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Class 12 Maths: questions from CBSE board papers, by chapter

Every question of the 4 CBSE board exam papers Shishya has solved (2026 Q.P. 65/1/1, 2025 Q.P. 65/1/1, 2024 Q.P. 65/1/1, 2023 Q.P. 65/1/1), listed under the NCERT chapter it was asked from. The question lines are in Shishya's own shorter words; each links to its step-by-step solution, and the paper page links CBSE's own question paper. Marks per chapter count every listed question, including both sides of an either/or choice.

Relations and Functions

12 questions, 31 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q32(a) · 5 marksOn the integers Z define R by: (x, y) ∈ R when a fixed prime p divides |x − y|. Is R an equivalence relation?
Asked in CBSE 2026 · 65/1/1 · Q32(b) · 5 marksShow f(x) = (3x + 2)/(5x − 3), from R − {3/5} to R − {3/5}, is one-one and onto.
Asked in CBSE 2025 · 65/1/1 · Q19 · 1 markAssertion: f: Z → Z, f(x) = 3x − 5 is bijective. Reason: a function that is one-one and onto is bijective.
Asked in CBSE 2025 · 65/1/1 · Q37(i) · 1 markOn 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)}. Which is reflexive and transitive but not symmetric?
Asked in CBSE 2025 · 65/1/1 · Q37(ii) · 1 markFrom R₁–R₅ on {1, 2, 3} above, which is reflexive and symmetric but not transitive?
Asked in CBSE 2025 · 65/1/1 · Q37(iii)(a) · 2 marksFrom R₁–R₅ on {1, 2, 3} above, which are symmetric yet neither reflexive nor transitive?
Asked in CBSE 2025 · 65/1/1 · Q37(iii)(b) · 2 marksWhich pairs must be added to R₂ = {(1,2),(1,3),(3,2)} on {1, 2, 3} to make it an equivalence relation?
Asked in CBSE 2024 · 65/1/1 · Q1 · 1 markf : R₊ → R (R₊ = non-negative reals), f(x) = 4x + 3. Is f one-one, onto, both or neither?
Asked in CBSE 2024 · 65/1/1 · Q26(a) · 3 marksOn A = {1, 2, 3, 4, 5}, R = {(x, y) : |x² − y²| < 8}. Test R for reflexivity, symmetry and transitivity.
Asked in CBSE 2024 · 65/1/1 · Q26(b) · 3 marksf : R → R, f(x) = ax + b; given f(2) = 3 and f(1) = 1, find f and test it for one-one and onto.
Asked in CBSE 2023 · 65/1/1 · Q21(a) · 2 marksf : A → B, f(x) = 2x, is a bijection and A = {1, 2, 3, 4}. Find B.
Asked in CBSE 2023 · 65/1/1 · Q32 · 5 marksOn R, (x, y) ∈ R when the product xy is irrational. Test R for reflexive, symmetric, transitive.

Inverse Trigonometric Functions

12 questions, 18 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q1 · 1 markIf y = 2 cos⁻¹x, what range of values can y take?
Asked in CBSE 2026 · 65/1/1 · Q25(a) · 2 marksSimplify tan⁻¹[(cos 2x − sin 2x)/(cos 2x + sin 2x)] for 0 < x < π/4.
Asked in CBSE 2026 · 65/1/1 · Q25(b) · 2 marksEvaluate tan(sin⁻¹1 − cos⁻¹(−1/2)).
Asked in CBSE 2025 · 65/1/1 · Q18 · 1 markA graph of a trigonometric function is shown; pick the graph of its inverse from four options.
Asked in CBSE 2025 · 65/1/1 · Q22 · 2 marksFind the value of tan⁻¹[2 sin(2 cos⁻¹(√3/2))].
Asked in CBSE 2024 · 65/1/1 · Q36(i) · 1 markA camera sits 5 m up a pole; a car is x m from the pole's base and sees the camera at elevation θ. Write θ in terms of the height and x.
Asked in CBSE 2024 · 65/1/1 · Q38(i) · 1 marksin is restricted to a set A onto [−1, 1] so that sin⁻¹ exists. Give such an interval A other than the principal branch.
Asked in CBSE 2024 · 65/1/1 · Q38(ii) · 1 markUsing principal values, find sin⁻¹(−1/2) − sin⁻¹(1).
Asked in CBSE 2024 · 65/1/1 · Q38(iii)(a) · 2 marksSketch y = sin⁻¹x on [−1, 1] with its principal value branch.
Asked in CBSE 2024 · 65/1/1 · Q38(iii)(b) · 2 marksFor f(x) = 2 sin⁻¹(1 − x), state its domain and its range.
Asked in CBSE 2023 · 65/1/1 · Q19 · 1 markA: f(x) = 2 sin⁻¹x + 3π/2 on [−1, 1] has range [π/2, 5π/2]. R: the principal branch of sin⁻¹x has range [0, π].
Asked in CBSE 2023 · 65/1/1 · Q21(b) · 2 marksFind sin⁻¹(sin 3π/4) + cos⁻¹(cos 3π/4) + tan⁻¹(1).

