Haryana Bseh Class 12 Mathematics — Real Previous-Year Papers

∫ (1)/(x(1+log x))dx is equal to: (a) x + log x + c (b) |x + log x| + c (c) log|1 + log x| + c (d) log(1+x) + c

∫ (sin⁶ x)/(cos⁸ x)dx is equal to:

The angle between the vectors hati + 3hatj + 3hatk and 3hati - 2hatj + hatk is: (a) 0° (b) 45° (c) 60° (d) 90°

Direction cosines of y-axis is ...........

The degree of the differential equation ((d²y)/(dx²))³ + ((dy)/(dx))² + sin((dy)/(dx)) + 1 = 0 is: (a) 3 (b) 2 (c) 1 (d) Not defined

If P(A) = (3)/(10), P(B) = (2)/(5) and P(A ∪ B) = (3)/(5), then P(B/A) is: (a) (1)/(4) (b) (1)/(3) (c) (5)/(12) (d) (7)/(12)

The probability of obtaining an even prime number on each dice, when a pair of dice is rolled, is: (a) 0 (b) (1)/(3) (c) (1)/(12) (d) (1)/(36)

2 4 \5 1 = 2x 4 \6 x , the possible value of x is/are: (a) 3 (b) √(3) (c) -√(3) (d) √(3), -√(3)

Let R be the relation in the set N given by R = \(a, b) : a = b - 2, b > 6\, choose the correct answer: (a) (2, 4) ∈ R (b) (3, 8) ∈ R (c) (6, 8) ∈ R (d) (8, 6) ∈ R

The vector equation of the line (x-5)/(7) = (y+4)/(7) = (6-z)/(2) is ..........

Area of the region bounded by the curve y² = 4x, y-axis and the line y = 3 is: (a) 2 (b) (9)/(4) (c) (9)/(3) (d) (9)/(2)

The general solution of the differential equation (dy)/(dx) = ex+y is: (a) ex + e-y = c (b) ex + ey = c (c) e-x + ey = c (d) e-x + e-y = c

Assertion (A): If a vector makes equal angle with co-ordinate axis then the direction cosines of the vector are ±((1)/(√3), (1)/(√3), (1)/(√3)). Reason (R): A vector makes α, β, γ angle with positive direction on x, y and z axis respectively, then their direction cosines are cosα, cosβ, cosγ. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (c) Assertion (A) is true, but Reason (R) is false. (d) Assertion (A) is false, but Reason (R) is true.

Assertion (A): If the events A and B are mutually exclusive events such that P(A) = 0.4, P(A ∪ B) = 0.6 and P(B) = P, then P = 0.2 Reason (R): Two events A and B are mutually exclusive events if P(A ∩ B) = P(A).P(B). (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (c) Assertion (A) is true, but Reason (R) is false. (d) Assertion (A) is false, but Reason (R) is true.

A doctor is to visit a patient. From past experience it is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively (3)/(10), (1)/(5), (1)/(10) and (2)/(5). The probabilities that he will be late are (1)/(4), (1)/(3) and (1)/(12), if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. Probability that he will not be late, when he comes by other means of transport.

(Continuing the doctor case study of Q.36) Name the theorem of probability used in (ii).

(Continuing the aluminium-box case study of Q.38) What will be the dimensions of the largest box?

(Continuing the aluminium-box case study of Q.38) What will be the side of the square removed to form the largest box?

Let T be the set of all triangles in a plane with R, a relation in T given by R = \(T₁, T₂) : T₁, T₂ are congruent\, show that R is an equivalence relation.

Find the value of x, y, z if A'A = I, where A = 0 2y z x y -z x -y z .

The volume of a cube is increasing at the rate of 8cm³/s. How fast is the surface area increasing when the length of edge is 12cm? **OR** Find the interval in which the function f, given f(x) = 2x² - 3x is, (a) increasing, (b) decreasing.

Find the general solution of the differential equation x(dy)/(dx) + 2y = x². **OR** Find the general solution of the differential equation ylog ydx - xdy = 0.

Three cards are drawn successively without replacement from a pack of 52 well shuffled cards. What is the probability that the first two cards are king and the third card drawn is an ace?

(Continuing the doctor case study of Q.36) When he arrives, he is late, what is the probability that he comes by train?

If f(x) is a continuous function defined on [0, a], then: ∫₀a f(x)dx = ∫₀a f(a-x)dx. On the basis of the above property of definite integral, answer the following: f(x) = (sin x - cos x)/(1+sin xcos x). Evaluate: ∫₀π/2 f(x)dx

(Continuing the definite-integral case study of Q.37) g(x) = log(1+tan x)dx. Evaluate: ∫₀π/4 g(x)dx

An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. On the basis of the above information, find the volume of the largest such box.

Express tan⁻¹(cos x)/(1-sin x), -(3π)/(2) < x < (π)/(2) in the simplest form. **OR** Find the value of: tan⁻¹(1)/(2)[sin⁻¹(2x)/(1+x²) + cos⁻¹(1-y²)/(1+y²)], |x| < 1, y > 0, xy < 1

Express the matrix A = 3 3 -1 \-2 -2 1 \-4 -5 2 as the sum of a symmetric and a skew-symmetric matrix.

Find (dy)/(dx), if xy = yx.

If y = e^acos⁻¹x, -1 < x < 1, show that: (1-x²)(d²y)/(dx²) - x(dy)/(dx) - a²y = 0

Evaluate: ∫ dfracxcos⁻¹x√(1-x²)dx **OR** Evaluate: ∫ e2xsin xdx

If veca = 2hati+2hatj+3hatk, vecb = -hati+2hatj+hatk and vecc = 3hati+hatj are such that veca + λvecb is perpendicular to vecc, then find the value of λ.

Solve the following system of equations by matrix method: 3x + 2y - 2z = 3 x + 2y + 3z = 6 2x - y + z = 2 **OR** If A = 2 -3 5 \3 2 -4 \1 1 -2 , find A⁻¹, and using A⁻¹ solve the system of equations: 2x - 3y + 5z = 11 3x + 2y - 4z = -5 x + y - 2z = -3

Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = -1 and x = 1. **OR** Find the area enclosed by the ellipse (x²)/(a²) + (y²)/(b²) = 1.

Find the shortest distance between the lines vecr = 6hati+2hatj+2hatk+λ(hati-2hatj+2hatk) and vecr = -4hati-hatk+μ(3hati-2hatj-2hatk). **OR** Find the vector equation of the line passing through the point (1, 2, -4) and perpendicular to the two lines (x-8)/(3) = (y+19)/(-16) = (z-10)/(7) and (x-15)/(3) = (y-29)/(8) = (z-5)/(-5).

Solve the following problem graphically: Minimize and Maximize Z = x + 2y Subject to the constraints x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200, x, y ≥ 0 **OR** Solve the following problem graphically: Maximize Z = x + y Subject to the constraints x + 4y ≤ 8, 2x + 3y ≤ 12, 3x + y ≤ 9, x, y ≥ 0