CBSE Board Paper 2026
80 Marks · 38 Questions
CBSE Class 12 Mathematics 2026 Main Paper 65/5/3 — Step-by-Step Solutions & Marking Scheme
Select any question below to view the complete worked solution and official CBSE stepwise marking breakdown.
Question 11M · Matrices
Which statement follows for a real square matrix A, given that A+Aᵀ is symmetric?
Question 21M · Matrices
Which entry condition characterizes diagonal matrices B=[bᵢⱼ]?
Question 31M · Determinants
Which statement is false when A is nonsingular?
Question 41M · Continuity And Differentiability
The function is (x²−4x−5)/(x+1) for x≠−1 and k at x=−1. Find k for continuity.
Question 51M · Inverse Trigonometric Functions
Which inverse trigonometric function has an incorrectly listed principal range?
Question 61M · Matrices
Given A=[[0,−3,4],[1,0,2]] and B=[[−3,0,1],[2,4,0]], find C if A+B+C=0.
Question 71M · Vector Algebra
Perpendicular strips have direction vectors (3,2,λ) and (2,−4,5). Find λ.
Question 81M · Probability
The printed data say 3P(A)=P(B)=3/5 and P(A | B)=2/3. Which listed union value does formal substitution give, and are…
Question 91M · Determinants
A=(3,1), B=(−2,1), C=(0,k) form a triangle of area 5. Find the possible k values.
Question 101M · Integrals
Evaluate \(\int\sqrt{(1+\cos x)/(1-\cos x)}\,dx\) where \(\cot(x/2)>0\).
Question 111M · Integrals
Find k if \(\int_0^1(6x^2-4x+k)\,dx=0\).
Question 121M · Vector Algebra
The position vector of (24,n) has length 25. Find n.
Question 131M · Differential Equations
Find the order and degree of \(y=(d^2y/dx^2)^2-\lambda\,dy/dx\).
Question 141M · Linear Programming
Find the maximum of Z=4x+3y over a polygon with vertices (0,0),(0,40),(20,40),(60,20),(60,0).
Question 151M · Applications Of Integrals
Find the area bounded by y=2x−4, the x-axis, x=0 and x=1.
Question 161M · Differential Equations
Solve \(dy/dx=e^{3x-y}\).
Question 171M · Continuity And Differentiability
Differentiate arccos((sin x+cos x)/√2) for −π/4<x<π/4.
Question 181M · Applications Of Derivatives
Find the absolute minimum of (x−2)²+5 on [−3,2].
Question 191M · Relations And Functions
Assertion: R={(1,1),(1,2),(2,1),(2,2),(3,3)} is an equivalence relation on {1,2,3}. Reason: A reflexive, symmetric and…
Question 201M · Linear Programming
Assertion: Minimizing Z=x+2y with 2x+y≥3, x+2y≥6, x,y≥0 has infinitely many minimizers, and the corners are (0,3) and…
Question 212M · Inverse Trigonometric Functions
Evaluate \(\tan[\cos^{-1}(\tan(3\pi/4))]\).
Question 222M · Three Dimensional Geometry
Find the acute angle between (x−2)/3=(y+5)/2=(1−z)/(−6) and (x−7)/1=y/2=(6−z)/(−2).
Question 232M · Applications Of Derivatives
Where is f(x)=x/3+3/x increasing for x≠0?
Question 242M · Continuity And Differentiability
Differentiate xˣ with respect to x ln x, for x>0. OR If y=P cos(ux)+Q sin(ux), with constants P,Q,u, show that y″+u²y=0.
Question 252M · Vector Algebra
For a=(2,−3,1), b=(0,4,−2), c=(3,0,2), find λ such that a+λb is perpendicular to c. OR Find the area of triangle ABC…
Question 263M · Applications Of Derivatives
The volume of a cube increases at a constant rate k. Prove that its surface-area rate is inversely proportional to…
Question 273M · Probability
A fair die is rolled. Let A={1,2,5}, B={3,5}, C={2,3,4,5}. Find P(A | C), P(C | A), P(A∩B | C) and P(A∪B | C). OR Draw…
Question 283M · Differential Equations
Solve (x−sin y)dy+tan y dx=0.
Question 293M · Vector Algebra
The points A=(55,−2), B=(5,8) and C=(a,−52) are collinear. Find a. OR For unit vectors a,b,c prove…
Question 303M · Integrals
Evaluate \(\int (x-\sin x)\,dx/(1-\cos x)\). OR Evaluate \(\int_0^2 dx/\sqrt{x^2+2x+3}\).
Question 313M · Linear Programming
Graphically maximize Z=2x/5+3y/10 subject to 2x+y≤1000, x+y≤800 and x,y≥0.
Question 325M · Matrices
One art, story and puzzle book each cost ₹21 in total. Four art, three story and two puzzle books cost ₹60. Six art…
Question 335M · Continuity And Differentiability
If \(y\sqrt{x^2+1}=\ln(\sqrt{x^2+1}-x)\), show that \((x^2+1)y\prime+xy+1=0\). OR Differentiate \(x^{\cot…
Question 345M · Applications Of Integrals
Sketch x²+y²=25. Find by integration the area of the minor segment cut off by x=5/2.
Question 355M · Three Dimensional Geometry
Find the foot from P=(0,2,3) to (−x−3)/(−5)=(1−y)/(−2)=(3z+12)/9 and the perpendicular length. OR The lines…
Question 364M · Applications Of Derivatives
An open cylindrical tumbler with base radius r, height h and volume V is made with fixed surface area S.
Question 374M · Probability
In a survey, vaccine shares for A₁,A₂,A₃ were 25%,35%,40%, respectively. Their conditional protection probabilities in…
Question 384M · Relations And Functions
A sustainability project selects four girls A={G₁,G₂,G₃,G₄} and three boys B={B₁,B₂,B₃}.