative
C) Zero
D) Imaginary
Ans : (C )
) If matrix [ ] is singular then is
14 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
A)2 B) C)0 D)
Ans : ( A)
) [ ] th n s
A) 4A-5I B) 4A+2I C) 4A+5I D) 4A-2I
Ans : (C )
) s ny n n m tr x s s l r th n | | | |wh r s
15 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
MCQs on Partial Differential Equation
1) The P.I. of ( √ is
(a)
(b)
(c)
(d)
Ans : (a)
2) The C.F. of partial differential equation =√ is
(a)
(b)
(c)
(d)
Ans : (b)
3) A first order partial differential equation (
(a) Linear equation
(b) Semi- linear equation
(c) Quasi-linear equation
(d) Non-linear equation
Ans : (a)
4) What is the C.F. of =0
(a)
(b) (y+2x) + (y-2x)
(c) (y-x) + (y+2x)
(d) (y+2x)+ (y-x)
Ans : (b)
5) is the homogeneous equation of degree
(a) 2
(b) 3
(c) 1
(d) 0
Ans : (c)
6) If -2 and -3 are two roots of partial differential equation, then its C.F. is
(a)
(b)
(c)
(d)
Ans : (c)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(7) The C.F. of partial differential equation
(a)
(b) )
(c)
(d)
Ans : (a)
(8) If 2,2,2 are three roots of partial differential equation, then its C.F. is
(a)
(b)
(c)
(d)
Ans : (b)
(8) The C.F. of partial differential equation is
(a)
(b)
(c)
(d)
Ans : (a)
9) The C.F. of partial differential equation is
(a)
(b)
(c)
(d)
Ans : (c)
10) The P.I. of partial differential equation is
(a) s
(b)
(c)
(d)
Ans : (d )
11) A first order partial differential equation
(a) Linear equation
(b) semi- linear equation
(c) Quasi-linear equation
(d) Non-linear equation
Ans : (a)
12) The C.F. of partial differential equations is
(a)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(b)
(c)
(d)
Ans : (a)
13) If 2,3,3 are three roots of the partial differential equation then its C.F. is
(a)
(b)
(c)
(d)
Ans : (b)
14) Which of the following is homogeneous equation of degree 2
(a)
(b)
(c)
(d)
Ans : (b)
15) One of the solutions of is
(a)
(b)
(c)
(d) None of the above
Ans : (a)
16) Solutions of partial differential equation by separation of
variables is given by:
(a) u(x,y)=
(b) u(x,y)=
(c) u(x,y) =
(d) u(x,y)=
Ans : (b)
17) The C.F. of is
(a)
(b)
(c)
(d)
Ans : (d )
18) The CF of is
(a)
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(b)
(c)
(d)
Ans : (b)
19) Solutions of partial differential equation by separation of variables is
given by:
( )
(a)
( )
(b)
(c)
(d)
Ans : (a)
20) The C.F. of is:
(a)
(b)
(c)
(d)
Ans : (d )
21) P.I. of is:
(a)
(b)
(c)
(d)
Ans : (c)
22) Solutions of partial differential equation by separation of variables is given
by:
(a)
(b)
(c)
(d)
Ans : (a)
23) When solving one-dimensional heat equation using a variable separable method we
get the solution if:
(a) k is positive
(b) k is negative
(c) k is zero
(d) It can be anything
Ans : (b)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
24) One solution of is
(a)
(b)
(c)
(d)
Ans : (a)
25) What is another name for heat equation:
(a) Induction equation (b) Condenser Equation
(c) Diffusion equation (d) Solar equation
Ans : (c)
26) The solution of first order partial differential equation is given by:
(a)
(b)
(c)
(d) All of the above.
Ans : (d )
27) While solving a partial differential equation using a variable separable method, we
get the ratio to a constant, which:
(a) can be +ve/-ve integer or zero
(b) can be +ve/-ve rational or zero
(c) must be a +ve integer
(d) must be a -ve integer
Ans : (b)
28) One of the solutions of is given by:
(a) (b)
(c) (d) None of these
Ans : (b)
29) The C.F. of is:
(a)
(b)
(c)
(d)
Ans : (d )
30) P.I. of is:
(a)
(b)
(c)
(d)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
Ans : (a)
31) Partial differential equation of one-dimensional heat equation is :
(a)
(b)
(c)
(d)
Ans : (a)
32) P.I. of is:
(a)
(b)
(c)
(d)
Ans : (d )
33) P.I. of is
(a)
(b)
(c)
(d)
Ans : (b)
34) For a partial differential equation in a function and two variables ; what
is the form obtained after separation of variables is applied:
(a)
(b)
(c)
(d)
Ans : (d )
35) The solution of partial differential equation by separation of
variables is given by:
(a)
(b)
(c)
(d)
Ans : (d )
36) One of the solution of is:
(a)
(b)
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(c)
(d) None of the above
Ans : (c)
37) If 1,1,4 are the three roots of the partial differential equation, then its C.F. is :
(a)
(b)
(c)
(d)
Ans : (a)
38) The C.F. of partial differential equation is:
(a)
(b)
(c)
(d)
Ans : (d )
39) If 2,3,4 are three roots of partial differential equation then its C.F. is:
(a)
(b)
(c)
(d)
Ans : (c)
40) --------------- s called Lagra ge’s equat o :
(a) (b)
(c) (d)
Ans : (a)
41) One of the solutions of is:
(a)
(b)
(c)
(d) None of the above
Ans : (a)
42) P.I. of partial differential equation is:
(a)
(b)
(c)
(d)
Ans : (c)
43) The first order partial differential equation is:
(a)Linear Equation
(b) Semi Linear Equation
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(c) Quasi liner equation
(d) Non-Linear Equation
Ans : (d )
44) The P.I. of ta is:
(a) ta
(b) ta
(c) ta
(d) ta
Ans : (d )
(45) What is the C.F. of
(a)
(b)
(c)
(d)
Ans : (b)
(45) What is the degree of homogenous partial differential equation
(a)
(b)
(c)
(d)
Ans : (b)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
MCQs on Lapalce Transform (Winter 2020)
Q.1) Find ,∫ -
A) B) C) D)
Ans : ( A)
Q. 2) L[ ] is
n 1 n n 1 n
A) n
B) n 1 C) n 1
D) n
s s s s
Ans : (C )
Q.3) Find the Laplace transform of
A) ( ) B) - ( ) C) ( ) C) ( )
Ans : ( B)
Q.4) If f (t) is periodic function with period T then L{ } is
A) ∫ f(u)du B) ∫ f(u)du
C) ∫ f(u)du D) ∫ f(u)du
Ans : ( A)
Q.5) , -n
A) B) C) D)
Ans : ( D)
Q.6) * + in positive when n is
A) Zero B) Negative integers C) Negative Rational D) Positive Integer
Ans : ( D)
Q.7) Laplace transform of 1 is
A) A) A) A)
Ans : ( A)
Q.8) If y(t) is then solution of -2 y = 4, y(0) = 1,then L[ ]=
A) -2 B) -2 C) -3 D) -2
Ans : ( B)
Q.9) [ ] is
A) t B) C) D)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Ans : ( B)
Q.10) Laplace transform of 3t is
A) B) C) D) [ ]
Ans : ( D)
Q.11) * +
A) sin(3t/2) B) 3sin(3t/2) C) sin (3t/2) D) sin (3t/2)
Ans : (C )
Q.12) { }
A) B) t sint C) D) t sint
Ans : (C )
Q.13) Laplace transform of ( )
A) B) [ ]
C) [ ]
D) [ ]
Ans : ( B)
Q.14) *, - +
A)[1-cos2(t-1)]u(t-1) B) [1-cos2(t-1)] C) [1-cos2t]u(t-1) D) [1-cos2(t+1)]u(t+1)
Ans : ( A)
Q.15) Laplace Transform of sinh 3t is
A) B) C) D)
Ans : ( D)
Q.16) Laplace Transform of 3t is
A) B) C) 1 D)
Ans : ( B)
Q.17) , - is
A) sint u(t) B) sin (t+π) C) sin(t+π) u(t+π) D) sin (t-π) u(t-π)
Ans : ( D)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.18) Laplace Transform of u(t-a) is
A) B) C) D)
Ans : (C )
Q.19) Inverse Laplace transform of * + is
A) * + B) * + C) * + D) * +
Ans : (C )
Q.20) If L(f(t)) = √ then L ( f(t))
A√ B) √ C) √ D) √
Ans : ( B)
Q.21) If f(t) = then L [f(t)] is
A) B) * + C) * + D) None of these
Ans : ( B)
Q.22) ∫ * + =
A) B) -4 C) D)
Ans : (C )
Q.23) * + is
A) Sin(t-4)u(t-4) B) cos4(t-2)u(t-2) C) cos (t-2)u(t-2 ) D) sin4(t-2)u(t-2)
Ans : ( B)
Q.24) If L[f(t)] =F(s)and L[g(t)]= G(s) then [ ] is
A) ∫ B) ∫
C) ∫ D) None of these
Ans : ( A)
Q.25) Given L[ √ ] = ⁄
then L * +=
√
A) B) C) D) None of these
√ √
Ans : ( A)
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.26) L ,∫ -=
A) B) C) D)
Ans : (C )
Q.26) ∫ sin 2t dt =
A) L { } B) C), - D) All are Correct
Ans : ( D)
Q.27) * +
A) B) C) D) none of these
Ans : (C )
Q.28) If f(t) is periodic function , where f(t+2)=f(t) then
Laplace transform of f(t) is
A) ∫ B) ∫
C) ∫ D) ∫
Ans : ( A)
Q.29) ∫
A)* + C)* + D)
Ans : (C )
Q.30) , can be written in term of unit step function as
A) e-tH (t)-e-t H(t+3) B)
C) D)
Ans : (C )
Q.31) L* +=
A) cot-1s B) tan-1s C) -1
s D) none of these
Ans : ( A)
Q.32) * +
A) * ( )+ B) * ( )+ C)* ( )+ D)* ( )+
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Ans : ( A)
Q.32) The inverse L.T. of is
A) [ ] B) A) [ ] C) [ ] D) [ ]
Ans : ( D)
Q.33) L[ sin2(t- ]=
A) B) C) D)
Ans : (C )
Q.34) Find the Laplace transform of
A) * + B) * + C) * + D) * +
Ans : ( A)
Q.35) Find the Laplace transform of t sinhat
A) B) C) D)
Ans : ( D)
Q.36) L *∫ +=
A) ( ) B) ( ) C) ( ) D) None of these
Ans : ( A)
Q.37) The Laplace transform of f(t)= is
A) B) C) D)
Ans : ( B)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.38) Laplace transform of function ,where a and b are constant are given by
A) B) C) D)
Ans : ( D)
Q.39) 2t with y(0)=,Y’(0)=0 is
A) B)
D) D)
Ans : (C )
{ } { }
–
A) B) C) D) * +
Ans : ( A)
{ }
B) C) D)
Ans : ( A)
{ } , then L{ }
√
B) C) D)
√ √ √ √
Ans : (C )
[ ]
A) B) C) D)
Ans : (C )
Q.44) Laplace transform of ∫
A) ( ) [ ] ( ) B) ( ) [ ] ( )
C) ( ) [ ] ( ) D) None of these
Ans : ( B)
Q.45) * +
√ √ √ √ √
A) ( ) B) ( ) C) ( ) D) ( )
√ √
Ans : (C )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
+y=0,
A) [ ] [ ] B) [ ] [ ]
C) [ ] [ ] D) None of these
Ans : ( B)
{ }
A) B)Zero C) D)1
Ans : ( A)
A) B) D)
Ans : ( A)
, -=
A) Cosh8t B) cos8t C) sinh8t D) sin8t
Ans : ( B)
[ ] { } , -
A) ∫ B) ∫
C) ∫ D) ∫
Ans : (C )
{ } ̅ { }
A) ̅ B) ∫ ̅ C) ̅ ̅
Ans : (C )
{ } ̅ { }
A) ̅ B) ∫ ̅ C) ̅ ̅
Ans : ( D)
{ } ∫
A) B) C) D) 0
Ans : (C )
is
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
A) B) C)
Ans : ( A)
{ }
A)sin u(t) B)sin (t+ C) D
Ans : ( D)
{∫ }
A) B) ⁄ D)
Ans : ( A)
∫ √
√
A)1/t B)1/2 C)1/3 D)1/u
Ans : ( B)
[ { }]
( ) ( ) ( ) ( )
A) B) C) D)
Ans : ( B)
{ }
A) B) C) D)
Ans : ( A)
{ }
A) B) C) D)
Ans : ( D)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
9 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
10 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
MCQs on Fourier Tranform
| |
Q.1) The Fourier sine Transform of
A)√ B) √ C) √ D) √
Ans : ( A)
Q.2) Fourier transform of dirac delta function 𝜹 (t) is given as .
A) 0 B) 1 C) 2π(W) D) π𝜹(W)
Ans : ( B)
Q.3) if { } = ̅ (s), then F{ }=
A) ̅ ( ) B) ̅
(s) C) ̅
(a) D) (̅ )
Ans : ( D)
Q.4) if f(x) = x2 then Fourier integral of f(x) is same as
A)Fourier sine transforms B) Fourier cosine transform C) Fourier sine integral D) Fourier
cosine integral
Ans : ( D)
Q.5) Fourier transform of
A) B) C) D) √
Ans : ( B)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.6) Fourier cosine transform of f(x) = 1,0≤x≤ 1 is
A) B) C) s D) √
√ √
Ans : ( D)
1, x 1
Q.7) The Fourier integral of
0 x 1
A) Fourier cosine integral B) fourier sin integral C) both a and b D) None
Ans : ( A)
Q.8) The Fourier sine transform of is
√ B) √ C) √ D) 1
Ans : (C )
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.9) If F[ ] F[ ] then F[ ]=?
A) B) C) D)
Ans : ( B)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.10) Fourier sine transform of is
A)√ B) √ C) √ D) √
Ans : ( B)
1, | x | 1
Q.11) Fourier sine transform of f ( x)
0, | x | 1
A)√ B) C) D) √
Ans : ( D)
Q.12) If Fc ( f ( x)) Fc (s) then of f (x) is
2 2
A)
Fc (s) cos(sx)ds
0
B)
F (s) cos(sx)dx
0
c
2 2
C)
Fc (s) cos(sx)ds
D)
F (s) cos(sx)dx
c
Ans : ( A)
Q.13) If f (x) is even function then Fourier transform of f (x) is
A) Fourier sine transform B) Fourier cosine transform
C) Fourier sine integral D) Fourier cosine integral
Ans : ( B)
Q.14) The Fourier transform of unit step function is given by
1 j
A) F ( j ) B) F ( j ) j C) F ( j ) D) F ( j )
j j
Ans : ( A)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.15) If the Fourier transform of f (t ) is F ( j ) the what is the Fourier transform of f (t )
A) F ( j ) B) F ( j ) C) F ( j ) D) None of these
Ans : ( D)
Q.16) The Fourier integral is useful for
A) Periodic Function B) Non-periodic Function
C) Logarithmic Function D) Discontinuous Function
Ans : ( D)
x sin mx
|x|
Q.17) If he Fourier transform e is √ then dx ?
0
1 x2
A) e m B) e 2 m C) 0 D) None of these
2
Ans : ( A)
cos xd
Q.18) The Fourier cosine integral of function e x the value of
0
1 2
A) ex B) ex C) ex D) ex
2 3 4 6
Ans : ( A)
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
MCQs on Matrix (Winter 2020)
Q.1) If if the characteristic equation of a matrix A then Eigen values of A
are______
( ) ( )
( ) ( )
Ans : ( D)
Q.2) Using Sylvester theorem the value of is for A [ ]
( )[ ] ( )[ ]
( )[ ] ( )[ ]
Ans : (C )
Q.3) Using Caley theorem , for the matrix A [ ] is calculate from
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Ans : ( B)
Q.4) Model matrix B corresponding to matrix A [ ] is________
( )[ ] ( )[ ]
( )[ ] ( )
Ans : (C )
Q.5) Examine the following system of vectors for linearly dependent:
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
Ans : ( A)
Q.6) Which of the following is condition for quadratic to canonical form
( ) ( )
( ) ( )
Ans : ( A)
Q.7) Inverse of orthogonal matrix is equal to
( ) its transpose ( ) its diagonal
( ) orthogonal ( )none
Ans : (C )
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.8) If the characteristic equation for the matrix A is 3 then the Eigen
value of the matrix A are
( ) ( )
( ) ( )
Ans : (C )
Q.9) Using Sylvester for matrix A [ ]
( ) [ ] ( ) [ ]
( ) [ ] ( ) [ ]
Ans : ( A)
Q.10) The characteristic equation for square matrix A is
( )| | ( )| |
( )| | ( )
Ans : ( A)
Q.11) On what value of the matrix ,A [ ] is an orthogonal
( ) ( )
√ √ √ √ √
( ) ( )
√ √ √ √ √ √
Ans : ( A)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.12) Investigate the vectors ( ) ( ) ( ) are
( ) Linearly dependent ( ) Linearly independent
( ) Orthogonal ( ) None of these
Ans : ( A)
Q.13) If 2,2,3 are Eigan value of a matrix [ ] then the determinant of A is
( ) ( ) ( ) ( )
Ans : ( D)
Q.14) Statement of Caley Hamilton theorem is
( ) Every square matrix satisfies its own characteristics equation
( ) The some of the Eigan value of matrix is sum of element of principle diagonal
( )The Eigan value of matrix A and its transpose are same.