Matrices

13 questions, 17 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q2 · 1 markWhich order is impossible for a row matrix: 2 × 1, 1 × 2, 1 × 1 or 1 × n?
Asked in CBSE 2026 · 65/1/1 · Q3 · 1 markFor matrices of suitable order, which transpose rules are true: (i) sum rule (A+B)' = A'+B'; (ii) (A−B)' equals B'−A'; (iii) (AB)' equals A'B'; (iv) (kAB)' equals kB'A' for a scalar k?
Asked in CBSE 2026 · 65/1/1 · Q6 · 1 markA and B are skew symmetric of the same order. Which of AB, AB + BA, (A + B)², A − B is skew symmetric?
Asked in CBSE 2025 · 65/1/1 · Q1 · 1 markA is the 3 × 3 diagonal matrix with diagonal entries −1, 1, 1. Find A⁻¹.
Asked in CBSE 2025 · 65/1/1 · Q9 · 1 markA is 3 × 3, B is the 3 × 1 column (−2, −5, −7) and C is the 1 × 3 row [9 8 7]. Which of AB, AC, BA is defined?
Asked in CBSE 2025 · 65/1/1 · Q13 · 1 markThe matrix with rows (7, 0, x), (0, 7, 0), (0, 0, y) is a scalar matrix. Find yˣ.
Asked in CBSE 2024 · 65/1/1 · Q2 · 1 markA matrix has 36 elements. How many different orders m × n are possible?
Asked in CBSE 2024 · 65/1/1 · Q5 · 1 markThe 2 × 2 matrix [x + y, 2; 5, xy] equals [6, 2; 5, 8]. Find 24/x + 24/y.
Asked in CBSE 2024 · 65/1/1 · Q18 · 1 markA, B are non-zero square matrices of equal order with (A + B)² = A² + B². What follows?
Asked in CBSE 2024 · 65/1/1 · Q32(b) · 5 marksA = [−1, a, 2; 1, 2, x; 3, 1, 1] has inverse [1, −1, 1; −8, 7, −5; b, y, 3]. Find (a + x) − (b + y).
Asked in CBSE 2023 · 65/1/1 · Q1 · 1 markA square matrix A satisfies A² − 3A + I = O, and A⁻¹ = xA + yI. Find x + y.
Asked in CBSE 2023 · 65/1/1 · Q4 · 1 markA = [3 4; 5 2] (rows separated by ;). If 2A + B = O (zero matrix), find B.
Asked in CBSE 2023 · 65/1/1 · Q14 · 1 markA = [0 1; −1 0] and (3I + 4A)(3I − 4A) = x²I. Find x.

Determinants

14 questions, 38 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q4 · 1 markΔ₁ has rows (1,0,0), (0,2,0), (0,0,3) and Δ₂ has rows (0,2,0), (1,0,0), (0,0,6). How are Δ₁ and Δ₂ related?
Asked in CBSE 2026 · 65/1/1 · Q5 · 1 markFind one value of x for which the determinant with rows (cos x, sin x) and (−cos x, sin x) equals 1.
Asked in CBSE 2026 · 65/1/1 · Q33(a) · 5 marksA has rows (0, 2, 1), (−2, −1, −2), (1, −1, 0). Find A⁻¹ and use it to solve the system: z − 2y = 7; −y − z + 2x = 8; x − 2y = 10.
Asked in CBSE 2026 · 65/1/1 · Q33(b) · 5 marksThe matrix with rows (3, −1, sin 3x), (−7, 4, cos 2x), (−11, 7, 2) is singular. Find all x in [0, π/2].
Asked in CBSE 2025 · 65/1/1 · Q6 · 1 markA is a 2 × 2 matrix with det A = 4. Find det(4 adj A).
Asked in CBSE 2025 · 65/1/1 · Q15 · 1 markA and B are invertible. Which is false: (A + B)⁻¹ = B⁻¹ + A⁻¹, (AB)⁻¹ = B⁻¹A⁻¹, adj A = |A| A⁻¹, or |A|⁻¹ = |A⁻¹|?
Asked in CBSE 2025 · 65/1/1 · Q35 · 5 marks180 students go into Sports, Music and Drama clubs. Sports = Music + Drama, and Music = half of Sports plus 20. Use the matrix method to find each club's size.
Asked in CBSE 2024 · 65/1/1 · Q12 · 1 markEvaluate the determinant with first row x + 1, x − 1 and second row x² + x + 1, x² − x + 1.
Asked in CBSE 2024 · 65/1/1 · Q19 · 1 markA: for A = [1, cos θ, 1; −cos θ, 1, cos θ; −1, −cos θ, 1], θ in [0, 2π], |A| lies in [2, 4]. R: cos θ always lies in [−1, 1].
Asked in CBSE 2024 · 65/1/1 · Q32(a) · 5 marksFor A = [1, −2, 0; 2, −1, −1; 0, −2, 1] find A⁻¹, then solve 2x − y − z = 8, x − 2y = 10, −2y + z = 7.
Asked in CBSE 2023 · 65/1/1 · Q2 · 1 markA is 2 × 2 with |A| = 2. Find the value of |4A⁻¹|.
Asked in CBSE 2023 · 65/1/1 · Q3 · 1 markFor a 3 × 3 matrix A, |adj A| = 64. What can |A| be?
Asked in CBSE 2023 · 65/1/1 · Q33(a) · 5 marksA = [1 2 −2; −1 3 0; 0 −2 1] and B⁻¹ = [3 −1 1; −15 6 −5; 5 −2 2]. Find (AB)⁻¹.
Asked in CBSE 2023 · 65/1/1 · Q33(b) · 5 marksUsing matrices, solve x + 2y + 3z = 6, 2x + z − y = 2, 3x + 2y − 2z = 3.

