CBSE Board Paper 2026
80 Marks · 38 Questions
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 — Step-by-Step Solutions & Marking Scheme
Select any question below to view the complete worked solution and official CBSE stepwise marking breakdown.
Question 11M · Vector Algebra
If (a+b)·(a−b)=198 and |a|=10|b|, choose the true statement.
Question 21M · Three Dimensional Geometry
For direction cosines (l₁,m₁,n₁) and (l₂,m₂,n₂), which formula gives the acute angle θ between the lines?
Question 31M · Three Dimensional Geometry
Which listed direction ratios are perpendicular to both (1,0,0) and (0,−1,0)?
Question 41M · 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 51M · Inverse Trigonometric Functions
Find the domain of arccos(4x+1).
Question 61M · Matrices
If A²=4A+3I and A⁻¹=xA+yI, find x+y from the inverse expression obtained using the matrix identity.
Question 71M · Matrices
If A and B are skew-symmetric of the same order, classify ABᵀ+BAᵀ.
Question 81M · Matrices
If [2 1]X=[3 4 5], find the order of X.
Question 91M · Matrices
For a nonzero square matrix A, suppose A²=A and (I−A)³=xA+I. Find x.
Question 101M · Determinants
If A adj(A)=3I₃, find det(2A).
Question 111M · Continuity And Differentiability
Find k for continuity of f(x)=x² sin(1/x) for x≠0 and f(0)=k(0+1).
Question 121M · Continuity And Differentiability
If y=2 arccos x, find dy/dx on −1<x<1.
Question 131M · Applications Of Derivatives
Find the rate of change of a sphere’s volume with respect to diameter when its radius is 5 cm.
Question 141M · Integrals
Evaluate \(\int\frac{\cos x}{\sqrt{1+\sin^2 x}}\,dx\).
Question 151M · Integrals
Which expression equals \(\int_{-1}^{1}(1-|x|)\,dx\)?
Question 161M · Applications Of Integrals
Which integral gives the area of the shaded region in the circle diagram?
Question 171M · 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 181M · Vector Algebra
Which inequality or equality holds for every pair of vectors a and b?
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 · 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 222M · Continuity And Differentiability
If \(x=e^{\sin^{-1}t}\) and \(y=e^{\cos^{-1}t}\), find dy/dx at \(t=1/\sqrt2\).
Question 232M · Applications Of Derivatives
Find where f(x)=x ln x increases on (0,∞).
Question 242M · 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 252M · Vector Algebra
Unit vectors a,b satisfy |a+2b|=|2a−b|. Find their angle.
Question 263M · Probability
Three patients undergo independent surgeries, each with 90% success probability. Find the probabilities of (i) exactly…
Question 273M · 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 283M · Integrals
Evaluate \(\int_0^{\pi/2}\frac{\sin^2x}{1+\sin 2x}\,dx\).
Question 293M · 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 303M · Differential Equations
Solve (x²−y²)dx+2xy dy=0. OR Solve \(\sin x\cos y\,dx+\cos x\sin y\,dy=0\), with \(y(0)=\pi/4\).
Question 313M · Continuity And Differentiability
If \((\sin x)^y=y^{\cos x}\), find dy/dx on branches with y>0 and sin x>0.
Question 325M · 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 335M · Relations And Functions
The printed problem specifies f:[0,∞)→A⊂ℕ, f(x)=4x²+12x+15. Examine whether it defines a function with that codomain…
Question 345M · Linear Programming
Graphically maximize Z=8x+8y for x≤20, 2x+3y≤120, 2x+y≤60 and x,y≥0.
Question 355M · Matrices
A draw awards 60 vouchers worth either ₹1000 or ₹500. The number of ₹1000 vouchers plus three times the number of ₹500…
Question 364M · 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 374M · 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 384M · Probability
In an entrance examination, 40% of candidates took a gap year after Class 12 and the others were regular students. Of…