CUET 2026 Mathematics May 19 Shift 1 Question Paper- Download Answer Key and Solution PDF

Let \(A\) be a non-singular \(3\times3\) matrix satisfying \[ A^3-6A^2+11A-6I=O. \]
If \[ B=A^2-5A+7I, \]
then find \(\det(B)\) given that \(\det(A)=6\).

• (A) \(1\)
• (B) \(8\)
• (C) \(27\)
• (D) \(64\)

If \[ \left| \begin{matrix} x+a & y & z \\ x & y+b & z \\ x & y & z+c \end{matrix} \right| = abc, \] where \(a,b,c \ne 0\), then find the value of \[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c}. \]

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(3\)

If \[ y = \left(\dfrac{x+1}{x-1}\right)^x, \]
then find \(\dfrac{dy}{dx}\).

• (A) \(y\left[\ln\left(\dfrac{x+1}{x-1}\right)-\dfrac{2x}{x^2-1}\right]\)
• (B) \(y\left[\ln\left(\dfrac{x+1}{x-1}\right)+\dfrac{2x}{x^2-1}\right]\)
• (C) \(y\left[\ln(x+1)-\ln(x-1)\right]\)
• (D) \(\dfrac{2y}{x^2-1}\)

Evaluate: \[ \int \dfrac{x^2+1}{x^4+1}\,dx \]

• (A) \(\dfrac{1}{\sqrt{2}}\tan^{-1}\left(\dfrac{x^2-1}{\sqrt{2}x}\right)+C\)
• (B) \(\tan^{-1}x+C\)
• (C) \(\dfrac{1}{2}\ln(x^2+1)+C\)
• (D) \(\dfrac{x}{x^2+1}+C\)

Let \[ \vec{a}=2\hat{i}-\hat{j}+\hat{k}, \qquad \vec{b}=\hat{i}+2\hat{j}-\hat{k}, \]
and \[ \vec{c}=\lambda\hat{i}+\mu\hat{j}+3\hat{k}. \]
If \[ [\vec{a}\ \vec{b}\ \vec{c}]=0 \]
and \[ \vec{c}\cdot(\vec{a}+\vec{b})=10, \]
then find the value of \(\lambda+\mu\).

• (A) \(1\)
• (B) \(2\)
• (C) \(3\)
• (D) \(4\)

Find the equation of the plane which passes through the point \[ (1,-2,3), \]
contains the line of intersection of the planes \[ x+y+z=1 \]
and \[ 2x-y+3z=4, \]
and is perpendicular to the plane \[ x-2y+2z+5=0. \]

• (A) \(5x-y+z-10=0\)
• (B) \(x+y+z-1=0\)
• (C) \(2x-y+3z-4=0\)
• (D) \(3x+y-z+2=0\)

Let \(A\) and \(B\) be two \(3\times3\) matrices such that \[ A^2-4A+3I=O \]
and \[ B=A^{-1}+2A. \]
Find the determinant of \(B\) if \(\det(A)=3\).

• (A) \(9\)
• (B) \(27\)
• (C) \(81\)
• (D) \(3\)

If \[ f(x)=\left(\frac{1+\sin x}{1-\sin x}\right)^{\tan x}, \]
then find \[ \lim_{x\to0}\frac{\ln f(x)}{x^2}. \]

• (A) \(1\)
• (B) \(2\)
• (C) \(4\)
• (D) \(0\)

If \[ \left| \begin{matrix} x+a & y & z
x & y+b & z
x & y & z+c \end{matrix} \right| = 2abc, \]
where \(a,b,c\neq0\), then find the value of \[ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}. \]

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(3\)

If \[ y= \left( x^{\sin x} \right)^{\tan x}, \]
then find \[ \dfrac{dy}{dx}. \]

• (A) \[ y\left[ \sec^2 x \,\sin x \,\ln x + \tan x \,\cos x \,\ln x + \dfrac{\tan x \,\sin x}{x} \right] \]
• (B) \ [y(\cos x + \sin x)]
• (C) \ [y \tan x]
• (D) \[ x^{\sin x} \]