Continuity and Differentiability

18 questions, 42 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q21(a) · 2 marksIs f continuous at x = 3, where f(x) = |x − 3|/(2(x − 3)) for x < 3 and f(x) = (x − 6)/6 for x ≥ 3?
Asked in CBSE 2026 · 65/1/1 · Q21(b) · 2 marksGiven √3(x² + y²) = 4xy, find dy/dx at the point (1/2, √3/2).
Asked in CBSE 2026 · 65/1/1 · Q34 · 5 marksGiven x = cos t and y = cos mt, prove (1 − x²) y'' − x y' + m²y = 0.
Asked in CBSE 2025 · 65/1/1 · Q5 · 1 markf(x) = |x| + |x − 1|. Decide its continuity and differentiability at x = 0 and x = 1.
Asked in CBSE 2025 · 65/1/1 · Q20 · 1 markAssertion: f(x) = 3x − 8 for x ≤ 5 and 2k for x > 5 is continuous at x = 5 when k = 5/2. Reason: continuity at a needs both one-sided limits equal to f(a).
Asked in CBSE 2025 · 65/1/1 · Q21(a) · 2 marksFind the derivative of 2^(cos²x) with respect to cos²x.
Asked in CBSE 2025 · 65/1/1 · Q21(b) · 2 marksGiven tan⁻¹(x² + y²) = a², find dy/dx.
Asked in CBSE 2025 · 65/1/1 · Q32(a) · 5 marksGiven √(1 − x²) + √(1 − y²) = a(x − y), prove dy/dx = √((1 − y²)/(1 − x²)).
Asked in CBSE 2025 · 65/1/1 · Q32(b) · 5 marksx = a(cos θ + log tan(θ/2)) and y = sin θ. Find d²y/dx² at θ = π/4.
Asked in CBSE 2024 · 65/1/1 · Q3 · 1 markf(x) = x² + 3 for x ≠ 0 and f(0) = 1. Where is f continuous and differentiable?
Asked in CBSE 2024 · 65/1/1 · Q13 · 1 markFind d/dx of sin(x²) at x = √π.
Asked in CBSE 2024 · 65/1/1 · Q21(a) · 2 marksIs f(x) = x²|x| differentiable at x = 0?
Asked in CBSE 2024 · 65/1/1 · Q21(b) · 2 marksGiven y = √(tan √x), show √x · dy/dx = (1 + y⁴)/(4y).
Asked in CBSE 2024 · 65/1/1 · Q27(a) · 3 marksGiven √(1 − x²) + √(1 − y²) = a(x − y), prove dy/dx = √((1 − y²)/(1 − x²)).
Asked in CBSE 2024 · 65/1/1 · Q27(b) · 3 marksFind dy/dx when y = (tan x)^x.
Asked in CBSE 2023 · 65/1/1 · Q13 · 1 markf(x) = 3x + 5 for x ≥ 2 and kx² for x < 2. For which k is f continuous?
Asked in CBSE 2023 · 65/1/1 · Q24 · 2 marksy = (x + √(x² − 1))². Prove (x² − 1)(dy/dx)² = 4y².
Asked in CBSE 2023 · 65/1/1 · Q38(ii) · 2 marksWith the same f and the value of a found, evaluate f″(1).

