Mathbee · CBSE Class 12 Maths

CBSE Board Paper 2026 80 Marks · 38 Questions

CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 — 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

Find the domain of arccos(2x−5).

Question 21M · Matrices

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

Question 31M · Matrices

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

Question 41M · Matrices

If the column (4,1,3)ᵀ multiplied by A gives [[−4,8,4],[−1,2,1],[−3,6,3]], find the order of A.

Question 51M · Matrices

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

Question 61M · Determinants

If B adj(B)=(1/3)I₃, find det(B⁻¹).

Question 71M · Continuity And Differentiability

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

Question 81M · Continuity And Differentiability

If y=arcsin x, find dy/dx on −1<x<1.

Question 91M · Applications Of Derivatives

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

Question 101M · Integrals

Evaluate \(\int\frac{dx}{2^x+2^{-x}}\).

Question 111M · Integrals

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

Question 121M · Applications Of Integrals

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

Question 131M · 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 141M · Vector Algebra

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

Question 151M · Vector Algebra

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

Question 161M · Three Dimensional Geometry

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

Question 171M · Three Dimensional Geometry

For lines with direction ratios (12,−3,9) and (4,q,−p), find (p,q) that makes them parallel.

Question 181M · 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 191M · 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 201M · Vector Algebra

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

Question 212M · Continuity And Differentiability

For x=t+1/t and y=t−1/t, find dy/dx at t=2.

Question 222M · Applications Of Derivatives

Find where f(x)=tan x−4x increases on (0,π/2).

Question 232M · 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 242M · Vector Algebra

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

Question 252M · 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 263M · 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 273M · Integrals

Evaluate \(\int_0^\pi\frac{\sin^{2026}x}{\sin^{2026}x+\cos^{2026}x}\,dx\).

Question 283M · 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 293M · Differential Equations

Solve \(2x^2y\prime=y^2+2xy\). OR Solve \((x+1)y\prime=2e^{-y}-1\) with \(y(0)=0\).

Question 303M · Continuity And Differentiability

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

Question 313M · Probability

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

Question 325M · Relations And Functions

Show that \(f:\mathbb R\to\mathbb R\), \(f(x)=x/\sqrt{1+x^2}\), is one-one but not onto.

Question 335M · Linear Programming

Graphically maximize Z=600x+400y subject to x+2y≤12, 4x+5y≥20, 2x+y≤12 and x,y≥0.

Question 345M · Matrices

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

Question 355M · 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 364M · 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…

Question 374M · Probability

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

Question 384M · 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…