CBSE Class 12 Mathematics 2025 — Set 65/1/1

If ∫ frac21/xx² dx = k · 21/x + C, then k is equal to (A) (-1)/(log 2) (B) -log 2 (C) -1 (D) (1)/(2)

The integrating factor of differential equation (x + 2y³) (dy)/(dx) = 2y is (A) e(y^2)/(2) (B) frac1√(y) (C) (1)/(y²) (D) e-(1)/(y^2)

If A = 7 0 x \0 7 0 \0 0 y is a scalar matrix, then yx is equal to (A) 0 (B) 1 (C) 7 (D) ± 7

The corner points of the feasible region in graphical representation of a L.P.P. are (2, 72), (15, 20) and (40, 15). If Z = 18x + 9y be the objective function, then (A) Z is maximum at (2, 72), minimum at (15, 20) (B) Z is maximum at (15, 20), minimum at (40, 15) (C) Z is maximum at (40, 15), minimum at (15, 20) (D) Z is maximum at (40, 15), minimum at (2, 72)

If A and B are invertible matrices, then which of the following is not correct ? (A) (A + B)⁻¹ = B⁻¹ + A⁻¹ (B) (AB)⁻¹ = B⁻¹A⁻¹ (C) adj (A) = |A| A⁻¹ (D) |A⁻¹| = |A|⁻¹

If the feasible region of a linear programming problem with objective function Z = ax + by, is bounded, then which of the following is correct ? (A) It will only have a maximum value. (B) It will only have a minimum value. (C) It will have both maximum and minimum values. (D) It will have neither maximum nor minimum value.

The area of the shaded region bounded by the curves y² = x, x = 4 and the x-axis is given by (A) ∫₀⁴ x dx (B) ∫₀² y² dy (C) 2 ∫₀⁴ √(x) dx (D) ∫₀⁴ √(x) dx

Assertion (A) : f(x) = 3x-8, x ≤ 5 \2k, x > 5 is continuous at x=5 for k = (5)/(2). Reason (R) : A function f is continuous at x=a if limx → a^- f(x) = limx → a^+ f(x) = f(a).

If vector veca = 3hati + 2hatj - hatk and vector vecb = hati - hatj + hatk, then which of the following is correct? (A) veca parallel vecb (B) veca perp vecb (C) |vecb| > |veca| (D) |veca| = |vecb|

∫-1¹ (|x|)/(x)dx,x ≠ 0 is equal to: (A) -1 (B) 0 (C) 1 (D) 2

The absolute maximum value of function f(x) = x³ - 3x + 2 in [0, 2] is: (A) 0 (B) 2 (C) 4 (D) 5

If veca + vecb + vecc = vec0, |veca| = √(37), |vecb| = 3 and |vecc| = 4, then angle between vecb and vecc is: (A) (π)/(6) (B) (π)/(4) (C) (π)/(3) (D) (π)/(2)

Assertion (A) : Let mathbbZ be the set of integers. A function f : mathbbZ → mathbbZ defined as f(x) = 3x - 5, ∀ x ∈ mathbbZ is a bijective. Reason (R) : A function is a bijective if it is both surjective and injective. (A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

Evaluate: tan⁻¹ [ 2 sin ( 2 cos⁻¹ frac√(3)2 ) ].

Find the area of a parallelogram whose diagonals are represented by veca = 2 hati - hatj + hatk and vecb = hati + 3 hatj - hatk.

Find the intervals in which the function f(x) = 5x(3)/(2) - 3x(5)/(2) is (i) increasing (ii) decreasing.

(a) Differentiate 2cos^2 x with respect to cos² x. OR (b) If tan⁻¹(x² + y²) = a², then find (dy)/(dx).

(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors veca = 3hati + hatj + 2hatk and vecb = 2hati - 2hatj + 4hatk. Determine the angle formed between the kite strings. Assume there is no slack in the strings. OR (b) Find a vector of magnitude 21 units in the direction opposite to that of overrightarrowAB, where A and B are the points A(2, 1, 3) and B(8, -1, 0) respectively.

The side of an equilateral triangle is increasing at the rate of 3 cm/s. Find the rate at which its area is increasing when the side is 15 cm.

