List of top Questions asked in UP Board XII

Let three vectors \(\vec a,\vec b\) and \(\vec c\) be 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, then find the value of \(|\vec a+\vec b+\vec c|\).

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

Show that \(\sin^{-1}\dfrac{12}{13}+\cos^{-1}\dfrac{4}{5}+\tan^{-1}\dfrac{63}{16}=\pi\).

Find the value of the determinant \(\begin{vmatrix}17&15&13\\9&-8&7\\-3&2&5\end{vmatrix}\).

If \(x=a\sin^3t,\ y=b\cos^3t\), then find \(\dfrac{dy}{dx}\) at the point \(t=\dfrac{\pi}{2}\).

Verify that the function \(y=a\cos x+b\sin x\), in which \(a,b\in R\), is a solution of the differential equation \(\dfrac{d^2y}{dx^2}+y=0\).

The length \(x\) of a rectangle is decreasing at the rate of 3 cm/min and the breadth \(y\) is increasing at the rate of 2 cm/min. When \(x=5\) cm and \(y=3\) cm, find the rate of change of the area of the rectangle.

If \(P(A)=0.8,\ P(B)=0.5\) and \(P(B\mid A)=0.4\), find \(P(A\cup B)\).

Show that the points \(A,B,C\), whose position vectors are respectively \(\vec a=3\hat i-4\hat j-4\hat k\), \(\vec b=2\hat i-\hat j+\hat k\) and \(\vec c=\hat i-3\hat j-5\hat k\), form the vertices of a right-angled triangle.

Prove that \(f(x)=\tan x\) is a continuous function.

A relation \(R\) is defined in \(N\times N\) as follows: \((a,b)\,R\,(c,d)\) if and only if \(ad=bc\). Prove that \(R\) is an equivalence relation.

If \(2\begin{bmatrix}x&5\\7&y-3\end{bmatrix}+\begin{bmatrix}3&-4\\1&2\end{bmatrix}=\begin{bmatrix}7&6\\15&10\end{bmatrix}\), find the values of \(x\) and \(y\).

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

The order of the differential equation \(x\left(\dfrac{d^3y}{dx^3}\right)^2+\sin x\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^4+y^5=0\) is

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

Show that \(\tan^{-1}\dfrac{2}{11}+\tan^{-1}\dfrac{7}{24}=\tan^{-1}\dfrac{1}{2}\).

Integrate \(x\log x\) with respect to \(x\).

If \(x=at^2,\ y=2at\), find the value of \(\dfrac{dy}{dx}\).

For which value of \(\lambda\) is the vector \(2\hat{i}+\hat{j}-\lambda\hat{k}\) perpendicular to the vector \(\hat{i}+4\hat{j}+\hat{k}\)?

The value of \(\cos(\sec^{-1}x+\text{cosec}^{-1}x)\), \(|x|\ge 1\) is

The value of the integral \(\displaystyle\int\dfrac{\sin^2x-\cos^2x}{\sin^2x\cos^2x}\,dx\) is

\(f:X \to Y\) will be an onto function if and only if its range is

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

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

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