Pre-Board Examination 2025-26
Class 12 – Mathematics
Section A (MCQs)
Time: 3 Hours Maximum Marks: 80
Section A: Choose the correct option. (20 × 1 = 20)
1. Consider the non-empty set consisting of children in a family and a relation R defined as a R b if a is brother of b. Then R is:
(a) Transitive but not symmetric
(b) Both symmetric and transitive
(c) Symmetric but not transitive
(d) Neither symmetric nor transitive
2. Principal value of tan⁻¹(−1) is:
3. Domain of function f(x) = sin⁻¹x is equal to:
(a) [0, 1]
(b) R
(c) [−1, 1]
(d) None of these
4. If A and B are invertible matrices, then the inverse of AB is equal to:
(a) AB
(b) BA
(c) A⁻¹B⁻¹
(d) B⁻¹A⁻¹
5. If order of matrix A is 2 × 3 and order of matrix B is 3 × 5, then order of matrix B′A′ is:
(a) 5 × 2
(b) 2 × 5
(c) 5 × 3
(d) 3 × 2
6. If A is an invertible square matrix of order 4, then |adj A| is equal to:
(a) |A|⁴
(b) |A|⁵
(c) |A|³
(d) |A|²
7. If a function is continuous at x = 4, then the value of k is:
8. If a given function is , then its value is:
9. The rate of change of the area of a circle with respect to its radius r at r = 6 cm/sec is:
(a) 10π cm²/sec
(b) 12π cm²/sec
(c) 8π cm²/sec
(d) 11π cm²/sec
10.
∫ from 0 to π/2 [ √(cos x) / ( √(cos x) + √(sin x) ) ] dx is equal to:
(a) π/4
(b) π/2
(c) π/6
(d) π/3
11.
∫ eˣ ( log x + 1/x ) dx is equal to:
(a) eˣ × 1/x + C
(b) eˣ log x
(c) eˣ + log x + C
(d) (eˣ log x) / x + C
12.
Degree of equation
d²y/dx² + 3 dy/dx + 2y = 0 is:
(a) 3
(b) 2
(c) 1
(d) 0
13.
The integrating factor of
x dy/dx − y = 2x² is:
(a) e⁻ˣ
(b) e⁻ʸ
(c) 1/x
(d) x
14.
If a is any vector then a · a is:
(a) 1
(b) 0
(c) |a|²
(d) None of these
15.
If
a = λi + 3j + 2k
and
b = i − j + 3k
are perpendicular to each other, then the value of λ is:
(a) 6
(b) 9
(c) 3
(d) −3
16.
Direction ratios of the line given by
(x − 1)/3 = (2y + 6)/10 = (1 − z)/ −7
are:
(a) <3, 10, −7>
(b) <3, −5, 7>
(c) <3, 5, 7>
(d) <3, 5, −7>
17.
Direction cosines of x-axis are:
(a) <1, 0, 0>
(b) <0, 1, 0>
(c) <0, 1, 1>
(d) <0, 1, 1>
18.
The maximum value of
Z = 3x + 5y
subject to the constraints
x + y ≤ 4, x ≥ 0, y ≥ 0
is:
(a) 0
(b) 12
(c) 20
(d) 32
19.
If E is any event then P(E) belongs to:
(a) [0, 1]
(b) (0, 1)
(c) [−1, 1]
(d) (−1, 1)
20.
If P(A) = 1/2 and P(B) = 0 then P(A | B) is:
(a) 0
(b) 1/2
(c) Not defined
(d) 1
10.
\[
\int_{0}^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} \, dx
\]
is equal to:
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{3} \)
11.
