List of top Questions asked in UP Board XII

(a) Prove that the function \( f(x) = |x - 1| \) is continuous at \( x = 1 \):
(c) The value of the integral: \[ \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{dx}{1+x^2} \]
(d) The value of the expression: \[ \hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} \]
Solve by matrix method the system of equations \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \] \
The value of \( \int \sin^2 x \, dx \) is:
Prove that: \[ \int_0^{\pi/2} \sin 2x \tan^{-1} (\sin x) \,dx = \left(\frac{\pi}{2} - 1\right) \]
Evaluate: \[ I = \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \]
Solve the differential equation: \[ \frac{dy}{dx} + \frac{y}{x} = x^2, \quad y = 1 \text{ when } x = 1 \]
Solve the differential equation: \[ (\tan^{-1} y - x) \, dy = (1 + y^2) \, dx \]
Solve the system of equations by matrix method: \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \]

If matrix \[ A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 3 & -3 \\ -2 & -4 & -4 \end{bmatrix}, \] then find \( A^{-1} \).

If \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}, \] where \( n \in \mathbb{N} \).

Find the perpendicular unit vectors on the vectors \[ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \quad \text{and} \quad \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} \] and find the sine of the angle between them.
Find the area of the circle \( x^2 + y^2 = a^2 \).
Find two positive numbers whose sum is 15 and the sum of their squares is minimum.
Find the shortest distance between the lines \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}) \]

Find the minimum value of  ( z = x + 3y ) under the following constraints: 

• x + y ≤ 8
• 3x + 5y ≥ 15
• x ≥ 0, y ≥ 0

Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

If \( R \) is the relation "less than" from \( A = \{1,2,3,4,5\} \) to \( B = \{1,4,5\} \), find the set of ordered pairs corresponding to \( R \). Also, define this relation from \( B \) to \( A \).

Find the value of: \[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) \]
Find the vector equation of the line which passes through the point \( (5,2,-4) \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).
Find the probability of 53 Sundays in a leap year.
Differentiate \( y = (\cos x)^{\sin x \).}
A die is thrown once. The number on the die is a multiple of 3 is denoted by \( E \), and the number on the die is even is denoted by \( F \). Are \( E \) and \( F \) independent events?
Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]