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

If the function \(f:N\to N\) is defined as \(f(x)=x^2\), then \(f\) is:

• (A) One-one and onto
• (B) One-one but not onto
• (C) Neither one-one nor onto
• (D) Many-one and onto

The relation \(R=\{(a,b): a\le b^2\}\) is defined in the set of real numbers. Then \(R\) is:

• (A) Reflexive and symmetric but not transitive
• (B) Reflexive and transitive but not symmetric
• (C) Symmetric and transitive but not reflexive
• (D) Not reflexive, not symmetric and not transitive

If vectors \(5\hat i-\lambda\hat j+2\hat k\) and \(2\hat i+3\hat j+4\hat k\) are perpendicular to each other, then the value of \(\lambda\) is:

• (A) 3
• (B) 4
• (C) 6
• (D) 0

If \(x-y=\pi\), then the value of \(\dfrac{dy}{dx}\) is:

• (A) 0
• (B) \(\pi\)
• (C) 1
• (D) \(x\)

Direction cosines of the \(x\)-axis are:

• (A) (1, 0, 0)
• (B) (0, 1, 0)
• (C) (0, 0, 1)
• (D) (0, 0, 0)

Find the principal value of \(\sin^{-1}\left(\dfrac{1}{\sqrt2}\right)\).

Find the value of \(\displaystyle\int_2^3\dfrac1x\,dx\).

Find the angle between vectors \(\hat i-2\hat j+3\hat k\) and \(3\hat i-2\hat j+\hat k\).

Direction ratios of a line are 2, \(-1\) and \(-2\). Find the direction cosines of the line.

If \(P(A)=\dfrac7{13}\), \(P(B)=\dfrac9{13}\) and \(P(A\cap B)=\dfrac4{13}\), then find \(P\left(\dfrac AB\right)\).

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

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

If \(f(x)=\begin{cases}x+2, & x\ne0\\1, & x=0\end{cases}\), then prove that the function is not continuous at \(x=0\).

Find the vector and Cartesian equations of a line which passes through the point \((1,2,3)\) and is parallel to the vector \(2\hat i+3\hat j+2\hat k\).

Find the general solution of the differential equation \(\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}\).

If \(y=5\cos x-3\sin x\), then prove that \(\dfrac{d^2y}{dx^2}+y=0\).

Find the projection of vector \(\hat i+3\hat j+7\hat k\) on vector \(7\hat i-\hat j+8\hat k\).

If \(P(A)=\dfrac13\), \(P(B)=\dfrac12\) and \(P(A\cup B)=\dfrac23\), then show that \(A\) and \(B\) are independent events.

Prove that \(\tan^{-1}\dfrac15+\tan^{-1}\dfrac17+\tan^{-1}\dfrac13+\tan^{-1}\dfrac18=\dfrac\pi4\).

If \(\cos y=x\cos(a+y)\) and \(\cos a\ne\pm1\), then prove that \(\dfrac{dy}{dx}=\dfrac{\cos^2(a+y)}{\sin a}\).

Solve the differential equation: \((1+x^2)\,dy+2xy\,dx=\cot x\,dx\).

Maximize \(Z=10x+3y\) by the graphical method under the following constraints: \(x\ge0;\ y\ge0;\ 5x+3y\le15;\ 2x+5y\le10\).

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

Find the value of the determinant \(\begin{vmatrix}a&b&c\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}\).

Evaluate \(\displaystyle\int\dfrac{x^2+1}{x^2-5x+6}\,dx\).

Prove that vectors \(2\hat i-\hat j+\hat k\), \(\hat i-3\hat j-5\hat k\) and \(3\hat i-4\hat j-4\hat k\) are vertices of a right-angled triangle.

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

If a die is thrown three times, then find the probability of getting at least one odd number.

If \(A=\begin{bmatrix}1&2&3\\2&4&5\\3&5&6\end{bmatrix}\), then find \(A^{-1}\).

Solve the following system of equations by using the Matrix Method: \[ x-y+2z=7 \] \[ 3x+4y-5z=-5 \] \[ 2x-y+3z=12 \]

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

Solve the differential equation \((x-y)\dfrac{dy}{dx}=x+2y\).

Find the value of: \(\displaystyle\int_0^{\pi/2}\log\sin x\,dx\).

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