Mathbee · CBSE Class 12 Maths

CBSE Board Paper 2026 80 Marks · 38 Questions

CBSE Class 12 Mathematics 2026 Main Paper 65(B) — Step-by-Step Solutions & Marking Scheme

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

Question 11M · Relations And Functions

Classify f:ℝ→ℝ, f(x)=x³.

Question 21M · Matrices

How many 2×3 matrices have every entry equal to 1 or 2?

Question 31M · Matrices

For A=[[cos 2α,−sin 2α],[sin 2α,cos 2α]], which listed α makes A+Aᵀ=I?

Question 41M · Determinants

A triangle with vertices (3,−7),(6,3),(k,3) has area 15. Find all k.

Question 51M · Applications Of Derivatives

Which listed interval has increasing x³e⁻³ˣ?

Question 61M · Continuity And Differentiability

Find f(0)=k for continuity of f(x)=sin(7x)/(5x) away from zero.

Question 71M · Continuity And Differentiability

Differentiate arctan((1−cos x)/sin x) on regular intervals.

Question 81M · Integrals

Evaluate \(\int\cos2x\,dx/(\sin^2x\cos^2x)\).

Question 91M · Integrals

Evaluate \(\int dx/(x^2+6x+10)\).

Question 101M · Differential Equations

Solve \(dy/dx=e^{y-x}\).

Question 111M · Differential Equations

Find the integrating factor for (1+y²)dx=(arctan y−x)dy.

Question 121M · Vector Algebra

For unit vectors a,b, when is a+2b also a unit vector?

Question 131M · Vector Algebra

Find the parallelogram area when its diagonals are (1,1,−2) and (1,−3,4).

Question 141M · Linear Programming

For corners (0,2),(6,0),(6,8),(0,5), find max Z−min Z for Z=4x+6y.

Question 151M · Linear Programming

If two distinct corners attain the same minimum of a linear objective over a convex feasible region, how many points…

Question 161M · Probability

If P(A | B)=P(B | A) and P(A∩B)>0, what follows?

Question 171M · Probability

Find the probability both fair dice show odd prime numbers.

Question 181M · Probability

With independent equally likely boy/girl outcomes for three children, find P(eldest is a boy | at least one boy).

Question 191M · Applications Of Derivatives

Assertion: e⁻ˣ decreases on ℝ. Reason: A function with negative derivative on an interval decreases there.

Question 201M · Probability

Assertion: Independent events A,B with P(B)>0 satisfy P(A | B)=P(A). Reason: Independence gives P(A∩B)=P(A)P(B).

Question 212M · Matrices

For A=[[3,1],[2,1]], find a,b such that A²+aA+bI=0. OR Using determinants, find the line through (−1,−1) and (1,3).

Question 222M · Applications Of Derivatives

A spherical balloon’s volume grows at 20 cm³/s. Find its surface-area rate at radius 4 cm. OR Find increasing…

Question 232M · Integrals

Evaluate \(\int (x-4)e^x\,dx/(x-2)^3\).

Question 242M · Vector Algebra

Find a unit vector perpendicular to AB and BC, with A=(3,−1,2), B=(1,−1,−3), C=(4,−3,1).

Question 252M · Three Dimensional Geometry

Find the reflection of (2,−1,5) in r=(11,−2,−8)+t(10,−4,−11).

Question 263M · Inverse Trigonometric Functions

Evaluate \(\arcsin(\sin(2\pi/3))+\arctan(\tan(5\pi/6))+\arccos(\cos(13\pi/6))\). OR Show f:ℕ→ℕ, f(n)=n²+n+1 is one-one…

Question 273M · Integrals

Evaluate \(\int x\,dx/\sqrt{7-6x-x^2}\). OR Evaluate \(\int 2\,dx/[(1-x)(1+x^2)]\).

Question 283M · Differential Equations

Solve \(xy\prime=y-x\tan(y/x)\). OR Solve \(y\prime=e^{x+y}+e^{x-y}\).

Question 293M · Integrals

Evaluate \(\int_0^\pi x\,dx/(1+\sin x)\).

Question 303M · Vector Algebra

For nonzero mutually perpendicular vectors a,b,c of equal norm, show that a+b+c is equally inclined to them.

Question 313M · Linear Programming

For feasible corners (0,10),(5,5),(15,15),(0,20), find p,q>0 such that Z=px+qy is maximal at both (15,15) and (0,20).

Question 325M · Matrices

Find the inverse of A=[[1,−1,1],[2,1,−3],[1,1,1]], then solve A(x,y,z)ᵀ=(4,0,2)ᵀ. OR For A=[[3,2],[7,5]]…

Question 335M · Applications Of Derivatives

Find dimensions of the closed circular cylinder of volume 200 cm³ with minimum surface area. OR Show that a cone of…

Question 345M · Applications Of Integrals

Find the area bounded by y=3x+2, the x-axis and x=−1, x=1.

Question 355M · Three Dimensional Geometry

Find the shortest distance between r=(1,2,3)+λ(1,−3,2) and r=(4,5,6)+μ(2,3,1).

Question 364M · Relations And Functions

Two friends play a dice game with outcome set B={1,2,3,4,5,6}. Define R on B by xRy when y is divisible by x.

Question 374M · Continuity And Differentiability

A class is studying dy/dx=(dy/dt)/(dx/dt) for parametrically defined curves on regular branches.

Question 384M · Probability

Bag I contains 5 red and 6 black balls. Bag II contains 3 red and 4 black balls. All specified draws and transfers are…