( )None of these
Ans : ( A)
Q.15) If [ ] is a non-singular square matrix and is the Eigen value of [ ] then the Eigen value
of is ____
( ) ( ) ( ) ( )
Ans : (C )
Q.16) If A [ ] then the matrix represented by is equal to ___
( ) ( )
( ) ( )
Ans : ( B)
Q.17) The characteristic equation of matrix A [ ]is ______
( ) ( )
( ) ( )
Ans : ( A)
Q.18) If A [ ] then s t n __________
( )I ( ) ( ) ( )1
Ans : ( A)
Q.19) Which of the following statement is true?
A) Every matrix satisfies its own characteristics equation.
B) Every matrix satisfies characteristics equation.
C) Every square matrix satisfies its own characteristic equation.
D) Every square matrix satisfies characteristic equation.
Ans : (C )
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.19) Inverse of the matrix [ ]
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( A)
Q.20) If [ ] then what is
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : (C )
Q.21) [ ] and [ ] then is
A) [ ]
B) [ ]
C) [ ]
D) None of these
Ans : (C )
Q.22) Eigen values of the matrix [ ] are
A)
B)
C)
D)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Ans : ( A)
Q.23) If [ ] then the characteristic equation is
A)
B)
C)
D)
Ans : ( A)
Q.24) If [ ] then [ ]
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( D)
Q.25) The characteristics equation of the matrix A of order 3x3 is Using
Caley Hamilton theorem simplified form of expression
is
A) 5A+3I
B) -5A+3I
C) 5A-3I
D) None of these
Ans : (C )
Q.26) The relation between the vector ; ( ) ( ) ( ) is
A)
B)
C)
D)
Ans : ( A)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.27) For an orthogonal matrix [ ] is
√
A) [ ]
√
B) [ ]
√
C) [ ]
√
D)
Ans : (C )
Q.28) The characteristics equation of the matrix [ ] is
A)
B)
C)
D)
Ans : ( A)
Q.30) The set of vectors are said to be linearly dependent then there exists
scalars not all zeros, such that
A) True
B) False
C) Could be either
D) None
Ans : ( A)
Q.31) If are eigen values of a matrix [A] then
A) Determinant of A
B) Zero
C) Sum of the elements of principal diagonal
D) None
Ans : (C )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.32) If A is the given matrix and be the eigen values of [ ] then the
Canonical form is _________
A)
B)
C)
D)
Ans : ( B)
Q.33) If 2,0,3 are Eigen values of a matrix [A] then the inverse of a A is exists
A) True
B) False
C) Could either
D) None
Ans : ( B)
Q.34) The linearly dependency of vector ( ) ( ) ( )
A)
B)
C)
D) None
Ans : (C )
Q.35) The functional relation of the vector ( ) ( ) ( )
A)
B)
C)
D) None
Ans : (C )
Q.36) The Eigen values of [ ] are
A) -3,-3,5
B) 1,2,3
C) 2,2,8
D) 0,3,15
Ans : ( A)
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.37) What are the eigen values of the diagonal matrix [ ]
A) 0,1,2
B) -1,0,2
C) -1,1,0
D) -1,1,2
Ans : ( D)
Q.38) The characteristic equation of a matrix [ ] is
A)
B)
C)
D)
Ans : ( B)
Q.39) By Sylv st r’s th or m ( ) ∑ ( ) ( ) where ( )
[ ]
A)
[ ]
[ ]
B)
[ ]
[ ]
C)
[ ]
D) None
Ans : (C )
Q.40) The eigen values of [ ] are
A) 3,3,5
B) 1,2,3
C) 2,-2,8
D) 0,3,15
Ans : ( D)
Q.41) s non z ro n v tor o th n s lso n n v tor wh r s n
non z ro onst nt
) ru B) ls ) oul th r D) None
Ans : ( A)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) r n v u o m tr x [ ] th n
A) Determine of A B) Zero
C) Sum of the element of principle diagonal D) None
Ans : ( A)
) th v n m tr x n th n v lu s o n
x
y
[ ] th n th non l From is ________
z
A) B) x y z ) D) x y z
Ans : ( B)
) th v n s y th n =?
A) A=[ ] B) A=[ ] C) A=[ ] D) A=[ ]
Ans : ( A)
)Solv n th qu t on x y z x y z
y r m r s rul th v lu o x y and z are respectively
A)1,2,3 B)1,2,-2 C)1,2,2 D)1,-2,2
Ans : (C )
) s squ r m tr x o or r h v n l n rly n p n nt n tor th n non s n ul r m tr x
B can be found such that
A)Canonical form B)Diagonal form C) Quadratic form D)None
Ans : ( B)
) Sp tr l r us o th m tr x [ ] is
A)2 B) 4 C) -6 D) 5
Ans : ( D)
) v tor o m tr x [ ]
A)[ ] B)[ ] C)[ ] D)[ ]
Ans : ( B)
) th or m n us to n
A) nv rs o m tr x B) Power of matrix
C) Any power of matrix D) All of the above
Ans : ( D)
)The largest eigen value for the matrix [ ] is
A)3 B)1 C)4 D)2
Ans : ( A)
9 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)Coefficient matrix obtained from the quadratic form is
A) Real symmetric matrix
B) Skew symmetric matrix
C) Orthogonal matrix
D) Modal matrix
Ans : ( A)
)Eigen vectors of a real symmetric matrix A are orthogonal if the eigen values are ______
A Repeated B Non-Repeated
C Complex D None of these
Ans : ( B)
)Which of the following is linear polynomial of matrix A?
A) A2 + 5A + I B) A2 + I C) A + 5I D) None of these
Ans : (C )
)If Q is an orthogonal matrix of the orthogonal eigen vectors then:
(a) (b) (c) (d) None
Ans : (a)
)If 1,2,3 are the eigen values of A, then the eigen values of are:
(a) 4,-12,20 (b) -4,12,-20 (c) -4,-12,20 (d) -4,-12,-20
Ans : (d )
)If [ ] , then is
(a) [ ]
(b) [ ]
(c) [ ]
(d) [ ]
Ans : (c)
)If [ ] , then =?
(a) [ ]
(b) [ ]
(c) [ ]
(d) [ ]
Ans : (a)
10 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)The matrix [ ] satisfy which of the following equations:
(a)
(b)
(c)
(d)
Ans : (b)
)The eigen vector of the matrix [ ] for eigen value is
(a) ( )
(b) ( )
(c) ( )
(d) ( )
Ans : (a)
)If the transformation of , , is regular
then
(a) [ ] (b) [ ] (c) [ ] (d) [ ]
Ans : (a)
)The eigen values of differential equation , are:
(a) 1,5 (b) 3,5 (c) 8,9 (d) -1,-5
Ans : (a)
)The sum of the Eigen values of the matrix A = [ ]
a) 2 b) 1 c) -2 d) -1
Ans : (c)
) The correct set of Eigen Values of [ ]
a)(-3,4,2) b) (0,7) c) (1,4,7) d) (-3,0,1)
Ans : (a)
) If A= [ ]
a)A+3I+2 b) (A+I)(A+2I) = 0 c) d) 0
Ans : (b)
11 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) If Y=AX is an orthogonal linear transformation then the matrix A is
a)non singular b) orthogonal c)singular d)none of these
Ans : (b)
)On what value of scalars the vectors: ( ) ( ) ( ) are
linearly dependent.
a)-1,-1,1 b)1,2,3 c)1,-1,2 d)-1,-1,3
Ans : (a)
)The system of equations AX=B are consistent if
a)Rank of A Rank of augmented matrix b)Rank of A=Rank of augmented matrix
c)Rank of A d) Rank of A
Ans : (b)
) If is the characteristics equation for the matrix A then the eigen
values of A are
A)3,3,5 B) C)3,-3,5 D)
Ans : ( D)
) If [ ] th n non l orm o
A) B)
) )
Ans : ( A)
) Eigen values of the matrix A=[ ]
a)1 and 6 b)-1 and -6 c)1 and -6 d)-1 and 6
Ans : (b)
) The sum of the Eigen values of the matrix A = [ ]
a) 5 b) 7 c) 4 d) none of these
Ans : (b)
)If A=[ ]
a) b) c) d)
Ans : (c)
12 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)If A=[ ] then Eigen values of are
a)-1,-9,-4 b)1,9,4 c)-1,-3,2 d)1,3,-2
Ans : (b)
) If A=[ ] ( sum of minor of diagonal element)
a)12 b)24 c)6 d)36
Ans : (d )
) If [ ] is an Eigen vector of matrix [ ] then Eigen value corresponding to
the given Eigen vector is
A) 1
B) -1
C) 3
D) -3
Ans : (C )
) Which of the following pair of vectors are linearly dependent?
A) [ ] ,[ ]
B) [ ] [ ]
C) [ ] ,[ ]
D) [ ] ,[ ]
Ans : ( A)
) The trace & determinant of a 2*2 matrix is -2 & -35 respectively eigen values are
A) -30 & -5
B) -37 & -1
C) -7 & 5
D) 17.5 & -2
Ans : (c)
) If A is an orthogonal matrix then
A) is an orthogonal
B) is an orthogonal
C) ( )
D) All are correct
Ans : ( D)
13 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) If eigen values of A are 1,2,3 the eigen values of matrix are
A)
B)
C)
D) 1,8,27
Ans : ( D)
) If [ ] then what will be the value of y Sylv st r’s th or m
A)
B)
C)
D) 0
Ans : ( A)
) The eigen value of the following matrix are [ ]
A)
B)
C)
D)
Ans : ( A)
) If the characteristics equation of a matrix A is its diagonal
form is
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( D)
) Atleast one Eigen value of a singular matrix is
A) Positive
B) Negative
C) Zero
D) Imaginary
Ans : (C )
) If matrix [ ] is singular then is
14 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
A)2 B) C)0 D)
Ans : ( A)
) [ ] th n s
A) 4A-5I B) 4A+2I C) 4A+5I D) 4A-2I
Ans : (C )
) s ny n n m tr x s s l r th n | | | |wh r s
15 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
MCQs on Z-Transform
Q1. The Z transform of is
( )
A)
( )
B)
( )
C)
( )
D)
Ans.(C )
Q2. One sided Z transform for the sequence ( ) is given by
A) ∑ ( )
B) ∑ ( )
C) ∑ ( )
D) ∑ ( )
Ans.(B)
Q3. Inverse Z transform of ( )
A)
B)
C)
D)
Ans.(C )
Q4. Z* ( )+
A) ( ) ( )
B) ( )
C) ( )
D) None of these
Ans.(D)
Q5. 0 1 is
A)
B) ( )
C) ( )
D)
Ans.(C )
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
( ) ( )
Q6. If * ( )+ ( ) then 2 3 ∫ hence 2 3
A) . /
B) . /
C) . /
D) . /
Ans.(A)
Q7. 2. / 3
A)
B)
C)
D)
Ans.(A)
Q8. The z-transform of a sequence x(n) which is given as ( ) ∑ ( ) is known
as
A) Uni-lateral Z-transform
B) Bi-lateral Z-transform
C) Tri-lateral Z-transform
D) None of these
Ans.(B)
Q9. Z transform of the sequence * +
A) + + + ……
B) + + + +……
C) ∑
D) All are correct
Ans.(D)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
Q10. For the difference equation + + with . Then Y(z) takes the
value
A) ( ) ( )
B) ( ) ( )
C) ( ) ( )
D) ( )( )
Ans.(A)
( )
Q11. If * ( )+ ( ) then ∫
A) * ( )+
( )
B) 2 3
( )
C) 2 3
( )
D) 2 3
Ans.(B)
Q12. The Z-transform of the unit step function ( ) is
A)
B)
C) ( )
( )
D)
Ans.(B)
( )
Q13. 2( )
3
A)
B)
C)
D)
Ans.(D)
Q14. 2( )( )
3
A)
. /
B) [ ]
. /
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
C)
D) Both A and B
Ans.(D)
Q15. 2 3
A)
B)
√
C)
√ ( √ )
D)
√ ( √ )
Ans.(A)
Q16. If * + then * +
A)
B) ( )
C) ( )
D) None of these
Ans.(B)
Q17. 2 3
A)
B)
C)
D)
Ans.(C )
Q18. Inverse Z transform of the function ( )( )
A) ( )
B) ( )
C)
D) ( )
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
Ans.(B)
Q19. If * ( )+ ( ) then * ( )+
A) ( )
B) ( )
C) ( )
D) * ( )+
Ans.(B)
Q20. If ( ) then
A)
B)
C)
D)
Ans.(C )
Q21. The Z-transform of the time function ( ) is
A)
B)
C) ( )
( )
D)
Ans.(B)
Q22. What is the set of all values of z for which X(z) attains a finite value?
A) Radius of divergence
B) Radius of convergence
C) Feasible solution
D) None of these
Ans.(B)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
Q23. Z transform of
A) ( )
B)
C)
D)
Ans.(C )
Q24. * ( )+
A)
B)
C)
D)
Ans.(A)
Q25. The inverse Z-transform of the ,| | | |
A) 2( )
B)
C)
D)
Ans.(A)
Q26. If * ( )+ ( ), * + * + then what is the value of * +
A) ( )
B) ( )
C)
D) ( )
Ans.(A)
Q27. The Z transform of is
A) ( )
B)
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
C) ( )
D) ( )
Ans.(D)
Q28. * +
A)
B)
C)
D)
Ans.(A)
Q29. The value of 2 3
A) ( )
B) ( )
C)
D)
Ans.(C )
Q30. The value of * +
A)
B)
C)
D)
Ans.(B)
Q31. The region of convergence of the z transform of a
A) | |
B) | |
C) (real part of z)>0
D) (Real part of z)<0
Ans.(C )
Q32. * +
A)
B)
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
C)
D)
Ans.(A)
Q33. * +
A) ( )
B)
C)
D)
Ans.(A)
Q34. If * ( )+ ( ) then * ( ) ( )+
A) ( )
B) ( )
C) ( )
D) ( )
Ans.(A)
Q35. The inverse Z transform of ( )
is
A)
B)
C)
D)
Ans.(A)
Q36. The inverse Z transform of ( )
is
A)
B)
C)
D)
Ans.(C )
Q37)Residue at pole of order is
A) ,( )
*( ) ( )+-
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Z-Transform | RTMNU Winter-2020
B) ,( )
*( ) ( )+-
C) ,( )
*( ) ( )+-
D) ,( )
*( ) ( )+-
Ans : ( A)
9 Mathematics-Ganit Sangrah | Satish Tiwari
M – 3 CSE Unit – 4
STATISTICAL TECHNIQUES
1. For two correlated variables x and y, if coefficient of correlation between x and y is 0.8014 ,
variance of x and y are 16 and 25 respectively. Then the covariance between x and y is
A) 162.08 B) 16.028 C) 160.28 D) 16.208
2. A curve of a symmetrical distribution is known as
A) Normal Curve B) Cusp C) Negative Curve D) Positive Curve
3. If the values of two correlated variables move in the opposite direction,
i.e. one increasing and other decreasing _________
A) The correlation is said to be linear
B) The correlation is said to be non-linear
C) The correlation is said to be positive
D) The correlation is said to be negative
4. A process by which we estimate the value of dependent variable on the basis
of one or more independent variables is called
A) Correlation B) Regression C) Residual D) None of these
5. The median of the data given below is 31 , 37 , 43 , 42 , 25 , 46 , 45 , 39 , 32
A) 39 B) 42 C) 32 D) 43
6. Which of the following divides a group of data into four equal parts ?
A) Deciles B) Percentiles C) Quartiles D) Standard Deviation
7. The mean deviation for the data 5 , 3 , 7 , 8 , 4 , 9 from mean is
A) 2 B) 3 C) 4 D) 7
8. The first quartile Q 1 for the data 20 , 30 , 25 , 23 , 22 , 32 , 36 is
A) 20 B) 22 C) 25 D) 32
9. The mode from the following data is
Age 0-6 6-12 12-18 18-24 24-30 30-36 36-42
Frequency 6 11 25 35 18 12 6
A) 22.20 B) 2.20 C) 20.22 D) 21.21
10. The mode of the data 0 , 1 , 6 , 7, 2 , 3 , 7 , 6 , 6 , 2 , 6 , 0 , 5 , 6 , 0 is
A) 0 B) 7 C) 2 D) 6
11. The value of third decile D 3 for 40 , 42 , 45 , 48 , 50 , 52 , 55 , 56 , 57 is
A) 42 B) 45 C) 48 D) 40
12. The mean of the series 3 , 5 , 3 , 7 , 2 , 5 , 3 is equal to
A) 2 B) 3 C) 4 D) None
13. What is the correct formula for mean deviation for ungrouped data ?
n n n n
xi 1
i x x
i 1
i x x
i 1
i x x
i 1
i x
A) B) C) D)
n n 1 n 1 n2
14. The mean of the following data is
Numbers 8 10 15 20
Frequency 5 8 8 4
A) 15 B) 14.2 C) 12.4 D) 12.8
15. The harmonic mean of 4 , 8 , 16 is
A) 8.687 B) 6.857 C) 5.857 D) 4.758
16. The 8th decile for the data 20 , 30 ,25 , 23 , 22 , 32 , 36 is equal to
A) 30 B) 20 C) 25 D) 32
17. The correct formula for Karl Pearson‟s coefficient of correlation r is
A) r
( x x) ( y y ) B) r
( x x) ( y y )
( x x) ( y y )
2 2
( x x) ( y y )
2 2
C) r
( x x) ( y y ) D) r
( x x) ( y y )
( x) ( y )
2 2
( x) ( y )
2 2
18. The geometric mean of 2 , 4 , 8 , 16 , 32 is equal to
A) 32 B) 64 C) 8 D) 16
19. The standard deviation for the data 9 ,7 , 10 , 8 , 9 , 7 , 8 , 9 is equal to
A) 1.12 B) 1.06 C) 1.006 D) 1
20. Compute the coefficient of Skewness from the following data:
x 6 7 8 9 10 11 12
f 3 6 9 13 8 5 4
A) 1.61 B) 1.43 C) 0 D) 2
21. In a negatively skewed distribution
A) Mean > Mode > Median B) Mode > Median > Mean
C) Mean > Median > Mode D) Mode< Median > Mean
22. If for a distribution the difference of first quartile and median is greater than
difference of median and third quartile then the distribution is classified as
A) Positively skewed B) Absolute open ended
C) Negatively skewed D) Not skewed at all
23. If two regression coefficients are 0.1 and 0.9 then the value of coefficient
of correlation is
A) 0.3 B) 0.3 C) 0.9 D) 0.9
24. If x 10 , y 90 , x 3 , y 12 and rxy 0.8 then the regression equation of
x on y is given by
A) y 3.2 x 5.8 B) x 3.2 y 5.8
C) x 8 0.2 y D) y 8 0.2 x
25. If r12 0.6 , r13 0.8 , r23 0.3 , 1 8 , 2 9 , 3 5 then b12.3 ?
A) 0.35 B) 1.09 C) 0.89 D) 0.45
26. The multiple correlation coefficient R3.12 is given by which of the following ?
A) R3.12
r
13 2 r23 2 2r12 r13 r23
1 r12
2
B) R3.12
r
13 2 r23 2 2r12 r13 r23
1 r12
2
C) R3.12
r
13 2 r23 2 2r12 r13 r23
1 r12
2
D) R3.12
r 13 2 r23 2 2r12 r13 r23
1 r12
2
27. If r12 0.25 , r13 0.35 , r23 0.45 then find R2.13
A) 1.46 B) 2.46 C) 0.46 D) 3.46
28. Calculate the coefficient of correlation between x and y series from the
( x x) ( y y) ( x x) ( y y) 122
2
following data: 2
136 , 138 ,
A) 0.98 B) 0.89 C) 0.98 D) 0.89
29. The coefficient of correlation from the following data is
x 1 2 3 4 5
y 2 5 3 8 7
A) 0.245 B) - 0.895 C) 0.9876 D) 0.8062
y
30. If x 10 , y 22 and r 2.2738 then the equation of line of regression y on
x
x is equal to
A) y 2.2738 x 44.738 B) y 2.2738 x 44.738
D) x 2.2738 y 44.738 D) x 2.2738 y 44.738
31. Find the value of median from the following data :
Class Interval 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50
Frequency 5 10 12 15 18
A) 30 B) 35 C) 32 D) 40
32. Calculate first quartile from the following data:
Class Interval 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75
Frequency 2 8 20 25 10 5
A) 50 B) 56.625 C) 60 D) 62.25
33. Find the value of 9th Decile D 9 from the following data :
Class Interval 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100
Frequency 10 20 20 15 15 20
A) 90 B) 98 C) 95 D) 92
34. Find mode from the following data :
Class Interval 0-4 4-8 8 - 12 12 - 16 16 - 20 20 - 24
Frequency 10 12 18 7 5 8
A) 8.25 B) 7.89 C) 9.41 D) 10.78
35. The first four moments about origin for a distribution are 1 ' 1.5 , 2 ' 17
3 ' 30 and 4 ' 108 then the coefficient of skewness is
A) 0.4924 B) 0.4924 C) 0.3456 D) 0.5648
36. Which of the following is true regarding “ Regression ”and “ Correlation ” ?
where y is dependent variable and x is independent variable.
A) The relationship is symmetric between x and y in both.
B) The relationship is not symmetric between x and y in both.
C) The relationship is not symmetric between x and y in case of
correlation but in case of regression it is symmetric.
D) The relationship is symmetric between x and y in case of
correlation but in case of regression it is not symmetric.
37. R3.12 is known as the multiple correlation coefficient in which
A) x3 is dependent variable and x1 , x2 are the independent variables
B) x3 is independent variable and x1 , x2 are the dependent variables
C) x1 , x2 , x3 all are the independent variables
D) x1 , x2 , x3 all are the dependent variables
38. If there is no skewness in the distribution then which of the following is true ?
A) Third quartile Mean = Median First quartile
B) Third quartile Median = Median First quartile
C) Third quartile Mode = Median First quartile
D) Third quartile Median = Median First quartile
39. If mean = 29.6 , mode = 27.52 and Standard deviation = 6.5 , then the value
of Karl – Pearson‟s coefficient of Skewness is
A) 0.32 B) 3.2 C) 0.45 D) 4.5
40. Which of the following is true
A) The multiple correlation lies between 0 and 1
B) If R1.23 = 0, then r12 = r13 = 0
C) R1.23 = R1.32
D) All of the above
M – 3 CSE Unit – 5
Stochastic Process and Sampling Techniques
(1)
The set of possible values of any individual member of a random process is called
(A) Random Variable
(B) State Space
(C) Sample Function
(D) None of These
Ans: B
(2)
If s and t are fixed, then the stochastic process {X(s, t)} is a
(A) Number
(B) Random Variable
(C) Single Time Function
(D) None of These
Ans: A
(3)
The mean of a random process {X(t)} is
(A) E{X(t)}
(B) E{X(t+1)}
(C) E{X2(t)}
(D) E{X2(t+1)}
Ans: A
(4)
The auto-correlation Rx(t1, t2) of a random process {X(t)} is
(A) E{X(t)}
(B) E{X(t1) X(t2)}
(C) E{X(t1) X(t2) X(t3)}
(D) E{X2(t)}
Ans: B
(5)
The correlation coefficient x (t1, t2 ) of a random process {X(t)} is
(A) [Rx(t1, t2)] / [Cx(t1, t2)]
(B) [Cx(t1, t2)] / [ Cx (t1, t1) C y (t2 , t2 ) ]
(C) [Cx(t1, t2)] / [ Cx (t1, t1) Ry (t2 , t2 ) ]
(D) [Cx(t1, t2)] / [ Rx (t1, t1) Ry (t2 , t2 ) ]
Ans: B
(6)
The cross-correlation Rxy(t1, t2) of two random processes
{X(t)} and {Y(t)} is
(A) E{X(t1) X(t2)}
(B) E{Y(t1) Y(t2)}
(C) E{X(t1) Y(t2)}
(D) None of These
Ans: C
(7)
The cross-covariance Cxy(t1, t2) of two random processes
{X(t)} and {Y(t)} is
(A) Rxy(t1, t2) - E{X(t1)}
(B) Rxy(t1, t2) - E{Y(t1)}
(C) Rxy(t1, t2) - E{X2(t1)} E{Y2(t2)}
(D) Rxy(t1, t2) - E{X(t1)} E{Y(t2}
Ans: D
(8)
The cross-correlation coefficient xy (t1, t2 ) of a random process {X(t)} is
(A) [Rxy(t1, t2)] / [Cxy(t1, t2)]
(B) [Cxy(t1, t2)] / [ Cxx (t1, t1) C yy (t2 , t2 ) ]
(C) [Cxy(t1, t2)] / [ Cxx (t1, t1) R yy (t2 , t2 ) ]
(D) [Cxy(t1, t2)] / [ Rxx (t1, t1) R yy (t2 , t2 ) ]
Ans: B
(9)
If the mean of a random process {X(t)} is constant and the auto-correlation depends only on time
difference, then it is called
(A) Weakly Stationary Random Process
(B) Covariance Stationary Random Process
(C) Wide Sense Stationary Random Process
(D) All of The Above
Ans: D
(10)
4
The auto-correlation function of a stationary process is Rx (t ) 25 .
1 6 t 2
Then the mean is
(A) 2
(B) 3
(C) 4
(D) 5 Ans: D
(11)
If X(t) = A cos(w0t+x) is a random process, where A and w0 are constants and x is uniformly
distributed random variable over (0, 2 ) , then E{X(t)} is
(A) 0
(B) 1
(C) 2
(D) 3 Ans: A
(12)
In a Markov process, the future behaviour depends only on the
(A) Present State
(B) Future State
(C) Past State
(D) None of The Above Ans: A
(13)
A square matrix whose each row sums to one is called a
(A) Right Stochastic Matrix
(B) Left Stochastic Matrix
(C) Doubly Stochastic Matrix
(D) None of The Above Ans: A
(14)
A square matrix whose each column sums to one is called a
(A) Right Stochastic Matrix
(B) Left Stochastic Matrix
(C) Doubly Stochastic Matrix
(D) None of The Above Ans: B
(15)
A square matrix whose each row and column sums to one is called a
(A) Right Stochastic Matrix
(B) Left Stochastic Matrix
(C) Doubly Stochastic Matrix
(D) None of The Above Ans: C
(16)
0.6 0.2 0.2
The transition matrix of an insect zooming in three states is given by P 0.4 0 0.6 .
0 0.8 0.2
Then P(X2 = 3 / X0 = 1) is
(A) 0.28
(B) 0.34
(C) 0.57
(D) 0.71
Ans: A
(17)
1 / 2 1/ 4 1/ 4
Consider a Markov chain with 3 states whose transition matrix is given by P 1 / 3 0 2 / 3 .
1 / 2 1/ 2 0
If P(X1 = 1) = P(X1 = 2) = 1/4, then P(X1 = 3, X2 = 2, X3 = 1) is
(A) 1/4
(B) 1/6
(C) 1/12
(D) None of The Above
Ans: C
(18)
A sequence of values X0, X1, X2, ………., Xt is called a
(A) Trajectory
(B) Projectile
(C) Random Distribution
(D) Continuous Distribution
Ans: A
(19)
The unique fixed probability vector for the stochastic matrix
0 1 0
P 0 0 1 is
1 / 2 1 / 2 0
(A) [1/5, 2/5, 2/5]
(B) [1/3, 1/3, 1/3]
(C) [1/2, 1/2, 0]
(D) [2/7, 3/7, 2/7]
Ans: A
(20)
A group of individuals under study is called
(A) Sample
(B) Population
(C) Parameter
(D) Statistic
Ans: B
(21)
In a random sample, which of the following is true ?
(A) Each member of the population has an equal chance of selection
(B) The first member of the population has more chance of selection
(C) The last member of the population has more chance of selection
(D) None of The Above Ans: A
(22)
Which of the following represent parameters ?
(A) Mean
(B) Mean x
(C) Standard Deviation
(D) Both (A) and (C)
Ans: D
(23)
Which logic does a sampling theory represent ?
(A) Logic of Quantifiers
(B) Logic of Semantics
(C) Logic of Induction
(D) None of The Above
Ans: C
(24)
A non-probability sampling in which researchers rely on their own judgement while choosing
members of the population is called
(A) Purposive Sampling
(B) Random Sampling
(C) Stratified Sampling
(D) Systematic Sampling
Ans: A
(25)
A statistical method involving the selection of elements from an ordered sampling frame is called
(A) Purposive Sampling
(B) Random Sampling
(C) Stratified Sampling
(D) Systematic Sampling
Ans: D
(26)
Which of the following represents standard error ?
(A) / n
(B) / N
(C) 2 / n
(D) 2 / N
Ans: A
(27)
A sample size of 25 is picked up at random from a population which is normally distributed with
mean 100 and variance 36. Then P{x 99} is
(A) 0.2033
(B) 0.4136
(C) 0.5837
(D) 0.6941
Ans: A
(28)
Which of the following represents sample variance ?
xi x 2
(A) i 1
n 1
n
xi x 2
(B) i 1
n 1
xi x 2
i 1
(C)
n 1
n
xi x 2
i 1
(D)
n 1
Ans: B
(29)
A research hypothesis constructed from literature review is a
(A) Null Hypothesis
(B) Alternate Hypothesis
(C) Both (A) and (B)
(D) None of The Above
Ans: B
(30)
In hypothesis testing type I error is
(A) H0 is rejected where it should have been accepted
(B) H0 is accepted where it should have been rejected
(C) Both (A) and (B)
(D) None of The Above
Ans: A
(31)
In hypothesis testing type II error is
(A) H0 is rejected where it should have been accepted
(B) H0 is accepted where it should have been rejected
(C) Both (A) and (B)
(D) None of The Above
Ans: B
(32)
Which of the following represent confidence coefficients ?
(A) 1.96
(B) 2.58
(C) Both (A) and (B)
(D) None of The Above
Ans: C
(33)
If is the significance level and C is the confidence level, then which of the following is true ?
(A) C 2 1
(B) C 2 1
(C) C 1
(D) C 1
Ans: D
(34)
A one-tailed test in a normal distribution, determines the area under
(A) Left Tail of Mean
(B) Right Tail of Mean
(C) One of The Tails
(D) All of The Above
Ans: D
(35)
In a two-tailed test, which of the following is true ?
(A) It tests whether a sample is greater or less than a range of values
(B) If sample falls in either of the critical regions, H a is accepted
(C) It is used to find significance at 5% level
(D) All of The Above
Ans: D
(36)
The „t‟ distribution is used to test the significance of
(A) Mean of a sample
(B) Difference between means of two small samples
(C) Correlation coefficient
(D) All of the above
Ans: D
(37)
If x1, x2, x3, ......, xn are the members of a random sample drawn from a population with mean , and
x is the mean of the sample, then the value of „t‟ is
(x ) n
(A) t
s
(x ) n
(B) t
s
( x x )2
(C) t
n 1
( x x )2
(D) t Ans: A
n 1
(38)
In the „t‟ test, if the calculated value of „t‟ is less than the tabulated value, then the null hypothesis is
(A) Accepted
(B) Rejected
(C) Further investigation required
(D) None of the above Ans: A
(39)
The mean life time of a sample of 100 LED bulbs produced by a company is 1570 hours with a
standard deviation of 120 hours. If the average life of bulbs is 1600 hours, then the value of „t‟ is
(A) 0.9
(B) 1.7
(C) 2.5
(D) 3.4
Ans: C
(40)
The value of chi-square for the following data is
O: 14, 18, 12, 11, 15, 14, 14
E: 14, 14, 14, 14, 14, 14, 14
(A) 2.14
(B) 3.78
(C) 5.52
(D) 6.71
Ans: A
Probability MCQs | RTMNU Winter-2020
MCQs on Probalility
RTMNU Winter 2020
Q.1. Consider the experiment of throwing two dice. Let X denote the sum on two dice. Then
P(1<X<8) is.
(A) 1/12 (B) 2/7 (C) 3/4 (D) 21/36
Ans.(D)
Q.2. If F x is the distribution function of r.v. X then Pa X b will be.
(A) F a F b (B) F b F a (C) a b (D) f a f b
Ans.(B)
1
;a x b
Q.3. The mean for the distribution f x b a
0 ; otherwise is
(A) a b (B)
a b (C)
b
(D)
a
2 2 2
Ans.(B)
Q.4. Which if the following represents Poisson distribution
x e x
(A) P X x (B) P X x
x! x!
(C) P X x x e (D) None of these
Ans.(A)
Q.5. The joint probability of two D.R.Vs X and Y is given as
1 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
C 2 x y , x 0,1,2, y 0,1,2,3
f x, y
0, otherwise
Conditional probability function f(x=1 | y=2) is
(A)1/4 (B)1/20 (C)1/3 (D)4/3
Ans.(A)
Q.6. If X is a discrete random variable, then function f(x) is
(A) Distribution function (B) Probability Mass function
(C) Density function (D) None of these
Ans.(B)
Q.7. For a poisson distribution, if mean(m)=1, then P(1) is?
(A) 1/e (B) e (C) e/2 (D) Indeterminate
Ans.(A)
Q.8. Find the mean and variance of X?
X 0 1 2 3 4
F(X) 1/9 2/9 3/9 2/9 1/9
(A) 2, 4/3 (B) 3, 4/3 (C) 2, 2/3 (D) 3, 2/3
Ans.(A)
2 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.9. A random variable assuming only a finite number of values is called
(A) Discrete variable (B) Random variable
(C) Constant (D) Continuous variable
Ans.(A)
Q.10. Three companies A,B and C supply 25%, 35% and 40% of the notebooks to a school. Past
experience shows that 5%, 4% and 2% of the notebooks produced by these copies are defective.
If a notebooks was found to be defective, what is the probability that the notebooks was supplied
by A?
44 25 13 11
(A) (B) (C) (D)
69 69 24 24
Ans.(B)
x 5 / 2, 0 x 1
Q.11. Find the mean of a random variable X if f(x)= 2 x, 1 x 2
0,
otherwise
(A)3.5 (B)3.75 (C)0.3035 (D)2.75
Ans.(C )
Q.12. The probability that a student knows the correct answer to a multiple choice question is
2/3. If the student does not know the answer, then the student guesses the answer. The
probability of the guessed answer being correct is 1/4. Given that the student has answered the
question correctly, the conditional probability that the student knows the correct answer is
2 3 5 8
(A) (B) (C) (D)
3 4 6 9
Ans.(D)
3 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.13. Suppose four coins are tossed, the values that of a random variable H (No. of heads) can
take are:
(A) 1,2,3,4 (B) 0,1,2,3,4 (C) 0,1,2,3 (D) 0,1
Ans.(B)
Q.14. If E and F are two events associated with the same sample space of a random experiment
then PE | F is given by
P E F P E F
(A) provided P F 0 (B) provided P F 0
P F P F
P E F P E F
(C) (D)
P F P E
Ans.(A)
4 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.15. If an unbiased coin is tossed once, then the two events head and tail are
(A) Mutually exclusive (B) Equally likely
(C) Exhaustive (D) All of the above
Ans.(D)
Q.16. A card is drawn from a pack of 52 cards. What is the probability of getting a king of a
black suit?
(A) 1/26 (B) 1/52 (C) 3/26 (D) 7/52
Ans.(A)
Q.17.The events when we have no reason to believe that one is more likely to occur than the
other is called:
(A) Equally likely events (B) Independent events
(C) Dependent events (D) Not equally likely events
5 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Ans.(D)
Q.18. Let X and Y be two independent random variables. Which one of the relations between
expectation(E), variance(Var) and covariance(Cov) given below is FALSE?
(A) E(XY)=E(X).E(Y) (B) Cov(X,Y)=0
(C) Var(X+Y)=Var(X)+Var(Y) (D)E X 2 .Y 2 =E X 2 /E Y 2
Ans.(D)
1
2 ; probabilit y 3
1
Q.19. If X is a random variable defined by X 1 ; probabilit y then the expectation of X is
6
3 ; probabilit y 1
2
(A) 1/3 (B) 2/3 (C) 4/3 (D)7/3
Ans.(D)
ke x , for x > 0
Q.20. Find the Expectation of a random variable X if F X
0, otherwise
(A) 0 (B) 1 (C) 2 (D) 3
Ans.(B)
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Probability MCQs | RTMNU Winter-2020
Q.21. In a simultaneous toss of two coins, the probability of getting no tail is:
1 1
(A) (B) (C) 2 (D) 1
4 2
Ans.(A)
Q.22. E(X) = = V(X) = 2 is for which distribution?
(A)Bernoulli’s (B) Binomial (C) Poisson’s (D) Normal
Ans.(C )
Q.23. If ‘m’ is the mean of a Poisson distribution, then variance is given by
(A) m 2 (B) m1/ 2 (C) m (D) m / 2
Ans.(C )
Q.24. The expectation of a random variable X (continuous & discrete) is given by
(A) Xf ( X ), Xf ( X ) (B) X 2
f ( X ), X 2 f ( X )
(C) f ( X ), f ( X ) (D) Xf ( X ), Xf ( X
2 2
)
Ans.(A)
7 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.25. A bag ‘A’ contains 2 white and 3 red balls and a bag ‘B’ contains 4 white and 5 red balls.
One ball is drawn at random from one of the bags is found to be red. The probability that it was
from bag B is:
27 3 25 5
(A) (B) (C) (D)
52 5 52 9
Ans.(C )
Q.26. The joint probability of two D.R.Vs X and Y is given as
C 2 x y , x 0,1,2, y 0,1,2,3
f x, y
0, otherwise
Then value of constant C is
(A) 7/42 (B) 5/42 (C) 1/42 (D) 11/42
8 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Ans.(C )
Q.27. What is moment generating function?
(A) M X t E etX (B) M X t E e tX
(C) M X t E e 2tX (D) M X t E et
Ans.(A)
Q.28. E(X) = npq is for which distribution?
(A)Bernoulli’s (B) Binomial (C) Poisson’s (D) Normal
Ans.(A) Ans.(B)
Q.29. An um contains five balls two balls are drawn and found to be white the probability that all
the balls are white is:
1 3 1 2
(A) (B) (C) (D)
2 10 10 5
Ans.(A)
9 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.30. If the random variable X is binomially distributed, then the variance of X is
(A) n (B) p (C) q (D) npq
Ans.(D)
Q.31. Each of three identical jewellary boxes have two drawers. In each drawer of the first box
there is a gold watch. In each drawer of the second box there is a silver watch. In one of the
drawers of the third box there is gold watch while in the other there is a silver watch. If we select
a box at random and open one of the drawers and find it to contain a silver watch, then the
probability that the other drawer has gold watch is
(A) 2/3 (B) 1 (C) 0 (D) 1/3
10 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Ans.(A)
Q.32. If X is a continuous random variable then P(a < x <b) is
a b 1
(A) f ( x)dx
b
(B) f ( x)dx
a
(C) f ( x)dx
0
(D) None of these
Ans.(B)
x 2 1 ;0 x 1
Q.33. Is the function F(x) = a distribution function ?
0 ; otherwise
(A) Yes (B) No (C) Insufficient Data (D) None of these
Ans.(B)
Q.34. In a Poisson distribution, the mean and standard deviation are equal.
(A) True (B) False
11 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Ans.(B)
ex ; x 0
Q.35. The moment generating function f(x) = is
0 ; otherwise
(A) 1/(1+t) (B) 1+t (C) t (D) 1/(1-t)
Ans.(D)
kx if 0 x 2
Q.36. If the PDF of a continuous r.v. is given as f ( x) .
0 otherwise
find the value of k.
1 1
(A)1 (B) (C) (D) None of these
2 4
Ans.(A)
Q.37. For a continuous random variable X, the probability density function f(x) represents
(A) The probability at a given value of X, (B) The area under the curve at X,
(C) The area under the curve to the right of X, (D) The height of the function at X,
Ans.(B)
Q.38. Suppose you roll two dice. What is the probability the sum is 8?
12 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
(A) 5/36 (B) 1/36 (C) 7/36 (D) 11/36
Ans.(A)
Q.39. Poisson distribution is applied for
(A) Continuous Random Variable (B) Discrete Random Variable
(C) Irregular Random Variable (D) Uncertain Random Variable
Ans.(A)
13 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
x : 0 1 2
Q.40. Value of k in the following PMF is
f ( x) : k k 2k
1 3 1
(A) (B) (C) (D) None of these
2 2 4
Ans.(C )
x, 0 x 1
Q.41. Find the Moment Generating Function of f(x) = 2 x, 1 x 2
0, otherwise
2 2 2 2
et 1 e t 1 e 2t 1 e 2t 1
(A) (B) (C) (D) 2
t t t t
Ans.(A)
Q.42. Which one of the following statements are true
(A) The maximum value of f x is 1, (B) The maximum value of F x is 1,
(C) Both A and B, (D) None of these
Ans.(C )
Q.43. Random variables X and Y have the joint PDF
14 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
2e x2 y , x 0, y 0
f x, y
0, otherwise
Marginal function of X given Y and Marginal density function of Y given X are
(A) and (B) and (C) ) and (D) ) and
Ans.(C )
Q.44. In a Poisson distribution, if ‘n’ is the number of trails and ‘p’ is the probability of success,
then the mean value is given by?
(A) (B) ( ) (C) ( ) (D) m p
Ans.(A)
Q.45. An industrial firm uses three hotels for its clients. From the past experience it is known
that 20%,50%,30% of the clients are assigned rooms at hotel palace, hotel ganapati and hotel
kundan plaza respectively. If the fault in plumbing is 5%,4%,8% of the rooms at hotel palace,
hotel ganapati and hotel kundan respectively, then the probability that a person with a room
having fault plumbing was assigned accommodation at hotel kundan plaza is.
(A) 4/9 (B) 2 (C) 1 (D) 2/3
Ans.(A)
Q.46. The distribution function F(x) of a continuous random variables is
x
(A) F x f v dv (B) F x f v dv
1
(C) F x f v dv (D) None of these
0
Ans.(A)
15 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.47. If f(x) is a mass function, then its corresponding random variable is
(A) Continuous Random Variable (B) Discrete Random Variable
(C) Probability Function (D) None of these
Ans.(B)
Q.48. The mathematical expectation of constant K is
(A) K (B) 1 (C) 0 (D) -1
Ans.(A)
0 ;x 0
Q.49. If F(x) =
1 e
x
2
;x 0
is the distribution function, then the value of F(2) is
(A) 0.7476 (B) 1 (C) 0 (D) 0.3234
Ans.(A)
Q.50. If F(x) is the distribution function of random variable X then P(a X b) is the
distribution function, then the value of F(2) is
(A) F (a) F (b) (B) F (b) F (a) (C) a b (D) f (b) f (a)
Ans.(B)
Q.51. Which of the following represents the normal distribution
1
(A) f (x) constant (B) f ( x) e ( x ) / 2 2
, x
2
2
1
(C) f ( x) e ( x ) / 2 2
, x
2
(D) None of these
Ans.(B)
16 Mathematics-Ganit Sangrah | Satish Tiwari
Probability MCQs | RTMNU Winter-2020
Q.52. The mileage in thousands of miles which car owners get with a certain kind of type is a
random variable having probability density function
1 x / 20 ,x 0
e
f ( x) 20
0 ,x 0
The probability that one of these tyres will last atmsot 10,000 miles
(A) 0.3935 (B) 0.1481 (C) 0.2231 (D) None of these
Ans.(A)
17 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
MCQs on Fourier Series
Ans : (b)
( | |
Ans : (b)
3)
d)
Ans : (c)
b) an even
d) trigonometric
Ans : (a)
b) an odd
d) logarithmic
Ans : (c)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
(
( (
b) 0
( (
c) d) None of these
Ans : (b)
7) ( {
(
a) 0 b) 1 c) 2 d) 4
Ans : (a)
8) ( {
( (
a) 0, b) 0, c) 0,0 d) None of these
Ans : (c)
9)
a) First and second quadrant b) Second and third quadrant
c) First and third quadrant d) Third and fourth quadrant
Ans : (a)
10) ( {
( (
a)
Ans : (d )
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
11) Fourier series of a function f(x) converges to f(x) if x is a point of
a) Continuity b) Discontinuity c) Both continuity and discontinuity d) None of
these
Ans : (b)
12) The Fourier series for f(x)=| | (
a) Sine terms b) Cosine terms c) Both sine and cosine terms d) None of
these
Ans : (b)
13) Any waveform can be expressed in Fourier series if
a) Sampling conditions are satisfied b) Dirichlet conditions are satisfied
c) Maxwells` conditions are satisfied d) Leibnitzs` conditions satisfied
Ans : (b)
14) In a Fourier series for f(x) = | | in (-
( (
a) 1 b) c) d) 0
Ans : (d )
15) For a function f(x) having Fourier expansion
∑ ∫ ( ∑ is
called
a) Dirichlets` identity b) Eulers` identity c) Parsevals` identity d) None of
these
Ans : (a)
16) The constant term in the Fourier series for function f(x) = - 2 in (-2, 2) is
a)
Ans : (d )
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
17) Find the value of
f(x)=1 in
a) b) c) d)
Ans : (a)
18) What is the value of , if f(x) = in the interval 0 to 5
a) 25/2 b) 125/2 c) 625/2 d) 625/5
Ans : (b)
19) Find the sum of using Fourier series expansion if
( {
a) b) c) d)
Ans : (a)
20 W D ’
a) f(x) is periodic b) f(x) is single valued
c) f(x) has infinite number of discontinuities d) f(x) is infinite
Ans : (d )
21 W D ’
a) (1/2) [f( ) – f( ] b) (1/2) [f( ) + f( ]
c) [f( ) + f( ] d) [f( ) – f( ]
Ans : (b)
22) Find the value of
f(x)=x in the interval
a) b) c) d)
Ans : (d )
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
23) The graph of even function is symmetric about
a) origin b) x-axis c) y-axis d) none of these
Ans : (c)
24) The Fourier series expansion of f(x) = x in ( , ) is
a) ∑ b) ∑ (
c) ∑ ( d) none of these
Ans : (c)
25) The function f(x) = { is
a) Even function b) Containing cosine terms only in Fourier series
expansion
c) Both A and B d) None of these
Ans : (c)
26) In the Half Range Fourier Cosine series of f(x) = xsin x in(0 is equal
to _______
a)
Ans : (b)
27) ( | |
( (
a) 0, b) 0, c) 0,0 d) None of these
Ans : (c)
28) In the Half Range Fourier Sine series of f(x) = lx- in(0 is equal to
_______
a)
Ans : (c)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Series | RTMNU Winter-2020
Q29) f(x)= { is
a) an even b) an odd c) neither even nor odd d) trigonometric
Ans : (b)
30) The Fourier series expansion of an even function contains
a) only cosine terms b) cosine terms and a constant
c) only sine terms d) sine terms and a constant
Ans : (b)
31) What is the Fourier series expansion of the function f(x) in the interval (c, c+2 ?
a) ∑ ∑ b) ∑ ∑
c) ∑ ∑ d) ∑ ∑
Ans : (a)
32 W D ’ Fourier series expansion
a) f(x) is periodic b) f(x) is single valued
c) f(x) has finite number of discontinuities d) f(x) is finite.
Ans : (d )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
MCQs on Fourier Tranform
| |
Q.1) The Fourier sine Transform of
A)√ B) √ C) √ D) √
Ans : ( A)
Q.2) Fourier transform of dirac delta function 𝜹 (t) is given as .
A) 0 B) 1 C) 2π(W) D) π𝜹(W)
Ans : ( B)
Q.3) if { } = ̅ (s), then F{ }=
A) ̅ ( ) B) ̅
(s) C) ̅
(a) D) (̅ )
Ans : ( D)
Q.4) if f(x) = x2 then Fourier integral of f(x) is same as
A)Fourier sine transforms B) Fourier cosine transform C) Fourier sine integral D) Fourier
cosine integral
Ans : ( D)
Q.5) Fourier transform of
A) B) C) D) √
Ans : ( B)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.6) Fourier cosine transform of f(x) = 1,0≤x≤ 1 is
A) B) C) s D) √
√ √
Ans : ( D)
1, x 1
Q.7) The Fourier integral of
0 x 1
A) Fourier cosine integral B) fourier sin integral C) both a and b D) None
Ans : ( A)
Q.8) The Fourier sine transform of is
√ B) √ C) √ D) 1
Ans : (C )
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.9) If F[ ] F[ ] then F[ ]=?
A) B) C) D)
Ans : ( B)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.10) Fourier sine transform of is
A)√ B) √ C) √ D) √
Ans : ( B)
1, | x | 1
Q.11) Fourier sine transform of f ( x)
0, | x | 1
A)√ B) C) D) √
Ans : ( D)
Q.12) If Fc ( f ( x)) Fc (s) then of f (x) is
2 2
A)
Fc (s) cos(sx)ds
0
B)
F (s) cos(sx)dx
0
c
2 2
C)
Fc (s) cos(sx)ds
D)
F (s) cos(sx)dx
c
Ans : ( A)
Q.13) If f (x) is even function then Fourier transform of f (x) is
A) Fourier sine transform B) Fourier cosine transform
C) Fourier sine integral D) Fourier cosine integral
Ans : ( B)
Q.14) The Fourier transform of unit step function is given by
1 j
A) F ( j ) B) F ( j ) j C) F ( j ) D) F ( j )
j j
Ans : ( A)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Fourier Transform | RTMNU Winter-2020
Q.15) If the Fourier transform of f (t ) is F ( j ) the what is the Fourier transform of f (t )
A) F ( j ) B) F ( j ) C) F ( j ) D) None of these
Ans : ( D)
Q.16) The Fourier integral is useful for
A) Periodic Function B) Non-periodic Function
C) Logarithmic Function D) Discontinuous Function
Ans : ( D)
x sin mx
|x|
Q.17) If he Fourier transform e is √ then dx ?