Solve the following Linear Programming Problem graphically: Constraints: x - y ≥ 0 x - 2y ≥ -2 x ≥ 0, y ≥ 0 Maximize Z = x + 2y.

Draw the graph of y = |x + 3| and using integration, find the area of the region bounded by the curve and the x-axis between x = -6 and x = 0.

(a) Find: ∫ (x + sin x)/(1 + cos x)dx OR (b) Evaluate: ∫₀π/4 fracdxcos³ x √(2sin 2x)

(a) Verify that the lines given by vecr = (1-λ)hati + (λ - 2)hatj + (3 - 2λ)hatk and vecr = (μ + 1)hati + (2μ - 1)hatj - (2μ + 1)hatk are skew lines. Hence, find the shortest distance between the lines. OR (b) During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by vecB = 2hati + 8hatj, vecW = 6hati + 12hatj and vecF = 12hati + 18hatj respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.

(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows: X: 0, 2, 4, 5 P(X): p, 2p, 3p, p where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday. OR (b) For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.

Find the absolute maximum and absolute minimum values of the function f(x) = 2x³ - 15x² + 36x + 1 in [1, 5].

A school wants to allocate its students into three clubs with the following conditions: Sports, Music and Drama: * The number of students in the Sports club is equal to the sum of the number of students in the Music and Drama clubs. * The number of students in the Music club is 20 more than half the number of students in the Sports club. * The total number of students in all three clubs is 180. Using matrix method, find the number of students in different clubs.

(a) If √(1 - x²) + √(1 - y²) = a(x - y), then prove that (dy)/(dx) = √((1 - y²)/(1 - x²)). OR (b) If x = a(cosθ + logtan(θ)/(2)) and y = sinθ, then find (d²y)/(dx²) at θ = (π)/(4).

(a) Find the image A' of the point A(1, 6, 3) in the line (x)/(1) = (y-1)/(2) = (z-2)/(3). Also, find the equation of the line joining A and A'. OR (b) Find a point P on the line (x+5)/(1) = (y+3)/(4) = (z-6)/(-9) such that its distance from the point Q(2, 4, -1) is 7 units. Also, find the equation of the line joining P and Q.

A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be x metres and with parallel to the partition be y metres. Answer the following questions based on the above information: (i) Find the equation formed by the total material of the boundary and parallel division to be used, using x and y. (ii) Write the area of the solar panel as a function of x. (iii) (A) Find the critical point of the area function. Using the second derivative test, find the critical point at which the area is maximum. Also find the maximum area. OR (iii) (B) Using the first derivative test, find the area of the maximum region bounded by 300 m of boundary material, where parallel division is also taken into account.

A class teacher is keen to assess the learning of the concept of 'relations' taught to her students. She writes the following five relations defined on each set A = \1, 2, 3\: R₁ = \(2, 3), (3, 2)\ R₂ = \(1, 2), (1, 3), (3, 2)\ R₃ = \(1, 2), (2, 1), (1, 1)\ R₄ = \(1, 1), (1, 2), (3, 3), (2, 2)\ R₅ = \(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)\ Students are asked to answer the following questions for the above relations: (i) Find the relation which is reflexive and transitive but not symmetric. (ii) Find the relation which is reflexive and symmetric but not transitive. (iii) (A) Find the relation which is symmetric but neither reflexive nor transitive. OR (iii) (B) To make relation R₂ an equivalence relation, write the pairs that need to be added.

A bank offers loans to its customers at various types of interest rates, such as fixed rate, floating rate and variable rate. From the past data of the bank, it is known that customers obtain loans at fixed rate, floating rate and variable rate with probabilities 10\%, 20\% and 70\% respectively. After taking a loan, a customer may repay the loan or may default in repaying the loan. From the bank's data, it is found that the probabilities of a person defaulting on repayment after taking a loan at fixed rate, floating rate and variable rate are 5\%, 3\% and 1\% respectively. Based on the above, answer the following questions: (i) What is the probability that a customer defaults in repaying the loan after taking a loan? 2 (ii) A customer defaults in repaying the loan after taking a loan. What is the probability that he had taken the loan at variable rate of interest? 2