CBSE Class 12 Mathematics 2026 — Set 65/3/1

The value of k for which the function f(x) = x² sin (1)/(x), x ≠ 0 k(x+1), x = 0 is a continuous function, is: (A) (1)/(4) (B) 2 (C) (1)/(2) (D) 0

If sin⁻¹ x = y, then (dy)/(dx) is: (A) cos⁻¹ x (B) cos y (C) (1)/(1-x²) (D) sec y

The rate of change of the volume of a sphere with respect to its diameter, when its radius is 5 cm, is: (A) 400π cm³/cm (B) 100π cm³/cm (C) 50π cm³/cm (D) 25π cm³/cm

∫ fracdx2x + 2-x is equal to: (A) tan⁻¹(2x) + C (B) tan⁻¹(2-x) + C (C) fractan⁻¹(2x)log 2 + C (D) (log 2)tan⁻¹(2x) + C

∫-1¹ (1-|x|) dx is equal to: (A) 2 ∫₀¹ (1+x) dx (B) 2 ∫-1⁰ (1+x) dx (C) 0 (D) 2 ∫-1⁰ (1-x) dx

The area of the shaded region of the circle given below (see figure) is equal to: (A) ∫₁³ √(9-y²) dy (B) 2 ∫₁³ √(9-y²) dy (C) ∫₀³ √(9-x²) dx (D) 2 ∫₀³ √(9-x²) dx

(dy)/(dx) = F(x, y) will be a homogeneous differential equation for which of the following functions? (i) F(x, y) = 3x + 2y (ii) F(x, y) = sin (y)/(x) + log y - log x (iii) F(x, y) = ey/x + 1 (iv) F(x, y) = √(x² + y²) - y (A) (i) and (ii) (B) (i), (ii) and (iii) (C) (ii), (iii) and (iv) (D) (ii) and (iii)

For any two vectors veca and vecb, which of the following statements is always true? (A) veca · vecb ≤ |veca| |vecb| (B) |veca + vecb| ≥ |veca| + |vecb| (C) |veca - vecb| = |veca| - |vecb| (D) |veca × vecb| ≥ |veca| |vecb|

If (veca + vecb) · (veca - vecb) = 198 and |veca| = 10|vecb|, then: (A) |veca| = √(2) (B) |vecb| = √(2) (C) |vecb| = 10√(2) (D) |veca| = frac10√(2)

If l₁, m₁, n₁ and l₂, m₂, n₂ are direction cosines of lines L₁ and L₂ respectively and θ is the acute angle between them, then: (A) cos θ = l₁ l₂ + m₁ m₂ + n₁ n₂ (B) sin θ = l₁ l₂ + m₁ m₂ + n₁ n₂ (C) tan θ = (l₁)/(l₂) + (m₁)/(m₂) + (n₁)/(n₂) (D) cos θ = |l₁ l₂ + m₁ m₂ + n₁ n₂|

Direction ratios of lines L₁ and L₂ are langle 12, -3, 9 rangle and langle 4, q, -p rangle respectively. The values of p and q for which L₁ and L₂ are parallel are respectively: (A) -1, 3 (B) 3, 1 (C) -3, -1 (D) -1, -3

If E and F are two independent events such that P(E) = (3)/(10), P(E ∪ F) = (1)/(2), then P(E|F) - P(F|E) is equal to: (A) (2)/(7) (B) (3)/(35) (C) (1)/(70) (D) (1)/(7)

Questions number 19 and 20 are Assertion-Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes: (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): A particular solution of the differential equation (dy)/(dx) = ex+y is ex + e-y = -2. Reason (R): The general solution of the differential equation (dy)/(dx) = ex+y is ex + e-y = C.

Assertion (A): Vectors veca and (-2veca), where veca ≠ vec0, are collinear vectors. Reason (R): veca · (-2veca) = 0.

If x = t + (1)/(t) and y = t - (1)/(t), then find (dy)/(dx) at t = 2.

Find the sub-interval of (0, (π)/(2)) in which the function f(x) = tan x - 4x is increasing.

(a) Find the value of sin [ cot⁻¹( √(2) cos(tan⁻¹ 1) ) ]. OR (b) A relation R on A = \1, 2, 3\ is defined as R = \(1, 1), (3, 3), (1, 2)\. Is R a symmetric relation? Justify. Write the smallest relation R₁ such that R ∪ R₁ becomes an equivalence relation on the set \1, 2, 3\.

