CBSE Board Paper 2026
80 Marks · 38 Questions
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 — Step-by-Step Solutions & Marking Scheme
Select any question below to view the complete worked solution and official CBSE stepwise marking breakdown.
Question 11M · Inverse Trigonometric Functions
Which inverse trigonometric function does the graph represent?
Question 21M · Matrices
Find the transpose of the 2×2 matrix with entries aᵢⱼ=(i+2j)²/3.
Question 31M · Continuity And Differentiability
How many points of (0,3) make f(x)=⌊x⌋ nondifferentiable?
Question 41M · Applications Of Integrals
Find the area between y=x and the x-axis from x=0 to x=2.
Question 51M · Continuity And Differentiability
Find k for continuity at zero if f(x)=sin x/x+cos x for x≠0 and f(0)=k.
Question 61M · Applications Of Derivatives
A cylindrical tank of radius 10 m receives sand at 314 m³/h. Find the approximate rate of increase of sand depth.
Question 71M · Inverse Trigonometric Functions
Evaluate arcsec(√2)+2 arccsc(−√2) on the principal branches.
Question 81M · Determinants
If (2,3), (0,4) and (p,2) are collinear, find p.
Question 91M · Continuity And Differentiability
Find the derivative of \(e^{e^x}\).
Question 101M · Integrals
Evaluate \(\int dx/\sqrt{25-16x^2}\).
Question 111M · Integrals
If \(\int_0^1 dx/(e^x+e^{-x})=\tan^{-1}e+k\), find k.
Question 121M · Differential Equations
Find the sum of order and degree of \(y=x\frac{dy}{dx}+3[2-(dy/dx)^2]\).
Question 131M · Linear Programming
For p,q>0, every point of segment AB in the shaded region maximizes Z=px+qy. Find the relation between p and q.
Question 141M · Vector Algebra
Find the area of triangle A=(0,1,1), B=(2,0,−1), C=(1,0,3).
Question 151M · Vector Algebra
If (3,−2,5)×(4,p,q)=0, find p and q.
Question 161M · Differential Equations
Which equation cannot be written as a first-order linear equation in one variable as a function of the other by…
Question 171M · Linear Programming
Classify the region 3x+y≥3, 2x−y≥−5 and x,y≥0.
Question 181M · Probability
An item is available in shops A, B and C with probabilities 4/5, 3/4 and 2/3. Assuming independent availability, find…
Question 191M · Matrices
Assertion: Square matrices with AB and BA defined need not satisfy AB=BA. Reason: Two diagonal matrices of the same…
Question 201M · Relations And Functions
Assertion: f:ℕ→ℕ, f(x)=x³+2 is injective but not onto. Reason: For every natural y there is no natural x with x³+2=y.
Question 212M · Continuity And Differentiability
Show that f(x)=cos x/(π/2−x) for x≠π/2 and f(π/2)=1 is continuous at π/2. OR Test differentiability at 2 for f(x)=x−1…
Question 222M · Vector Algebra
Find a vector of length 5 perpendicular to both (4,−1,3) and (−2,1,−2). OR A unit vector makes angles π/4, π/3 and…
Question 232M · Applications Of Derivatives
Find the intervals where f(x)=x/3+3/x increases, for x≠0.
Question 242M · Vector Algebra
In parallelogram PQRS, PQ=(2,3,1) and QR=(1,−2,1). Find unit vectors in the directions PR and SQ.
Question 252M · Inverse Trigonometric Functions
Evaluate \(\tan^{-1}(-1/\sqrt3)+\cot^{-1}(1/\sqrt3)+\tan^{-1}(\sin(-\pi/2))+\tan^{-1}(\tan(2\pi/3))\).
Question 263M · Three Dimensional Geometry
Find the points on (x−2)/3=(1−y)/2=(z−3)/2 at distance √2 from (1,2,3). OR Find the shortest distance between…
Question 273M · Matrices
For A=[[2,0,1],[2,1,3],[1,−1,0]], compute A²−5A+4I.
Question 283M · Probability
A debate is held with probability 1/3 and a quiz with probability 2/3. For debate choose two from team A (4 girls, 6…
Question 293M · Linear Programming
Graphically maximize Z=20x+10y under x+2y≤28, 3x+y≤24, x≥2 and y≥0.
Question 303M · Relations And Functions
Is f:ℝ∖{3}→ℝ∖{1}, f(x)=(x−2)/(x−3), one-one and onto? OR For a fixed positive integer n, determine whether xRy if n…
Question 313M · Continuity And Differentiability
If \(xy=e^{x-y}\), find dy/dx. OR Differentiate…
Question 325M · Integrals
Evaluate \(\int x\,dx/[(x-1)(x^2+4)]\). OR Evaluate \(\int_0^1 x\tan^{-1}x\,dx/(1+x^2)^{3/2}\).
Question 335M · Applications Of Integrals
Use integration to find the area bounded by y=5x+4, y=0, x=−1 and x=1.
Question 345M · Differential Equations
Solve \(ye^y\,dx=(y^3+2xe^y)\,dy\) with y(0)=1. OR Solve (x³−3xy²)dx=(y³−3x²y)dy.
Question 355M · Three Dimensional Geometry
Find the distance from P=(1,2,3) to (x−6)/3=(y−7)/2=(z−7)/(−2). Find the perpendicular line through P and its foot.
Question 364M · Matrices
A cuboid’s length plus breadth exceeds its height by 3 cm. Twice its length plus three times its breadth plus its…
Question 374M · Probability
Box I contains 30 red caps, Box II contains 20 red and 10 green caps, and Box III contains 30 green caps. Box i is…
Question 384M · Applications Of Derivatives
Orange juice fills a conical cup of depth 15 cm and top radius 5 cm at 0.1 cm³/s. The juice has depth h and surface…