Punjab Pseb Class 12 Mathematics — Real Previous-Year Papers

If [[x-2y, 0], [5, x]] = [[-3, 0], [5, 3]], then y is equal to: (a) 1 (b) 3 (c) 2 (d) 4

If the order of the matrix A is 2×3 then the order of the matrix (A')' is: (a) 2×3 (b) 3×2 (c) 2×2 (d) 3×3

If A = 0, 1, 4, 9, 16, 25, ...... then function defined by f: Z → A, f(x) = x² is: (a) one-one but not onto (b) onto but not one-one (c) one-one and onto (d) neither one-one nor onto

Domain of function cosec⁻¹ is: (a) [-1, 1] (b) R - (-1, 1) (c) R (d) (-1, 1)

Principal value of cos⁻¹(1/2) is: (a) π/2 (b) π/3 (c) π/4 (d) π/6

If f(x) = tan 5x / 4x, x ≠ 0; m² - 1, x = 0 is continuous at x = 0 (m > 0) then value of m is: (a) 9/4 (b) 5/4 (c) 25/16 (d) 3/2

If y = e^(log x) then dy/dx is: (a) log x - x (b) x e^(log x) (c) 1 (d) e^(log x) log x

If y = sec x then, at x = π/4, (dy/dx)² is equal to: (a) 1 (b) √2 (c) 2 (d) 4

Direction ratios of the straight line vector r = 2i - 3j + k + m(9i - 2j + 5k) are: (a) <2, -3, 1> (b) <9, 2, 5> (c) <-2, 3, -1> (d) <9, -2, 5>

Vector equation of the line (x-5)/-4 = (y-3)/5 = (z+3)/-8 is: (a) vector r = 4i - 5j - 8k + μ(5i + 3j - 3k) (b) vector r = -4i + 5j + 8k + μ(5i + 3j - 3k) (c) vector r = 5i + 3j - 3k + μ(4i - 5j - 8k) (d) vector r = 5i + 3j - 3k + μ(-4i + 5j - 8k)

Maximum value of Z = 5x + 3y + 2, subject to the constraints x + y ≤ 7, x, y ≥ 0, is on the point. (a) (7, 0) (b) (0, 7) (c) (3, 4) (d) (4, 3)

If P(A) = 1/2, P(B) = 3/8 and P(A∪B) = 27/40 then P(A/B) is equal to: (a) 2/5 (b) 8/15 (c) 2/3 (d) 5/8

Ajay and Meera are contesting for two vacancies in a company. Probability of selection of Ajay is 7/9 and that of Meera is 4/7. What is the probability that both will be rejected? (a) 61/63 (b) 6/63 (c) 41/63 (d) 28/63

Using determinants find the equation of a line which passes through the points (2, -3) and (-5, 6).

Find the critical points of function f(x) = 2x³ - 15x² + 36x - 19.

Find the angle between vectors vector a = 5i - j + 7k and vector b = 9i + 4j - k.

Evaluate ∫ (1 + tan²x) / (tan²x + 6 tan x - 7) dx.

Find the values of a and b if the function f(x) = ax² + b, x > 2 ; 2, x = 2 ; 2ax - b, x < 2 is continuous at x = 2.

Using integration, find the area bounded by the circle x² + y² = 9 in the first quadrant.

Satnam is pouring water in his cylindrical flask at the rate of 400 cc/s but flask is leaking at the rate of 4 cc/s. Will the water level increase or decrease in the flask? Also find the rate of change of water level if radius of the flask is 3 cm.

A relation R = (x, y): x ≤ y² where x, y ∈ R is defined on the set of real numbers R. Show that this relation is neither reflexive nor symmetric nor transitive.

If A = [[2, 3], [1, 2]] and f(x) = x² - 4x + 1 then find f(A). Also find A⁻¹ using the value of f(A).

Solve the following linear programming problem graphically: Maximise and minimise Z = 3x + 2y subject to the constraints 4x + y ≥ 8, x + y ≤ 8, x - y ≥ 0, x ≥ 0, y ≥ 0.

Solve: x log x (dy/dx) + y = (2/x) log x. **OR** Solve: x² dy - (3x² + xy + y²) dx = 0; given that y = 1 when x = 1.

Bag I contains 4 red balls and 7 white balls. Bag II contains 5 red balls and 6 white balls. One of the bags is chosen at random and a ball is drawn from it. Find the probability of drawing: (i) a red ball. (ii) a white ball. Which ball is more likely to be drawn?

If f(x) = (tan x)x, g(x) = x^(tan x), then find that amongst f(x) and g(x), which function changes less rapidly with respect to the independent variable x, when x = π/4. Also find the difference between f'(π/4) and g'(π/4). (Take π = 3.14 and loge(π/4) = -0.2) **OR** If y = log(x + √(x² + 1)) then find dy/dx and d²y/dx². Also show that dy/dx > d²y/dx² at x = 1.

Evaluate ∫ 1 / ((x-1)(x-2)(x+3)) dx. **OR** Evaluate ∫ e^(3x) cos 5x dx.

(a) [4 marks] Prove that for any two non-zero vectors vector a and vector b, |vector a + vector b| ≤ |vector a| + |vector b|. Also write the name of this inequality. (b) [2 marks] Find the projection of vector 3i - 2j - 7k on the vector 9i + j + 7k. **OR** [6 marks] Find the shortest distance between the following pairs of lines: vector r = 2i + j - 3k + μ(i - 7j + 2k) and vector r = 5i - j - 4k + λ(6i - j + k).

[6 marks] A window is in the form of a rectangle surmounted by a semi-circle. If perimeter of window is 20 m then find the dimensions of the window so that it can admit maximum light through the whole opening. **OR** (a) [3 marks] Evaluate ∫ 1/(1 + tan x) dx. (b) [3 marks] Evaluate ∫ sin x sin 2x sin 3x dx.

If double of Ravneet's present age is subtracted from the sum of double of Manisha's present age and Navdeep's present age then we get 4. Five years ago, if we subtract the sum of Manisha's age and Navdeep's age from the double of Ravneet's age then we get 8. After five years, if we subtract double of Manisha's age from the sum of Ravneet's and Navdeep's age then we get 1. Using matrices, find the present ages of Manisha, Ravneet and Navdeep. **OR** (a) [4 marks] Express [[8, 3], [7, 5]] as the sum of a symmetric matrix and a skew-symmetric matrix. (b) [2 marks] Using determinant, find the area of a triangle whose vertices are (3, -4), (-5, 9) and (2, 7).