0
1 x2
A) e m B) e 2 m C) 0 D) None of these
2
Ans : ( A)
cos xd
Q.18) The Fourier cosine integral of function e x the value of
0
1 2
A) ex B) ex C) ex D) ex
2 3 4 6
Ans : ( A)
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
MCQs on Lapalce Transform (Winter 2020)
Q.1) Find ,∫ -
A) B) C) D)
Ans : ( A)
Q. 2) L[ ] is
n 1 n n 1 n
A) n
B) n 1 C) n 1
D) n
s s s s
Ans : (C )
Q.3) Find the Laplace transform of
A) ( ) B) - ( ) C) ( ) C) ( )
Ans : ( B)
Q.4) If f (t) is periodic function with period T then L{ } is
A) ∫ f(u)du B) ∫ f(u)du
C) ∫ f(u)du D) ∫ f(u)du
Ans : ( A)
Q.5) , -n
A) B) C) D)
Ans : ( D)
Q.6) * + in positive when n is
A) Zero B) Negative integers C) Negative Rational D) Positive Integer
Ans : ( D)
Q.7) Laplace transform of 1 is
A) A) A) A)
Ans : ( A)
Q.8) If y(t) is then solution of -2 y = 4, y(0) = 1,then L[ ]=
A) -2 B) -2 C) -3 D) -2
Ans : ( B)
Q.9) [ ] is
A) t B) C) D)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Ans : ( B)
Q.10) Laplace transform of 3t is
A) B) C) D) [ ]
Ans : ( D)
Q.11) * +
A) sin(3t/2) B) 3sin(3t/2) C) sin (3t/2) D) sin (3t/2)
Ans : (C )
Q.12) { }
A) B) t sint C) D) t sint
Ans : (C )
Q.13) Laplace transform of ( )
A) B) [ ]
C) [ ]
D) [ ]
Ans : ( B)
Q.14) *, - +
A)[1-cos2(t-1)]u(t-1) B) [1-cos2(t-1)] C) [1-cos2t]u(t-1) D) [1-cos2(t+1)]u(t+1)
Ans : ( A)
Q.15) Laplace Transform of sinh 3t is
A) B) C) D)
Ans : ( D)
Q.16) Laplace Transform of 3t is
A) B) C) 1 D)
Ans : ( B)
Q.17) , - is
A) sint u(t) B) sin (t+π) C) sin(t+π) u(t+π) D) sin (t-π) u(t-π)
Ans : ( D)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.18) Laplace Transform of u(t-a) is
A) B) C) D)
Ans : (C )
Q.19) Inverse Laplace transform of * + is
A) * + B) * + C) * + D) * +
Ans : (C )
Q.20) If L(f(t)) = √ then L ( f(t))
A√ B) √ C) √ D) √
Ans : ( B)
Q.21) If f(t) = then L [f(t)] is
A) B) * + C) * + D) None of these
Ans : ( B)
Q.22) ∫ * + =
A) B) -4 C) D)
Ans : (C )
Q.23) * + is
A) Sin(t-4)u(t-4) B) cos4(t-2)u(t-2) C) cos (t-2)u(t-2 ) D) sin4(t-2)u(t-2)
Ans : ( B)
Q.24) If L[f(t)] =F(s)and L[g(t)]= G(s) then [ ] is
A) ∫ B) ∫
C) ∫ D) None of these
Ans : ( A)
Q.25) Given L[ √ ] = ⁄
then L * +=
√
A) B) C) D) None of these
√ √
Ans : ( A)
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.26) L ,∫ -=
A) B) C) D)
Ans : (C )
Q.26) ∫ sin 2t dt =
A) L { } B) C), - D) All are Correct
Ans : ( D)
Q.27) * +
A) B) C) D) none of these
Ans : (C )
Q.28) If f(t) is periodic function , where f(t+2)=f(t) then
Laplace transform of f(t) is
A) ∫ B) ∫
C) ∫ D) ∫
Ans : ( A)
Q.29) ∫
A)* + C)* + D)
Ans : (C )
Q.30) , can be written in term of unit step function as
A) e-tH (t)-e-t H(t+3) B)
C) D)
Ans : (C )
Q.31) L* +=
A) cot-1s B) tan-1s C) -1
s D) none of these
Ans : ( A)
Q.32) * +
A) * ( )+ B) * ( )+ C)* ( )+ D)* ( )+
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Ans : ( A)
Q.32) The inverse L.T. of is
A) [ ] B) A) [ ] C) [ ] D) [ ]
Ans : ( D)
Q.33) L[ sin2(t- ]=
A) B) C) D)
Ans : (C )
Q.34) Find the Laplace transform of
A) * + B) * + C) * + D) * +
Ans : ( A)
Q.35) Find the Laplace transform of t sinhat
A) B) C) D)
Ans : ( D)
Q.36) L *∫ +=
A) ( ) B) ( ) C) ( ) D) None of these
Ans : ( A)
Q.37) The Laplace transform of f(t)= is
A) B) C) D)
Ans : ( B)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
Q.38) Laplace transform of function ,where a and b are constant are given by
A) B) C) D)
Ans : ( D)
Q.39) 2t with y(0)=,Y’(0)=0 is
A) B)
D) D)
Ans : (C )
{ } { }
–
A) B) C) D) * +
Ans : ( A)
{ }
B) C) D)
Ans : ( A)
{ } , then L{ }
√
B) C) D)
√ √ √ √
Ans : (C )
[ ]
A) B) C) D)
Ans : (C )
Q.44) Laplace transform of ∫
A) ( ) [ ] ( ) B) ( ) [ ] ( )
C) ( ) [ ] ( ) D) None of these
Ans : ( B)
Q.45) * +
√ √ √ √ √
A) ( ) B) ( ) C) ( ) D) ( )
√ √
Ans : (C )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
+y=0,
A) [ ] [ ] B) [ ] [ ]
C) [ ] [ ] D) None of these
Ans : ( B)
{ }
A) B)Zero C) D)1
Ans : ( A)
A) B) D)
Ans : ( A)
, -=
A) Cosh8t B) cos8t C) sinh8t D) sin8t
Ans : ( B)
[ ] { } , -
A) ∫ B) ∫
C) ∫ D) ∫
Ans : (C )
{ } ̅ { }
A) ̅ B) ∫ ̅ C) ̅ ̅
Ans : (C )
{ } ̅ { }
A) ̅ B) ∫ ̅ C) ̅ ̅
Ans : ( D)
{ } ∫
A) B) C) D) 0
Ans : (C )
is
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
A) B) C)
Ans : ( A)
{ }
A)sin u(t) B)sin (t+ C) D
Ans : ( D)
{∫ }
A) B) ⁄ D)
Ans : ( A)
∫ √
√
A)1/t B)1/2 C)1/3 D)1/u
Ans : ( B)
[ { }]
( ) ( ) ( ) ( )
A) B) C) D)
Ans : ( B)
{ }
A) B) C) D)
Ans : ( A)
{ }
A) B) C) D)
Ans : ( D)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
9 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Laplace Transform | Winter-2020
10 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
MCQs on Matrix (Winter 2020)
Q.1) If if the characteristic equation of a matrix A then Eigen values of A
are______
( ) ( )
( ) ( )
Ans : ( D)
Q.2) Using Sylvester theorem the value of is for A [ ]
( )[ ] ( )[ ]
( )[ ] ( )[ ]
Ans : (C )
Q.3) Using Caley theorem , for the matrix A [ ] is calculate from
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Ans : ( B)
Q.4) Model matrix B corresponding to matrix A [ ] is________
( )[ ] ( )[ ]
( )[ ] ( )
Ans : (C )
Q.5) Examine the following system of vectors for linearly dependent:
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
Ans : ( A)
Q.6) Which of the following is condition for quadratic to canonical form
( ) ( )
( ) ( )
Ans : ( A)
Q.7) Inverse of orthogonal matrix is equal to
( ) its transpose ( ) its diagonal
( ) orthogonal ( )none
Ans : (C )
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.8) If the characteristic equation for the matrix A is 3 then the Eigen
value of the matrix A are
( ) ( )
( ) ( )
Ans : (C )
Q.9) Using Sylvester for matrix A [ ]
( ) [ ] ( ) [ ]
( ) [ ] ( ) [ ]
Ans : ( A)
Q.10) The characteristic equation for square matrix A is
( )| | ( )| |
( )| | ( )
Ans : ( A)
Q.11) On what value of the matrix ,A [ ] is an orthogonal
( ) ( )
√ √ √ √ √
( ) ( )
√ √ √ √ √ √
Ans : ( A)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.12) Investigate the vectors ( ) ( ) ( ) are
( ) Linearly dependent ( ) Linearly independent
( ) Orthogonal ( ) None of these
Ans : ( A)
Q.13) If 2,2,3 are Eigan value of a matrix [ ] then the determinant of A is
( ) ( ) ( ) ( )
Ans : ( D)
Q.14) Statement of Caley Hamilton theorem is
( ) Every square matrix satisfies its own characteristics equation
( ) The some of the Eigan value of matrix is sum of element of principle diagonal
( )The Eigan value of matrix A and its transpose are same.
( )None of these
Ans : ( A)
Q.15) If [ ] is a non-singular square matrix and is the Eigen value of [ ] then the Eigen value
of is ____
( ) ( ) ( ) ( )
Ans : (C )
Q.16) If A [ ] then the matrix represented by is equal to ___
( ) ( )
( ) ( )
Ans : ( B)
Q.17) The characteristic equation of matrix A [ ]is ______
( ) ( )
( ) ( )
Ans : ( A)
Q.18) If A [ ] then s t n __________
( )I ( ) ( ) ( )1
Ans : ( A)
Q.19) Which of the following statement is true?
A) Every matrix satisfies its own characteristics equation.
B) Every matrix satisfies characteristics equation.
C) Every square matrix satisfies its own characteristic equation.
D) Every square matrix satisfies characteristic equation.
Ans : (C )
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.19) Inverse of the matrix [ ]
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( A)
Q.20) If [ ] then what is
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : (C )
Q.21) [ ] and [ ] then is
A) [ ]
B) [ ]
C) [ ]
D) None of these
Ans : (C )
Q.22) Eigen values of the matrix [ ] are
A)
B)
C)
D)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Ans : ( A)
Q.23) If [ ] then the characteristic equation is
A)
B)
C)
D)
Ans : ( A)
Q.24) If [ ] then [ ]
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( D)
Q.25) The characteristics equation of the matrix A of order 3x3 is Using
Caley Hamilton theorem simplified form of expression
is
A) 5A+3I
B) -5A+3I
C) 5A-3I
D) None of these
Ans : (C )
Q.26) The relation between the vector ; ( ) ( ) ( ) is
A)
B)
C)
D)
Ans : ( A)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.27) For an orthogonal matrix [ ] is
√
A) [ ]
√
B) [ ]
√
C) [ ]
√
D)
Ans : (C )
Q.28) The characteristics equation of the matrix [ ] is
A)
B)
C)
D)
Ans : ( A)
Q.30) The set of vectors are said to be linearly dependent then there exists
scalars not all zeros, such that
A) True
B) False
C) Could be either
D) None
Ans : ( A)
Q.31) If are eigen values of a matrix [A] then
A) Determinant of A
B) Zero
C) Sum of the elements of principal diagonal
D) None
Ans : (C )
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.32) If A is the given matrix and be the eigen values of [ ] then the
Canonical form is _________
A)
B)
C)
D)
Ans : ( B)
Q.33) If 2,0,3 are Eigen values of a matrix [A] then the inverse of a A is exists
A) True
B) False
C) Could either
D) None
Ans : ( B)
Q.34) The linearly dependency of vector ( ) ( ) ( )
A)
B)
C)
D) None
Ans : (C )
Q.35) The functional relation of the vector ( ) ( ) ( )
A)
B)
C)
D) None
Ans : (C )
Q.36) The Eigen values of [ ] are
A) -3,-3,5
B) 1,2,3
C) 2,2,8
D) 0,3,15
Ans : ( A)
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
Q.37) What are the eigen values of the diagonal matrix [ ]
A) 0,1,2
B) -1,0,2
C) -1,1,0
D) -1,1,2
Ans : ( D)
Q.38) The characteristic equation of a matrix [ ] is
A)
B)
C)
D)
Ans : ( B)
Q.39) By Sylv st r’s th or m ( ) ∑ ( ) ( ) where ( )
[ ]
A)
[ ]
[ ]
B)
[ ]
[ ]
C)
[ ]
D) None
Ans : (C )
Q.40) The eigen values of [ ] are
A) 3,3,5
B) 1,2,3
C) 2,-2,8
D) 0,3,15
Ans : ( D)
Q.41) s non z ro n v tor o th n s lso n n v tor wh r s n
non z ro onst nt
) ru B) ls ) oul th r D) None
Ans : ( A)
8 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) r n v u o m tr x [ ] th n
A) Determine of A B) Zero
C) Sum of the element of principle diagonal D) None
Ans : ( A)
) th v n m tr x n th n v lu s o n
x
y
[ ] th n th non l From is ________
z
A) B) x y z ) D) x y z
Ans : ( B)
) th v n s y th n =?
A) A=[ ] B) A=[ ] C) A=[ ] D) A=[ ]
Ans : ( A)
)Solv n th qu t on x y z x y z
y r m r s rul th v lu o x y and z are respectively
A)1,2,3 B)1,2,-2 C)1,2,2 D)1,-2,2
Ans : (C )
) s squ r m tr x o or r h v n l n rly n p n nt n tor th n non s n ul r m tr x
B can be found such that
A)Canonical form B)Diagonal form C) Quadratic form D)None
Ans : ( B)
) Sp tr l r us o th m tr x [ ] is
A)2 B) 4 C) -6 D) 5
Ans : ( D)
) v tor o m tr x [ ]
A)[ ] B)[ ] C)[ ] D)[ ]
Ans : ( B)
) th or m n us to n
A) nv rs o m tr x B) Power of matrix
C) Any power of matrix D) All of the above
Ans : ( D)
)The largest eigen value for the matrix [ ] is
A)3 B)1 C)4 D)2
Ans : ( A)
9 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)Coefficient matrix obtained from the quadratic form is
A) Real symmetric matrix
B) Skew symmetric matrix
C) Orthogonal matrix
D) Modal matrix
Ans : ( A)
)Eigen vectors of a real symmetric matrix A are orthogonal if the eigen values are ______
A Repeated B Non-Repeated
C Complex D None of these
Ans : ( B)
)Which of the following is linear polynomial of matrix A?
A) A2 + 5A + I B) A2 + I C) A + 5I D) None of these
Ans : (C )
)If Q is an orthogonal matrix of the orthogonal eigen vectors then:
(a) (b) (c) (d) None
Ans : (a)
)If 1,2,3 are the eigen values of A, then the eigen values of are:
(a) 4,-12,20 (b) -4,12,-20 (c) -4,-12,20 (d) -4,-12,-20
Ans : (d )
)If [ ] , then is
(a) [ ]
(b) [ ]
(c) [ ]
(d) [ ]
Ans : (c)
)If [ ] , then =?
(a) [ ]
(b) [ ]
(c) [ ]
(d) [ ]
Ans : (a)
10 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)The matrix [ ] satisfy which of the following equations:
(a)
(b)
(c)
(d)
Ans : (b)
)The eigen vector of the matrix [ ] for eigen value is
(a) ( )
(b) ( )
(c) ( )
(d) ( )
Ans : (a)
)If the transformation of , , is regular
then
(a) [ ] (b) [ ] (c) [ ] (d) [ ]
Ans : (a)
)The eigen values of differential equation , are:
(a) 1,5 (b) 3,5 (c) 8,9 (d) -1,-5
Ans : (a)
)The sum of the Eigen values of the matrix A = [ ]
a) 2 b) 1 c) -2 d) -1
Ans : (c)
) The correct set of Eigen Values of [ ]
a)(-3,4,2) b) (0,7) c) (1,4,7) d) (-3,0,1)
Ans : (a)
) If A= [ ]
a)A+3I+2 b) (A+I)(A+2I) = 0 c) d) 0
Ans : (b)
11 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) If Y=AX is an orthogonal linear transformation then the matrix A is
a)non singular b) orthogonal c)singular d)none of these
Ans : (b)
)On what value of scalars the vectors: ( ) ( ) ( ) are
linearly dependent.
a)-1,-1,1 b)1,2,3 c)1,-1,2 d)-1,-1,3
Ans : (a)
)The system of equations AX=B are consistent if
a)Rank of A Rank of augmented matrix b)Rank of A=Rank of augmented matrix
c)Rank of A d) Rank of A
Ans : (b)
) If is the characteristics equation for the matrix A then the eigen
values of A are
A)3,3,5 B) C)3,-3,5 D)
Ans : ( D)
) If [ ] th n non l orm o
A) B)
) )
Ans : ( A)
) Eigen values of the matrix A=[ ]
a)1 and 6 b)-1 and -6 c)1 and -6 d)-1 and 6
Ans : (b)
) The sum of the Eigen values of the matrix A = [ ]
a) 5 b) 7 c) 4 d) none of these
Ans : (b)
)If A=[ ]
a) b) c) d)
Ans : (c)
12 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
)If A=[ ] then Eigen values of are
a)-1,-9,-4 b)1,9,4 c)-1,-3,2 d)1,3,-2
Ans : (b)
) If A=[ ] ( sum of minor of diagonal element)
a)12 b)24 c)6 d)36
Ans : (d )
) If [ ] is an Eigen vector of matrix [ ] then Eigen value corresponding to
the given Eigen vector is
A) 1
B) -1
C) 3
D) -3
Ans : (C )
) Which of the following pair of vectors are linearly dependent?
A) [ ] ,[ ]
B) [ ] [ ]
C) [ ] ,[ ]
D) [ ] ,[ ]
Ans : ( A)
) The trace & determinant of a 2*2 matrix is -2 & -35 respectively eigen values are
A) -30 & -5
B) -37 & -1
C) -7 & 5
D) 17.5 & -2
Ans : (c)
) If A is an orthogonal matrix then
A) is an orthogonal
B) is an orthogonal
C) ( )
D) All are correct
Ans : ( D)
13 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
) If eigen values of A are 1,2,3 the eigen values of matrix are
A)
B)
C)
D) 1,8,27
Ans : ( D)
) If [ ] then what will be the value of y Sylv st r’s th or m
A)
B)
C)
D) 0
Ans : ( A)
) The eigen value of the following matrix are [ ]
A)
B)
C)
D)
Ans : ( A)
) If the characteristics equation of a matrix A is its diagonal
form is
A) [ ]
B) [ ]
C) [ ]
D) [ ]
Ans : ( D)
) Atleast one Eigen value of a singular matrix is
A) Positive
B) Negative
C) Zero
D) Imaginary
Ans : (C )
) If matrix [ ] is singular then is
14 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Matrix | RTMNU Winter-2020
A)2 B) C)0 D)
Ans : ( A)
) [ ] th n s
A) 4A-5I B) 4A+2I C) 4A+5I D) 4A-2I
Ans : (C )
) s ny n n m tr x s s l r th n | | | |wh r s
15 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
MCQs on Partial Differential Equation
1) The P.I. of ( √ is
(a)
(b)
(c)
(d)
Ans : (a)
2) The C.F. of partial differential equation =√ is
(a)
(b)
(c)
(d)
Ans : (b)
3) A first order partial differential equation (
(a) Linear equation
(b) Semi- linear equation
(c) Quasi-linear equation
(d) Non-linear equation
Ans : (a)
4) What is the C.F. of =0
(a)
(b) (y+2x) + (y-2x)
(c) (y-x) + (y+2x)
(d) (y+2x)+ (y-x)
Ans : (b)
5) is the homogeneous equation of degree
(a) 2
(b) 3
(c) 1
(d) 0
Ans : (c)
6) If -2 and -3 are two roots of partial differential equation, then its C.F. is
(a)
(b)
(c)
(d)
Ans : (c)
1 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(7) The C.F. of partial differential equation
(a)
(b) )
(c)
(d)
Ans : (a)
(8) If 2,2,2 are three roots of partial differential equation, then its C.F. is
(a)
(b)
(c)
(d)
Ans : (b)
(8) The C.F. of partial differential equation is
(a)
(b)
(c)
(d)
Ans : (a)
9) The C.F. of partial differential equation is
(a)
(b)
(c)
(d)
Ans : (c)
10) The P.I. of partial differential equation is
(a) s
(b)
(c)
(d)
Ans : (d )
11) A first order partial differential equation
(a) Linear equation
(b) semi- linear equation
(c) Quasi-linear equation
(d) Non-linear equation
Ans : (a)
12) The C.F. of partial differential equations is
(a)
2 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(b)
(c)
(d)
Ans : (a)
13) If 2,3,3 are three roots of the partial differential equation then its C.F. is
(a)
(b)
(c)
(d)
Ans : (b)
14) Which of the following is homogeneous equation of degree 2
(a)
(b)
(c)
(d)
Ans : (b)
15) One of the solutions of is
(a)
(b)
(c)
(d) None of the above
Ans : (a)
16) Solutions of partial differential equation by separation of
variables is given by:
(a) u(x,y)=
(b) u(x,y)=
(c) u(x,y) =
(d) u(x,y)=
Ans : (b)
17) The C.F. of is
(a)
(b)
(c)
(d)
Ans : (d )
18) The CF of is
(a)
3 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(b)
(c)
(d)
Ans : (b)
19) Solutions of partial differential equation by separation of variables is
given by:
( )
(a)
( )
(b)
(c)
(d)
Ans : (a)
20) The C.F. of is:
(a)
(b)
(c)
(d)
Ans : (d )
21) P.I. of is:
(a)
(b)
(c)
(d)
Ans : (c)
22) Solutions of partial differential equation by separation of variables is given
by:
(a)
(b)
(c)
(d)
Ans : (a)
23) When solving one-dimensional heat equation using a variable separable method we
get the solution if:
(a) k is positive
(b) k is negative
(c) k is zero
(d) It can be anything
Ans : (b)
4 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
24) One solution of is
(a)
(b)
(c)
(d)
Ans : (a)
25) What is another name for heat equation:
(a) Induction equation (b) Condenser Equation
(c) Diffusion equation (d) Solar equation
Ans : (c)
26) The solution of first order partial differential equation is given by:
(a)
(b)
(c)
(d) All of the above.
