List of top Mathematics Questions

The critical points of the function \( f(x) = (x-3)^3(x+2)^2 \) are:

The second term of a G.P. is \( \frac{1}{2} \). If the product of first five terms is 32, then the common ratio of the G.P. is:

The value of \[ \left(\frac{10i}{(2-i)(3-i)}\right)^{2024} \] is equal to:

The axis of a parabola is parallel to the y-axis and its vertex is at \((5, 0)\). If it passes through the point \((2, 3)\), then its equation is:

The sum of the series \( \frac{1}{2^{10}} + \frac{1}{2^{11}} + \cdots + \frac{1}{2^{19}} \) is equal to:
The modulus of the complex number \[ \frac{(1 + i)^{10} (2 - i)^6}{(2i - 4)^4} \] is equal to:
Three coins are tossed simultaneously. Then the probability that exactly two tails appear is:
Let \( A \) and \( B \) be two sets each containing more than one element. If \( n(A \times B) = 155 \), then \( n(A) \) is equal to:
Let

\[ f(x) = \begin{cases} x\left( \frac{\pi}{2} + x \right), & \text{if } x \geq 0 \\ x\left( \frac{\pi}{2} - x \right), & \text{if } x < 0 \end{cases} \]

Then \( f'(-4) \) is equal to:
There are 3 different mathematics books and 4 different physics books in a shelf. Then the number of ways these books can be arranged so that the mathematics books are together is:

For a hyperbola, the vertices are at \( (6, 0) \) and \( (-6, 0) \). If the foci are at \( (2\sqrt{10}, 0) \) and \( -2\sqrt{10}, 0) \), then the equation of the hyperbola is:

If \(\sec \theta + \tan \theta = 2 + \sqrt{3}\), then \(\sec \theta - \tan \theta\) is:

The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:

If a line makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the positive directions of the x, y, and z-axis respectively, then \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) equals:

If \( a = \tan^{-1}\left(\frac{4}{3}\right) \) and \( b = \tan^{-1}\left(\frac{1}{3}\right) \), where \( 0<a, b<\frac{\pi}{2} \), then \( a - b \) is:

If \( 0 \leq x \leq 5 \), then the greatest value of \( \alpha \) and the least value of \( \beta \) satisfying the inequalities \( \alpha \leq 3x + 5 \leq \beta \) are, respectively,

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors. The angle between \( \vec{a} \) and \( \vec{b} \) is \( 30^\circ \), the angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \) is \( 45^\circ \). If \( |\vec{b}| = \sqrt{6} \) and \( |\vec{c}| = 2\sqrt{2} \), then \( |\vec{b} + \vec{c}| \) is:

The integral \(\int e^x \sqrt{e^x} \, dx\) equals:

Let \( f(x) = \begin{cases} x^2 - \alpha, & \text{if } x < 1 \\ \beta x - 3, & \text{if } x \geq 1 \end{cases} \). If \( f \) is continuous at \( x = 1 \), then the value of \( \alpha + \beta \) is:

The centre of the hyperbola \(16x^2 - 4y^2 + 64x - 24y - 36 = 0\) is at the point:

The vectors \(\vec{a} = 4\mathbf{i} - 3\mathbf{j} - \mathbf{k}\) and \(\vec{b} = 3\mathbf{i} + 2\mathbf{j} + \lambda\mathbf{k}\) are perpendicular to each other. Then the value of \(\lambda\) is equal to:

The period of the function \( f(x) = \sin\left( \frac{3x}{2} \right) \) is equal to:

The value of \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos^2 x}{1 + 2^{-x}} \, dx \] is equal to:

The coefficient of \( x^{17} \) in

\[ (1 - x)^{13} (1 + x + x^2)^{12} \] is:
Real part of \[ \left( \frac{1+i}{1-i} \right) \left( \frac{2+i}{2-i} \right) \] is: