List of top Questions asked in UP Board XII

Show that \( f(x) = |x| \, \textbf{is continuous at} \, x = 0.\)

If \[ P(A) = \frac{3}{13}, P(B) = \frac{5}{13}, \text{and} P(A \cap B) = \frac{2}{13}, \] \(\text{then find the value of }\) \( P(B|A) \).

If \[ P(A) = 0.4 \, \text{and} \, P(B) = 0.5, \, \text{also, A and B are independent events, then find} \] (i) \( P(A \cup B) \) and (ii) \( P(A \cap B) \).

Find the minimum value of \[ Z = 50x + 70y \] \(\text{under the following constraints by graphical method:}\) \[ 2x + y \geq 8, \] \[ x + 2y \geq 10, x \geq 0, y \geq 0. \]

If \[ A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] \(\text{then prove that}\) \[ A \cdot \text{adj}(A) = |A| \cdot I. \text{ Also, find } A^{-1}. \]

Solve the following system of equations by matrix method: \[ 2x + y - z = 1, \] \[ 3x - 2y + 3z = 8, \] \[ 4x - 3y + 2z = 4. \]
Evaluate: \[ \int_{-1}^{3/2} |x \sin (\pi x)| \, dx. \]
Find the differential coefficient of \[ y = x^x + (\cos x)^{\tan x}. \]
Prove that every differentiable function is continuous. Examine continuity and differentiability of the function \[ f(x) = |x + 2| \text{at} x = -2. \]
There are three children in a family. If it is known that at least one child is a girl among them, find the probability that all three children are girls.
Find the area of the region bounded by the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]
Find the particular solution of the differential equation \[ \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, (x \neq 0) \] given that \( y = 0 \) if \( x = \frac{\pi}{2} \).
Find the shortest distance between two lines: \[ \mathbf{r_1} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda (2\hat{i} + 3\hat{j} + 6\hat{k}) \] and \[ \mathbf{r_2} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu (2\hat{i} + 3\hat{j} + 6\hat{k}) \]
Prove that the function \[ f(x) = \tan^{-1} (\sin x + \cos x), x > 0 \text{ is always increasing function on } (0, \frac{\pi}{4}). \]
Evaluate: \[ \int \sqrt{x^2 - a^2} \, dx. \]
Prove that: \[ \left| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} \right| = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \]
If \( y = e^{a \cos^{-1} x}, -1 \leq x \leq 1 \), then prove that \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - a^2 y = 0 \).
Show that the points \( A(2, 3, 4) \), \( B(-1, -2, 1) \) and \( C(5, 8, 7) \) are collinear.
Evaluate: \[ \int \sqrt{x^2 + 2x + 5} \, dx \]
Let a relation \( R = \{(a, b) : (a - b) \text{ is a multiple of 5} \} \) be defined on the set \( \mathbb{Z} \) (set of integers). Prove that \( R \) is an equivalence relation.
If \[ A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}, B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}, \] and \[ 2A + 3X = 5B, \text{then find the matrix} \, X. \]
If \( x = a \cos^2 t, y = b \sin^2 t \), then find \( \frac{dy}{dx} \).
Find the equation of tangent at the point \( (am^2, am^3) \) on the curve \[ ay^2 = x^3. \]
If \[ \begin{bmatrix} x + z \\ y + z \\ x + y + z \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \\ 9 \end{bmatrix}, \] then find the value of \( x \), \( y \), and \( z \).
Find the general solution of \[ \frac{dy}{dx} = \frac{x - 1}{2 + y}. \]