Mathbee · CBSE Class 12 Maths

CBSE Board Paper 2026 80 Marks · 38 Questions

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

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

Question 11M · Three Dimensional Geometry

Which listed direction cosines belong to the line x=1, y=z? The printed symmetric form is (1−x)/0=y=z.

Question 21M · Linear Programming

What is the linear function that a linear programming problem maximizes or minimizes?

Question 31M · Linear Programming

Which pair gives the nontrivial constraints for the shaded feasible region?

Question 41M · Probability

For events A and B with P(A)≠0 and P(B)≠1, find P(A′ | B′).

Question 51M · Relations And Functions

On A={1,2,3}, classify R={(1,3),(3,3),(1,1),(2,2),(3,1)}.

Question 61M · Matrices

For same-order square matrices A and B, which statements always hold? (i) (A+B)(A−B)=A²−B². (ii) AB=BA. (iii)…

Question 71M · Matrices

If [[1,a,b],[−1,2,c],[0,5,3]] is symmetric, find 3a+b+c.

Question 81M · Matrices

For A=[[tan x,cot x],[−cot x,tan x]], solve A+Aᵀ=2I for x∈[0,π/2] where A is defined.

Question 91M · Matrices

For a square matrix A, which expression equals (3A)⁻¹ when it exists?

Question 101M · Determinants

The determinant of [[−1,−2,5],[−2,a,−1],[0,4,2a]] is −86. Find the sum of the possible a values.

Question 111M · Continuity And Differentiability

If x+y=xy, find dy/dx.

Question 121M · Integrals

Evaluate \(\int\frac{dx}{\sec x+\tan x}\).

Question 131M · Applications Of Derivatives

Which statement holds for f(x)=x+1/x, x≠0?

Question 141M · Applications Of Integrals

Which expression gives the area bounded by y=x² and y=16?

Question 151M · Differential Equations

Choose the listed general solution for x²dy+y²dx=0.

Question 161M · Differential Equations

Find the listed integrating factor for 2x y′−y=3.

Question 171M · Vector Algebra

If |a|=5 and −2≤λ≤1, find the sum of the largest and smallest values of |λa|.

Question 181M · Vector Algebra

Which vector of magnitude 3 makes equal angles with the x- and y-axes and is perpendicular to the z-axis?

Question 191M · Vector Algebra

Assertion: |a×b|²+(a·b)²=|a|²|b|². Reason: |a×b|=(a·b) tan θ for θ≠π/2, where θ is the angle between the vectors.

Question 201M · Three Dimensional Geometry

Assertion: A line can have direction cosines (1,1,1). Reason: cos θ=1 is possible for θ=0.

Question 212M · Three Dimensional Geometry

Find the point on x=(y−1)/2=(z−2)/3 whose y-coordinate equals three times its x-coordinate.

Question 222M · Relations And Functions

Is f: ℝ∖{3}→ℝ, f(x)=(x−2)/(x−3), onto? OR Is f: ℤ×ℤ→ℤ×ℤ, f(x,y)=(2y,3x), injective?

Question 232M · Continuity And Differentiability

If \(x=e^{\sin^{-1}t}\) and \(y=e^{\cos^{-1}t}\), find dy/dx at \(t=1/\sqrt2\).

Question 242M · Applications Of Derivatives

Find the absolute maximum of cos x+sin²x on [0,π]. OR The volume of a solid hemisphere increases at a uniform rate…

Question 252M · Vector Algebra

For AB=(0,1,1) and AC=(3,−1,4), prove AD=(AB+AC)/2 where D is the midpoint of BC, then find |AD|.

Question 263M · Probability

On a stormy day a sniper is three times as likely to hit as to miss with each shot. For two shots, find the…

Question 273M · Integrals

Evaluate \(\int\frac{3x-1}{\sqrt{x^2-4x}}\,dx\).

Question 283M · Integrals

Evaluate \(\int_{\pi/12}^{5\pi/12}\frac{dx}{1+\sqrt{\cot x}}\). OR Evaluate…

Question 293M · Integrals

Find F if F′(x)=1/(eˣ+1) and F(0)=ln(1/2).

Question 303M · Differential Equations

Solve \(xy\prime=y-x\sin^2(y/x)\), given \(y(1)=\pi/6\). OR Solve \(y\ln y\,\frac{dx}{dy}+x=\frac2y\).

Question 313M · Linear Programming

Graphically maximize Z=4500x+5000y subject to x+y≤250, 25x+40y≤7000 and x,y≥0.

Question 325M · Applications Of Derivatives

Find the intervals of increase and decrease for ln(sin x+cos x) on (0,π/2). OR Revolve a rectangle of perimeter 30 cm…

Question 335M · Inverse Trigonometric Functions

Find the domain of q(x)=arccos(4x²−3), solve q(x)=0, and find the range of 3q(x)−π.

Question 345M · Three Dimensional Geometry

Find the intersection of the line through A=(1,2,3), B=(5,8,11) and r=(4,1,0)+t(5,2,1). Find the line through the…

Question 355M · Matrices

For P=[[1,−1,0],[2,3,4],[0,1,2]] and Q=[[2,2,−4],[−4,2,−4],[2,−1,5]], compute QP. Use matrices to solve x−y=3…

Question 364M · Continuity And Differentiability

A racing track has equation y=f(x), where f(x)=x⁴−4x²+4 for 0≤x<3 and f(x)=x²+40 for x≥3.

Question 374M · Probability

In a group of 50 smokers, 30 are male. Lung complications occur in 170 per 1000 male smokers and 120 per 1000 female…

Question 384M · Applications Of Integrals

A 3 m wide track surrounds an elliptical ground given by 9x²+16y²=144. The outer edge meets the positive x- and y-axes…