Application of Derivatives

26 questions, 45 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q7 · 1 markLeast value of f(x) = x³ − 12x on [0, 3].
Asked in CBSE 2026 · 65/1/1 · Q22 · 2 marksPerfume in an inverted conical bottle (semi-vertical angle π/6) decreases at 1 mm³/min. How fast is the perfume level falling when it is 10 mm deep?
Asked in CBSE 2026 · 65/1/1 · Q36(i) · 1 markA delivery firm has 5000 subscribers at ₹300 a year; each ₹1 rise loses 10 subscribers. After a rise of ₹x, how many leave?
Asked in CBSE 2026 · 65/1/1 · Q36(ii) · 1 markSame firm (5000 subscribers, ₹300 fee, 10 lost per ₹1 rise): write revenue R(x) after a fee rise of ₹x.
Asked in CBSE 2026 · 65/1/1 · Q36(iii)(a) · 2 marksWith R(x) = (5000 − 10x)(300 + x), find x giving maximum revenue.
Asked in CBSE 2026 · 65/1/1 · Q36(iii)(b) · 2 marksWith R(x) = (5000 − 10x)(300 + x), find where R increases and decreases in (0, 5000).
Asked in CBSE 2025 · 65/1/1 · Q8 · 1 markFind the absolute maximum of f(x) = x³ − 3x + 2 on [0, 2].
Asked in CBSE 2025 · 65/1/1 · Q24 · 2 marksOn which intervals is f(x) = 5x^(3/2) − 3x^(5/2) (i) increasing, (ii) decreasing?
Asked in CBSE 2025 · 65/1/1 · Q26 · 3 marksAn equilateral triangle's side grows at 3 cm/s. How fast is its area growing when the side is 15 cm?
Asked in CBSE 2025 · 65/1/1 · Q33 · 5 marksFind the absolute maximum and minimum of f(x) = 2x³ − 15x² + 36x + 1 on [1, 5].
Asked in CBSE 2025 · 65/1/1 · Q36(i) · 1 markA rectangular solar installation on a roof uses 300 m of boundary material, including one partition parallel to one pair of sides. Sides perpendicular to the partition are x m, sides parallel to it y m. Write the material equation in x and y.
Asked in CBSE 2025 · 65/1/1 · Q36(ii) · 1 markFor the same installation (2x + 3y = 300), write the area as a function of x.
Asked in CBSE 2025 · 65/1/1 · Q36(iii)(a) · 2 marksFor A(x) = x(300 − 2x)/3, find the critical point, confirm a maximum by the second derivative test, and find the maximum area.
Asked in CBSE 2025 · 65/1/1 · Q36(iii)(b) · 2 marksUsing the first derivative test, find the largest area possible with the 300 m of material and the partition.
Asked in CBSE 2024 · 65/1/1 · Q4 · 1 markf is differentiable in the open interval (a, b) and continuous on its closure. Which condition makes f strictly increasing there?
Asked in CBSE 2024 · 65/1/1 · Q22 · 2 marksShow that f(x) = 4x³ − 18x² + 27x − 7 has no maximum or minimum.
Asked in CBSE 2024 · 65/1/1 · Q36(ii) · 1 markWith θ = tan⁻¹(5/x) as above, find dθ/dx.
Asked in CBSE 2024 · 65/1/1 · Q36(iii)(a) · 2 marksThe car moves away from the 5 m pole at 20 m/s. Find dθ/dt when it is 50 m from the pole.
Asked in CBSE 2024 · 65/1/1 · Q36(iii)(b) · 2 marksAnother car, 50 m from the 5 m pole, has its elevation angle changing at 3/101 rad/s. Find its speed.
Asked in CBSE 2023 · 65/1/1 · Q16 · 1 markf(x) = a(x − cos x) decreases strictly on all of R. Where must a lie?
Asked in CBSE 2023 · 65/1/1 · Q25 · 2 marksProve f(x) = 16 sin x/(4 + cos x) − x decreases strictly on (π/2, π).
Asked in CBSE 2023 · 65/1/1 · Q37(i) · 1 markA conical tank part (semi-vertical angle 45°) is full of water draining at 2 cm³/s. Express the water volume through the radius r.
Asked in CBSE 2023 · 65/1/1 · Q37(ii) · 1 markSame cone (V decreasing at 2 cm³/s). How fast does r change when r = 2√2 cm?
Asked in CBSE 2023 · 65/1/1 · Q37(iii)(a) · 2 marksSame cone. At r = 2√2 cm, at what rate does the wet curved surface shrink?
Asked in CBSE 2023 · 65/1/1 · Q37(iii)(b) · 2 marksSame cone. Find dh/dt at the moment the slant height is 4 cm.
Asked in CBSE 2023 · 65/1/1 · Q38(i) · 2 marksA roller-coaster path is f(x) = a(x − 3)(x + 1)(x + 9) and it meets the y-axis at (0, −1). Find a.

Integrals

23 questions, 57 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q8 · 1 markIf ∫ 3ax/(b² + c²x²) dx = A log|b² + c²x²| + K, find A.
Asked in CBSE 2026 · 65/1/1 · Q9 · 1 markEvaluate the integral of x³/(x² + 2|x| + 1) from x = −1 to 1.
Asked in CBSE 2026 · 65/1/1 · Q26 · 3 marksEvaluate the integral of x tan⁻¹x from 0 to 1.
Asked in CBSE 2026 · 65/1/1 · Q27(a) · 3 marksFind ∫ √((x + 2)/(x − 2)) dx.
Asked in CBSE 2026 · 65/1/1 · Q27(b) · 3 marksFind ∫ x²/((x² + 9)(x² + 16)) dx.
Asked in CBSE 2026 · 65/1/1 · Q28 · 3 marksLet I₁ = ∫ dx/(1 + cos 2x) over [−π/4, π/4] and I₂ = ∫ |x| dx over [−1/2, 1/2]. Prove I₁ − 4I₂ = 0.
Asked in CBSE 2025 · 65/1/1 · Q3 · 1 markEvaluate the integral of |x|/x from x = −1 to 1 (x ≠ 0).
Asked in CBSE 2025 · 65/1/1 · Q10 · 1 markIf the integral of 2^(1/x)/x² dx equals k · 2^(1/x) + C, find k.
Asked in CBSE 2025 · 65/1/1 · Q28(a) · 3 marksIntegrate (x + sin x)/(1 + cos x) with respect to x.
Asked in CBSE 2025 · 65/1/1 · Q28(b) · 3 marksEvaluate the integral from 0 to π/4 of dx / (cos³x · √(2 sin 2x)).
Asked in CBSE 2024 · 65/1/1 · Q6 · 1 markWhich integral over [a, b] always equals the integral of f(x) from a to b?
Asked in CBSE 2024 · 65/1/1 · Q23(a) · 2 marksIntegrate x√(1 + 2x) with respect to x.
Asked in CBSE 2024 · 65/1/1 · Q23(b) · 2 marksEvaluate the integral of sin√x / √x from 0 to π²/4.
Asked in CBSE 2024 · 65/1/1 · Q28(a) · 3 marksIntegrate x² / ((x² + 4)(x² + 9)) with respect to x.
Asked in CBSE 2024 · 65/1/1 · Q28(b) · 3 marksEvaluate the integral of |x − 1| + |x − 2| + |x − 3| from 1 to 3.
Asked in CBSE 2024 · 65/1/1 · Q33(a) · 5 marksEvaluate the integral of (sin x + cos x)/(9 + 16 sin 2x) from 0 to π/4.
Asked in CBSE 2024 · 65/1/1 · Q33(b) · 5 marksEvaluate the integral of sin 2x · tan⁻¹(sin x) from 0 to π/2.
Asked in CBSE 2023 · 65/1/1 · Q5 · 1 markThe derivative of f(x) is log x. Find f(x).
Asked in CBSE 2023 · 65/1/1 · Q6 · 1 markEvaluate the integral of sec²(x − π/6) from x = 0 to x = π/6.
Asked in CBSE 2023 · 65/1/1 · Q26 · 3 marksEvaluate the integral of log(sin x) − log(2 cos x) from 0 to π/2.
Asked in CBSE 2023 · 65/1/1 · Q27 · 3 marksIntegrate 1/[√x(√x + 1)(√x + 2)] with respect to x.
Asked in CBSE 2023 · 65/1/1 · Q31(a) · 3 marksEvaluate the integral of eˣ sin x from 0 to π/2.
Asked in CBSE 2023 · 65/1/1 · Q31(b) · 3 marksIntegrate 1/[cos(x − a) cos(x − b)] with respect to x.

Application of Integrals

9 questions, 21 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q10 · 1 markArea enclosed by y = x|x|, the x-axis and the lines x = −1 and x = 1.
Asked in CBSE 2026 · 65/1/1 · Q38(i) · 1 markA roundabout's boundary is C₁: x² + y² = 64 and its central pond is C₂: x² + y² = 4. Sketch both.
Asked in CBSE 2026 · 65/1/1 · Q38(ii) · 1 markWrite y as a function of x for C₁: x² + y² = 64 and C₂: x² + y² = 4.
Asked in CBSE 2026 · 65/1/1 · Q38(iii)(a) · 2 marksBy integration, find the area inside the roundabout boundary x² + y² = 64.
Asked in CBSE 2026 · 65/1/1 · Q38(iii)(b) · 2 marksBy integration, find the area of the pond x² + y² = 4.
Asked in CBSE 2025 · 65/1/1 · Q17 · 1 markWhich integral gives the area between y² = x, the line x = 4 and the x-axis, in the first quadrant (shown shaded)?
Asked in CBSE 2025 · 65/1/1 · Q31 · 3 marksSketch y = |x + 3| and use integration to find the area between it and the x-axis from x = −6 to x = 0.
Asked in CBSE 2024 · 65/1/1 · Q34 · 5 marksBy integration, find the area of the ellipse x²/16 + y²/4 = 1 lying between x = −2 and x = 2.
Asked in CBSE 2023 · 65/1/1 · Q35 · 5 marksBy integration, find the area enclosed by y = x², y = x + 2 and the x-axis.

Differential Equations

13 questions, 23 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q11 · 1 markIntegrating factor of R dx/dy + Px = Q, where P, Q, R are functions of y.
Asked in CBSE 2026 · 65/1/1 · Q12 · 1 markFind the order and the degree of d/dx (eʸ) = 0.
Asked in CBSE 2026 · 65/1/1 · Q29(a) · 3 marksGeneral solution of x² dy/dx = x² + xy + y².
Asked in CBSE 2026 · 65/1/1 · Q29(b) · 3 marksParticular solution of xy dy/dx = (x + 2)(y + 2) with y(1) = −1.
Asked in CBSE 2025 · 65/1/1 · Q4 · 1 markAmong y² − xy, x − 3y, sin²(y/x) + y/x and tan x − sec y, which one is NOT homogeneous in x and y?
Asked in CBSE 2025 · 65/1/1 · Q12 · 1 markFind the integrating factor of (x + 2y³) dy/dx = 2y.
Asked in CBSE 2024 · 65/1/1 · Q8 · 1 markFind the integrating factor of (1 − x²) dy/dx + xy = ax, where −1 < x < 1.
Asked in CBSE 2024 · 65/1/1 · Q14 · 1 markState the order and degree of [1 + (dy/dx)²]³ = d²y/dx².
Asked in CBSE 2024 · 65/1/1 · Q29 · 3 marksSolve x² dy/dx − xy = x² cos²(y/(2x)) given y = π/2 when x = 1.
Asked in CBSE 2023 · 65/1/1 · Q7 · 1 markFor d²y/dx² + (dy/dx)³ = sin y, add the order and the degree.
Asked in CBSE 2023 · 65/1/1 · Q15 · 1 markSolve x dy − (1 + x²) dx = dx in general form.
Asked in CBSE 2023 · 65/1/1 · Q28(a) · 3 marksSolve dy/dx + y sec²x = tan x sec²x with y(0) = 0.
Asked in CBSE 2023 · 65/1/1 · Q28(b) · 3 marksSolve x dy − y dx − √(x² + y²) dx = 0.