For two unit vectors veca and vecb, if |veca + 2vecb| = |2veca - vecb|, then find the angle between veca and vecb.

(a) If the lines (x-3)/(1) = (1-y)/(1) = (z+2)/(p) and (2-x)/(3) = (y+1)/(5) = (z+56)/(2p) are mutually perpendicular, then find the value(s) of p. OR (b) Find the vector equation of the line passing through the origin and perpendicular to both the lines vecr = 2hati - hatj + 2hatk + λ(3hati + 4hatj + 2hatk) and vecr = μ(hati - hatj + hatk).

(a) Find: ∫ fracdxx1/2 + x1/3 OR (b) Find: ∫ tan⁻¹((1 - x)/(1 + x)) dx

Evaluate: ∫₀π fracsin²⁰²⁶ xsin²⁰²⁶ x + cos²⁰²⁶ x dx

(a) Find: ∫ (cos x)/((2 + sin x)(4 + sin x)) dx OR (b) Find: ∫ (x + 3)/(x² + 4x + 5) dx

(a) Find the general solution of the differential equation 2x² (dy)/(dx) = y² + 2xy. OR (b) Find a particular solution of the differential equation (x+1)(dy)/(dx) = 2e-y - 1, given that y = 0 when x = 0.

If (sin x)y = ycos x, then find (dy)/(dx).

A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that Robotic Knee replacement surgeries have a 90\% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that: (i) exactly one surgery is successful, (ii) at most two surgeries are successful.

Show that the function f : mathbbR → mathbbR given by f(x) = fracx√(1 + x²) is one-one but not onto.

Solve the following Linear Programming Problem graphically: Maximise Z = 600x + 400y subject to the constraints x + 2y ≤ 12 4x + 5y ≥ 20 2x + y ≤ 12 x, y ≥ 0

(a) On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method. OR (b) Given that P = 2 -1 \3 4 , Q = 5 2 \7 4 and R = 2 5 \3 8 , find a matrix S such that PQ - RS is a null matrix.

(a) Write the equations of the given straight lines l₁ and l₂ in vector form. Hence, check whether the lines intersect or not. l₁: (x+3)/(-3) = (y-1)/(1) = (z-5)/(5) l₂: (x+1)/(-1) = (2-y)/(-2) = (z-5)/(5) OR (b) The opposite sides of a square are along the lines: vecr = hati + 2hatj - 4hatk + λ(2hati + 3hatj + 6hatk) vecr = 3hati + 3hatj - 5hatk + μ(2hati + 3hatj + 6hatk) If the direction ratios of the other pair of opposite sides of the square are langle -3, 6, p rangle, then find the area of the square and also the value of p.

Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown in the figure. Two ladders of length l₁ and l₂ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions: (i) Express p(x) = l₁² + l₂² in terms of x. (1) (ii) Find p'(x). (1) (iii) (a) Find the value of x for which l₁² + l₂² is minimum. (2) OR (iii) (b) If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum? (2)

A survey was conducted to find out the success rate of students who qualified the entrance examination by dropping a year after class XII. As per the data collected, 40% students appearing in the examination were dropouts and the remaining students were regular students of class XII. Of the dropouts, 5% qualify the examination while 10% of the regular students qualify the examination. Based on the above information, answer the following questions: (i) Find the probability that a student selected at random is a regular student. (1) (ii) A student is selected at random from a group of dropout students. What is the probability that the student will not qualify the examination? (1) (iii) (a) A student selected at random qualified the examination. Find the probability that the student is not a dropout. (2) OR (iii) (b) A student selected at random did not qualify the examination. Find the probability that the student was a regular student. (2)

There is a triangular park in the society. The park is divided into two sections as shown in the figure. In the region OAC, children are allowed to play games like cricket, football, while in the region AOB, activities which involve running are not allowed. The vertices of the triangular park ABC are A(0, 4), B(-2, 0) and C(3, 0). Based on the above information, answer the following questions: (i) Write the equation of the boundary line AB of the park. (1) (ii) Write the equation of the boundary line AC of the park. (1) (iii) (a) Using integration, find the area of region OAC, in which children are allowed to play cricket, football. (2) OR (iii) (b) Using integration, find the area of region AOB. (2)