CBSE Board Paper 2026
80 Marks · 38 Questions
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 — Step-by-Step Solutions & Marking Scheme
Select any question below to view the complete worked solution and official CBSE stepwise marking breakdown.
Question 11M · Continuity And Differentiability
Let \(f(x)=|x|/x\) for \(x\ne0\), with \(f(0)=0\). Which statement is incorrect?
Question 21M · Inverse Trigonometric Functions
Evaluate \(\sin^{-1}(\sin(3\pi/5))\).
Question 31M · Applications Of Derivatives
For \(f(x)=-|x+1|+5\), which statement is true?
Question 41M · Matrices
If \(A=\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\) satisfies \(A^T+A=I\), which listed…
Question 51M · Matrices
For a square matrix A, which statement is not generally true?
Question 61M · Determinants
If \(\begin{vmatrix}3x&x+2\\3(x-1)&x-2\end{vmatrix}=\begin{vmatrix}1&5\\3&3\end{vmatrix}\), find x.
Question 71M · Inverse Trigonometric Functions
Evaluate \(\tan^{-1}(-1)+\cos^{-1}(-1/2)+\sin^{-1}(1/2)\).
Question 81M · Matrices
If \(A=\begin{bmatrix}1&-2&3\\3&2&-1\end{bmatrix}\), \(B=\begin{bmatrix}2&3\\-1&2\\4&-5\end{bmatrix}\), and \(C=AB\)…
Question 91M · Determinants
For order-three matrices with \(\det A=4\), \(\det B=7\), find \(\det(2AB)\).
Question 101M · Integrals
If \(\int_0^1(9x^2-4x-2k)\,dx=0\), find k.
Question 111M · Applications Of Derivatives
A particle has position S(t)=2t³−6t²+8. Find its velocity when the acceleration vanishes.
Question 121M · Integrals
If \(\int_0^4 dx/(x+\sqrt{x})=k\log3\), find k.
Question 131M · Integrals
Evaluate \(\int e^x(2+\sin2x)/(1+\cos2x)\,dx\).
Question 141M · Continuity And Differentiability
Differentiate \(\ln(\sin x)\) with respect to \(\sqrt{\cos x}\) where both derivatives are defined and the denominator…
Question 151M · Probability
For independent events A and B, which statement is not generally correct?
Question 161M · Continuity And Differentiability
Differentiate \(\cos^{-1}(e^x)\) with respect to x.
Question 171M · Vector Algebra
If \((3,-2,4)\) and \((\lambda,1,0)\) are perpendicular, find \(\lambda\).
Question 181M · Probability
If P(A)=0.3, P(B)=0.7 and P(B | A)=0.5, find P(A∪B).
Question 191M · Relations And Functions
Assertion A. On the set of all books in a school library, let R relate books with equal page counts and S relate two…
Question 201M · Vector Algebra
Assertion A. Points A=(6,−7,0), B=(16,−19,−4), C=(1,−1,2) are collinear. Reason R. Three points are collinear when C…
Question 212M · Matrices
Find x if [[6,3x],[8,−6]]=[[6,x²+2],[8,x²−5x]].
Question 222M · Probability
Two cards are drawn uniformly without replacement from cards numbered 1 through 22. Find the probability that both…
Question 232M · Continuity And Differentiability
Find a,b for continuity of f(x)=2 for x≤3, ax+b for 3<x<5, and 5 for x≥5. OR If \(2y^3+y=2x\), prove…
Question 242M · Linear Programming
Shade the feasible region 4x+y≥20, 2x+3y≥30, x,y≥0 and list its corners.
Question 252M · Relations And Functions
Thumb pins on a graph sheet are related when their distance is less than 3 cm. Determine whether this defines an…
Question 263M · Three Dimensional Geometry
Find the line through (1,−1,2) parallel to the x-axis and its shortest distance to r=(2,1,−1)+t(3,−5,2). OR For…
Question 273M · Continuity And Differentiability
The printed question sets \(y=\sqrt{\ln(\sin(x^2/3+1))}\) and asks to show \(y\prime=x\cot(x^2/3+1)/(3y)\). Derive the…
Question 283M · Applications Of Derivatives
Find the product of maximum and minimum values of cos x−(1/2)cos 2x on [0,π/2].
Question 293M · Relations And Functions
On positive natural numbers, \(R=\{(x,y):x+2y=25\}\). Find its domain and range and decide whether it is an…
Question 303M · Linear Programming
Minimize \(Z=4x+y\) subject to \(x+y\ge3\), \(x+2y\ge4\), \(x,y\ge0\), graphically.
Question 313M · Matrices
Express \(A=\begin{bmatrix}2&4\\5&6\end{bmatrix}\) as a sum of a symmetric and a skew-symmetric matrix. OR Using…
Question 325M · Integrals
Evaluate \(\int_{-\pi}^{\pi}2x(1+\sin x)/(1+\cos^2x)\,dx\).
Question 335M · Applications Of Integrals
Using integration, find the area enclosed by \(y=x^2\) and \(y=2\). OR Using integration, find the area inside…
Question 345M · Differential Equations
Solve \(y+\frac{d}{dx}(xy)=x(\sin x+\ln x)\), for x>0.
Question 355M · Vector Algebra
Nonzero vectors α,β,γ satisfy α×β=γ and β×γ=α. Show they are mutually perpendicular, norm β=1, and norm α=norm γ. Find…
Question 364M · Probability
Companies X,Y,Z have selection probabilities 1/3,2/9,4/9. Their conditional probabilities of poor construction quality…
Question 374M · Three Dimensional Geometry
A cuboid swimming pool has breadth line \((-x+2)/(-5)=y=z-3\). Students A and B swim along its length. Student A…
Question 384M · Applications Of Derivatives
A window consists of a rectangle of length x and breadth y metres, surmounted by an equilateral triangle on its…