List of top Statistical Science Questions on Real Analysis

\[ \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \] is an eigenvector of the matrix \[ \begin{pmatrix} 3 & 6 & 7 \\ 3 & 3 & 7 \\ 5 & 6 & 5 \end{pmatrix}. \] With respect to which eigenvalue?

Let \[ A= \begin{pmatrix} 1 & -2 & -9 & 5 \\ 0 & 1 & 2 & -6 \end{pmatrix}. \] Then, the solution set of the system of equations \[ Ax=0 \] is:

Let \[ X= \begin{pmatrix} 1 & a & 3 \\ 1 & 3 & 3 \\ 4 & 2 & 2 \end{pmatrix} \] be the adjoint of a $3 \times 3$ matrix $A$, and let \[ \det(A)=6. \] Then, the value of $a$ is:

The number of distinct real roots of \[ \begin{vmatrix} \sin x & \cos x \\ \cos x & \sin x \end{vmatrix} = 0 \] in the interval \[ -\frac{\pi}{4} \leq x \leq \frac{\pi}{4} \] is:

Rank of the matrix $A = \begin{pmatrix} 2 & 3 & -1 \\ -8 & -7 & 6 \\ 6 & -1 & -7 \end{pmatrix}$ is

Let \[ v_1 = \begin{pmatrix} 1 \\ -2 \\ 7 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix} \] be two vectors in $\mathbb{R}^3$. Then which of the following is true?

A first order differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$, where $n$ is a real number is called}

The value of $\lim_{x \to -1} \frac{x^{10} - 1}{x + 1}$ is given by}

If $\beta(m, n)$, $\Gamma(m)$ and $\Gamma(n)$ for $m > 0, n > 0$ represents beta and gamma functions respectively, then which one of the following is correct?

The value of $\int (x^2 + 1)^{50} \cdot 2x \, dx$ is given by}

Equation of a straight line which passes through the point P(-1, 2) such that the part of the line between the 2 axes is divided internally by the point P in the ratio 3 : 4 is :

Out of 60 children in a colony, it was found that 25 like cricket, 26 like football and 26 like badminton. 9 like both cricket and badminton, 11 like both cricket and football and 8 like both football and badminton. 8 children don't like any of the 3 sports. Then the number of children who like all the 3 sports is:

Which of the following statements is false?

For which values of $x$, the complex numbers $\cos x + i \sin 2x$ and $\sin x - i \cos 2x$ are conjugates of each other?

The cubic equation $4x^3 + 3x^2 + 2x + 1 = 0$ has:

The modulus and argument of complex number $z = \frac{1}{(2\sqrt{3} + 2i)^4}$ is}

The Area of triangle whose vertices are iota, $w$ and $w^2$, where $w$ and $w^2$ are non real cube roots of unity, is :

The least upper bound (lub) and greatest lower bound (glb) of the set $\left\{1 + \frac{(-1)^n}{n} \; ; \; n \in \mathbb{N}\right\}$ are:

For $f(x) = [x^2 + 1]^{[x + 1]}$, where $[x]$ denotes the greatest integer less than or equal to $x$, which of the following is not true:

For the series with $n^{th}$ term $x_n = \frac{n}{(n+1)(n+2)}$, the following is true:

For the sequence $(x_n)$, if $x_1 > 1$ and $x_{n+1} = 2 - \frac{1}{x_n}$, $n \geq 1$, which of the following is true: