List of top Mathematics Questions asked in UP Board XII

Solve the system of equations by matrix method:
\( 3x - 2y + 3z = 8 \)
\( 2x + y - z = 1 \)
\( 4x - 3y + 2z = 4 \)
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?
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 \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).
Differentiate \( \tan^{-1}\left(\frac{\sin x}{1 + \cos x}\right) \) with respect to x.
There are 4 white and 2 black balls in a bag and in another bag 3 white and 5 black balls. Find the probability of getting both black balls if a ball is drawn from each bag.
Find the unit vector perpendicular to each of the vectors \( (\vec{a} + \vec{b}) \) and \( (\vec{a} - \vec{b}) \) where \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \).
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.
Find the degree of the differential equation \( xy \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) = 2 \)
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 \( gof \neq fog \).
Prove that the function f(x) = |x| is continuous at x = 0.
Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.
The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be
The value of expression \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k} \) is
(b) At which point is the slope of the line \( y = x + 1 \) equal to the slope of the curve \( y^2 = 4x \)?
(d) If vectors \( \mathbf{5i - \lambda j + 2k} \) and \( \mathbf{2i + 3j + 4k} \) are perpendicular, find \( \lambda \):
Evaluate: \[ I = \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \]

There are 10 black and 5 white balls in a bag. Two balls are taken out, one after another, and the first ball is not placed back before the second is taken out. Assume that the drawing of each ball from the bag is equally likely. What is the probability that both the balls drawn are black? 

Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]
Find the probability of 53 Sundays in a leap year.
If \( y = \frac{x^2 + 3x + 4}{e^x \cos x} \), find \( \frac{dy}{dx} \).

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

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

Solve the differential equation \[ \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \]

Find the value of \[ \int \frac{\sec^2 2x}{(\cot x - \tan x)^2} \, dx. \] 

(c) If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A \mid B) = \frac{2}{5} \), then find \( P(A \cap B) \).