Ans : (d )
27) While solving a partial differential equation using a variable separable method, we
get the ratio to a constant, which:
(a) can be +ve/-ve integer or zero
(b) can be +ve/-ve rational or zero
(c) must be a +ve integer
(d) must be a -ve integer
Ans : (b)
28) One of the solutions of is given by:
(a) (b)
(c) (d) None of these
Ans : (b)
29) The C.F. of is:
(a)
(b)
(c)
(d)
Ans : (d )
30) P.I. of is:
(a)
(b)
(c)
(d)
5 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
Ans : (a)
31) Partial differential equation of one-dimensional heat equation is :
(a)
(b)
(c)
(d)
Ans : (a)
32) P.I. of is:
(a)
(b)
(c)
(d)
Ans : (d )
33) P.I. of is
(a)
(b)
(c)
(d)
Ans : (b)
34) For a partial differential equation in a function and two variables ; what
is the form obtained after separation of variables is applied:
(a)
(b)
(c)
(d)
Ans : (d )
35) The solution of partial differential equation by separation of
variables is given by:
(a)
(b)
(c)
(d)
Ans : (d )
36) One of the solution of is:
(a)
(b)
6 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(c)
(d) None of the above
Ans : (c)
37) If 1,1,4 are the three roots of the partial differential equation, then its C.F. is :
(a)
(b)
(c)
(d)
Ans : (a)
38) The C.F. of partial differential equation is:
(a)
(b)
(c)
(d)
Ans : (d )
39) If 2,3,4 are three roots of partial differential equation then its C.F. is:
(a)
(b)
(c)
(d)
Ans : (c)
40) --------------- s called Lagra ge’s equat o :
(a) (b)
(c) (d)
Ans : (a)
41) One of the solutions of is:
(a)
(b)
(c)
(d) None of the above
Ans : (a)
42) P.I. of partial differential equation is:
(a)
(b)
(c)
(d)
Ans : (c)
43) The first order partial differential equation is:
(a)Linear Equation
(b) Semi Linear Equation
7 Mathematics-Ganit Sangrah | Satish Tiwari
MCQs on Partial Differential Equation | RTMNU Winter-2020
(c) Quasi liner equation
(d) Non-Linear Equation
Ans : (d )
44) The P.I. of ta is:
(a) ta
(b) ta
(c) ta
(d) ta
Ans : (d )
(45) What is the C.F. of
(a)
(b)
(c)
(d)
Ans : (b)
(45) What is the degree of homogenous partial differential equation
(a)
(b)
(c)
(d)
Ans : (b)
8 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020
1. The real part of the complex no. z = x + iy is given by
̅
A) – ̅ B)
̅
B) D) ̅
Ans : ( A)
2. If then real part of is
A) –
B) –
C)
D) –
Ans : ( A)
dz
3. The value of z 2dz , where C is a circle given by | z 2 | 1
c
A) 0
B)
C)
D)
Ans : ( B)
z2
4. The value of c z 4 1dz using Cauchy’s Integral around the circle
where is
A) B)
C) D)
Ans : ( B)
5. If and f(z) is analytic function then is
A)
B)
C)
D) None of these
Ans : ( B)
1
6. The value of 1 z
c
2
dz where C is the contour |z - = 1 is
A). B)
C) D)
Ans : ( B)
1 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
7. The complex no. z x iy which satisfy the equation lie on
A) A circle with as the center & redius 1
B) A circle with as the center & radius 1
C)
D)
Ans : ( B)
8. An analytic function of a complex variable is expressed as . If
then
A)
B)
C)
D)
Ans : (C )
9. Product of complex no.s & is
A)
B)
C)
D)
Ans : ( D)
10. Using Cauchy’s Integral theorem, the value of the integral (integration being taken in
z3 6
contour clock-wise direction) c 3z i dz ,C is is
A)
B)
C)
D)
Ans : ( A)
1
11. The value of the contour integral 2
| z i| 2
z 4
dz in the positive sence is
A)
B)
C)
D)
Ans : ( B)
2 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
12. For an analysis function is given by Then
A)
B)
C)
D)
Ans : ( D)
13. Cauchy-Reimann Equation in cartesian form are
A)
B)
C)
D)
Ans : ( D)
3z 4
14. The value of the integral z
c
2
4z 5
dz where C is the circle is given by
A) 0
B) ⁄
C) ⁄
D) 1
Ans : ( A)
15. If is a pole of order 1, the residue of at is given by
A) li
B) li
C) li
D) li
Ans : (C )
16. If where are real, then the value of is
A) 1
√
B)
C)
D)
Ans : ( A)
17. If then the poles of are
A)
B)
C)
D)
3 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
Ans : ( B)
z4
18. The value of z
c
2
2z 5
dz , where C is a circle given by is
A) 0
B)
C)
D)
Ans : ( A)
19. Which of the following is the real part of the function at
A) -3
B) 3
C) 5
D) 2
Ans : ( B)
20. If is analytic inside and on the boundary of a closed curve C, except at some poles,
then
A) f ( z)dz 0
c
B) f ( z)dz 0
c
C) f ( z)dz 2i (sum of the residue at the poles within C)
c
D) f ( z)dz (sum of the residue at the poles within C)
c
Ans : (C )
21. If is an analytic function then is
A)
B)
C)
D)
Ans : ( B)
22. The function log is not analytic at point
A)
B)
C)
D)
Ans : ( A)
4 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
23. If is a pole of order I is an Analytic function inside & on the closed curve C,
‘a’ is any point within C then the Cauchy’s Integral For ula is given by
f ( z) 2i ( n )
A) ( z a)
c
n 1
dz
n!
f (a)
f ( z) 2i ( n )
B) ( z a)
c
n 1
dz
n!
f (a)
f ( z) 2i ( n )
C) ( z a)
c
n 1
dz
n!
f (a)
f ( z) 2i
D) ( z a)
c
n 1
dz
(n 1)!
f ( n ) (a)
Ans : (C )
24. For the function of a complex variable z, the point is
A) A pole of order 3
B) A pole of order 2
C) A pole of order 1
D) Not a singularity
Ans : ( B)
25. If then which of the poles will lies inside the region
A) only
B) only ⁄
C) ⁄
D) ⁄
Ans : ( B)
26. If then the residue of at z=3 is
A) ⁄
B) 0
C) 4
D) ⁄
Ans : ( A)
27. If then the residue of at z=2 is
A) 0
B) -2
C) -3
D) 1
Ans : (C )
5 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
28. For the function of complex variable z, the point z=0 is
A) A pole of order 3
B) A pole of order 2
C) A pole of order 1
D) Not a singularity
Ans : ( A)
29 .If where are real, then the value of is
A) 1
√
B)
C)
D)
Ans : ( A)
can be represented as
A)
B)
C)
D)
Ans : ( D)
31. Which of the following is/are used to solve Line Integrals?
A) Cauchy’s Integral For ula
B) Cauchy’s Integral Theore
C) Cauchy’s Residue Theore
D) All of there
Ans : ( D)
32.The residue of the function at z=2 is
A) -1/32
B) -1/16\
C) 1/16
D) 1/32
Ans : ( A)
32. The analytic function has singularities at
A) 1 & -1
B) 1&i
C) 1 & -i
D) i & -i
Ans : ( D)
6 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
34. The residue of a complex function at its poles are
A)
B)
C)
D)
Ans : (C )
35. The residue of the function at it’s pole is
A)
B)
C)
D)
Ans : ( A)
36. where √ is given by
A) 0
B) ⁄
C)
D) 1
Ans : ( B)
37. The value of the integral ∫ evaluated around the circle is _______
A)
B)
C)
D)
Ans : ( B)
38. C-R equation in polar form are
A)
B)
C)
D)
Ans : ( A)
39. Which of the following is a type of singularity
A) Isolated Singularity
B) Removable Singularity
C) Essential Singularity
7 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
D) All of these
Ans : ( D)
40. If then which of the poles will lies inside the region
A)
B)
C)
D)
Ans : ( D)
41. If then the pole of are
A)
B)
C)
D)
Ans : ( A)
42. A closed curve represents a ________
A) Ellipse
B) Circle
C) Triangular Region
D) Rectangular Region
Ans : ( B)
43. If then the pole of f(z) are
A)
B)
C)
D)
Ans : ( A)
44. If is analytic inside and on the boundary of a closed curve C, then f ( z)dz ...
c
A)
B)
C) 1
D) 0
Ans : ( D)
8 Mathematics-Ganit Sangrah | Satish Tiwari
[MCQs on Complex Analysis (Functions of Complex Variables) Winter-2020]
45. If then the residue of at z=-1 is
A) ⁄
B) 3
C) 0
D) ⁄
Ans : ( B)
46. If then real part of is
A) –
B) –
C)
D) –
Ans : ( D)
47. A function is analytic function if _______
A) Real part of f(z) is analytic
B) Imaginary part of f(z) is analytic
C) Both real & imaginary part of f(z) is analytic
D) None of these
Ans : (C )
48. The value of ∮ C is a circle given by |z| = 3, is
A) 0 B) 2 C) 2 D) 2
Ans : ( B)
9 Mathematics-Ganit Sangrah | Satish Tiwari
UNIT-II: NUMERICAL METHODS FOR DIFFERENTIAL EQUATION
𝑑𝑦
1) If 𝑑𝑥 = 𝑓(𝑥, 𝑦) 𝑎𝑡 𝑦(𝑥0 ) = 𝑦0 , then the first approximate value of 𝑦1 by Euler’s method is
A) 𝑦1 = 𝑦0 + ℎ𝑓(𝑥0 , 𝑦0 ) B) 𝑦1 = 𝑦0 + 4ℎ𝑓(𝑥0 , 𝑦0 )
C) 𝑦1 = 𝑦0 − ℎ𝑓 (𝑥0 , 𝑦0 ) D) 𝑦1 = 𝑦0 + 2ℎ𝑓(𝑥0 , 𝑦0 )
Ans: A
𝑑𝑦
2) If 𝑑𝑥 = 𝑓(𝑥, 𝑦) 𝑎𝑡 𝑦(𝑥0 ) = 𝑦0 then the first approximate value of 𝑦1 by Euler’s modified method is
(1) 3ℎ (1) ℎ
A) 𝑦1 = 𝑦0 + [𝑓 (𝑥0 , 𝑦0 ) + 𝑓(𝑥0 + ℎ , 𝑦1 )] B) 𝑦1 = 𝑦0 + 2 [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 + ℎ , 𝑦1 )]
2
(1) ℎ (1) ℎ
C) 𝑦1 = 𝑦0 + 2 [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 , 𝑦1 )] D) 𝑦1 = 𝑦0 + 2 [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 , 𝑦1 )]
Ans: B
3) Which of the following represents Taylor’s series expansion of 𝑦 = 𝑓(𝑥 )𝑎𝑡 𝑥 = 𝑎
𝑥2 2 𝑥2
A) 𝑦(𝑥) = 𝑥1 𝑦0′ + 𝑦0′′ + ⋯ B) 𝑦(𝑥 ) = 𝑦(0) + 𝑥𝑦 ′ (0) + 𝑦 ′′ (0) + ⋯
2! 2!
𝑥2 𝑥2
C) 𝑦(𝑥 ) = 𝑥𝑦 ′ (0) + 𝑦 ′′ (0) + ⋯ D) 𝑦(𝑥 ) = 1 + 𝑥𝑦 ′ (0) + 𝑦 ′′ (0) + ⋯
2! 2!
Ans: B
𝑑𝑦
4) If 𝑑𝑥 = 𝑥 2 𝑦 − 1 , 𝑦(0) = 1 then the value of 𝑦 ′′′ (0) is
A) -1 B) -2
C) 2 D) 3
Ans: C
5) In which of the following method, we approximate the curve of solution by the tangent in each interval?
A) Picard’s method B) Euler’s method
C) Newton’s method D) Runge-Kutta method
Ans: B
𝑑𝑦
6) Using Euler’s method for = 𝑙𝑜𝑔10 (𝑥 + 𝑦) , 𝑦(1) = 2 , the value of 𝑦(1.2) is
𝑑𝑥
A) 2.0854 B) 2.384
C) 2.0954 D) 1.288
Ans: C
𝑑𝑦
7) Using Euler’s method for = 𝑥 + √𝑦 , 𝑦(0) = 1 , the value of 𝑦(0.2) is
𝑑𝑥
A) 0.4 B) 0.2
C) 1.2 D) 0.2421
Ans: C
8) Taylor series method will be very useful to give some ______ for powerful numerical methods.
A) Initial value B) Final value
C) Initial starting value D) Middle value
Ans: C
9) Find (𝑥0 , 𝑦0 ) , given that 𝑦 ′ = 𝑥 + 𝑦 , 𝑦(0) = 2 using Taylor series formula
A) (1,2) B) (2,1)
C) (0,2) D) (2,0)
Ans: C
10) 𝑦𝑛+1 = 𝑦𝑛 + ℎ 𝑓(𝑥𝑛 , 𝑦𝑛 ) is the iterative formula for
A) Euler’s method B) Taylor series method
C) Picard’s method D) Milne’s method
Ans: A
𝑑𝑦
11) Using Euler’s method for = 𝑦, 𝑦(0) = 1 the value of 𝑦(0.1) is
𝑑𝑥
A) 1.125 B) 1.105
C)1.128 D) 2.235
Ans: B
𝑑𝑦
12) Using Euler’s method for = 𝑦 + 𝑒 𝑥 , 𝑦(0) = 0 , the value of 𝑦(0.2) is
𝑑𝑥
A) 0.4 B) 0.2421
C) 1.2 D) 0.2
Ans: D
13) Which of the following methods is not used to find the solution of first order D.E.?
A) Euler’s Method B) Euler’s Modified Method
C) Taylor’s Series Method D) Regula Falsi Method
Ans: D
19 Q. The Taylor’s series of 𝑓(𝑥) is
𝑥2
a) 𝑓 (𝑥 + ℎ) = 𝑓(ℎ) + 𝑥𝑓 ′ (ℎ) + 𝑓"(ℎ) + ⋯ ⋯ ⋯
2!
ℎ2
b) 𝑓 (𝑥 + ℎ) = 𝑓(𝑎) + ℎ𝑓 ′ (𝑎) + 2!
𝑓"(𝑎) + ⋯ ⋯ ⋯
𝑥2
c) 𝑓 (𝑥 ) = 𝑓 (0) + 𝑥𝑓 ′ (0) + 𝑓"(0) + ⋯ ⋯ ⋯
2!
𝑎2 ℎ2
d) 𝑓 (𝑥 ) = 𝑓 (0) + ℎ𝑎𝑓(0) + 𝑓"(0) + ⋯ ⋯ ⋯
2!
Ans. c
dy
20Q. If f ( x, y ) at y(x0) = y0, then Predictor value
dx
y4 by Milne’s corrector formula is
2ℎ 4ℎ
a) 𝑦4 = 𝑦2 + (𝑓2 + 4𝑓3 + 𝑓4 (1) ) b) 𝑦4 = 𝑦0 + (2𝑓1 − 𝑓2 + 2𝑓3 )
3 3
ℎ 4ℎ
c) 𝑦4 (1) = 𝑦2 + 3 (𝑓2 + 4𝑓3 + 𝑓4 (1) ) d) 𝑦4 (1) = 𝑦0 + (2𝑓1 − 𝑓2 + 2𝑓3 )
3
Ans: b
dy
21Q.If f ( x, y ) then the value of k at x = x1 by Runge-Kutta method is
dx
1
a) y1 y0 k1 2k2 2k3 k4 b) 𝑘 = (𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4 )
6
1 1
c) 𝑘 = 6 (𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4 ) d) 𝑦1 = 𝑦0 + 6 (𝑘1 − 2𝑘2 + 2𝑘3 − 𝑘4 )
Ans: c
22Q.While solving the ordinary differential equation
𝑑𝑦
= 𝑥𝑦 + 𝑦 2 by Milne’s Predictor & Corrector
𝑑𝑥
method, given y (0) = 1, h = 0.1, then f0 =
a) 0.1 b) 0.2
c) 1 d) 1.5
Ans: c
d2y
23Q. If xy 4 y 0 using Runge-Kutta method, it is first reduced to
dx 2
dy dz dy dz
a) z, xy 4 y b) xy 4 y, z
dx dx dx dx
dy dz dy dz
c) z , xy 4 y d) z, 4 xy y
dx dx dx dx
Ans: c
𝑑𝑦 𝑥 2 +𝑦 2
24Q.To solve the ordinary differential equation 𝑑𝑥 = 10
by Runge - Kutta method of fourth order, given y (0) = 1,
h = 0.1, then k1 =
a) 0 b) 0.01 c) 0.02 d) 0.1
Ans: b
25Q. Euler Method for solving the differential equation
dy
f ( x, y ) is specified by
dx
a) y n1 y n h f ( xn , y n )
b) y n1 h f ( xn , y n )
c) y n1 y n h f ( xn1 , y n1 )
d) y n1 y n f ( xn , y n )
Ans: a
26Q. Which of the following method is one step method?
a) Taylor’s series method
b) Euler’s modified method
c) Runge Kutta method
d) Milne’s predictor corrector method
Ans: a
𝑑2 𝑦 dy
27Q. For solving D. E. = f (x, y, ) using Runge Kutta method ,
𝑑𝑥 2 dx
it is first reduced to
a) first order D. E. b) simultaneous linear equation
c) simultaneous first order D. E. d) Higher order O. D. E.
Ans c
28Q. While solving the ordinary differential equation
𝑑𝑦
= 𝑙𝑜𝑔10 (𝑥 + 𝑦) by Euler’s modified method, given y (1) = 2,
𝑑𝑥
h = 0.2, then predictor y1 =
a) 1.0.954 b) 2.0954
c) 2 d) None of these
Ans: b
29Q. Runge Kutta method is used to find
approximate solution of …….
a) Simultaneous differential equations
b) First order first degree differential equation
c) Second order differential equation
d) All options are correct
Ans: d
𝑑𝑦
30Q. Consider the initial value problem 𝑑𝑥 = 2 + ⌊√𝑥𝑦⌋ , y (0) = 1, find y(1.6).
The above problem can be solved in how many iterations?
a) 1 b) 2
c) 3 d) all of these
Ans d
***
MCQ U-I Numerical Methods
1. If the root of the equation x log 10 x 1.2 0 lying between 2.7 and 2.8 then
first approximation to root using Regulafalsi Method is given by
(a) 2.7202 (b)2.7406 (c) 2.7605 (d)2.7806
Ans. (b)
2.The formula used for solving the equation using Regula Falsi method is
𝑎𝑓(𝑏)−𝑏𝑓(𝑎)
𝑥= 𝑓(𝑏)−𝑓(𝑎)
a)True b) False
c) None of above d) 0
Ans (a)
3. Find the positive root of the equation 3x-cosx-1=0 using Regula Falsi method and
correct up to 4 decimal places.
a)0.6701 b)0.5071 c)0.6071 d)0.5701
Ans(c)
4. Newton – Raphson formula of successive approximation to find the approximate value of a root
of the equation f ( x) 0 is
f ( xn ) f ( xn )
(a) Xn-1 = xn (b) Xn-1 = xn
f ( xn ) f ( xn )
f ( xn ) f ( xn )
(c) Xn-1 = xn (d) Xn-1 = xn
f ( xn ) f ( xn )
Ans. (a)
5. Newton – Raphson method is used to calculate 3 65 by solving x3 = 65 . If x0 = 4 is taken as
Initial approximation , then the first approximation x1 is
(a) 65/16 (b) 131/32 (c) 191/48 (d) 193/48
Ans. (d)
6. If 3 is taken as an approximate root of the equation x 30 0 , then a better approximate by
3
Newton – Raphson method is
(a) 10/3 (b) 28/9 (c) 8/3 (d) 26/9
Ans. (b)
7. The order of convergence in Newton-Raphson method is
(a) 2 (b) 3 (c) 0 (d) None of these
Ans. (a)
8. According to Gauss –sedial method which process of representation of x,y,z is
correct for the equation x+7y-3z = -22, 5x-2y+3z =18,2x-y+6z = 22
1 1 1
(a)𝑥 = 5 {18}𝑦 = 7 {−22}, 𝑧 = 6 {22}
1 1 1
(b)𝑥 = 5 {18 + 2𝑦 − 3𝑧}𝑦 = 7 {−22 − 𝑥 + 3𝑧}, 𝑧 = 6 {22 − 2𝑥 + 𝑦}
1 1 1
(c)𝑥 = 5 {18 + 2𝑦 − 𝑧}𝑦 = 7 {−22 − 𝑥 + 𝑧}, 𝑧 = 6 {22 − 𝑥 + 𝑦}
1 1 1
(d)𝑥 = 5 {12𝑦 − 3𝑧}𝑦 = 7 {−𝑥 + 3𝑧}, 𝑧 = 6 {−2𝑥 + 𝑦}
Ans(b)
9 . Solve the following equations using Crout’s Method to find the value of x.
x+y+z = 7 : x+2y+3z=16 : x+3y+4z = 22
a) 3 b) 7 c) 0 d) 1
Ans( d)
10. Which is correct Lower triangular matrix for
3x 2 y 7 z 4, 2 x 3 y z 5, 3x 4 y z 7 using Crout’s Method.
2 7 2 7 2 7
1 3 3 1 3 3 3 0 0 1 3 3
11 11 5 11
(a) 0 1 (b) 0 1 (c) 2 0 (d) 0 1
5 5 3 5
0 0 1 0 0 1 3 2
8 0 0 1
5
Ans. (c)
𝑑𝑦
11. Taylor series for the differential equation 𝑑𝑥 = 𝑥 2 − 𝑦, 𝑔𝑖𝑣𝑒𝑛𝑦(0) = 1 𝑖𝑠
𝑥3 𝑥4 𝑥3 𝑥4
(a) 𝑦 = 1 − 𝑥 + − + − − − (b) 𝑦 = 1 + 𝑥 + + + −−−
3 4 3 4
𝑥3 𝑥4 𝑥3 𝑥4
(c) 𝑦 = 1 − 𝑥 − − − − − − (d) 𝑦 = 𝑥 + − + −−−
3 4 3 4
Ans(a)
12. Which of the following represents Taylors series expansion of 𝑦 = 𝑓 (𝑥 )𝑎𝑡𝑥 = 𝑎
𝑥2 2 𝑥2
a) 𝑦(𝑥 ) = 𝑥1 𝑦0′ + 𝑦0′′ + ⋯b)𝑦(𝑥 ) = 𝑦(0) + 𝑥𝑦 ′ (0) + 𝑦 ′′ (0) + ⋯
2! 2!
𝑥2 𝑥2
c) 𝑦(𝑥 ) = 𝑥𝑦 ′ (0) + 𝑦 ′′ (0) + ⋯d) 𝑦(𝑥 ) = 1 + 𝑥𝑦 ′ (0) + 𝑦 ′′ (0) + ⋯
2! 2!
Ans(b)
𝑑𝑦
13. If = 𝑥 2 𝑦 − 1 , 𝑦(0) = 1 then the value of 𝑦 ′′′ (0) is
𝑑𝑥
a) -1 b) -2 c) 2 d) 3
Ans(c)
𝑑𝑦
14. Eular’s modified method for the differential equation 𝑑𝑥 = 𝑓(𝑥, 𝑦), 𝑔𝑖𝑣𝑒𝑛𝑦(𝑥0 ) = 𝑦0
is given by
ℎ
(a)𝑦𝑛+1 = 2 {𝑓 (𝑥𝑛 , 𝑦𝑛 ) + 𝑓 (𝑥𝑛+1 , 𝑦𝑛+1 )
(b)𝑦𝑛+1 = 𝑦𝑛 + {𝑓 (𝑥𝑛 , 𝑦𝑛 ) + 𝑓 (𝑥𝑛+1 , 𝑦𝑛+1 )
ℎ
(c)𝑦𝑛+1 = 𝑦𝑛 + 2 {𝑓 (𝑥𝑛 , 𝑦𝑛 ) + 𝑓(𝑥𝑛+1 , 𝑦𝑛+1 )
ℎ
(d)𝑦𝑛+1 = 𝑦𝑛 + 2 {𝑓 (𝑥𝑛+1 , 𝑦𝑛+1 )
Ans c
𝑑𝑦
15. If 𝑑𝑥 = 𝑓 (𝑥, 𝑦) 𝑎𝑡 𝑦(𝑥0 ) = 𝑦0 then the first approximate value of 𝑦1
by Eulers modified method is
(1) 3ℎ
a) 𝑦1 = 𝑦0 + [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 + ℎ , 𝑦1 )]
2
(1) ℎ
b) 𝑦1 = 𝑦0 + 2 [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 + ℎ , 𝑦1 )]
(1) ℎ
c) 𝑦1 = 𝑦0 + 2 [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 , 𝑦1 )]
(1) ℎ
d) 𝑦1 = 𝑦0 + 2 [𝑓(𝑥0 , 𝑦0 ) + 𝑓(𝑥0 , 𝑦1 )]
Ans(b)
𝑑𝑦
16. To solve the ordinary differential equation 3𝑑𝑥 + 5𝑦 2 = 𝑠𝑖𝑛𝑥 , 𝑦(0) = 5
by Euler’s method, you need to rewrite the equation as
𝑑𝑦 𝑑𝑦 1
a)𝑑𝑥 = 𝑠𝑖𝑛𝑥 − 5𝑦 2 , 𝑦(0) = 5 b)𝑑𝑥 = 3 (𝑠𝑖𝑛𝑥 − 5𝑦 2 ) , 𝑦(0) = 5
𝑑𝑦 1 𝑑𝑦 1
c) = 5 (𝑠𝑖𝑛𝑥 − 5𝑦 2 ) , 𝑦(0) = 5 d)𝑑𝑥 = 3 (𝑐𝑜𝑠𝑥 − 5𝑦 2 ) , 𝑦(0) = 5
𝑑𝑥
Ans(b)
𝑑𝑦
17. Using Euler’s method for𝑑𝑥 = 𝑙𝑜𝑔10 (𝑥 + 𝑦) , 𝑦(1) = 2 , the value of 𝑦(1.2) is
a) 2.0854 b) 2.384 c) 2.0954 d) 1.288
Ans(c)
𝑑𝑦
18. Using Euler’s method for𝑑𝑥 = 𝑥 + √𝑦 , 𝑦(0) = 1 , the value of 𝑦(0.2) is
a) 0.4 b) 0.2 c) 1.2 d) 0.2421
Ans(c)
19. Euler’s modified formula is given by,
(a) y n 1 y0
h
f ( xn , yn ) f ( xn1 , yn1 )(b) y n1 yn h f ( xn , yn ) f ( xn1 , yn1 ) (c)
2 2
y n 1 yn
h
f ( xn , yn ) f ( xn1 , yn1 ) (d) y n1 y0 f ( xn , yn ) f ( xn1 , yn1 )
2
Ans.(c)
20. What is the value of k1 using Runge-Kutta Method of fourth order
dy y x
for the differential equation , y (0) 1 and h 0.2
dx y x
(a) 0.1 (b)0.2 (c) 0.3 (d) 0.4
Ans.(b)
𝑑𝑦
21. If = 𝑓(𝑥, 𝑦), then the value of y at x = 𝑥1 by Runge-Kutta four order method is:
𝑑𝑥
a) 𝑦 = 𝑦0 + (𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4 ) b) 𝑦 = 𝑦1 + (𝑘1 − 2𝑘2 + 2𝑘3 − 𝑘4 )
1 ℎ
c) 𝑦1 = 𝑦0 + 6 (𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4 ) d)𝑦1 = 𝑦0 + 6 (𝑘1 − 2𝑘2 + 2𝑘3 − 𝑘4 )
Ans(c)
𝑑𝑦
22. If = 𝑥𝑦 + 𝑦 2 𝑎𝑡𝑦(0) = 1 and 𝑘1 = 0.1, 𝑘2 = 0.1155, 𝑘3 = 0.1172, 𝑘4 = 0.13598,
𝑑𝑥
then the first approximate value of 𝑦 at x=0.1 by Rungekutta forth order formula is
a) 1.277 b) 1.1359 c) 1.1168 d) 1.1270
Ans(c)
𝑑𝑦
23. If𝑑𝑥 = 𝑓 (𝑥, 𝑦) 𝑎𝑡 𝑦(𝑥0 ) = 𝑦0 , then the first approximate value of 𝑦4
by Milnes corrector formula is
(1) ℎ (1) (1) ℎ (1)
a) 𝑦4 = 𝑦2 − 3 (𝑓2 + 4𝑓3 + 𝑓4 )b) 𝑦4 = 𝑦2 + 3 (𝑓2 + 4𝑓3 + 𝑓4 )
(1) 4ℎ (1) (1) ℎ (1)
c) 𝑦4 = 𝑦0 + (𝑓2 − 4𝑓3 + 𝑓4 )d) 𝑦4 = 𝑦1 + 4 3 (𝑓2 − 4𝑓3 − 𝑓4 )
3
Ans(b)
dy 1
24. Given (1 x 2 ) y 2 , given y (0) 1, y (0.1) 1.06, y (0.2) 1.12, y (0.3) 1.21,
dx 2
f (0,1) 0.5, f (0.1,1.06) 0.5674, f (0.2,1.12) 0.6522, f (0.3,1.21) 0.7980,
then by Milne’s predictor method y (0.4) is to be obtained as
(a) 1.2772 (b) 1.2798 (c) 1.3808 (d) 1.4043
Ans.(a)
𝑑𝑦
25. If = 𝑓 (𝑥, 𝑦)and y (𝑥0 ) =𝑦0 , then the value of 𝑦4 at x = 𝑥4 by Milnes Predictor formula is:
𝑑𝑥
4ℎ ℎ
a) 𝑦4 = 𝑦0 + (2𝑓1 − 𝑓2 + 2𝑓3 )b) 𝑦4 = 𝑦0 + (2𝑓1 − 𝑓2 + 2𝑓3 )
3 3
4ℎ ℎ
c) 𝑦4 = 𝑦0 − (2𝑓1 − 𝑓2 + 2𝑓3 )d)𝑦4 = 𝑦0 − 3 (2𝑓1 − 𝑓2 + 2𝑓3 )
3
Ans(a)
26. Using Milnes predictor-corrector method solve the following differential equation
𝑑𝑦
= 2𝑒 𝑥 − 𝑦𝑎𝑡𝑥 = 0.4 𝑔𝑖𝑣𝑒𝑛𝑦(0) = 2 , 𝑦(0.1) = 2010, 𝑦(0.2)
𝑑𝑥
= 2.040, 𝑦(0.3) = 2.090, 𝑦(0.4) =
(a)2.1621 (b)2.1001 (c)2.0010 (d)2
Ans(a)
−4 5
27. Largest eigen value of the matrix A=[ ] is
1 2
(a) 3, (b) 0 (c) -3 (d) 2
Ans (c)
Multiple Choice Questions on U-VI Probability Distribution
1. In a Binomial distribution , if ' n ' is the is the number of trials and ' p ' is the
probability of success, then the mean value is given by ___________
A) n p B) n C) p D) n p 1 p
2. In a Binomial Distribution, if p , q and n are probability of success, failure
and number of trials respectively then variance is given by __________
A) n p B) n p q C) n p 2 q D) n p q 2
3. The probability of getting between 2 heads to 4 heads in 10 tosses of fair
coin using Binomial distribution is
A) 2.215 B) 1.587 C) 0.366 D) 0.712
4. Out of 800 families with 5 children each , how many would you expect to
have 3 boys using Binomial distribution if equal probabilities for boys and
girls is assumed ?
A) 100 B) 200 C) 300 D) 250
5. If the random variable X has the probability function
x e
f ( x) P( X x) , x 0 ,1, 2 , .... where is given positive constant ,
x!
then X is said to be
A) Binomially distributed B) Normally distributed
C) Poisson distributed D) None of these
6.In a Poisson distribution, if mean 1 then P ( X 1) ?
A) 1 e B) e 2 C) e 2 D) e
7. If is the mean of Poisson distribution , then P( X 0 ) is equal to
A) e B) e C) e D) e
8. The discrete probability distribution in which the outcome is very small
with a very small period of time is classified as
A) Binomial distribution B) Cumulative distribution
C) Normal distribution D) Poisson distribution
9. If the outcome of a discrete random variable follow a Poisson distribution ,
then which of the following is true ?
