Practice set

Grade: XI

Practice following questions.

Logic, Set, and Real Number System.

1. For any statements p and q prepare a truth table for q ∧ {∼ ( p ∨q ) } .

2. Define a statement. If p and q are two statements then construct truth table for ∼ ( p ∨q ) .
3. Define tautology, conditional and biconditional statement. If p and q are two statements then
prove that p ∨∼ ( p ∧q ) is a tautology.
4. Define converse, inverse and contradiction of a statement. Prove that the statement
( p ∧q)∧∼( p∨ q) is a contradiction.
5. Define logically equivalent statements. If p, q, and r are three statements then prove that the
statement ( p ∧q ⇒ r ¿ ⟺[ p ⇒(q ⇒ r)]is a tautology.
6. Define Difference of two sets. If A, B, and C are three non-empty sets then prove that
A−(B ∩C)=( A−B)∪( A−C).
7. What do you mean by symmetric difference of any two sets? For non-empty sets A, B, and C
prove that A−( B ∪ C )=( A−B ) ∩ ( A−C ) .
8. If A and B are two subsets of U then prove that
i. A ∪ B= A ∩ B ii. A ∩ B= A ∪ B
9. Rewrite the following using absolute value sign:
i. −7 ≤ x ≤ 4 ii. −4 ≤3 x ≤5
10. Rewrite the following without using absolute value sign:
i. |3 x +2|≤1 ii. |5 x−3|≤ 4
11. If x and y are any two real numbers then prove that |x + y|≤|x|+¿ y∨¿ .

Function

1. If f : R → R be defined by f ( x )=9 x −1, x ∈ R , examine whether f is one-one.

2. If f : R → R be defined by f ( x )=4 x +3 , x ∈ R , show that f is one-one and onto. Also find
−1
f ( x) .
3. Find the domain of the function a) y= √ x−2 b) y= √ 1− x2
4. If f : R → R be defined by f ( x )=cx+ d , c ≠ 0∧d are real numbers, show that f is one-one and
onto. Find f −1 ( x) . Also, show that fo f −1=x .
5. Define composite function of two functions f ∧g . Let f : R → R and g : R → R be defined by
f ( x )=x +2∧g ( x )=4 x−1 , x ∈ R . Find ( fog ¿ ( x )∧(gof )(x ). Are fog ¿ x∧(gof )(x ) are
3

commutative? Are they one-one?

6. Define domain and range of the function. Find domain and range of the following functions
a) y= √ 21−4 x−x 2 c) y=x 3
b) y= √ x 2 +4 x −21 d) y=5 x 2

Curve Sketching

1. Show that following functions are neither even nor odd.

a) y=x + x 2 b) y=ax+ b
2. Sketch the graph of y=x 2−2 x−3 indicating its specific features.
3. Sketch the graph of y=x 2 +2 x+ 3 indicating its specific features.
4. Sketch the graph of y= ( x −1 )( x−3 )( x−5 ) indicating its specific features.
5. Sketch the graph of y= ( x −1 )3 indicating its specific features.

Sequence and Series

1. If A.M., G.M., and H.M. are arithmetic mean, Geometric mean, and Harmonic mean respectively
between two unequal positive real numbers then prove that
i. ( G . M . ) 2= A . M . × H . M . ii. A . M .>G . M .> H . M .
a b c
2. If a, b, and c are in H.P then prove that , , are in H.P.
b+c c +a a+ b
3. If A be A.M. and H be H.M. between two real numbers a and b then prove that
a− A b−B A
× =
a−H b−H H
4. Find the sum of infinite geometric series,
1 1 1 1 1 1
i. + + + + + +… … … … …
5 7 5 2 72 53 7 3
3 5 3 5 3 5
ii. − + − + − +……………
4 42 43 4 4 45 4 6
5. A ball is dropped from a height of 48 M and rebounds two third of the distance it falls. If it
continues to fall and rebound in same way, how far will it travel before coming to rest?
6. One side of the equilateral triangle is 24 cm. The midpoints of the side are joined to form another
triangle whose mid points in turn are joined to still form another triangle. This process continues
indefinitely, Find the sum of the perimeters of all triangles.

