List of top Mathematics Questions on Mathematical Reasoning

The value of \[ \int_{-100}^{100} \frac{x+x^3+x^5}{1+x^2+x^4+x^6}\,dx \] is:

Let \( f:[0,1]\to\mathbb{R} \) and \( g:[0,1]\to\mathbb{R} \) be defined as: \[ f(x)= \begin{cases} 1, & x \text{ rational}\\ 0, & x \text{ irrational} \end{cases} \quad g(x)= \begin{cases} 0, & x \text{ rational}\\ 1, & x \text{ irrational} \end{cases} \] Then:

Let \( f(x) \) be continuous on \( [0,5] \) and differentiable in \( (0,5) \). If \( f(0)=0 \) and \( |f'(x)| \le \frac{1{5} \) for all \( x \in (0,5) \), then \( \forall x \in [0,5] \):}

The value of the integral \[ \int_{3}^{6} \frac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}} \, dx \] is:

A function \( f : \mathbb{R} \to \mathbb{R} \) satisfies \[ f\!\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y) + f(0)}{3} \quad \text{for all } x,y \in \mathbb{R}. \] If \( f'' \) is differentiable at \( x = 0 \), then \( f \) is:

Question:
If \[ f(x) = \begin{cases} x^2 + 3x + a, & x \le 1 \\ bx + 2, & x > 1 \end{cases} \] for \( x \in \mathbb{R} \), and the function is everywhere differentiable, then:

The set of points of discontinuity of the function \[ f(x) = x - [x], \quad x \in \mathbb{R} \] is:

Let \[ A = \begin{bmatrix} 5 & 5\alpha & \alpha \\ 0 & \alpha & 5\alpha \\ 0 & 0 & 5 \end{bmatrix} \] If \[ |A|^2 = 25 \] then \( |\alpha| \) equals:

If \( a, b, c \) are positive real numbers each distinct from unity, then the value of the determinant \[ \begin{vmatrix} 1 & \log_a b & \log_a c \\ \log_b a & 1 & \log_b c \\ \log_c a & \log_c b & 1 \end{vmatrix} \] is:

Let \( f(x)=\max\{x+[x],\, x-[x]\ \), where \( [x] \) is the greatest integer \( \le x \). Then} \[ \int_{-3}^{3} f(x)\,dx \] has the value:

If \[ f(x)=\int_0^{\sin^2 x}\sin^{-1}\!\sqrt{t}\,dt, \quad g(x)=\int_0^{\cos^2 x}\cos^{-1}\!\sqrt{t}\,dt, \] then the value of \( f(x)+g(x) \) is:

If \( f \) is the inverse function of \( g \) and \( g'(x) = \dfrac{1{1 + x^n} \), then the value of \( f'(x) \) is:}

Let \[ f_n(x) = \tan\frac{x}{2}(1+\sec x)(1+\sec 2x)\cdots(1+\sec 2^{n-1}x), \] then:

Evaluate \[ \lim_{x \to 0} \frac{\tan\!\left(\lfloor -\pi^2 \rfloor x^2\right) - x^2 \tan\!\left(\lfloor -\pi^2 \rfloor\right)}{\sin^2 x} \]

If \( x=-1 \) and \( x=2 \) are extreme points of \[ f(x) = \alpha \log|x| + \beta x^2 + x \quad (x \ne 0), \] then:

If for a matrix \( A \), \( |A| = 6 \) and \[ \operatorname{adj}A = \begin{bmatrix} 1 & -2 & 4
4 & 1 & 1
-1 & k & 0 \end{bmatrix}, \] then \( k \) is equal to: