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

The function \(f: R \to R\) defined by \(f(x) = 3x + 5,\ \forall x \in R\) is:

• (A) \(f\) is one-one onto
• (B) \(f\) is many-one onto
• (C) \(f\) is one-one but not onto
• (D) \(f\) is neither one-one nor onto

The value of \(\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})\) is:

• (A) \(\pi\)
• (B) \(\dfrac{-\pi}{2}\)
• (C) \(0\)
• (D) \(2\sqrt{3}\)

If \(A\) is any matrix and \(K\) is any constant, then \((KA)'\) is:

• (A) \(K'A'\)
• (B) \(A'K'\)
• (C) \(KA'\)
• (D) \(KA\)

If the radius of a circle is \(r = 6\) cm, then the rate of change of Area with respect to radius is:

• (A) \(10\pi\)
• (B) \(12\pi\)
• (C) \(8\pi\)
• (D) \(11\pi\)

The value of \(\displaystyle\int_0^{2/3} \frac{dx}{4+9x^2}\) is:

• (A) \(\dfrac{\pi}{6}\)
• (B) \(\dfrac{\pi}{12}\)
• (C) \(\dfrac{\pi}{24}\)
• (D) \(\dfrac{\pi}{4}\)

Evaluate: \(\displaystyle\int_{-\pi/2}^{\pi/2} (x^3 + x\cos x)\, dx\)

Find the area of the region bounded by the curve \(y=\sin x\), \(x=0\) and \(x=2\pi\).

Find the order and degree of the differential equation \(\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + \sin\left(\dfrac{dy}{dx}\right) + 1 = 0\).

Find the area of the parallelogram whose adjacent sides are given by \(\vec a = 3\hat i+\hat j+4\hat k\) and \(\vec b = \hat i-\hat j+\hat k\).

If \(A^2=A\), then simplify \((I+A)^2 - 7A\), where \(A\) is a square matrix.

If the ordered pairs \((2x-3,-5)\) and \((x,y-1)\) are equal, then find the value of \(x\) and \(y\).

Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).

Differentiate \(\log(\log x)\), \(x>1\) with respect to \(x\).

Prove that on \(R\), the function \(f(x)=e^{2x}\) is increasing.

Prove that \((\vec a+\vec b)\cdot(\vec a+\vec b) = |\vec a|^2+|\vec b|^2\) if and only if \(\vec a\) and \(\vec b\) are perpendicular. It is given that \(\vec a\ne0,\ \vec b\ne0\).

Find the direction cosines of the \(x\), \(y\) and \(z\) axes.

If \(P(A)=\dfrac{7}{13}\), \(P(B)=\dfrac{9}{13}\) and \(P(A\cap B)=\dfrac{4}{13}\), then find \(P(A/B)\).

Solve: \(\displaystyle\int \cos^2 x\, dx\)

Evaluate: \(\displaystyle\int_0^1 \frac{\tan^{-1}x}{1+x^2}\, dx\)

Three vectors \(\vec a, \vec b\) and \(\vec c\) satisfy the condition \(\vec a+\vec b+\vec c=0\). If \(|\vec a|=3, |\vec b|=4\) and \(|\vec c|=5\), then find the value of \(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).

Find the angle between the lines \(\vec r = 3\hat i+2\hat j-4\hat k+\lambda(\hat i+2\hat j+2\hat k)\) and \(\vec r = 5\hat i-2\hat j+\mu(3\hat i+2\hat j+6\hat k)\).

Minimize \(z=3x+2y\) under the following constraints by graphical method: \(x+y\ge8,\ x\ge0,\ 3x+5y\le15,\ y\ge0\).

If \(A=\begin{bmatrix}2&3\\1&-4\end{bmatrix},\ B=\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\), then find \((AB)^{-1}\).

Write \(\tan^{-1}\left(\dfrac{\cos x-\sin x}{\cos x+\sin x}\right)\), \(\dfrac{-\pi}{4}<x<\dfrac{3\pi}{4}\) in the simplest form.

If the function \(f:\left[0,\dfrac{\pi}{2}\right]\to R\) is given by \(f(x)=\cos x\) and the function \(g:\left[0,\dfrac{\pi}{2}\right]\to R\) is given by \(g(x)=\sin x\), then prove that \(f\) and \(g\) are one-one, but \(f+g\) is not one-one.

Is the function \(f(x)=\begin{cases}x+5,&x\le1\\x-5,&x>1\end{cases}\) continuous at \(x=1\)?

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

A family has two children. If it is known that at least one child is a boy, then find the probability of both children being boys.

Find the general solution of the differential equation \(x\cos\left(\dfrac yx\right)\dfrac{dy}{dx} = y\cos\left(\dfrac yx\right)+x\).

Evaluate: \(\displaystyle\int_0^\pi \frac{x\,dx}{a^2\cos^2x+b^2\sin^2x}\)

Find the area bounded by the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

Prove that the volume of the largest right circular cone that can be inscribed in a sphere of radius \(R\) is \(\dfrac{8}{27}\) of the volume of the sphere.

Solve the system of linear equations by Matrix Method: \(2x-3y+5z=11\), \(3x+2y-4z=-5\), \(x+y-2z=-3\).

If \(A=\begin{bmatrix}1&2&3\\3&-2&1\\4&2&1\end{bmatrix}\), then show that \(A^3-23A-40I=0\).