Matrices and Determinants

| |
1 x x3
3
1. Prove that, 1 y y
3
1 z z

| |
1 1 1
2. Prove that, α β γ
αγ γα αβ

| |
a b c
2 2 2
3. Prove that, a b c
3 3 3
a b c

| |
a+l m n
4. Prove that, l a+ m n
l m a+ n

| |
1 1 1
5. Prove that, a b c
3 3 3
a b c

Complex Number
1. If z 1 and z 2 are two complex numbers then prove that
i. |z 1 + z 2|≤|z 1|+|z 2|
ii. |z 1−z 2|≤|z 1|+| z2|
2−36 i
2. Define conjugate of a complex number. Find the square root of .
2+3 i
8+6 i
3. Define complex number with examples. Find the square root of .
1+i
4. Find the square root of −5+12 i.
5. Find the square root of z=7−24 i .
3−2 i
6. If x−iy= then prove that x 2+ y 2=1.
3+2 i
Quadratic equations

1. Prove that the quadratic equation ( a+ b )2 x2−2 ( a2−b 2 ) x + ( a−b )2=0have equal roots.
2. If ( 1+m2 ) x 2+2 cmx + ( c 2−a 2 )=0 have equal roots then prove that c 2=a2 (1+m 2).
3. If one root of a x 2 +bx +c=0 be twice of another then prove that 2 b2=9 ac .
4. If one root of the quadratic equation x 2+ px +q=0 be square of another then prove that
3
p −( 3 p−q ) q+ q=0
5. Form a quadratic equation with the rational coefficients whose one root is
i. √ 3−4 ii. 4 +3 √ −1
6. Find the value of k so that following equation have common root.
2 2
i. x + 4 x +3=0 and x + 5 x + k .
2 2
ii. x + kx +2=0 and x −4 x+3 . s

Trigonometry

−1 5
1. Find the value of cos ⁡(sin )
13
5 −1 −1 −π
).
i. cos ⁡(sin ii. tan sin ⁡( )
13 2
2. a. If cos−1 x +cos−1 y=θ show that x 2−2 xycosθ+ y 2=sin2 θ
b. If cos−2 x +cos−1 y +cos−1 z=π show that x 2+ y 2+ z 2 +2 xyz=1

3. solve: a. √ 3 cosθ+ sinθ=√ 2

b. sinx−cosx=− √ 2

4. solve: a. sin 2 θ=cos 3θ , where 0<θ <2 π

b. cosx + sinx=cos 2 x+ sin 2 x

5.

Straight Line
1. If P and P2 are length of perpendiculars drawn from the points (cosθ , sinθ) and
(−secθ , cosecθ ) on the line xxsecθ+ ycosecθ=0 respectively then prove that
4 2
2
−P 2=4 .
P1
2. Find the length of perpendicular drawn from (−3 ,−5 ) on the line x + y−6=0 .
3. Find the equation of bisectors of the angle between the lines x−2 y=0 and 2 y−11 x=6 . Also
show that bisectors are perpendicular to each other.
4. Show that the length of perpendicular from the point ( x 1 , y 1 ) on the straight line
xcosα + ysinα =p is ±(x ¿ ¿ 1 cos α + y 1 sin α− p).¿
5. Prove that the product of perpendiculars drawn from the points ±( √ a 2−b2 , 0) on a straight line
x y
cosθ+ sin θ=1 is b 2.
a b
6. Find the bisector of the acute angle between the lines 3 x−4 y +3=0∧12 x−5 y−1=0.

Pair of straight line

1. Determine two straight lines represented by 6 x 2−xy−12 y 2−8 x +29 y−14=0 .

2. Prove that the straight lines ( x 2 + y 2 ) sin2 α= ( xcosθ− ysinθ )2 include an angle 2 α .
3. Find the angle between the pair of straight lines x 2+ 6 xy+ 9 y 2+ 4 x +12 y−5=0
4. Find the condition under which the second-degree equation
2 2 2
a x +2 hxy +b y +2 hx+2 f y + c=0 represents pair of straight lines.
5. Prove that the bisectors of an angle between the pair of straight lines a x 2 +2 hxy +b y 2=0 is h ¿.
6. Show that the equation 6 x 2+ 7 xy−20 y 2 + x+ 14 y=0 represent a pair of straight line.

Coordinate in space

1. Show that the points ( 2 , 3 ,−1 ) (4, 5, 0), (6, 2, 6), and (0, 4, 1) taken in order are the vertices of
square.
2. Show that the points ( 4 , 7 ,6 ) (2, 3, 2), (1, -2, -1), and (3,2,3) taken in order are the vertices of
parallelogram.
3. If P (2,4,3), Q (x, y, z) and R (-2, 2, 3) be three points such that R divides PQ in the ratio of 2:1
internally then find the coordinates of Q.
4. Find the ratio and division point in which the line segment joining points are
i. (-3,4, -8) and (5 6, 4) divides XY-plane.
ii. (1, -1, 2) and (2, 1, -1) divides XZ-plane.
5. If a line makes an angle of α , β ,∧γ with X, Y, and Z-axes respectively then prove that
2 2 2
i. sin α + sin β +sin γ=2
ii. cos 2 α +cos 2 β+ cos 2 γ =−1
6. Find the angle between two lines whose DCS are given by
1 1 1 1 1 1
i. ,− , and - , , ii. (1, 0, 0) and (0, -1, 0)
√3 √ 3 √3 √3 √ 3 √ 3
7. Find the direction cosines ( l , m, n ) of two lines whose direction cosines are related by equations
4 l+ 3 m−2 n=0 and lm−mn+ nl=0. Also find the angle between them.
8. If direction cosines l , m, n of two lines connected by equations l+m+n=0 and
2 lm−mn+2 nl=0 then find the angle between two lines.
 −1 1
9. Show that acute angle between any two diagonals of cube is cos .
3
Vectors

