List of top Mathematics Questions asked in UP Board XII

Find the angle between the pair of lines \( \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \).
Show that the given function \( f(x) = \cos x \) is increasing in \( (\pi, 2\pi) \).
The volume of a cube is increasing at the rate of 9 cm\(^3\)/s. If the length of its edge is 10 cm, then its surface area is increasing with which rate?
If \( [a_{ij}] = 2i - j \), then determine a matrix A of order \( 2 \times 3 \).
If \( y = x^{x^{x^{.... \text{ad inf}}}} \), then prove that \( x\frac{dy}{dx} = \frac{y^2}{1 - y \log x} \).
Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).
The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is :
Find the principal value of \( \sec^{-1}(-\sqrt{2}) \).
Find the direction cosines of a line joining two points \( (-2, 4, -5) \) and \( (1, 2, 3) \).
If P(A) = 0.3, P(B) = 0.4, then find P(A/B) if A and B are independent events.
The order of the differential equation \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y \cos x = 0 \) is :
If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then find the value of \( \alpha \).
Solve the system of equations by matrix method:
\( 3x - 2y + 3z = 8 \)
\( 2x + y - z = 1 \)
\( 4x - 3y + 2z = 4 \)
Prove that the semi-vertical angle of a cone with given slant height and maximum volume is \( \tan^{-1}(\sqrt{2}) \).
Find a particular solution of the differential equation \( (x - y) (dx + dy) = dx - dy \) when \( y = -1 \) if \( x = 0 \).
Integrate: \( \int \left(\frac{2 + \sin 2x}{1 + \cos 2x}\right) e^x dx \)
The value of \( \int_1^{\sqrt{3}} \frac{dx}{1 + x^2} \) will be :
Find the shortest distance between the lines \( \vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = (2\hat{i} + \hat{j} - \hat{k}) + \mu(3\hat{i} + \hat{j} + 2\hat{k}) \).
The radius of an air bubble is increasing at the rate of \( \frac{1}{2} \) cm/s. At what rate is the volume of the bubble increasing while the radius is 1 cm?
Minimize Z = 3x + 2y by graphical method under the following constraints:
x + y \( \ge \) 8,
3x + 5y \( \le \) 15,
x \( \ge \) 0, y \( \ge \) 0
The probability of solving a question by the three students A, B, C are respectively \( \frac{3}{10}, \frac{1}{5} \) and \( \frac{1}{10} \). Find the probability of solving the question.
If three vectors \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) satisfying the condition \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \). If \( |\vec{a}| = 3 \), \( |\vec{b}| = 4 \) and \( |\vec{c}| = 2 \), then find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).
If the Cartesian equation of a line is \( \frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2} \), then find its equation in vector form.
Differentiate \( \tan^{-1}\left(\frac{\sin x}{1 + \cos x}\right) \) with respect to x.
If \( R_1 \) and \( R_2 \) be two equivalence relations on a set A, then prove that \( R_1 \cap R_2 \) be also an equivalence relation.