Mathbee · CBSE Class 12 Maths

CBSE Board Paper 2026 80 Marks · 38 Questions

CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 — Step-by-Step Solutions & Marking Scheme

Select any question below to view the complete worked solution and official CBSE stepwise marking breakdown.

Question 11M · Continuity And Differentiability

If y=arctan x, find dy/dx.

Question 21M · Applications Of Derivatives

Find the rate of change of a sphere’s volume with respect to diameter when its radius is 5 cm.

Question 31M · Integrals

Evaluate \(\int dx/\sqrt{e^{-2x}-1}\).

Question 41M · Integrals

Which expression equals \(\int_{-1}^{1}(1-|x|)\,dx\)?

Question 51M · Applications Of Integrals

Which integral gives the area of the shaded region in the circle diagram?

Question 61M · Differential Equations

For which right-hand sides is dy/dx=F(x,y) homogeneous? (i) 3x+2y, (ii) sin(y/x)+ln y−ln x, (iii) exp(y/x)+1, (iv)…

Question 71M · Vector Algebra

Which inequality or equality holds for every pair of vectors a and b?

Question 81M · Vector Algebra

If (a+b)·(a−b)=198 and |a|=10|b|, choose the true statement.

Question 91M · Three Dimensional Geometry

For direction cosines (l₁,m₁,n₁) and (l₂,m₂,n₂), which formula gives the acute angle θ between the lines?

Question 101M · Three Dimensional Geometry

For direction ratios (1,−2,3) and (−2,p,−6), find p for parallel lines.

Question 111M · Probability

Independent events E and F satisfy P(E)=3/10 and P(E∪F)=1/2. Find P(E | F)−P(F | E).

Question 121M · Inverse Trigonometric Functions

Find the domain of arcsin(1−2x).

Question 131M · Matrices

If A²=4A+3I and A⁻¹=xA+yI, find x+y from the inverse expression obtained using the matrix identity.

Question 141M · Matrices

If A and B are skew-symmetric of the same order, classify ABᵀ+BAᵀ.

Question 151M · Matrices

If B[1 2 3]=[[1,2,3],[0,1,1],[2,0,1]], what order must B have?

Question 161M · Matrices

For a nonzero square matrix A, suppose A²=A and (I−A)³=xA+I. Find x.

Question 171M · Determinants

If A adj(A)=2026 I₃, find det(adj(A)).

Question 181M · Continuity And Differentiability

Find k for continuity of f(x)=x² sin(1/x) for x≠0 and f(0)=k(0+1).

Question 191M · Vector Algebra

Assertion: For a≠0 the vectors a and −2a are collinear. Reason: a·(−2a)=0.

Question 201M · Differential Equations

Assertion: \(e^x+e^{-y}=-2\) is a particular solution of \(dy/dx=e^{x+y}\). Reason: \(e^x+e^{-y}=C\) gives the general…

Question 212M · Inverse Trigonometric Functions

Evaluate \(\sin[\cot^{-1}(\sqrt2\cos(\tan^{-1}1))]\). OR On {1,2,3}, is R={(1,1),(3,3),(1,2)} symmetric? Find the…

Question 222M · Vector Algebra

Unit vectors a,b satisfy |a+2b|=|2a−b|. Find their angle.

Question 232M · Three Dimensional Geometry

Find p if (x−3)/1=(1−y)/1=(z+2)/p and (2−x)/3=(y+1)/5=(z+56)/(2p) are perpendicular. OR Find the vector equation of…

Question 242M · Continuity And Differentiability

If \(x=e^{t+1/t}\) and \(y=e^{t-1/t}\), find dy/dx at t=−2.

Question 252M · Applications Of Derivatives

Find where \(f(x)=x^2e^{-x}\) increases on (0,∞).

Question 263M · Integrals

Evaluate \(\int\frac{\cos x\,dx}{(2+\sin x)(4+\sin x)}\). OR Evaluate \(\int\frac{x+3}{x^2+4x+5}\,dx\).

Question 273M · Differential Equations

Solve (x²+y²)dy=xy dx. OR Solve \(y\prime-3y\cot x=\sin 2x\), with \(y(\pi/2)=2\).

Question 283M · Continuity And Differentiability

If \((\sin x)^y=y^{\cos x}\), find dy/dx on branches with y>0 and sin x>0.

Question 293M · Probability

Three patients undergo independent surgeries, each with 90% success probability. Find the probabilities of (i) exactly…

Question 303M · Integrals

Evaluate \(\int\frac{dx}{x^{1/2}+x^{1/3}}\). OR Evaluate \(\int\tan^{-1}\!\left(\frac{1-x}{1+x}\right)dx\).

Question 313M · Integrals

Evaluate \(\int_{-\pi/2}^{\pi/2}\frac{\cos^2x}{2^x+1}\,dx\).

Question 325M · Matrices

A draw awards 60 vouchers worth either ₹1000 or ₹500. The number of ₹1000 vouchers plus three times the number of ₹500…

Question 335M · Three Dimensional Geometry

Write (x+3)/(−3)=(y−1)/1=(z−5)/5 and (x+1)/(−1)=(2−y)/(−2)=(z−5)/5 in vector form and check whether the lines…

Question 345M · Relations And Functions

Show that f:[0,∞)→[−5,∞), f(x)=4x²+4x−5 is one-one and onto. Find p≥0 with f(p)=3.

Question 355M · Linear Programming

Graphically maximize Z=12x+18y subject to x+y≤1200, x−2y≥0, x+3y≥600 and x,y≥0.

Question 364M · Probability

In an entrance examination, 40% of candidates took a gap year after Class 12 and the others were regular students. Of…

Question 374M · Applications Of Integrals

A triangular park has vertices A=(0,4), B=(−2,0) and C=(3,0). O is the origin. Segment AO divides the park into OAC…

Question 384M · Applications Of Derivatives

Two vertical poles, 22 m and 16 m high, stand on opposite sides of a 20 m wide road. Two ladders of lengths l₁ and l₂…