CBSE Board Paper 2026
80 Marks · 38 Questions
CBSE Class 12 Mathematics 2026 Supplementary Paper 65(B)/7 — Step-by-Step Solutions & Marking Scheme
Select any question below to view the complete worked solution and official CBSE stepwise marking breakdown.
Question 11M · Determinants
If det([[3,x,−4],[5,−6,0],[−7,8,1]])=0, find x.
Question 21M · Matrices
Find the order of [5 4 1][[1,2,3],[3,0,1],[1,4,5]](2,0,1)ᵀ.
Question 31M · Matrices
Classify B=[[0,0,7],[0,7,0],[−7,0,0]].
Question 41M · Continuity And Differentiability
How many real points are excluded from continuity of (3−x)/(3x²−x³) as printed?
Question 51M · Continuity And Differentiability
Find the derivative of \(y=e^{\sin\sqrt{x}}\) for x>0.
Question 61M · Integrals
Evaluate \(\int x\sin^3(x^2)\cos(x^2)\,dx\).
Question 71M · Differential Equations
Solve \(dy/dx=e^{x-y}\).
Question 81M · Differential Equations
Find the integrating factor for x dy+(2y−x³)dx=0 on x≠0.
Question 91M · Three Dimensional Geometry
Find the direction cosines of AB for A=(3,5,1), B=(1,1,5).
Question 101M · Vector Algebra
Choose a unit vector perpendicular to (−1,1,1) and (−1,−1,−1).
Question 111M · Three Dimensional Geometry
Find λ if (1−x)/(−3)=(2y−2)/8=(z+6)/5 is parallel to the line with direction (−6,λ,−10).
Question 121M · Three Dimensional Geometry
Find λ if (x−3)/(3λ)=(y+2)/1=(3−z)/7 and (4−x)/3=(y−5)/(2λ)=(z−1)/2 are perpendicular.
Question 131M · Vector Algebra
Find the scalar projection of (2,−1,1) on (1,−2,1).
Question 141M · Linear Programming
If px+y takes minimum 14 at (3,2), find p.
Question 151M · Linear Programming
In a linear programming problem with a feasible polygon and an attained optimum, where can an optimum be found?
Question 161M · Probability
Find the probability of at least one head in three independent fair coin tosses.
Question 171M · Probability
Roll a fair die and independently draw from a standard 52-card deck. Find the probability of a die result above four…
Question 181M · Probability
If P(A)=0.5, P(B)=0.7 and P(B | A)=0.6, find P(A∪B).
Question 191M · Determinants
Assertion: A skew-symmetric 3×3 matrix has determinant zero. Reason: Every square matrix of order two has…
Question 201M · Three Dimensional Geometry
Assertion: Angles 45°,90°,45° to the positive axes give direction cosines (1/√2,0,1/√2). Reason: Direction cosines are…
Question 212M · Matrices
Multiply [[2,1],[3,2],[−1,1]] by [[1,0,1],[−1,2,1]]. OR If Aᵀ=[[2,4],[3,6],[1,5]] and B=[[1,1,2],[−1,3,2]], find Aᵀ−Bᵀ.
Question 222M · Applications Of Derivatives
The side of an equilateral triangle grows at 3 cm/s. Find its area rate when the side is 10 cm.
Question 232M · Integrals
Evaluate \(\int\sec^3(3x+2)\tan(3x+2)\,dx\).
Question 242M · Vector Algebra
If a+b+c=0 and |a|=3, |b|=5, |c|=7, find the angle between a and b. OR Find the area of the parallelogram with…
Question 252M · Three Dimensional Geometry
Find vector and Cartesian equations of the line through (1,−2,−3) parallel to the line joining (1,−1,4) and (2,1,2).
Question 263M · Inverse Trigonometric Functions
Evaluate \(\cos^{-1}(\cos(2\pi/3))+\sin^{-1}(\sin(2\pi/3))+\tan^{-1}(\tan(2\pi/3))\).
Question 273M · Applications Of Derivatives
Find increasing and decreasing intervals of 1−12x−9x²−2x³. OR Find the maximum and minimum of 3x⁴−8x³+12x²−48x+25 on…
Question 283M · Integrals
Evaluate \(\int e^x\,dx/\sqrt{5-4e^x-e^{2x}}\). OR Evaluate \(\int x^2\,dx/[(x^2+4)(x^2+9)]\).
Question 293M · Integrals
Evaluate \(\int_1^3(|x-1|+|x-2|+|x-3|)\,dx\).
Question 303M · Differential Equations
Solve (3xy+y²)dx+(x²+xy)dy=0. OR Solve x(1+y²)dx−y(1+x²)dy=0 through (1,0).
Question 313M · Linear Programming
Find the maximum and minimum of Z=4x+3y on the bounded feasible region with corners (25,0),(16,16),(0,24).
Question 325M · Relations And Functions
On ordered pairs of positive natural numbers define (a,b)R(c,d) when ad=bc. Prove R is an equivalence relation. OR…
Question 335M · Continuity And Differentiability
If x=sin t and y=sin(pt), prove (1−x²)y″−xy′+p²y=0 on regular branches.
Question 345M · Applications Of Integrals
Use integration to find the area of \(x^2/4+y^2/3=1\) between x=−2 and x=2.
Question 355M · Probability
A uniformly random card is lost from a standard deck. Two cards drawn from the rest are both spades. Find the…
Question 364M · Matrices
Dealers P and Q sell motorcycle types A and B. In 2022 P sold 160 A and 70 B; Q sold 100 A and 40 B. In 2023 P sold…
Question 374M · Applications Of Derivatives
A rectangular garden uses a house wall as one side. There are 300 m of fencing for the other three sides. Let x be…
Question 384M · Three Dimensional Geometry
A glass box has corners A=(1,2,3), B=(2,2,3), C=(2,4,3), D=(1,4,3), A′=(1,2,6), B′=(2,2,6), C′=(2,4,6), D′=(1,4,6).