\[
\int e^x\left(\log x + \frac{1}{x}\right) dx
\]
is equal to:
(a) \( e^x \cdot \frac{1}{x} + C \)
(b) \( e^x \log x \)
(c) \( e^x + \log x + C \)
(d) \( \frac{e^x \log x}{x} + C \)
12. Degree of equation
\[
\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0
\]
is:
13. The integrating factor of
\[
x\frac{dy}{dx} - y = 2x^2
\]
is:
(a) \( e^{-x} \)
(b) \( e^{-y} \)
(c) \( \frac{1}{x} \)
(d) \( x \)
14. If \( \vec{a} \) is any vector then \( \vec{a} \cdot \vec{a} \) is:
(a) 1
(b) 0
(c) \( |\vec{a}|^2 \)
(d) None of these
15. If
\[
\vec{a} = \lambda \hat{i} + 3\hat{j} + 2\hat{k}
\]
and
\[
\vec{b} = \hat{i} - \hat{j} + 3\hat{k}
\]
are perpendicular to each other, the value of \( \lambda \) is:
16. Direction ratios of the line given by
\[
\frac{x-1}{3} = \frac{2y+6}{10} = \frac{1-z}{-7}
\]
are:
(a) <3, 10, −7>
(b) <3, −5, 7>
(c) <3, 5, 7>
(d) <3, 5, −7>
17. Direction cosines of x-axis are:
(a) <1, 0, 0>
(b) <0, 1, 0>
(c) <0, 1, 1>
(d) <0, 1, 1>
18. The maximum value of
\[
Z = 3x + 5y
\]
subject to the constraints
\[
x + y \le 4, \quad x \ge 0, \quad y \ge 0
\]
is:
(a) 0
(b) 12
(c) 20
(d) 32
19. If E is any event then P(E) belongs to:
(a) [0, 1]
(b) (0, 1)
(c) [−1, 1]
(d) (−1, 1)
20. If
\[
P(A)=\frac{1}{2}, \quad P(B)=0
\]
then \( P(A|B) \) is:
(a) 0
(b) \( \frac{1}{2} \)
(c) Not defined
(d) 1
Pre-Board 2025-26 - Mathematics
2. Using determinants, find the equation of line passing from the points (1, 4) and (-1, 2).
3. If y = xsin-1x then, Find dy/dx.
4. Find the intervals in which the function f(x) = 6 - 9x - 2x2 is strictly decreasing.
5. Find ∫ (sin3x + cos3x) / (sin2x . cos2x) dx.
6. Find the area of the region bounded by y2 = 4x, x = 1, x = 4 and the x-axis in the First quadrant.
7. Find the general solution of the differential equation dy/dx = [y(x2 - 1)] / [x(y2 - 1)].
8. Find area of parallelogram whose adjacent sides are given by the vector a = i + j - k and b = 2i - j + 3k.
9. Show that the function f: R → R defined by f(x) = (2x - 1) / 3, x ∈ R is one-one and onto. Also find the inverse of f.
10. If A =
and I =
then prove that (A - 4I)(A - 5I) = 0.
11. If f(x) = |x| sin(1/x) if x ≠ 0, and f(x) = 0 if x = 0, then discuss continuity of f(x) at x = 0.
(Or)
If y = e
ax Sin bx prove that d
2y/dx
2 - 2a(dy/dx) + (a
2 + b
2)y = 0.
12. Evaluate ∫ dx / [(x + 1)(x + 2)(x + 3)]
(Or)
Evaluate ∫ e
4x Cos 7x dx.
13. Solve the differential equation: (3xy + y
2)dx = (x
2 + xy)dy
(Or)
Solve the differential equation: dy/dx - 2y = 3x.
14. Minimize z = 3x + 2y subject to the constraints x + y ≥ 8, 3x + 5y ≤ 15, x ≥ 0, y ≥ 0.
15. Bag I contains 5 red and 6 black balls, bag II contains 4 red and 7 black balls. One bag is chosen at random and a ball is drawn which is found to be red. Find the probability that it was drawn from the bag II.
16. Solve the following system of equations by matrix method:
2x - 3y + 5z = 11
3x + 2y - 4z = -5
x + y - 2z = -3
(Or)
a) Express the matrix
as sum of symmetric and skew-symmetric matrices.
b) If A =
and B =
. Verify that (AB)' = B'A'.
17. Prove that the volume of the largest cone that can be inscribed in a sphere of radius r is 8/27 of the volume of the sphere.
(Or)
Evaluate ∫
0π/2 log Sinx dx.
18. Find the shortest distance between the lines: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 2)/3 = (y - 3)/4 = (z - 5)/5.
(Or)
a) Find scalar projection of
a = 2i - j - 3k on
b = 3i - 5j + k.
b) Find the angle between vectors
a +
b and
a -
b if
a = 2i - j + 3k and
b = 3i - j - 2k.
