List of top Mathematics Questions

If \[ A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] and \(\theta = \frac{2\pi}{7}\), then \(A^{100} = A \times A \times \ldots\) (100 times) is equal to:
If the slope of the tangent drawn at any point \((x, y)\) on the curve \(y = f(x)\) is \(y = 6x^2 + 10x - 9\) and \(f(2) = 0\), then \(f(-2) = \)?
If the period of the function \[ f(x) = 2\cos(3x+4) - 3\tan(2x-3) + 5\sin(5x) - 7 \] is \( k \), then \( f(x) \) is periodic with fundamental period \( k \). Find \( k \).

If \( \sqrt{5} - i\sqrt{15} = r(\cos\theta + i\sin\theta), -\pi < \theta < \pi, \) then

\[ r^2(\sec\theta + 3\csc^2\theta) = \]

A unit vector in XY-plane making an angle \(45^\circ\) with \(\hat{i} + \hat{j}\) and an angle \(60^\circ\) with \(3\hat{i} - 4\hat{j}\) is:
Let \( z \) be a non-zero complex number such that \[ z = \frac{16}{\bar{z}}. \] Then the locus of \( z \) is:
(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:
If \[ f(x) = 3x^{15} - 5x^{10} + 7x^5 + 50\cos(x - 1), \] then \[ \lim_{h \to 0} \frac{f(1 - h) - f(1)}{h^2 + 3h} \] is:
Let \( a, b, c \) be positive numbers. If \( a + b + c \geq K \left[ (a + b)(b + c)(c + a) \right]^{1/3} \), then the maximum value of \( K \) is:
Let \( n(A) = m \) and \( n(B) = n \), if the number of subsets of \( A \) is 56 more than that of subsets of \( B \), then \( m + n \) is equal to:
Let \( f(x) \) be a polynomial function satisfying \[ f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right). \] If \( f(4) = 65 \) and \( I_1, I_2, I_3 \) are in GP, then \( f'(I_1), f'(I_2), f'(I_3) \) are in:
Let \( f(x) \) be defined as: \[f(x) = \begin{cases} 3 - x, & x<-3 \\ 6, & -3 \leq x \leq 3 \\ 3 + x, & x>3 \end{cases}\]
Let \( \alpha \) be the number of points of discontinuity of \( f(x) \) and \( \beta \) be the number of points where \( f(x) \) is not differentiable. Then, \( \alpha + \beta \) is:

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{dx}{\sec^2 x + (\tan^{2022} x - 1)(\sec^2 x - 1)} \]

Let \( z = x + iy \), where \( y>0 \). If \( z + \overline{z} = 6 \) and \( |z| + | \overline{z} | = 10 \), then \( z = \)

Let the foci of a hyperbola $ H $ coincide with the foci of the ellipse $ E : \frac{(x - 1)^2}{100} + \frac{(y - 1)^2}{75} = 1 $ and the eccentricity of the hyperbola $ H $ be the reciprocal of the eccentricity of the ellipse $ E $. If the length of the transverse axis of $ H $ is $ \alpha $ and the length of its conjugate axis is $ \beta $, then $ 3\alpha^2 + 2\beta^2 $ is equal to:

The vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a $\triangle ABC$. The length of the median through $A$ is:

When \( |x| < 2 \), the coefficient of \( x^2 \) in the power series expansion of

\[ \frac{x}{(x-2)(x-3)} \]

is:

If \( z = 1 + i \), then the maximum value of \( |z + 12 + 9i| \) is
By shifting the origin to the point \( (h,5) \) by the translation of coordinate axes, if the equation \[ y = x^2 - 9x^2 + cx - d \] transforms to \( Y = X^2 \), then \( \left( \frac{d - c}{h} \right) \) is: \
The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad \text{where } p > 5, \] is \( \frac{\sqrt{3}}{2} \). Then the value of \( p \) is
Evaluate $$ \cot \left( \sum_{n=1}^{50} \tan^{-1} \left( \frac{1}{1 + n + n^2} \right) \right).= $$
Evaluate the integral: $\int_{-\pi}^{\pi} x^2 \sin(x) \, dx$
If the mirror image of the point $P(3, 4, 9)$ in the line $\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14(\alpha + \beta + \gamma)$ is:
If $y = \tan^{-1} \frac{x}{1+2x^2} + \tan^{-1} \frac{x}{1+6x^2} + \tan^{-1} \frac{x}{1+12x^2}$, then $\left(\frac{dy}{dx}\right)_{x=\frac{1}{2}} =$
If lines $\frac{x - 1}{-3} = \frac{y - 2}{2k} = \frac{z - 3}{2}$ and $\frac{x - 1}{3k} = \frac{y - 5}{1} = \frac{z - 6}{-5}$ are mutually perpendicular, then $k$ is equal to: