List of top Mathematics Questions

The area of the region bounded by the curves $y_1(x) = x^4 - 2x^2$ and $y_2(x) = 2x^2$, $x \in \mathbb{R}$, is
 

Let \( a = \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \cdots + \frac{n-1}{n^2} \right) \) and \( b = \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n} \right). \) Which of the following is/are true?

Consider the following system of linear equations: 
\[ \begin{cases} x + y + 5z = 3, \\ x + 2y + mz = 5, \\ x + 2y + 4z = k. \end{cases} \] 

The system is consistent if 
 

The function \( x^4 e^{-2x^2/3} \) (for \( x > 0 \)) has a maximum at a value of \( x \) equal to ...........
(Round off to two decimal places)

The correct statement regarding the determinants (Det) of matrices R, S and T is 


 

A bouncing ball is dropped from an initial height of \( h \) meters above a flat surface. Each time the ball hits the surface, it rebounds a distance \( r \times h \) meters and it bounces indefinitely. Consider the value of \( h = 5 \) meters and \( r = \frac{1}{3} \). The total vertical distance (up and down) travelled (in meters) by the ball is ...........

The average value of function \( f(x) = \sqrt{9 - x^2} \) on \([-3, 3]\), rounded off to TWO decimal places, is ............ 
 

The length of transverse and conjugate axes in a hyperbola are 6 and 8, respectively. The eccentricity of the hyperbola, rounded off to TWO decimal places, is .......

The median of Y in the following data is ............. 
\[ \begin{array}{|c|c|c|c|c|c|c|} \hline \text{Serial number} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline Y & 22 & 12 & 10 & 14 & 16 & 20 \\ \hline \end{array} \]

The solution to the integral \( \int_0^1 2y\sqrt{1 + y^2} \, dy \), rounded off to TWO decimal places, is ............ 
 

Which of the following curve/straight line equations will pass through the origin when plotted on a graph?
If a coin is tossed three times, what is the probability that NO two successive tosses show the same face?
The curve \( y = x^4 - 4x^3 + 4x^2 - 4 \) has tangents parallel to the x-axis at the following points \( (x, y) \)
Which one of the following functions has a discontinuity in the second derivative at \(x = 0\), where \(x\) is a real variable?
If \(P\) and \(Q\) are Hermitian matrices, which of the following is/are true?
(A matrix \(P\) is Hermitian if \(P = P^{\dagger}\), where the elements \(p_{ij}^{\dagger} = p_{ji}^{*}\))
Evaluate: \(\displaystyle \lim_{x \to 0^+} x^x\)

Let $f : [-1,3] \to \mathbb{R}$ be a continuous function such that $f$ is differentiable on $(-1,3)$, $|f'(x)| \le \dfrac{3}{2}$ for all $x \in (-1,3)$, $f(-1) = 1$ and $f(3) = 7$. Then $f(1)$ equals .................
 

Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?

For real constants $a$ and $b$, let \[ M = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ a & b \end{bmatrix} \] be an orthogonal matrix. Then which of the following statements is/are always TRUE?

Let the sequence $\{x_n\}_{n \ge 1}$ be given by $x_n = \sin \dfrac{n\pi}{6}$, $n = 1, 2, \ldots$. Then which of the following statements is/are TRUE?
 

Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always
 

Let $T: \mathbb{R}^3 \to \mathbb{R}^4$ be a linear transformation. If $T(1,1,0) = (2,0,0,0)$, $T(1,0,1) = (2,4,0,0)$, and $T(0,1,1) = (0,0,2,0)$, then $T(1,1,1)$ equals
 

The value of the integral \[ \int_{0}^{2} \int_{0}^{\sqrt{2x - x^2}} \sqrt{x^2 + y^2} \, dy \, dx \] is

Let $\{a_n\}_{n \ge 1}$ be a sequence of real numbers such that $a_1 = 1, a_2 = 7$, and $a_{n+1} = \dfrac{a_n + a_{n-1}}{2}$, $n \ge 2$. Assuming that $\lim_{n \to \infty} a_n$ exists, the value of $\lim_{n \to \infty} a_n$ is
 

Consider the following system of linear equations: 
\[ \begin{cases} ax + 2y + z = 0 \\ y + 5z = 1  \\ by - 5z = -1 \end{cases} \] 

Which one of the following statements is TRUE?