Maharaja Surajmal Institute of Technology
Mid term Exam(Online)
Subject: Applied Mathematics-I Paper Code: BS111
Max Marks: 30
Q1. The rank of the matrix is 2, then x is
[ ]
1 5 x
5 1 −1
1 2 1
a) 1
b) 2
c) 3
d) 0
Ans c) (2 marks)
Q2 Consider the system of linear equations
( a+ 3 ) x + y +2 z=a
a x + ( a−1 ) y+ z=a
3 ( a+1 ) x +a y + ( α +3 ) z=3
The set of values of a for which the above system is inconsistent is
a) {all real numbers except 0 and 1}
b) {0,1}
c) {0}
d) {1}
Ans b) (3 marks)
Q3 If the sum of the eigenvalues of the matrix is 5, then the value of k is
[ ]
2 1 0
0 4 3
0 0 k
a) 0
b) 2
c) -2
d) -1
Ans d) (1 marks)
Q4. Which one of the following is an eigenvalue of the given matrix?
[ ]
−2 2 −3
2 1 −6
−1 −2 0
a) -3
b) 0
c) 1
d) -1
Ans a) (2 marks)
Q5. Which of the following statement is true?
a) A matrix has only one eigenvector corresponding to an eigenvalue
b) A matrix may have many eigenvectors corresponding to an eigenvalue
c) A matrix may have no eigenvector corresponding to an eigenvalue
d) None of these
Ans b) (1 marks)
Q6. A skew symmetric matrix is
a) A square matrix and all the diagonal elements may not be not zero.
b) A non square matrix and all the diagonal elements are zero.
c) A square matrix and all the diagonal elements are zero
d) None of these
Ans c) (1 marks)
Q7. An orthogonal matrix is
a) Either rectangular or square matrix
b) Only square matrix
c) Only non square matrix
d) Transpose of a non square matrix
Ans b) (1 marks)
Q8. The following symmetric matrix is obtained from which quadratic form
[ ]
0 1 1
1 0 −1
1 −1 0
2 2 2
a) x + y + z −2 yz
2 2 2
b) x + y + z +2 yz
c) 2 xy−2 yz +2 zx
2 2 2
d) 2 x +2 y +2 z −2 yz
Ans c) (1 marks)
Q9. Which one of following is eigenvector of the given matrix?
[ ]
−2 2 −3
2 1 −6
−1 −2 0
'
a) ( 6 0 2)
b) ( 0 0 0 )'
c) ( 1 2 −1 )'
d) None of these
Ans c) (2 marks)
Q10. If λ is a characteristic root of A, then λ+ k is a characteristic root of
a) A+ kI
b) A−kI
c) A
d) None of these
Ans a) (1 marks)
∂(u , v )
Q11. If u=x+ y and v=xy , then is
∂(x , y )
a) xy
b) x + y
c) x− y
x
d)
y
Ans c) (2 marks)
Q12. A pair of functions which are functionally independent is
a) u=x+ y , v =x− y
b) u=x2 + y 2 , v=x 2− y 2
c) u=e x+ y , v=log ( x− y)
d) u=e x− y , v=log ( x− y )
Ans d) (2 marks)
Q13. What is the minimum value of f ( x 2+ 4 y 2 +2 z 2) on the plane x−2 y + z=15 ?
a) 60
b) 70
c) 80
d) 90
Ans 90 d) (3 marks)
Q13. Which among the following correctly defines Leibnitz rule of a function given by
b
f ( α )=∫ f (x , α )dx where a and b are constants?
a
b
∂
a) f ( α )=
'
∫ f (x , α ) dx
∂α a
b
d
b) f ( α )=
'
∫ f ( x , α ) dx
dα a
b
∂
c) f ( α )=∫
'
f (x , α ) dx
a ∂α
b
d
d) f ( α )=∫
'
f ( x , α ) dx
a dα
Ans c) (1 marks)
∂u
Q14. If u=x y , then is
∂y
a) 0
y−1
b) yx
y
c) x log x
y
d) x log y
Ans c) (1 marks)
Q 15. If f xx ( a , b ) , B=f xy ( a , b ) , C=f yy (a , b), then f (x , y ) will have a minimum at (a ,b) if
2
a) f x =0 , f y =0 , AC < B and A< 0
2
b) f x =0 , f y =0 , AC=B and A> 0
2
c) f x =0 , f y =0 , AC > B and A> 0
2
d) f x =0 , f y =0 , AC > B and A< 0
Ans c) (1 marks)
dz
Q16. If z=u2 + v 2 and u=a t 2 , v =2 at , find .
dt
a) 4 a2 t(t 2 +2)
b) 4 a2 (t 2 +2)
c) 4 t(t 2 +2)
d) (t 2+ 2)
Ans a) (2 marks)
∂(u , v) ∂( x , y)
Q17. If J 1= and J 2= then J 1 J 2=… …
∂( x , y) ∂(u , v)
a) 0
b) 1
c) -1
d) 2
Ans b) 1 (1 marks)
u ∂(x , y )
Q18. If x=uv , y= , then is
v ∂(u , v )
a) −2 u/v
b) −2 v /u
c) 0
d) 1
Ans a) (2 marks)