Vector Algebra

20 questions, 31 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q13 · 1 markFind p so that the vectors (1, 2, 3) and (2, −p, 1) are perpendicular.
Asked in CBSE 2026 · 65/1/1 · Q14 · 1 markFind m so that the points (−1, −1, 2), (2, m, 5) and (3, 11, 6) are collinear.
Asked in CBSE 2026 · 65/1/1 · Q15 · 1 markVectors a, b have lengths 8 and 3, and the cross product has length 12. Find the absolute value of their dot product.
Asked in CBSE 2026 · 65/1/1 · Q23 · 2 marksWith P(1, 3, 2) and Q(−1, 0, 8), find the vector along QP whose magnitude is 14.
Asked in CBSE 2026 · 65/1/1 · Q24 · 2 marksA parallelogram has adjacent sides given by the vectors (3, −2, 2) and (1, 0, 2). Give its diagonal vectors and their lengths.
Asked in CBSE 2025 · 65/1/1 · Q2 · 1 markFor a = 3î + 2ĵ − k̂ and b = î − ĵ + k̂, which holds: a ∥ b, a ⊥ b, |b| > |a| or |a| = |b|?
Asked in CBSE 2025 · 65/1/1 · Q11 · 1 marka + b + c = 0 with |a| = √37, |b| = 3, |c| = 4. Find the angle between b and c.
Asked in CBSE 2025 · 65/1/1 · Q23 · 2 marksA parallelogram has diagonals a = 2î − ĵ + k̂ and b = î + 3ĵ − k̂. Find its area.
Asked in CBSE 2025 · 65/1/1 · Q25(a) · 2 marksTwo crossing kite strings lie along a = 3î + ĵ + 2k̂ and b = 2î − 2ĵ + 4k̂ (taut). Find the angle between them.
Asked in CBSE 2025 · 65/1/1 · Q25(b) · 2 marksA = (2, 1, 3), B = (8, −1, 0). Find the vector of length 21 pointing opposite to AB.
Asked in CBSE 2025 · 65/1/1 · Q29(b) · 3 marksBowler B = 2î + 8ĵ, wicket keeper W = 6î + 12ĵ, leg slip F = 12î + 18ĵ are collinear. In what ratio does W divide BF?
Asked in CBSE 2024 · 65/1/1 · Q7 · 1 markUnit vectors â and b̂ make angle θ with sin θ = 3/5. Find â · b̂.
Asked in CBSE 2024 · 65/1/1 · Q15 · 1 markFind the vector that starts at B(3, −4, 7) and ends at A(2, −3, 5).
Asked in CBSE 2024 · 65/1/1 · Q24 · 2 marksNon-zero vectors a, b satisfy (a + b) ⊥ a and (2a + b) ⊥ b. Prove |b| = √2 |a|.
Asked in CBSE 2024 · 65/1/1 · Q25 · 2 marksParallelogram ABCD has AB = 2î − 4ĵ + 5k̂ and diagonal DB = 3î − 6ĵ + 2k̂. Find AD and the area of ABCD.
Asked in CBSE 2023 · 65/1/1 · Q8 · 1 markFind p so that 2î + pĵ + k̂ is at right angles to −4î − 6ĵ + 26k̂.
Asked in CBSE 2023 · 65/1/1 · Q9 · 1 markEvaluate (î × ĵ)·ĵ + (ĵ × î)·k̂.
Asked in CBSE 2023 · 65/1/1 · Q10 · 1 marka + b = î and a = 2î − 2ĵ + 2k̂. Find |b|.
Asked in CBSE 2023 · 65/1/1 · Q22 · 2 marksFind every vector of length 3√3 parallel to î + ĵ + k̂.
Asked in CBSE 2023 · 65/1/1 · Q23(a) · 2 marksA, B, C have position vectors a, b, c and lie on one line in that order, with vector AC = (5/4) vector AB. Write c using a and b.

Three Dimensional Geometry

15 questions, 42 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q16 · 1 markLength of the perpendicular from (2, 5, 7) to the line x/1 = y/0 = z/0.
Asked in CBSE 2026 · 65/1/1 · Q20 · 1 markAssertion: line 1 (x = py + q, z = ry + s) is perpendicular to line 2 (z = r'y + s', x = p'y + q') when pp' + rr' = 1. Reason: lines r = a₁ + λb₁ and r = a₂ + μb₂ are perpendicular if b₁·b₂ = 0.
Asked in CBSE 2026 · 65/1/1 · Q35 · 5 marksAre the lines (x − 1)/2 = (y − 2)/3 = (z − 3)/4 and (x − 4)/5 = (y − 1)/2 = z parallel? If yes, find the gap between them; if not, find where they meet (if they do).
Asked in CBSE 2025 · 65/1/1 · Q29(a) · 3 marksShow that r = (1 − λ)î + (λ − 2)ĵ + (3 − 2λ)k̂ and r = (μ + 1)î + (2μ − 1)ĵ − (2μ + 1)k̂ are skew, then find the shortest distance between them.
Asked in CBSE 2025 · 65/1/1 · Q34(a) · 5 marksFind the image A′ of A(1, 6, 3) in the line x/1 = (y − 1)/2 = (z − 2)/3, and the equation of line AA′.
Asked in CBSE 2025 · 65/1/1 · Q34(b) · 5 marksOn the line (x + 5)/1 = (y + 3)/4 = (z − 6)/(−9), find P with PQ = 7 where Q = (2, 4, −1); then find the equation of PQ.
Asked in CBSE 2024 · 65/1/1 · Q9 · 1 markA line has direction cosines √3k, √3k, √3k. Find k.
Asked in CBSE 2024 · 65/1/1 · Q16 · 1 markHow far is P(a, b, c) from the y-axis?
Asked in CBSE 2024 · 65/1/1 · Q20 · 1 markA: no line can be perpendicular to all three coordinate axes. R: for angles α, β, γ with the axes, cos²α + cos²β + cos²γ = 1.
Asked in CBSE 2024 · 65/1/1 · Q35 · 5 marksP′(1, 0, 7) is the image of P(x, y, z) in the line x/1 = (y − 1)/2 = (z − 2)/3. Find P.
Asked in CBSE 2023 · 65/1/1 · Q11 · 1 markFind the direction cosines of the line (x − 1)/2 = (1 − y)/3 = (2z − 1)/12.
Asked in CBSE 2023 · 65/1/1 · Q20 · 1 markA: the line joining (1, 2, 3) to (3, −1, 3) is (x − 3)/2 = (y + 1)/3 = (z − 3)/0. R: the two-point form of a line in space.
Asked in CBSE 2023 · 65/1/1 · Q23(b) · 2 marksLine 1: x = 2 + 2λ, y = 1 + 7λ, z = −3 − 3λ. Line 2: x = −2 − μ, y = 8 + 2μ, z = 5 + 4μ. Are they perpendicular?
Asked in CBSE 2023 · 65/1/1 · Q34(a) · 5 marksA line goes through (1, 2, −4) parallel to AB, where A(3, 3, −5), B(1, 0, −11). Give its vector and Cartesian forms, then the distance between it and line AB.
Asked in CBSE 2023 · 65/1/1 · Q34(b) · 5 marksWrite the line joining A(1, 2, 3) to B(3, 5, 9); then find its points lying 14 units from B.

Linear Programming

12 questions, 20 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q17 · 1 markFor Z = 5x + 7y over the feasible region shown, find (maximum of Z) − (minimum of Z).
Asked in CBSE 2026 · 65/1/1 · Q18 · 1 markWhat is the degree of the objective function in a linear programming problem?
Asked in CBSE 2026 · 65/1/1 · Q30 · 3 marksMinimise Z = 13x − 15y graphically subject to: x + y at most 7; 2x − 3y + 6 non-negative; x and y non-negative.
Asked in CBSE 2025 · 65/1/1 · Q14 · 1 markAn LPP has corner points (2, 72), (15, 20) and (40, 15), with Z = 18x + 9y. Where are the maximum and minimum of Z?
Asked in CBSE 2025 · 65/1/1 · Q16 · 1 markFor Z = ax + by, what can be said about its optimum values when the feasible region is bounded?
Asked in CBSE 2025 · 65/1/1 · Q27 · 3 marksGraphically maximise Z = x + 2y with x − y ≥ 0, x − 2y ≥ −2, x ≥ 0, y ≥ 0.
Asked in CBSE 2024 · 65/1/1 · Q10 · 1 markWhat kind of function does a linear programming problem optimise?
Asked in CBSE 2024 · 65/1/1 · Q17 · 1 markHow many corner points does the region x ≥ 0, y ≥ 0, x + y ≥ 4 have?
Asked in CBSE 2024 · 65/1/1 · Q30 · 3 marksBy graph, maximise Z = 500x + 300y subject to x, y ≥ 0, 4x + 5y ≥ 20, 2x + y ≤ 12 and x + 2y ≤ 12.
Asked in CBSE 2023 · 65/1/1 · Q17 · 1 markAn LPP's feasible region has corners (2, 72), (15, 20), (40, 15). Where is z = 18x + 9y largest and smallest?
Asked in CBSE 2023 · 65/1/1 · Q18 · 1 markHow many corner points does the region y ≥ 0, x ≥ 0, y ≤ x, x − 2y + 2 ≥ 0 have?
Asked in CBSE 2023 · 65/1/1 · Q29 · 3 marksMaximise z = 6x + 3y graphically subject to x + 5y ≥ 115, 4x + y ≥ 80, 3x + 2y ≤ 150 with x, y ≥ 0.

