List of top Mathematics Questions

Simplify the expression: \[ \sec^2 x + 5 \tan x + 5 \]

Let \( \alpha, \beta \) be real numbers such that \( \pi<(\alpha - \beta)<3\pi \). If: \[ \sin \alpha + \sin \beta = \frac{-21}{65}, \quad \cos \alpha + \cos \beta = \frac{-2}{65} \] then the value of \( \cos\left( \frac{\beta - \alpha}{2} \right) \) is:

All pairs \( (x, y) \) that satisfy the inequality \[ 2\sqrt{\sin^2x - 2\sin x + 5} \cdot \frac{1}{4^{\sin^2 y}} \leq 1 \] also satisfy the equation:

If \( ^9C_3 + ^9C_5 = ^{10}C_r \) for some \( r \in \mathbb{N} \), then \( r = \) ?

If \( ^{2n}C_3 : ^nC_3 = 12 : 1 \), then \( n = \) ?

Given 5 different green toys, 4 different blue toys, and 3 different red toys, how many combinations of toys can be chosen taking at least one green and one blue toy?

There are 4 oranges, 5 apples, and 7 mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is:

The number of solutions of the equations \[ x + y + z = 1,\quad x^2 + y^2 + z^2 = 1,\quad x^3 + y^3 + z^3 = 1 \] is:

The number of real values of \( m \) such that the equation \[ x^2 + (2m + 1)x + m = 0 \] has equal roots is:

Evaluate: \[ \sum_{k=1}^{6} \left[ \sin\left( \frac{2\pi k}{7} \right) - i \cos\left( \frac{2\pi k}{7} \right) \right] \]

If \( (x - iy)^{1/3} = a - ib \), then the value of \( \frac{x}{2a} + \frac{y}{2b} \) is:

If \( x = -5 + 2\sqrt{-4} \), then the value of \( x^4 + 9x^3 + 35x^2 - x + 4 \) is:

If \( \alpha, \beta \) are the roots of \( x^2 - 10x - 8 = 0 \), with \( \alpha>\beta \), and \( a_n = \alpha^n - \beta^n \), then the value of: \[ \frac{a_{10} - 8a_8}{5a_9} \] is:

If the rank of the matrix \[ A = \begin{bmatrix} 1 & 2 & 1 & -1 \\ -1 & 2 & 3 & 5 \\ 0 & 1 & k & k \end{bmatrix} \] is 2 and \( k \) is a real number, then \( k \) is a root of which of the following quadratic equations?

If the solution of the system of simultaneous equations \[ \frac{1}{x} + \frac{2}{y} - \frac{3}{z} = 1, \quad \frac{2}{x} - \frac{4}{y} + \frac{3}{z} = 1, \quad \frac{3}{x} + \frac{6}{y} - \frac{6}{z} = 4 \] is \( x = \alpha, y = \beta, z = \gamma \), then the value of \( \alpha^2 + \gamma^2 \) is:

A true statement among the following identities is:

The range of the real valued function \( f(x) = \frac{x^2 + x + 1}{x} \) is

If \( A = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 8 & 0 \\7 & 1 \end{bmatrix} \), and \( A^3 = B \), then \( x = \) ?

If \( b \) and \( c \) are non-zero real numbers, and \[ A = \begin{bmatrix} 1 & b & c \\ b & 2 & 3 \\ c & 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & b & c \\ -b & 0 & 2 \\ -c & -2 & 0 \end{bmatrix}, \] then \( \det(A + B) = \) ?

If a function \( f: \mathbb{R} - \{l\} \to \mathbb{R} - \{m\} \) defined by \( f(x) = \frac{x+3}{x-2} \) is a bijection, then \( 3l + 2m = \) ?

In a game. a child will win Rs 5 if he gets all heads or all tails when three coins are tossed simultaneously and he will lose Rs 3 for all other cases. The expected amount to lose in the game is

If $f(x)=\begin{cases} x+a, & x \leq 0 \\ |x-4|, & x\gt0\end{cases}$ and 
$g(x)= \begin{cases}x+1 & , x\lt0 \\ (x-4)^2+b, & x \geq 0\end{cases}$ 
are continuous on $R$, then $(gof) (2)+(fog)(-2)$ is equal to:

Let \(\alpha, \beta\) and \(\gamma\) be three positive real numbers Let \(f ( x )=\alpha x ^5+\beta x ^3+\gamma x , x \in R\) and \(g: R \rightarrow R\) be such that \(g(f(x))=x\) for all \(x \in R\) If \(a_1, a_2, a_3, \ldots, a_n\) be in arithmetic progression with mean zero, then the value of \(f\left(g\left(\frac{1}{n} \displaystyle\sum_{i=1}^n f\left(a_i\right)\right)\right)\)is equal to :

Consider the equation $\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty) $ Which of the following statements is/are TRUE ?