Himachal Hpbose Class 12 Mathematics — Real Previous-Year Papers

The function 'f' given by f(x) = log(cos x) is strictly decreasing on (a) (0, π/2) (b) (π/2, π) (c) (3π/2, 2π) (d) None of the above

∫ tan x dx equals (a) log|sin x| + C (b) log|cos x| + C (c) log|sec x| + C (d) log|cot x| + C

Order and degree of differential equation (y''')² + (y'')³ + (y')⁴ + y⁵ = 0 is (a) Order = 2, Degree = 3 (b) Order = 3, Degree = 2 (c) Order = 1, Degree = 4 (d) Order = 3, Degree = 3

The Integrating factor of differential equation (1 − y²) dy/dx + y·x = ay is (a) 1/(y²−1) (b) 1/√(y²−1) (c) 1/(1−y²) (d) 1/√(1−y²)

Dot Product of two vectors is defined as (a) a·b = |a||b| sin θ (b) a·b = |a||b| cos θ (c) a·b = |a||b| sin θ n̂ (d) None of the above

Direction cosines of x-axis are (a) <0, 0, 0> (b) <1, 1, 1> (c) <0, 0, 1> (d) <1, 0, 0>

Assertion (A): If a line has direction ratios −18, 12, −4 then its direction cosines are −9/11, 6/11, −2/11. Reason (R): If a line has direction ratios a, b, c then its direction cosines are a/√(a²+b²+c²), b/√(a²+b²+c²), c/√(a²+b²+c²). (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) Both Assertion (A) and Reason (R) are false.

Assertion (A): If P(A) = 0.8, P(B) = 0.5 and P(B/A) = 0.4 then P(A∩B) = 0.32 Reason (R): Conditional Probability of 'B' when event A has occurred is given by P(B/A) = P(A∩B) / P(A) (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) Both Assertion (A) and Reason (R) are false.

Assertion (A): Two events A and B are such that P(A) = 1/4, P(B) = 1/2 and P(A∩B) = 1/8 then two events A and B are independent. Reason (R): Two events are independent if the probability of occurrence of one does not affect the probability of occurrence of other and P(A∩B) = P(A) + P(B) − P(A∪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) Both Assertion (A) and Reason (R) are false.

Show that the relation R in the set A of all the books in a library of a college, given by R = (x, y) : x and y have same number of pages is an equivalence relation.

Solve the equations for x, y, z and t if 2·[[x, z], [y, t]] + 3·[[1, −1], [0, 2]] = 3·[[3, 5], [4, 6]]

Find the equation of line joining the points (3, 1) and (9, 3) using determinants.

Find all points of discontinuity of 'f' where 'f' is defined by f(x) = x/|x|, if x ≠ 0; f(x) = 0, if x = 0

Evaluate ∫ 1/√(8 + 3x − x²) dx

By using the properties of definite Integral Evaluate ∫₀⁴ |x − 1| dx

Find the area bounded by the curve y² = 4x, y-axis, and the line y = 3.

Solve the differential equation (x² − y²) dx + 2xy dy = 0 **OR** Solve the differential equation dy/dx + sec x · y = tan x

Show that the vectors 2î − ĵ + k̂, î − 3ĵ − 5k̂ and 3î − 4ĵ − 4k̂ form the vertices of a right angled triangle.

Find the area of parallelogram whose adjacent sides are given by the vectors a = 3î + ĵ + 4k̂ and b = î − ĵ + k̂.

Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently. Find the probability that (i) The problem is solved (ii) Exactly one of them solves the problem.

Two groups are competing for position on Board of directors of a corporation. The probabilities of winning of first and second groups will be 0.6 and 0.4 respectively. Further if the first group wins, the probabilities of introducing a new product is 0.7 and corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group. **OR** Event A and B are such that P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. State whether A and B are independent.

Write in the simplest form: tan⁻¹((3a²x − x³) / (a³ − 3ax²)), a > 0; −a/√3 ≤ x ≤ a/√3 **OR** Prove that: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).

Find dy/dx, if y = cos⁻¹((1 − x²) / (1 + x²)), 0 < x < 1 **OR** Differentiate w.r.t. 'x' (log x)^(cos x)

Solve the system of linear equations by using matrix method. 3x − 2y + 3z = 8 2x + y − z = 1 4x − 3y + 2z = 4

Prove that the volume of largest cone that can be inscribed in a sphere of radius 'R' is 8/27 of the volume of the sphere. **OR** Show that the function y = log(1 + x) − 2x/(2 + x), x > −1, is an increasing function of 'x' throughout its domain.

Find the shortest distance between the lines: r = (1 − t) î + (t − 2) ĵ + (3 − 2t) k̂ and r = (s + 1) î + (2s − 1) ĵ − (2s + 1) k̂ **OR** Find the angle between two lines: r = (3î + ĵ − 2k̂) + λ(î − ĵ − 2k̂) and r = (2î − ĵ − 56k̂) + μ(3î − 5ĵ − 4k̂)

Solve the following Linear Programming Problem (LPP), graphically. Minimise and Maximise Z = x + 2y Subject to: x + 2y ≥ 100 2x − y ≤ 0 2x + y ≤ 200 x, y ≥ 0