List of top Mathematics Questions

If \(A\) is a square matrix such that \(A^2 = A\), then \((I - A)^3\) is:
In a rough sketch, mark the region bounded by \( y = 1 + |x + 1| \), \( x = -2 \), \( x = 2 \), and \( y = 0 \). Using integration, find the area of the marked region.
If \( A = \begin{bmatrix} 2 & 1
3 & 4 \end{bmatrix} \), then the determinant of matrix \( A \) is:
If \( 5 \sin^{-1} \alpha + 3 \cos^{-1} \alpha = \pi \), then \( \alpha = ? \)
If \( |\vec{a}| = 10, |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 12 \), then the value of \( |\vec{a} \times \vec{b}| \) is:
\[ \int \frac{\tan^2 \sqrt{x}}{\sqrt{x}} \, dx \text{ is equal to:} \]
Find the maximum slope of the curve \( y = x^3 + 3x^2 + 9x - 30 \).

Let \( \vec{a} \) and \( \vec{b} \) be two co-initial vectors forming adjacent sides of a parallelogram such that:
\[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] Find the area of the parallelogram.

The domain of the function \( f(x) = \cos^{-1}(2x) \) is:
The equation of the circle passing through the point \( (1, 2) \) and touching both axes is:
The integrating factor of the differential equation \( \frac{dy}{dx} + y = \frac{1 + y}{x} \) is:
Find the coordinates of the point which divides the line segment joining A(3, –2) and B(5, 4) in the ratio 2:3.

Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.

An amount of ₹ 10,000 is put into three investments at the rate of 10%, 12% and 15% per annum. The combined annual income of all three investments is ₹ 1,310, however, the combined annual income of the first and second investments is ₹ 190 short of the income from the third. Use matrix method and find the investment amount in each at the beginning of the year.

The probability that a student buys a colouring book is 0.7, and a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find:
(i) The probability that she buys both the colouring book and the box of colours.
(ii) The probability that she buys a box of colours given she buys the colouring book.
A fruit box contains 6 apples and 4 oranges. A person picks out a fruit three times with replacement. Find:
(i) The probability distribution of the number of oranges he draws.
(ii) The expectation of the number of oranges.

Prove that:
\( \tan^{-1}(\sqrt{x}) = \frac{1}{2} \cos^{-1}\left( \frac{1 - x}{1 + x} \right), \quad x \in [0, 1] \)

Find the value of λ, if the points A(−1,−1,2), B(2,8,λ), C(3,11,6) are collinear.

A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum.
On the basis of the above information, answer the following questions :
Find \( \frac{dS}{dx} \).

Find: \[ I = \int (\sqrt{\tan x} + \sqrt{\cot x}) dx. \]
Find the general solution of the differential equation \[ x^2 \frac{dy}{dx} = x^2 + xy + y^2 \] OR
If \( \log_2 3 = p \), express \( \log_8 9 \) in terms of \( p \):
Find the equation of the tangent line to the curve \( y = x^2 - 3x + 2 \) at the point \( (1, 0) \).

Four students of class XII are given a problem to solve independently. Their respective chances of solving the problem are: \[ \frac{1}{2},\quad \frac{1}{3},\quad \frac{2}{3},\quad \frac{1}{5} \] Find the probability that at most one of them will solve the problem.

Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is: