List of top Questions asked in UP Board XII

If \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be functions such that \(f(x)=\cos x\) and \(g(x)=3x^3\), then find \(f\circ g\).

Prove that \(3\cos^{-1}x=\cos^{-1}(4x^3-3x),\ x\in\left[\dfrac12,1\right]\).

Prove that the given function \(f(x)=3x+17\) is increasing on \(\mathbb{R}\).

If \(A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\) and \(B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}\), then find the value of \((2A-B)\).

Find the projection of vector \((\hat i-\hat j)\) on vector \((\hat i+\hat j)\).

Find the integrating factor of the differential equation \(x\dfrac{dy}{dx}+2y=x^2\ (x\neq0)\).

If \(X=\{1,2,3\}\) and \(Y=\{a,b\}\), then find the number of functions from \(X\) to \(Y\).

If \(P(A)=\dfrac12\), \(P(B)=\dfrac13\) and \(A,B\) are independent events, find the value of \(P(A/B)\).

Write \(\tan^{-1}\left(\dfrac{x}{\sqrt{a^2-x^2}}\right),\ |x|<a\) in simplest form.

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

The function \(f:Z\to Z\) defined by \(f(x)=x^3\ \forall x\in Z\) is:

The value of \(\displaystyle\int x\cos x\,dx\) is:

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

\(A=[a_{ij}]_{m\times n}\) is a square matrix if:

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

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

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

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.

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

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

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

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

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.

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 \(\mu=\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a\).