List of top Questions asked in UP Board XII

Find the unit vector perpendicular to each of the vectors (\( \vec{a} + \vec{b} \)) and (\( \vec{a} - \vec{b} \)) where \[\vec{a} = \hat{i} + \hat{j} + \hat{k}, \, \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}.\]

Show that the function \( f(x) = 7x^2 - 3 \) is an increasing function when \( x>0 \).

The radius of an air bubble is increasing at the rate of \(\frac{1}{2} \, \text{cm/s}\). At what rate is the volume of the bubble increasing while the radius is 1 cm?

If three vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) satisfying the condition \(\vec{a} + \vec{b} + \vec{c} = 0\). If \(|\vec{a}| = 3\), \[|\vec{b}| = 4 \text{ and } |\vec{c}| = 2, \text{ then find the value of } \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}.\] 5

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

If \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) be functions defined by \( f(x) = \cos x \) and \( g(x) = 3x^2 \) respectively, then prove that \( g \circ f \neq f \circ g \).

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

If \( P(A) = 0.12, P(B) = 0.15 \) and \( P(B/A) = 0.18 \), then find the value of \( P(A \cap B) \).

Find the degree of the differential equation \[xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \left( \frac{dy}{dx} \right) = 2\]

Prove that the function \( f(x) = |x| \), is continuous at \( x = 0 \).

A relation \( R = \{(a, b) : a = b - 1, b \geq 3\ \) is defined on set \( N \), then}

The modulus function \( f : \mathbb{R} \to \mathbb{R}^+ \) given by \( f(x) = |x| \) is

The value of expression \(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\) is

The degree of differential equation \[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{\frac{1}{3}} \text{ is} \]

The value of \( \displaystyle \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

Differentiate between a strike and a lockout.

What is alienation?

What is rural society?

Identify two negative social effects of Green Revolution.

How did planters live?

Define Nation.

What is class system?

State the main features of Panchayati Raj System.

Give two characteristics of modernization.