A)The mean equals the variance
B) The mean equals the standard deviation
C)The median equals the variance
D) The median equals the standard deviation
10. In a Poisson distribution, if mean 3 then P ( X 2 ) ?
A) 0.3456 B) 1.2450C) 0.2241 D) 0.0417
11. In a normal distribution ,the highest point on the curve occurs at the mean
, which is also the
A) Median and mode B) Geometric mean and harmonic mean
C) Lower and upper quartiles D) Variance and standard deviation
12. Normal Distribution is applied for ___________
A) Continuous Random Variable B) Discrete Random Variable
C) Uncertain Random Variable D) None of these
13. Normal Distribution is symmetric is about ___________
A) Variance B) Mean
C) Standard deviation D) Covariance
14. The graph of standard normal curve is always
A) Flat B) Bell shaped C) Circular D) Spiked
15. In a normal distribution , if Z is the standardized variable then Z = ?
X X
A) B)
X X
C) D)
16. The limiting form of the Binomial distribution as n and p 0 , in
such a way that n p , where is fixed positive number is
A) Normal distribution B) Poisson distribution
C) Exponential distribution D) Uniform distribution
17. The range of normal distribution is
A) 0 ton B) 0 to C) 1 to + 1 D) to
18. For a Binomial distribution the relation between mean and variance is
A) Mean < Variance B) Mean> Variance
C) Mean = Variance D) None of these
19. The shape of the normal curve depends on its ________
A) Mean deviation B) Standard deviation
C) Quartile deviation D) Correlation
20. In Binomial distribution , if 8 is the number of trials for an event and
probability of success is 0.5 , then mean is
A) 2 B) 4 C) 6 D) 8
21. In Binomial distribution , if 4 is the number of trials for an event and
probability of failure is 0.5 , then variance is
A) 0 B) 2 C) 1 D) 4
22. In Poisson distribution , if P ( 1 ) P ( 2 ) then mean is
A) 2 B) 1 2 C) 1 D) 3
23. The normal probability density function curve is symmetrical about the
mean i.e. the area to the right of the mean is the same as the area to
the left of the mean. This means P X P X is equal to
A) 0 B) 1 C) 0.5 D) 0.25
24. Which of the following is not property of a Binomial distribution ?
A) Probability of success remains constant
B) n is fixed
C) Successive trials are dependent
D) It has two parameters
25. The Binomial probability distribution is symmetrical when
A) p q B) p q C) p q D) n p n p q
26. Mean and variance of standard normal distribution is
A) 1 & 0 B) 0 & 1 C) 0 & D) & 1
27. Which of the following statement is correct ?
A) The mean of the Poisson distribution (with parameter ) equals the
mean of the Exponential distribution (with parameter ) only
when 1
B) Poisson distribution and Noramal distribution are having same
characteristics.
C) The Exponential distribution is continuous and defined in the
interval ,
D) The Binomial distribution has equal mean and variance only when
p 0.5
28. Which of the following distribution is suitable to measure the length of
time that elapses between the arrival of cars at a petrol station pump ?
A) Binomial B) Normal C) Exponential D) Poisson
29. The theorem which states that as the sample size increases the
sampling distribution must approach the normal distribution is
classified as
A) Limited approximation theorem
B) Primary limit theorem
C) Central mass theorem
D)Central limit theorem
e x , x 0
30. A continuous random variable having density function f ( x)
0 , x 0
is said to be
A) Exponentially distributed
B) Binomially distributed
C)Normally distributed
D) None of these
31. Assume that, you usually get 2 phone calls per hour , calculate the
probability, that a phone call will come within the next hour.
A) 0.3935 B) 1 C) 0.01 D) 2.413
32. Which of the following statements are true about exponential
distribution ?
A) Exponential distribution is a continuous probability distribution that
often concerns the amount of time until some specific event happens.
B)The key property of the exponential distribution is memoryless as the
past has no impact on its future behavior.
C) The mean of the exponential distribution is 1/λ and the variance of the
exponential distribution is 1/λ2.
D) All of above statements are true.
33. If X is a normally normally distributed variable with mean μ = 30 and
standard deviation σ = 4 then P(x < 40) is
A) 0.9938 B) 0.3944 C) 0.2345 D) 0.5
34. If only 3 students came to attend class today, find the probability for
exactly 4 students to attend the classes tomorrow using Poisson
distribution for e = 2.71828 (approximately)
A)0.16803 B) 0.73452 C) 1.25789 D) 4.56850
35. If a coin is tossed 10 times then what is the chances of getting exactly 6 heads
using Binomial distribution ?
A)0.362144 B)0.205078 C)0.360678 D) None
1. For two correlated variables x and y , if coefficient of correlation between x
and y is 0.8014 , variance of x and y are 16 and 25 respectively. Then the
covariance between x and y is
A) 162.08 B) 16.028 C) 160.28 D) 16.208
2. If two regression lines are 5 y − 8 x + 17 = 0 and 2 y − 5 x + 14 = 0 then the mean
values of x and y i.e. x , y are
A) x = 4 , y = 3 B) x = 3 , y = 4
C) x = 1 , y = 2 D) x = 2 , y = 3
3. If the values of two correlated variables move in the opposite direction,
i.e. one increasing and other decreasing _________
A) The correlation is said to be linear
B) The correlation is said to be non-linear
C) The correlation is said to be positive
D) The correlation is said to be negative
4. A process by which we estimate the value of dependent variable on the basis
of one or more independent variables is called
A) Correlation B) Regression C) Residual D) None of these
5. The median of the data given below is 31 , 37 , 43 , 42 , 25 , 46 , 45 , 39 , 32
A) 39 B) 42 C) 32 D) 43
6. Which of the following divides a group of data into four equal parts ?
A) Deciles B) Percentiles C) Quartiles D) Standard Deviation
7. The mean deviation for the data 5 , 3 , 7 , 8 , 4 , 9 from mean is
A) 2 B) 3 C) 4 D) 7
8. The first quartile Q 1 for the data 20 , 30 , 25 , 23 , 22 , 32 , 36 is
A) 20 B) 22 C) 25 D) 32
9. The mode from the following data is
Age 0-6 6-12 12-18 18-24 24-30 30-36 36-42
Frequency 6 11 25 35 18 12 6
A) 22.20 B) 2.20 C) 20.22 D) 21.21
10. The mode of the data 0 , 1 , 6 , 7, 2 , 3 , 7 , 6 , 6 , 2 , 6 , 0 , 5 , 6 , 0 is
A) 0 B) 7 C) 2 D) 6
11. The value of D 3 for 40 , 42 , 45 , 48 , 50 , 52 , 55 , 56 , 57 is
A) 42 B) 45 C) 48 D) 40
12. The mean of the series 3 , 5 , 3 , 7 , 2 , 5 , 3 is equal to
A) 2 B) 3 C) 4 D) None
13. What is the correct formula for mean deviation for ungrouped data ?
n n n n
x
i =1
i −x xi =1
i −x x
i =1
i −x x
i =1
i −x
A) B) C) D)
n n +1 n −1 n+2
14. The mean of the following data is
Numbers 8 10 15 20
Frequency 5 8 8 4
A) 15 B) 14.2 C) 12.4 D) 12.8
15. The harmonic mean of 4 , 8 , 16 is
A) 8.687 B) 6.857 C) 5.857 D) 4.758
16. The 8th decile for the data 20 , 30 ,25 , 23 , 22 , 32 , 36 is equal to
A) 30 B) 20 C) 25 D) 32
17. The correct formula for Karl Pearson’s coefficient of correlation r is
A) r = ( x − x) ( y − y ) B) r = ( x + x) ( y + y )
( x − x) ( y − y )
2 2
( x + x) ( y + y )
2 2
C) r = D) r =
( x − x) ( y − y ) ( x − x) ( y − y )
( x) ( y )
2 2
( x) ( y )
2 2
18. The geometric mean of 2 , 4 , 8 , 16 , 32 is equal to
A) 32 B) 64 C) 8 D) 16
19. The standard deviation for the data 9 ,7 , 10 , 8 , 9 , 7 , 8 , 9 is equal to
A) 1.12 B) 1.06 C) 1.006 D) 1
20. Compute the coefficient of Skewness from the following data:
x 6 7 8 9 10 11 12
f 3 6 9 13 8 5 4
A) 1.61 B) 1.43 C) 0 D) 2
21. In a negatively skewed distribution
A) Mean > Mode > Median B) Mode > Median > Mean
C) Mean > Median > Mode D) Mode< Median > Mean
22. If for a distribution the difference of first quartile and median is greater than
difference of median and third quartile then the distribution is classified as
A) Positively skewed B) Absolute open ended
C) Negatively skewed D) Not skewed at all
23. If two regression coefficients are − 0.1 and − 0.9 then the value of coefficient
of correlation is
A) − 0.3 B) 0.3 C) − 0.9 D) 0.9
24. If x = 10 , y = 90 , x = 3 , y = 12 and rxy = 0.8 then the regression equation of
x on y is given by
A) y = 3.2 x + 5.8 B) x = 3.2 y + 5.8
C) x = −8 + 0.2 y D) y = −8 + 0.2 x
25. If r12 = 0.6 , r13 = 0.8 , r23 = 0.3 , 1 = 8 , 2 = 9 , 3 = 5 then b12.3 = ?
A) 0.35 B) 1.09 C) 0.89 D) 0.45
26. The multiple correlation coefficient R3.12 is given by which of the following ?
A) R3.12 =
(r
13 )2 + (r23 )2 − 2(r12 )(r13 )(r23 )
1 − (r12 )
2
B) R3.12 =
(r
13 )2 + (r23 )2 + 2(r12 )(r13 )(r23 )
1 − (r12 )
2
C) R3.12 =
(r
13 )2 + (r23 )2 + 2(r12 )(r13 )(r23 )
1 + (r12 )
2
D) R3.12 =
(r 13 )2 − (r23 )2 + 2(r12 )(r13 )(r23 )
1 − (r12 )
2
27. If r12 = 0.25 , r13 = 0.35 , r23 = 0.45 then find R2.13
A) 1.46 B) 2.46 C) 0.46 D) 3.46
28. Calculate the coefficient of correlation between x and y series from the
( x − x) ( y − y) ( x − x) ( y − y) = 122
2
following data: 2
= 136 , = 138 ,
A) 0.98 B) 0.89 C) − 0.98 D) − 0.89
29. The coefficient of correlation from the following data is
x 1 2 3 4 5
y 2 5 3 8 7
A) 0.245 B) - 0.895 C) 0.9876 D) 0.8062
y
30. If x = 10 , y = 22 and r = −2.2738 then the equation of line of regression y on
x
x is equal to
A) y = − 2.2738 x + 44.738 B) y = − 2.2738 x − 44.738
D) x = − 2.2738 y − 44.738 D) x = − 2.2738 y + 44.738
31. Find the value of median from the following data :
Class Interval 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50
Frequency 5 10 12 15 18
A) 30 B) 35 C) 32 D) 40
32. Calculate first quartile from the following data:
Class Interval 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75
Frequency 2 8 20 25 10 5
A) 50 B) 56.625 C) 60 D) 62.25
33. Find the value of 9th Decile D 9 from the following data :
Class Interval 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 - 100
Frequency 10 20 20 15 15 20
A) 90 B) 98 C) 95 D) 92
34. Find mode from the following data :
Class Interval 0 - 4 4-8 8 - 12 12 - 16 16 - 20 20 - 24
Frequency 10 12 18 7 5 8
A) 8.25 B) 7.89 C) 9.41 D) 10.78
35. The first four moments about origin for a distribution are 1 ' = −1.5 , 2 ' = 17
3 ' = −30 and 4 ' = 108 then the coefficient of skewness is
A) 0.4924 B) − 0.4924 C) 0.3456 D) 0.5648
36. Which of the following is true regarding “ Regression ”and “ Correlation ” ?
where y is dependent variable and x is independent variable.
A) The relationship is symmetric between x and y in both.
B) The relationship is not symmetric between x and y in both.
C) The relationship is not symmetric between x and y in case of
correlation but in case of regression it is symmetric.
D) The relationship is symmetric between x and y in case of
correlation but in case of regression it is not symmetric.
37. R3.12 is known as the multiple correlation coefficient in which
A) x3 is dependent variable and x1 , x2 are the independent variables
B) x3 is independent variable and x1 , x2 are the dependent variables
C) x1 , x2 , x3 all are the independent variables
D) x1 , x2 , x3 all are the dependent variables
38. If there is no skewness in the distribution then which of the following is true ?
A) Third quartile − Mean = Median − First quartile
B) Third quartile − Median = Median − First quartile
C) Third quartile − Mode = Median − First quartile
D) Third quartile Median = Median First quartile
39. If mean = 29.6 , mode = 27.52 and Standard deviation = 6.5 , then the value
of Karl – Pearson’s coefficient of Skewness is
A) 0.32 B) 3.2 C) 0.45 D) 4.5
40. If two regression equations of the variables x and y are x = 19.13 + 087 y and
y = 11.64 + 0.50 x then the value of coefficient of correlation is
A) 0.88 B) 0.77 C) 0.55 D) 0.66
RTMNU Question Bank
Branch CSE Unit-II Matrices
(1) If 𝜆1 , 𝜆2 , 𝜆3 , … … … . . 𝜆𝑛 are the Eigen values of matrix A, then the Eigen values of 𝐴−1
are
(A) 𝜆1 , 𝜆1 , 𝜆1 … … … … . . 𝜆1
1 2 3 𝑛
(B) 𝜆1 , 𝜆2 , 𝜆3 , … … … . . 𝜆𝑛
(C)𝜆1 1 1
+ 1, 𝜆 + 1, 𝜆 + 1 … … … … . . 𝜆 + 1.
1
1 2 3 𝑛
(D) 𝜆1 1 1
− 1, 𝜆 − 1, 𝜆 − 1 … … … … . . 𝜆 − 1.
1
1 2 3 𝑛
(2) The vectors (1, 2, 3), (2, -2, 6) are
(A) Linearly Independent
(B) Linearly dependent
(C) Both dependent and independent
(D) None of these
6 − 2 2
(3) If 2 , 2 ,8 are the Eigen values of the matrix A = − 2 3 − 1 then A = ?
2 − 1 3
(A) 30 (B) 12 (C) 32 (D) None of these
2 2 1
(4) What is the characteristic equation of the matrix 𝐴 = [1 3 1] ?
1 2 2
(A) 𝜆3 + 5𝜆2 + 11𝜆 − 7 = 0 (B) 𝜆3 − 7𝜆2 + 11𝜆 − 5 = 0
(C) 𝜆3 − 7𝜆2 − 11𝜆 + 5 = 0 (D) 𝜆3 − 5𝜆2 − 11𝜆 − 7 = 0
2 1
(6) If 𝐴 = [−3 ].Then adjoint of matrix [𝜆𝐼 − 𝐴] is…………….
4
𝜆+2 1 𝜆−2 −1
(A) [ ] (B) [ ]
−3 𝜆+4 −3 𝜆−4
𝜆−2 1 𝜆−2 −1
(C) [ ] (D) [ ]
−3 𝜆+4 3 𝜆−4
(7) The matrix constructed by placing the Eigen vectors together is called
(A) Diagonalized matrix (B) Singular matrix
(C) Modal matrix (D) None of the above
(8) Eigen values of a matrix 𝐴 = [32 2
] are 5 and 1. What are the eigen values of matrix
3
𝐴2 ?
(A) 1 and 25 (B) 6 and 4
(C) 5 and 1 (D) 2 and 10
2 1 3
(9) The sum of Eigen values of the matrix [1 5 1] is
3 1 3
(A) 5 (B) 7 (C) 9 (D) 10
(10) Are the following vectors linearly dependent? 𝑋1 = [3,2,7], 𝑋2 = [2,4,1] , 𝑋3 = [1, −2,6]
(A) Dependent (B) Independent
(C) Can’t say (D) None of these
(11) A square matrix A such that 𝐴 = −𝐴𝑇 is called a …………
(A) Symmetric matrix (B) Skew symmetric matrix
(C) Orthogonal matrix (D) Diagonal matrix
(12) Any non-zero vector X is said to be a eigen vector of a matrix A if there exist a
number λ such that
(A) 𝐴𝑋 + 𝜆𝑋 = 0 (B) 𝐴2 𝑋 = −𝜆𝑋
(C) 𝐴2 𝑋 = 𝜆𝑋 (D) 𝐴𝑋 = 𝜆𝑋
1 0 1
(13) To find largest eigen value by iteration method for matrix 𝐴 = [1 7 1 ] , initial
3 1 12
approximation to the eigen vector is given by
1 1
(A) 𝑋 (0) = [1] (B) 𝑋(0) = [1]
1 0
1 0
(C) 𝑋 (0) = [0] (D) 𝑋(0) = [0]
0 0
(14) Eigen values of a matrix 𝐴 = [3 2
] are 5 and 1. Then diagonal form 𝐵−1 𝐴𝐵 of
2 3
matrix A is?
(A) 𝐵−1 𝐴𝐵 = [1 0
] (B) 𝐵−1 𝐴𝐵 = [5 0
]
0 1 0 1
(C) 𝐵−1 𝐴𝐵 = [0 5
] (D) 𝐵−1 𝐴𝐵 = [−5 0
]
1 0 0 −1
(15) If the characteristic equation of matrix A is 3 − 18 2 + 45 = 0 then A = ?
(A) 0 (B) 45 (C) 18 (D) None of these
3 4
(16) If 𝐴=[ ] then eigen value of matrix A are
4 −3
(A) 0 , 5 (B) 5 , 5 (C) 5 , - 5 (D) 3 , 5
1 2
(17) If 𝐴=[ ] then eigen values of A2 are
−1 4
(A) 2 , 3 (B) 4 , 9 (C) 2 , 9 (D) 0 , 9
(18) Using Sylvester’s theorem polynomial P( A) = C0 I + C1 A + C 2 A 2 + − − − + C k A k ,
where A is a square matrix of order ‘n’ having distinct eigen values , can
be expressed as
n n
(A) P( A) = P(r ) Z (r ) (B) P( A) = P(r ) + Z (r )
r =1 r =1
n n
P ( r )
(C) P( A) = P(r ) − Z (r ) (D) P( A) =
r =1 r =1 Z ( r )
(19) If B is the non-singular modal matrix for given square matrix A then the
Matrix B −1 AB is always a ------
(A) Square Matrix (B) Null Matrix
(C) Diagonal Matrix (D) Upper Triangular Matrix
(20) In Sylvester’s theorem the eigen value of matrix A are always
(A) Zero (B) Equal (C) Unity (D) Distinct
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