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

If \(\dfrac{dy}{dx}=e^{x+y}\), then its solution is:

• (A) \(e^{-y}=e^{x}+c\)
• (B) \(e^{x}+e^{-y}=c\)
• (C) \(e^{-x}-e^{-y}=c\)
• (D) \(e^{-x}+e^{-y}=c\)

\(\sin(\tan^{-1}x)\), \(|x|<1\) is equal to:

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

The interval in which the function \(f(x)=x^2-4x+6\) is increasing, is:

• (A) \((2,10)\)
• (B) \((2,\infty)\)
• (C) \((-2,\infty)\)
• (D) \((0,\infty)\)

The function \(f(x)=2x\), \(x\in R\) is:

• (A) one-one but not onto
• (B) one-one and onto
• (C) many-one and onto
• (D) many-one but not onto

If \(3P(A)=P(B)=\dfrac{5}{13}\) and \(P(A/B)=\dfrac{2}{5}\), then \(P(A\cup B)\) will be:

• (A) \(\dfrac{20}{39}\)
• (B) \(\dfrac{16}{39}\)
• (C) \(\dfrac{11}{39}\)
• (D) \(\dfrac{14}{39}\)

If \(f:R\to R\) and \(g:R\to R\) are defined by \(f(x)=\cos x\) and \(g(x)=3x^2\) respectively, find \(gof\).

If a line makes \(90^\circ\), \(60^\circ\) and \(30^\circ\) with the \(x\), \(y\) and \(z\)-axes respectively in the positive direction, then find its direction cosines.

Find the value of \(\tan^{-1}\sqrt3-\sec^{-1}(-2)\).

Find the value of the integral \(\displaystyle\int x^2\tan(x^3+2)\,dx\).

If for a unit vector \(\vec a\), \((\vec x-\vec a)\cdot(\vec x+\vec a)=12\), then find \(|\vec x|\).

If \(\vec a=2\hat i+\hat j+3\hat k\) and \(\vec b=3\hat i+5\hat j-2\hat k\), find \(|\vec a\times\vec b|\).

Prove that the function \(f:R\to\{x\in R:-1<x<1\}\), where \(f(x)=\dfrac{2x}{1+|x|}\), \(x\in R\), is one-one.

If \(f(x)=\sin x\), \(x\in[0,\pi/2]\) and \(g(x)=\cos x\), \(x\in[0,\pi/2]\), prove that \((f+g)\) is not one-one.

If the lines \(\dfrac{x-1}{-3}=\dfrac{y-2}{2k}=\dfrac{z-3}{2}\) and \(\dfrac{x-1}{3k}=\dfrac{y-1}{1}=\dfrac{z-6}{-5}\) are mutually perpendicular, find the value of \(k\).

Find the Cartesian equation of the line parallel to the vector \(3\hat i+2\hat j-8\hat k\) and passing through the point \((5,2,-4)\).

If \(x\in\left(0,\dfrac{\pi}{4}\right)\), then prove that \(\cot^{-1}\left(\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)=\dfrac{x}{2}\).

Find the value of the determinant \(\begin{vmatrix}a & b & c\\ b & c & a\\ c & a & b\end{vmatrix}\).

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

Suppose that \(\vec a\), \(\vec b\) and \(\vec c\) are three vectors such that \(|\vec a|=3\), \(|\vec b|=4\), \(|\vec c|=5\), and each of them is perpendicular to the sum of the other two vectors. Find \(|\vec a+\vec b+\vec c|\).

Find the maximum value of the objective function \(Z=5x+10y\) by graphical method under the following constraints: \(x+2y\le120\), \(x+y\ge60\), \(x-2y\ge0\), \(x\ge0\), \(y\ge0\).

For what value of \(\lambda\), is the function \[f(x)=\begin{cases}\lambda(x^2-2x), & x\le0 \\ 4x+1, & x>0\end{cases}\] continuous at \(x=0\)?

If the position vectors of points \(A\), \(B\), \(C\) and \(D\) are respectively \(\hat i+\hat j+\hat k\), \(2\hat i+5\hat j\), \(3\hat i+2\hat j-3\hat k\) and \(\hat i-6\hat j-\hat k\), then find the angle between the lines \(AB\) and \(CD\). Prove that \(AB\) and \(CD\) are collinear.

Find the interval in which the function \(f(x)=\dfrac{4\sin x-2x-x\cos x}{2+\cos x}\) is strictly increasing and strictly decreasing.

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

Prove that if \(A\) and \(B\) are independent events, then the probability of happening of at least one of \(A\) or \(B\) is \(\left[1-P(A')P(B')\right]\).

Prove that the height of the right circular cone of maximum volume inscribed in a sphere of radius \(r\) is \(\dfrac{4r}{3}\).

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

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

Suppose that \(A=\begin{bmatrix}2 & -1\\3 & 4\end{bmatrix}\), \(B=\begin{bmatrix}5 & 2\\7 & 4\end{bmatrix}\), \(C=\begin{bmatrix}2 & 5\\3 & 8\end{bmatrix}\). Find the matrix \(D\) such that \(CD-AB=0\).

Solve the following system of equations by matrix method: \(\dfrac2x+\dfrac3y+\dfrac{10}z=4\), \(\dfrac4x-\dfrac6y+\dfrac5z=1\), \(\dfrac6x+\dfrac9y-\dfrac{20}z=2\).

Prove that the function \(y=Ae^{3x}\cos4x+Be^{3x}\sin4x\), where \(A\), \(B\) are arbitrary constants, is the solution of the differential equation \(\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+25y=0\).

Find \(\displaystyle\int\left[\sqrt{\tan x}+\sqrt{\cot x}\right]dx\).

Find the particular solution of the differential equation \(\dfrac{dy}{dx}+y\cot x=4x\csc x\), \((x\neq0)\), given that \(y=0\) when \(x=\dfrac{\pi}{2}\).

Find \(\displaystyle\int\dfrac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx\).

Find \(\displaystyle\int e^x\left(\dfrac{1+\sin x}{1+\cos x}\right)dx\).