This document contains 50 questions related to mathematics. The questions cover a range of topics including functions, trigonometry, calculus, matrices, probability, and linear programming. They involve tasks such as showing functions are injective or surjective, evaluating derivatives, solving differential equations, finding probabilities, expressing problems as linear programming problems, and more.

1. Question bank
Class XII
Mathematics
1. Let f: [- ] [-1, 1] be a function defined as f(x) = sinx. Show that f is invertible.
2. If f and g are functions from R R defined as f(x) = sin x and g(x) = find fog and
gof.
3. Let A = {1, 2, 3], let R be the relation defined as R = {(a, b): a divides b}. Show that
R is reflexive and transitive but not symmetric.
4. Show that f: R
*
R
*
defined as f(x) = x
2
is invertible. Also find f
-1
.
5. If f: R
+
R be a function defined as f(x) = x
2
+4x+6. Show that f is one one but not
onto. Find the range of S of f .Also show that f: R
+
S defined by f(x) =x
2
+4x+6 is
invertible. Now find f
-1
.
6. Let A = R x R and be the binary operation on A defined by (a, b) (c, d) = (ac-bd,
ad+bc). Show that is commutative. Show that (1, 0) is the identity element for .
7. Find the principal value of the following: (a) (b) .
8. Prove the following: (a) tan
-1
x + tan
-1
y = tan
-1
( (b) sin
-1
+ sin
-1
=cos
-1
(c)
2 tan
-1
+ tan
-1
= tan
-1
.
9. Simplify the following: (a) tan
-1
( (b) tan
-1
[ ].
10. Find the value of “x”: (i) sin (sin
-1
+ cos
-1
x) = 1 (ii) tan
-1
( ) = tan
-1
x.
11. Show that the value of is positive, zero, or negative according
as a 1, a=1 or a 1.
12. Show that = 2(a+b) (b+c) (c+a).
13. Evaluate .
14. Show that = sin (A-B) sin (B-C) sin(C-A).
15. Find the value of x so that the area of the Δ ABC whose vertices are A(x, 4), B (5, 4)
and C (2,-6), is 35 square units.
16. Show that [- (a+b+c)] is a root of the equation: = 0.
17. If A = , verify that A
2
- 4A – I= 0. Hence find A
-1
.
18. If A= , find k so that A
2
= kA + 2I.
19. Find the values of x, y and z if the matrix A = is such that AA‟=I.

2. 20. Find the inverse by elementary transformation: (i)
(ii) .
21. The angles of a triangle are in AP. If the greatest angle is twice the smallest angle,
find the angles of the triangle using matrices.
22. Solve using matrices: x +xy + yz = -5, x – xy – yz = 8, 3x - 2xy + yz = 1.
23. Discuss the continuity of the function f(x) = .
24. Test the continuity of f(x) = at x=0.
25. Find the derivative of each of the following functions with respect to „x‟: (i) y
= (ii) x
sinx
+(sinx)
x
(iii) tan
-1
(
26. If y= and x = find .
27. If y = 1+x + + + ….. , show that – y =0.
28. Find the equation of the tangent to the curve + = at (x0 , y0) and show that
the sum of its intercepts on axes is constant.
29. Water is filled in a swimming pool of base area 30x15 m
2
at the rate 90,000
liters/hour. Find the rate at which the water level rises.
30. Find the intervals in which the following functions are increasing or decreasing: (i)
f(x) =(x-1)
3
(x-2)
2
(ii)f(x) =sin
4
x + cos
4
x in [0, ] (iii) f(x) = sin3x in [0, ]
31. Show that a right triangle of given hypotenuse has maximum area, when it is an
isosceles triangle.
32. Find the coordinates of a point on the parabola y= x
2
+7x +2 which is nearest to the
straight line y=3x-3.
33. Find the point on the curve y = x- 3x+9x+6 at which slope of the tangent to the
curve is minimum .Also find the least slope.
34. Integrate with respect to x : (i) (ii) tanx tan2x tan3x (iii) (iv)
(v) (vi)
35. dx (ii) dx (iii) (iv)
36. Find the area bounded by the curve y
2
= 4x and x
2
=4y.
37. Draw the rough sketch of y= cosx and y= sinx as x varies from 0 to , and find the
area of the region enclosed by them and the x-axis.
38. Find the area of the region included between the parabola y
2
= x
2
– 4x + 5 and the
line y = x+1.
39. Solve the following differential equations: (i) cosy dy + cosx siny dx = 0 given that
y= and x= (ii) y- x = a (y
2
+ x
2
) when x = a and y = -a (iii) y dx + (x – y
3
) dy =0.
40. Show that the points (2, -1, 1), (1, 3, 2), (-1, 1, 4) are collinear.

3. 41. Find the equation of the plane parallel to the plane 2x-y+z-1 =0 and passing
through the point (1,-1, 2). Also find the distance of that plane from the origin.
42. Find the angles between the lines: = (2 - +3 ) + λ ( ) and ( ) +
μ( ).
43. Find the equations of the lines of shortest distance between the lines:
and .
44. An insurance company insured 2000 scooter drivers, 3000 car drivers, and 5000
truck drivers. The probability of an accident involving a scooter, a car and a truck
is 0.05, 0.03 and 0.02 respectively. One of the insured drivers meets with an
accident. What is the probability that he is a scooter driver?
45. By examining the chest x-ray, the probability that TB is detected when a person is
actually suffering from it is 0.99. The probability that the doctor diagnoses
incorrectly that a person has TB on the basis of x-ray is 0.001. In a certain city 1 in
1000 persons suffers from TB. A person is selected at random and is diagnosed to
have TB. What is the chance that he actually has TB?
46. Two cards are drawn from the well-shuffled pack of 52 cards. Find the mean and
variance for the number of face cards obtained.
47. How many times a man toss a fair coin so that the probability of having at least
one head is more than 90%.
48. If a young man rides his motor cycle at 25km/h, he has to spend Rs 2/km on petrol.
If he rides it at a faster speed of 40km/h, the petrol cost increases to Rs 5/km. He
has Rs 100 to spend on petrol and wishes to find what is the maximum distance he
can travel within 1 hour. Express it as LPP and solve it graphically.
49. Minimize z = 3x + 4y subject to the constraints: 4x + y ≥ 7, x + 3y ≥ 10, x, y≥ 0.
50. In a hostel, 60% of the students read Hindi newspaper, 40% read English
newspaper and 20% read both English and Hindi newspapers. A student is
selected at random.
(i)Find the probability that he reads neither Hindi nor English newspapers.
(ii) If he reads Hindi newspaper, find the probability that he reads English
newspaper.
(iii) If he reads English newspaper, find the probability that he reads Hindi
newspaper.