List of top Questions asked in UP Board XII- 2024

Define the 'wavefront' of a wave.

Ionisation energy of Hydrogen atom is 13.6 eV. In a state where \( n = 2 \), what will be ionisation energy of its electron?

Deduce the dimensional equation of self-inductance.

State Ampere's Circuital Law.

Write the equation for the relationship between specific conductivity (\( \sigma \)) and drift velocity (\( v_d \)).

Diffusion current in a p-n junction is greater than the drift current in magnitude:

The equation \( E = pc \), (where \( E \) and \( p \) are energy and momentum respectively) is valid:

An electromagnetic wave propagating through vacuum, described by \( E = E_0 \sin(kx - \omega t) \), \( B = B_0 \sin(kx - \omega t) \), then:

Differentiate \( y = x^x + (\cos x)^{\tan x} \) with respect to \( x \).

The probabilities of solving a question by \( A \) and \( B \) independently are \( \frac{1}{2} \) and \( \frac{1}{3} \) respectively. If both of them try to solve it independently, find the probability that:

Suppose that \( A = \{ 1, 2, 3 \} \), \( B = \{ 4, 5, 6, 7 \} \), and \( f = \{ (1, 4), (2, 5), (3, 6) \} \) be a function from \( A \) to \( B \). Then \( f \) is:

(e) The angle between the vectors \( 3\hat{i} - 2\hat{j} + \hat{k} \) and \( 2\hat{i} + \hat{j} + 3\hat{k} \) will be:

Find the shortest distance between the lines: \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}), \] \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}). \]

If \( A \) and \( B \) are independent events, where \( P(A) = \frac{3}{10} \), \( P(B) = \frac{6}{10} \), then find \( P(A \cup B) \) and \( P(A \cap B) \).

Find the shortest distance between two lines \( l_1 \) and \( l_2 \) whose vector equations are: \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] and \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}). \]

Prove that a relation \( R = \{(a, b) : (a-b) \) is a multiple of \( 5 \} \) is an equivalence relation in the set of integers \( \mathbb{Z} \).

(c) Prove that \[ \begin{vmatrix} 1 \omega & \omega^2 \\ \omega & \omega^2 1 \\ \omega^2 1 \omega \end{vmatrix} = 0 \], where \( \omega \) is a cube root of unity.

Find \( x \) and \( y \) if \[ 2 \left[ \begin{array}{cc} 1 & 3 \\ 0 & x \end{array} \right] + \left[ \begin{array}{cc} y & 0 \\ 1 & 2 \end{array} \right] = \left[ \begin{array}{cc} 5 & 6 \\ 1 & 8 \end{array} \right]. \]

The given events \( A \) and \( B \) are such that \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{2} \), and \( P(A \cap B) = \frac{1}{8} \); then find \( P(A' \cap B') \).

Show that the relation:

\[ R = \{(a, b) : (a - b) \text{ is a multiple of 5} \} \]

on the set \( \mathbb{Z} \) of integers is an equivalence relation.

Given:

\[ x \sqrt{1 + y} + y \sqrt{1 + x} + x = 0 \]

for \( -1 < x < 1 \), prove that:

\[ \frac{dy}{dx} = -\frac{1}{(1+x)^2}. \]

The differential coefficient of the \( \sin(x^2 + 5) \) with respect to \( x \) will be:

If

\[ A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix} \]

Then find \( AB \) and \( BA \).

Solve: