CBSE Board Paper 2026
80 Marks · 38 Questions
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 — 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| = 8, |b| = 3 and |a × b| = 12, then the value of |a · b| is
Question 21M · Three Dimensional Geometry
Find the perpendicular distance from (3,4,2) to the z-axis.
Question 31M · Linear Programming
The feasible region of a linear programming problem with objective function Z = 5x + 7y is shown below. The maximum…
Question 41M · Linear Programming
The degree of an objective function of a linear programming problem is
Question 51M · Inverse Trigonometric Functions
If y=sin⁻¹x+π, find the range of y.
Question 61M · Matrices
Which property must hold for every scalar matrix of order 3?
Question 71M · Matrices
Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)′ = A′ + B′ (ii) (A −…
Question 81M · Determinants
If Δ₁ = | 1 0 0; 0 2 0; 0 0 3 | and Δ₂ = | 0 2 0; 1 0 0; 0 0 6 |, then
Question 91M · Determinants
One of the values of x for which | cos x sin x; −cos x sin x | = 1 is
Question 101M · Matrices
For symmetric matrices A and B of equal order, classify AB−BA.
Question 111M · Applications Of Derivatives
Find the minimum of e raised to −x on [0,3].
Question 121M · Integrals
If ∫ (3ax)/(b² + c²x²) dx = A log| b² + c²x² | + K, then the value of A is
Question 131M · Integrals
The value of ∫₋₁¹ x³/(x² + 2|x| + 1) dx is
Question 141M · Applications Of Integrals
The area bounded by the curve y = x|x|, x-axis and the ordinates x = −1 and x = 1 is given by
Question 151M · Differential Equations
The integrating factor of differential equation R dx/dy + P x = Q, where P, Q, R are functions of y is
Question 161M · Differential Equations
Find the order and degree of \(\frac{d}{dx}(y^{\prime})^3+(y^{\prime})^3=1\).
Question 171M · Vector Algebra
The value of p for which vectors î + 2ĵ + 3k̂ and 2î − p ĵ + k̂ are perpendicular to each other is
Question 181M · Vector Algebra
The value of m for which the points with position vectors −î − ĵ + 2k̂, 2î + m ĵ + 5k̂ and 3î + 11ĵ + 6k̂ are…
Question 191M · Probability
Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that…
Question 201M · Three Dimensional Geometry
Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other…
Question 212M · Inverse Trigonometric Functions
Simplify: tan⁻¹( (cos2x − sin2x)/(cos2x + sin2x) ), 0 < x < π/4. OR Evaluate: tan( sin⁻¹1 − cos⁻¹(−1/2) ).
Question 222M · Continuity And Differentiability
Check whether function f(x) defined as f(x) = { |x−3|/[2(x−3)], x<3; (x−6)/6, x≥3 } is continuous at x=3 or not. OR If…
Question 232M · Inverse Trigonometric Functions
Simplify \(\sin^{-1}\sqrt{(1+\cos 2x)/2}\), for \(0<x<\pi/2\). OR Evaluate \(\cos(\sin^{-1}(-1)-\tan^{-1}(-\sqrt3))\).
Question 242M · Vector Algebra
Using vectors, find the area of the triangle with A=(1,2,3), B=(2,−1,4), and C=(4,5,−1).
Question 252M · Vector Algebra
Vectors a = 3î − 2ĵ + 2k̂ and b = î + 2k̂ represent the two adjacent sides of a parallelogram. Find the vectors…
Question 263M · Integrals
Evaluate \(\int_0^1 \ln(1+x^2)\,dx\).
Question 273M · Probability
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown…
Question 283M · Integrals
Find: ∫ √((x+2)/(x−2)) dx. OR Find: ∫ x²/((x²+9) (x²+16)) dx.
Question 293M · Integrals
If I₁ = ∫₋π/4^π/4 dx/(1+cos2x) and I₂ = ∫₋1/2^1/2 |x| dx, then show that I₁ − 4I₂ = 0.
Question 303M · Differential Equations
Solve \((y^2-x^2)\,dx=2xy\,dy\). OR Solve \((1+e^{2x})\,dy+(1+y^2)e^x\,dx=0\), with \(y(1)=0\).
Question 313M · Linear Programming
Solve the following linear programming problem graphically: Minimize Z = 13x − 15y subject to constraints x + y ≤ 7…
Question 325M · Three Dimensional Geometry
Show that the line through (0,4,1) and (2,3,−1) is parallel to the line through (4,5,0) and (2,6,2). Find their…
Question 335M · Relations And Functions
A relation R is defined on Z, the set of integers, as R = {(x, y) : |x − y| is divisible by a prime number ‘p’, x, y ∈…
Question 345M · Matrices
If A = [[0,2,1],[-2,-1,-2],[1,-1,0]], find A⁻¹ and use it to solve the system: −2y + z = 7, 2x − y − z = 8, x − 2y =…
Question 355M · Continuity And Differentiability
For \(x=a(\sin t-t\cos t)\) and \(y=b(\cos t+t\sin t)\), find the first and second derivatives of y with respect to x.
Question 364M · Applications Of Integrals
Roundabouts are often made on busy roads to ease the traffic and avoid red lights. One such roundabout is made such…
Question 374M · Applications Of Derivatives
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹300 per subscriber…
Question 384M · Probability
In an online jackpot, there is one first prize of ₹3,00,000, two second prizes of ₹2,00,000 each and three third…