Probability

23 questions, 45 marks across 4 papers

Asked in CBSE 2026 · 65/1/1 · Q19 · 1 markAssertion: for a fair die, the probability of a prime given an odd number is 2/3. Reason: for any events A, B, the conditional probability of A given B equals P(A ∪ B)/P(B).
Asked in CBSE 2026 · 65/1/1 · Q31(a) · 3 marksBag I: 3 red, 4 white; bag II: 8 red, 6 white. A die decides: below 3 → bag I, else bag II. One ball is drawn. P(red)?
Asked in CBSE 2026 · 65/1/1 · Q31(b) · 3 marksP(X or Y happens) = a and P(exactly one of X, Y happens) = b. Prove P(X') + P(Y') = 2 − 2a + b.
Asked in CBSE 2026 · 65/1/1 · Q37(i) · 1 markA jackpot sells 1,00,000 tickets with prizes: one ₹3,00,000, two ₹2,00,000 and three ₹50,000. What amounts can a ticket holder win?
Asked in CBSE 2026 · 65/1/1 · Q37(ii)(a) · 2 marksSame jackpot (1,00,000 tickets; 1 prize of ₹3,00,000, 2 of ₹2,00,000, 3 of ₹50,000): P(one ticket wins ₹2,00,000 or more)?
Asked in CBSE 2026 · 65/1/1 · Q37(ii)(b) · 2 marksSame jackpot (1,00,000 tickets, 6 prizes in all): P(one ticket wins nothing)?
Asked in CBSE 2026 · 65/1/1 · Q37(iii) · 1 markRohan holds one ticket of the above jackpot (6 winning tickets out of 1,00,000) and one ticket of another jackpot with win chance 1 in 1,00,000. P(he wins on exactly one ticket)?
Asked in CBSE 2025 · 65/1/1 · Q7 · 1 markE and F are independent with P(E) = 2/3 and P(F) = 3/7. Find P(E | not F).
Asked in CBSE 2025 · 65/1/1 · Q30(a) · 3 marksX (students absent on a Saturday) takes 0, 2, 4, 5 with probabilities p, 2p, 3p, p. (i) Find p (1 mark). (ii) Find the mean of X (2 marks).
Asked in CBSE 2025 · 65/1/1 · Q30(b) · 3 marksOf 3000 applicants, two-thirds are women. A man gets a distinction in the written test with probability 0.4, a woman with 0.35. Find P(a random applicant gets a distinction).
Asked in CBSE 2025 · 65/1/1 · Q38(i) · 2 marksBank loans are fixed, floating or variable rate with probabilities 10%, 20%, 70%; default rates are 5%, 3%, 1% respectively. Find P(a borrower defaults).
Asked in CBSE 2025 · 65/1/1 · Q38(ii) · 2 marksWith the same data, a borrower has defaulted. Find the probability the loan was at the variable rate.
Asked in CBSE 2024 · 65/1/1 · Q11 · 1 markGiven P(A | B) = P(A′ | B), which relation must hold?
Asked in CBSE 2024 · 65/1/1 · Q31 · 3 marksE, F are independent with P(E ∪ F) = 0.6 and P(not E) = 0.6. Find P(F) and P(Ē ∪ F̄).
Asked in CBSE 2024 · 65/1/1 · Q37(i) · 2 marksSevere, moderate and light turbulence are equally likely; a plane is late with chance 55%, 37%, 17% respectively. Find P(late).
Asked in CBSE 2024 · 65/1/1 · Q37(ii) · 2 marksIn the same setting, given the plane was late, find the probability the turbulence was moderate.
Asked in CBSE 2023 · 65/1/1 · Q12 · 1 markP(A|B) = 0.3, P(A) = 0.4, P(B) = 0.8. Find P(B|A).
Asked in CBSE 2023 · 65/1/1 · Q30(a) · 3 marksX takes 1, 2, 3 with probabilities k/2, k/3, k/6. Find (i) k, (ii) P(1 ≤ X < 3), (iii) E(X).
Asked in CBSE 2023 · 65/1/1 · Q30(b) · 3 marksA, B independent with P(A ∩ B′) = 1/4 and P(A′ ∩ B) = 1/6. Find P(A) and P(B).
Asked in CBSE 2023 · 65/1/1 · Q36(i) · 1 markVenn diagram of Yoga types A, B, C (see figure); P(C) = 0.44. Find x.
Asked in CBSE 2023 · 65/1/1 · Q36(ii) · 1 markSame Venn diagram, P(C) = 0.44. Find y.
Asked in CBSE 2023 · 65/1/1 · Q36(iii)(a) · 2 marksSame Venn diagram. Find P(C | B).
Asked in CBSE 2023 · 65/1/1 · Q36(iii)(b) · 2 marksSame Venn diagram. Probability that a person practises A or B and not C.

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