List of top Mathematics Questions asked in UP Board XII

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

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:

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

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

If \(y=\sin^{-1}x\), then show that \((1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}=0\).

Find the area of the circle \(x^2+y^2=r^2\).

Find the value of \(\displaystyle\int_0^{\pi}\dfrac{x\,dx}{a^2\cos^2x+b^2\sin^2x}\).

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

Prove that the radius of the largest (maximum surface area) right circular cylinder that can be inscribed in a cone is half of the radius of the cone.

Evaluate: \(\displaystyle\int_0^{\pi/2}\log\sin x\,dx\).

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

If vectors \(\vec a,\vec b,\vec c\) are mutually perpendicular and having equal magnitude, then show that the vector \(\vec a+\vec b+\vec c\) is equally inclined to the vectors \(\vec a,\vec b,\vec c\).

A circular disc having radius 3 cm is warmed and due to expansion its radius is increasing at the rate 0.05 cm/s. Find the increasing rate of its area when its radius is 3.2 cm.

Prove that the function defined by \(f(x)=\begin{cases}x^2\sin\dfrac1x, & x\neq0\\0, & x=0\end{cases}\) is continuous.

Find the minimum value of the linear programming problem \(Z=200x+500y\) under the following constraints: \(x+2y\ge10\), \(3x+4y\le24\), \(x\ge0,y\ge0\) by graphical method.

A die is thrown twice and found that the sum of the numbers is 6. Find the conditional probability of getting number 4 at least once.

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

Find the general solution of the differential equation \(\log\left(\dfrac{dy}{dx}\right)=3x+4y\).

Show that the line passing through the points \((1,-1,2)\) and \((3,4,-2)\) is perpendicular to the line passing through the points \((0,3,2)\) and \((3,5,6)\).

Find the magnitude of the vectors \(\vec a\) and \(\vec b\), if having equal magnitudes and angle \(60^\circ\) between them and their scalar product is \(\dfrac12\).

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

If \(A=\begin{bmatrix}3&-2\\4&-2\end{bmatrix}\) and \(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) and \(A^2=KA-2I\), then find the value of \(K\).

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

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