UP Board Class 12 Mathematics Question Paper 2026 (Set 324 CY) with Solutions

If \(f:R\to R\) is given by \(f(x)=(5-x^{5})^{1/5}\), then \(f\circ f(x)\) is equal to:

• (A) \(x^{1/5}\)
• (B) \(x\)
• (C) \(x^{5}\)
• (D) \(5-x^{5}\)

The value of \(\displaystyle\int_{2}^{2\sqrt3}\frac{dx}{4+x^{2}}\) is:

• (A) \(\dfrac{\pi}{12}\)
• (B) \(\dfrac{\pi}{18}\)
• (C) \(\dfrac{\pi}{24}\)
• (D) \(\dfrac{\pi}{6}\)

A relation \(R\) is defined in the set \(N\) as \(R=\{(x,y):y=x+5,\,y>5\}\). Then which of the following is correct?

• (A) \((6,10)\in R\)
• (B) \((3,6)\in R\)
• (C) \((4,9)\in R\)
• (D) \((2,6)\in R\)

The value of \(\displaystyle\int_{0}^{\pi/4}\sin^{3}2x\cos2x\,dx\) is:

• (A) \(\dfrac12\)
• (B) \(\dfrac14\)
• (C) \(\dfrac18\)
• (D) \(\dfrac1{16}\)

The probability of impossible events is:

• (A) \(0\)
• (B) \(\dfrac12\)
• (C) \(\dfrac13\)
• (D) \(\dfrac14\)

Find the value of \(\cos^{-1}\Big(\dfrac12\Big)+2\sin^{-1}\Big(\dfrac12\Big)\).

Write \(\cot^{-1}\!\Big(\dfrac{1}{\sqrt{x^{2}-1}}\Big)\), \(x>1\) in the simplest form.

If the direction ratios of a line are \(3,-2,-6\), then find its direction cosines.

Find the angle between the vectors \(\vec a=\hat i+\hat j-\hat k\) and \(\vec b=\hat i-\hat j+\hat k\).

Show that the function \(f(x)=x^{3}-6x^{2}+12x,\ x\in R\), is an increasing function on \(R\).

If two vectors \(\vec a=5\hat i-\hat j-3\hat k\) and \(\vec b=\hat i+3\hat j-5\hat k\), then show that \((\vec a+\vec b)\) and \((\vec a-\vec b)\) are perpendicular.

If \(A\) and \(B\) are independent events and \(P(A)=0.3\), \(P(B)=0.4\), then find (i) \(P(A\cap B)\), (ii) \(P(A\cup B)\).

Prove that the relation given by \(R=\{(1,1),(2,2),(3,3),(1,2),(2,3)\}\) in the set \(\{1,2,3\}\) is reflexive but neither symmetric nor transitive.

If \(\sin^{-1}(1-x)-2\sin^{-1}x=\dfrac{\pi}{2}\), then find the value of \(x\).

If \((\vec a+\vec b)\cdot(\vec a-\vec b)=8\) and \(|\vec a|=8|\vec b|\), then find \(|\vec a|\) and \(|\vec b|\).

Simplify: \(\cos\theta\begin{bmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{bmatrix}+\sin\theta\begin{bmatrix}\sin\theta & -\cos\theta\\\cos\theta & \sin\theta\end{bmatrix}\).

Simplify: \(\tan^{-1}\!\left(\dfrac{3a^{2}x-x^{3}}{a^{3}-3ax^{2}}\right)\).

Show that the points \(A(2,3,-4)\), \(B(1,-2,3)\) and \(C(3,8,-11)\) are collinear.

Find the minimum value of the objective function \(Z=3x+5y\) under the following constraints by the graphical method: \(x+3y\ge3,\ x+y\ge2,\ x\ge0,\ y\ge0\).

Find the minimum distance between the lines \(\vec r=\hat i+\hat j+\lambda(2\hat i-\hat j+\hat k)\) and \(\vec r=2\hat i+\hat j-\hat k+\mu(3\hat i-5\hat j+2\hat k)\).

If \(y=Ae^{mx}+Be^{nx}\), then show that \(\dfrac{d^{2}y}{dx^{2}}-(m+n)\dfrac{dy}{dx}+mny=0\).

Find the values of \(a\) and \(b\) for which the function defined by \(f(x)=\begin{cases}ax+1, & x\le3\\bx+3,& x>3\end{cases}\) is continuous at \(x=3\).

Find the value of the determinant \(\begin{vmatrix}18&22&-13\\7&-9&11\\-11&7&17\end{vmatrix}\).

Three vectors \(\vec a,\vec b,\vec c\) satisfy the condition \(\vec a+\vec b+\vec c=0\). If \(|\vec a|=3,|\vec b|=4\) and \(|\vec c|=2\), then find the value of \(\mu=\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).

Prove that the height of a cylinder of maximum volume, inscribed in a sphere of radius \(R\), is \(\dfrac{2R}{\sqrt3}\).

Differentiate: \(y=x^{\sin x}+(\cos x)^{\tan x}\).

Prove that if \(E\) and \(F\) are two independent events, then \(E\) and \(F'\) will also be independent.

Find the area surrounded by the line \(y=3x+2\), the \(x\)-axis, and the ordinates \(x=-1\) and \(x=1\).

Solve the following system of equations by matrix method: \(3x-2y+3z=8,\ 2x+y-z=1,\ 4x-3y+2z=4\).

Express the matrix \(B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\) in the form of the sum of a symmetric matrix and a skew-symmetric matrix.

Solve the differential equation \((x\,dy-y\,dx)\,y\sin\!\left(\dfrac{y}{x}\right)=(y\,dx+x\,dy)\,x\cos\!\left(\dfrac{y}{x}\right)\).

Find the value of \(\displaystyle\int_{-1}^{2}|x^{3}-x|\,dx\).

Find the value of \(\displaystyle\int_{1}^{4}\big(|x-1|+|x-2|+|x-3|\big)\,dx\).

Find the value of the integral \(\displaystyle\int\dfrac{x^{4}}{(x-1)(x^{2}+1)}\,dx\).

Show that the general solution of the differential equation \(\dfrac{dy}{dx}+\dfrac{y^{2}+y+1}{x^{2}+x+1}=0\) is \(x+y+1=A(1-x-y-2xy)\), in which \(A\) is a parameter.