List of top Mathematics Questions

For a complex number $z = x + iy$, where $x, y \in \mathbb{R}$, denote $\hat{z} = y + ix$. The locus of $z$ satisfying $|z + \hat{z}| = |z - \hat{z}|$ in the complex plane is

Amar, Akbar, and Anthony write a test independently. The probability that Amar passes the test and Akbar fails the test is $\frac{3}{20}$. The probability that Akbar passes the test and Anthony fails the test is $\frac{1}{4}$. The probability that Amar and Anthony both pass the test is $\frac{2}{5}$. Then the probability that at least one of Amar, Akbar and Anthony fails the test is

Let $m$ and $p$ be real numbers such that the polynomial $f(x) = x^2 + mx + p$ has two distinct negative rational roots. Then the polynomial $g(x) = x^2 - (m^2 - 2p)x + p^2$ has distinct

Let $S = \{(x, y) \mid x, y \in \mathbb{N}, 1 \le x \le 15, 1 \le y \le 20\}$ be a set. Let $\mathcal{R}$ be the equivalence relation on $S$ defined by $(x, y) \mathcal{R} (x', y')$ if and only if $x + y = x' + y'$. Then the number of equivalence classes of $\mathcal{R}$ on $S$ is

If $x \in (0, 1)$ and $\sin^{-1} x - \sin^{-1} \frac{1}{4} = \frac{\pi}{3}$, then $x$ equals

The parabola $y = -x^2 + 16$ intersects the $x$-axis at points A and B. Further, the parabola intersects the line $y = 7$ at points M and N. Then the area of the quadrilateral with vertices A, B, M and N is

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) = -f(x)$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $h'(x) = f(x)^2 + (f'(x))^2$ for all $x \in \mathbb{R}$. If $h(0) = 1$ and $h\left(\frac{1}{3}\right) = \frac{5}{3}$, then the value of $h(100)$ is

Suppose $\lim_{x \to -2} \frac{bx^2 + 15x + 15 + b}{x^2 + x - 2} = L$, where $b$ and $L$ are real numbers. Then

The domain of the real function $f(x) = \sin^{-1}\left(\log_2\left(\frac{x^2}{2}\right)\right)$ is

A possible solution of the system of equations \[ x^2 - 8xy + 16y^2 = 0 \] \[ (\log_{10} x)^2 + 2(\log_{10} x)(\log_{10} y) + (\log_{10} y)^2 = 4 \] is

A can complete a piece of work in 15 days and B can complete the same work in 10 days. In how many days will they complete the work together?

If the simple interest on a sum of money for 4 years at 8% per annum is Rs 1,920, find the principal.

A shopkeeper marks an article at Rs 2,500 and offers a discount of 20%. If the cost price of the article is Rs 1,700, what is the profit percentage earned?

The ratio of the ages of A and B is 5:7. After 8 years, the ratio becomes 7:9. What is B's present age?

A train travelling at 72 km/h crosses a platform of length 180 m in 24 seconds. What is the length of the train?

The differential equation is \[ \frac{dy}{dx}+y\tan x=\sec x \] and \(y(0)=1\). Then the value of \(y\left(\frac{\pi}{4}\right)\) is

Let \(a_1, a_2, a_3, \dots\) be a G.P. of increasing positive terms such that \(a_2 \cdot a_3 \cdot a_4=64\) and \(a_1 + a_3 + a_5 = \frac{813}{7}\). Then \(a_3 + a_5 + a_7\) is equal to:

The value of \[ \int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x^{11}}{1 - \sin\left(|x| + \frac{\pi}{6}\right)} \right) dx \] is equal to

Let O be the vertex of the parabola \(x^2=4y\) and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2:3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is:

Let \(f: R \to (0, \infty)\) be a twice differentiable function such that \(f(3) = 18\), \(f'(3)=0\) and \(f''(3) = 4\). Then \(\lim_{x \to 1} \log_e \left[ \frac{f(2+x)}{f(3)} \right]^{\frac{18}{(x-1)^2}}\) is equal to:

If \(x^2+x+1=0\), then the value of \(\left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \dots + \left(x^{25} + \frac{1}{x^{25}}\right)^2\) is:

If the coefficient of \( x \) in the expansion of \[ (ax^2 + bx + c)(1 - 2x)^{26} \] is \(-56\) and the coefficients of \( x^2 \) and \( x^3 \) are both zero, then \( a + b + c \) is equal to

For \(n\in\mathbb{N}\), if \[ y=ax^{n+1}+bx^{-n}, \] then \[ x^2\frac{d^2y}{dx^2} \] is equal to:

The eccentricity of the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\) is: