List of top Questions asked in UP Board XII

'Chipko Movement' is related to:

Which of the following is called the fourth pillar of democracy?

Who is the founder of Arya Samaj?

Which of the following is not the characteristic of culture?

Liberalization is related with:

Who was the originator of the ‘Cyclical Theory’?

The main basis for separating rural and urban communities is:

Who among the following is not a sociologist?

Solve: \(\frac{dy}{dx} = \frac{x+y+1}{2x+2y+3}\).

Solve: \((1+y^2) dx = (\tan^{-1} y - x) dy\).

If \( I = \int_{0}^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \), find the value of I.

For the two vectors \( \vec{a} \) and \( \vec{b} \), prove that \( |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}| \) when \( \vec{a} \neq \vec{0} \) and \( \vec{b} \neq \vec{0} \).

If \( I = \int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}} \), then prove that \( I = \frac{\pi}{12} \).

A car is started to move from a point P at time t = 0 and is stopped at the point Q. The distance x metre covered by the car in t second is given by \(x = t^2\left(3-\frac{t}{2}\right)\). Find the time required by the car to reach at the point Q and also find the distance between P and Q.

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

If x, y, z are three independent events, then prove that \(P(X \cap Y \cap Z) = P(X) \cdot P\left(\frac{Y}{X}\right) \cdot P\left(\frac{Z}{X \cap Y}\right)\).

Solve the following system of equations by matrix method :
2x + 3y + 3z = 5
x - 2y + z = -4
3x - y - 2z = 3

Without cover a box is formed by 6 m x 16 m rectangular steel sheet on cutting the squares of length x m from its each corner. Then find the maximum volume of the box.

Let A and B are independent events and P(A)=0.3 and P(B)=0.4 ; then find P(B/A).

Find the area of a triangle whose vertices are A(2, 2, 2), B(2, 1, 3) and C(3, 2, 1).

Let function \( f:N \to Y \) is defined as \( f(x)=4x+3 \) where \( Y=\{y \in N : y=4x+3 \text{ for } x \in N\} \). Prove that f is invertible, also find the inverse of the function f.

Find the minimum value of the objective function \( Z = -50x + 20y \) by graphical method under the following constraints :
\( 2x - y \geq -5 \)
\( 3x + y \geq 3 \)
\( 2x - 3y \leq 12 \)
\( x \geq 0, y \geq 0 \)

Prove that \( \sin^{-1} x = \tan^{-1} [x / \sqrt{1-x^2}] \).

Find the direction-cosines of the sum of the vectors \( \vec{a} = 3\hat{i} + 4\hat{j} - 3\hat{k} \) and \( \vec{b} = -2\hat{i} - 3\hat{j} + \hat{k} \).

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