1. If 3 i⃗ + ⃗j− ⃗k and λ i⃗ + ⃗j −⃗k are collinear, find λ .

2. Prove that the vectors a⃗ −2 ⃗b−3 ⃗c, −2 ⃗a +3 b−4 ⃗ ⃗ 2 ⃗c are coplanar.
⃗c and −b+
3. Show that 2 ⃗i +3 ⃗j+ 4 ⃗k and i⃗ −3 ⃗j +5 k⃗ and −9 ⃗j+6 ⃗k are collinear.
4. Determine whether the following vectors are linearly dependents or independents?
i. 5 i⃗ + 3 ⃗j−7 k⃗ , 6 i⃗ +2 ⃗j+9 ⃗k , and 3 i⃗ + 8 ⃗j+4 ⃗k
ii. i⃗ + ⃗j−2 k⃗ , 3 i⃗ −2 ⃗j+4 ⃗k , and 3 i⃗ −7 ⃗j +14 ⃗k
Measures of dispersion

1. Calculate the mean, standard deviation and coefficient of variance from following data

Marks 10 11 12 13 14
Frequenc 3 12 18 12 2
y
2. Calculate the variance and coefficient of variance from following data

Weight (K.G.) 0-10 10-20 20-30 30-40 40-50

Frequency 12 33 30 15 10
3. Calculate the coefficient of skewness based on mean, mode and standard deviation from
following data.

Income 10 12 14 16 20
Frequenc 5 8 15 7 5
y
4. Consider the following distribution.

Distribution A Distribution B
Arithmetic mean ( X ) 100 90
Median ( M d ) 90 80
Standard deviation ( σ ) 10 10
Is the distribution A same as distribution B regarding the coefficient of variation and coefficient of
skewness?
2
5. If Σ fx=110, Σ f x =1650∧M o=12.5 , find the Karl Peason’s coefficient of Skewness.

Probability

1. If three coins are tossed simultaneously, find the probability of turning all head.
2. An urn contains 4 white and 8 red balls. If two balls are drawn random, find the probability of
getting one of each color.
3. A card is drawn at random from a well shuffled pack of 52 cards. What is the probability that it is
heart or queen?
2 3
4. If P ( A )= ∧P ( B )= , find P ( A ∪ B ) .
3 5
3 1 1
5. If P ( A )= ∧P ( B )= ∧P ( A ∩ B ) = , find P ( A ∩B )∧P ¿)
8 2 4
Limits and Continuity
1. Evaluate: a) lim √ x−√ x−3
x→ ∞

b) lim √¿ ¿ ¿
x→ ∞

3
x −64
2. Evaluate: a) lim 2
x→ 4 x −16
2
4 x −1
b) lim1 2 x−1
x→
2

xcotθ−θcotx
3. Evaluate: a) lim
x →θ x−θ
xsinθ−θsinx
b) lim
x →θ x−θ

x− √8−x 2
4. Evaluate: a) lim
x →2 √ x 2−12−4
x−√ 2−x 2
b) lim
x →1 2 x−√ 2+ x 2
3x x
e −1 2 −1
5. Evaluate: a) lim b) lim
x x

{
x→ 4 x→ o

1
x when 0≤ x <
2
1 1
6. Show that the function f ( x )= 1 when x= at x= is discontinuous. Also redefine the
2 2
1
1−x when < x <2
2
1
function so that f ( x ) be continuous at x=
2

{
2 x+ 1 when x <1
7. A function f ( x ) be defined as f ( x )= 2 when x=1 is function continuous at x=1? If not
3 x whenx>1
why? How can the function so that f ( x ) be continuous at x=1 ?

8. When does the function f(x) is continuous at x = a.? Test the continuity of the function

{
2
2−x when x< 2
f ( x )= 3 when x=2 at x=2
x−4 when x> 2
9. Define the continuity of a function at a point. Test the continuity of the function

{
2 x +3 when x<1
f ( x )= 4 when x=1 at x=1
6 x−1 when x >1

The Derivatives

1. Find differential coefficient of following:

a) x 3 + y 3=3 axy b) a x 2 +2 hxy +b y 2=1
c) x + y=sin ( x + y ) d) x 2 y 2= ( x + y )
e) When x=tan t∧ y=sin t cos t f) when x=a ¿
g) e √ sin x h) e √ cos x
i) y=esin ⁡¿¿ j) y=ecos ⁡¿¿
2. Find from first principle the derivatives of

1 1
a¿ b)
√ x −1 √ x +a
c ¿ √ 3 x−2 d) √ 2 x −3

1 1
e¿ f)
√ ax +b √3 x−4
5
g ¿ ( ax +b ) h) ( 5 x+ 3 )9

3. Find from the first principle the derivatives of

a ¿ sin ( 2 x+ 3 ) b) cos (5 x +9 )

c ¿ √ sin 2 x d) √ cos 3 x

5x
e ¿ tan ( 3 x−4 ) f) cot
2
2
g ¿ sin x h) cos 2 x

4. Find from first principle the derivatives of

x
2 x+ 3
a¿e b)e 2
2 3
c ¿ log e x d) log e 3 x
dy
5. Find if
dx
2 3 5
a ¿ x y =( x+ y ) b) x 6 y 3=( x+ y )9

Application of Derivatives

1. Find the absolute maxima and absolute minima of the function f ( x )=3 x 2−6 x + 4 in [-1,2].
2. Find the interval in which the following functions are increasing or decreasing.
a) f ( x )=2 x 3−12 x 2+7 b) f ( x )=2 x 3−9 x 2 +12 x +90
3. Find the maxima and minima of the following functions. Also, find the point of inflection (if
exists).
a) f ( x )=2 x 3−15 x 2+ 36 x+5 b) f ( x )=x 3−3 x 2−9 x +27
c. f ( x )=x 3−6 x2 +9 x +5 d) f ( x )=2 x 3−9 x 2 +24 x +3
4. Show that following functions have neither maximum nor minimum value
a) x 3−6 x 2 +12 x +50
b) x 3−3 x 2+ 6 x+ 4
5. Define Absolute maxima absolute minima, local maxima, and local minima of a function. Show
that the function f ( x )=x 3−6 x+ 9 x−2has maximum value 2 at x=1.

Antiderivatives

1. Evaluate

x +3 2
x +7
a¿ ∫ dx b) ∫ dx
x−3 x +3

c ¿∫
dx cos √ x
d) ∫ dx .
xlo g e x 2 √x

e ¿∫ log e x dx f) ∫ 1−( 1
x
2 )
dx

3 1
g ¿∫ ( x +3 x ) dx h) ∫ ( 2 x +3 ) dx
2 2

1 0

2. Evaluate
3
a ¿ ∫ cose c x dx b) ∫ sec 3 x dx
c ¿∫ x e dx d) ∫ ( x 2 +1 ) e x dx .
2 ax

e ¿∫ x sin x dx f) ∫ e cosx dx
2 x

g ¿∫ e cos x sin 2 x dx
2 2
h) ∫ ese c x sec 2 x tan x dx

e ¿∫
a
2
dx f) ∫
√ a2−x 2 dx
x √ a 2−x 2
2
x2
3. Evaluate
ax
a ¿ ∫ e cosbx dx b) ∫e 2 x cos ( 3 x+ 4 ) dx
4. Evaluate
π
1
2
5. a ¿ ∫ sin ⁡¿ ¿ dx
0
b)
∫ 1+sinx
0

Application of antiderivatives

π
1. Find the area bounded by the curve y=cos x , at x=0∧x = .
2
2. Find the area bounded by the curve y=sin x , at x=0∧x=π .
3. Find the area under the circle x 2+ y 2=36 by using the integration method.
4. Find the area under the circle 16 x 2+ 9 y 2=144 by using the integration method.
2 2
x y
5. Find the area of the enclosed by following curve+ =1by using integration method.
9 4
6. Find the area between the curves a) y 2=16 x∧ y=x . b) y 2=4 x∧x=3 .

Numerical Computation

1. Find the roots of the equations x 3−8 x−4=0 which lies between 3 and 4 by using Newton
Raphson’s method correct to 4 decimal places.
2. Using bisection method solve the equation x 3−9 x +1=0 for the roots lying between 2 and 3,
correct to 2 significance figures.
3. Find the roots of the equations x 3 +3 x −5=0 which lies between 1 and 2 by using Newton
Raphson’s method correct to 3 decimal places.
π
4
4. Using Trapezoidal rule integrate
∫ sinx dx , n=4
0
1
1 dx
5. Using Simpson’s rule evaluate ∫ dx , in n=4
3 0 1+ x
2

π
6. Using Simpson’s rule evaluate ∫ sin x dx , n=6
0
4
7. Apply Simpson’s rule to evaluate ∫ e log e x dx , n=3
x

1
Best wishes!!!