List of top Mathematics Questions

An open hemispherical storage tank has radius $13$ m. Oil flows into the tank such that the depth $h$ of oil in the tank changes at the rate of $3$ m/hr. When $h=1$ m, the rate of change of the area of the top surface of the oil is:

If the equation of the tangent to the parabola $y^2=4ax$ at the point $(at^2,2at)$ is perpendicular to the line $x+y=1$, then the value of $t$ is:

If the mean and variance of the numbers $2, 4, 6, 8, 10$ are $\mu$ and $\sigma^2$ respectively, then the value of $\mu + \sigma^2$ is:

If the vectors \(\vec a=2\hat i+3\hat j-\hat k\) and \(\vec b=\hat i-2\hat j+4\hat k\), then \(\vec a\cdot\vec b\) is:

A company is migrating its database, and two software engineers, Ishaan and Kavya, take turns running a data-sync script that has a constant success rate of \(\dfrac38\) per attempt. If Ishaan initiates the first attempt and they persist until the migration is successful, what is the probability that Kavya is the one who initiates the successful sync?

Let the population of a species of birds surviving at a time \(t\) be governed by the differential equation \[ \frac{dp}{dt}-p=-100 \] If \(p(0)=50\), then \(p(-\ln2)\) is equal to:

\[ \int(\sin^6x+\cos^6x+3\sin^2x\cos^2x)\,dx= \]

If the two ends of the major axis of an ellipse are \((5,0)\) and \((-5,0)\) and one focus lies on the line \(3x-5y-9=0\), then its equation is:

If \[ \lim_{x\to0}\left(\frac{p\sin2x+1-\cos2x}{x+\tan x}\right)=1, \] then the value of $p$ is:

If the determinant of the matrix \[ \begin{vmatrix} 1 & 2 \\ 3 & x \end{vmatrix} \] is equal to $-2$, then the value of $x$ is:

If the matrix \[ M= \begin{bmatrix} x+5 & a & -4 \\ -2 & 0 & b \\ c & 6 & y+1 \end{bmatrix} \] is a skew-symmetric matrix, then the value of \[ ab+c^2-xy \] is:

The domain of the function \[ f(x)=\sin^{-1}(\sqrt{x-1}) \] is:

Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \[ f(x)=x^3-3x+1 \] Then the function \(f\) is:

The function \(f(x)=|x|+|x-1|\) is:

The area enclosed by the curve \(y=-x^2\) and the line \(x+y+2=0\) is:

A student needs to buy notebooks (\(n\)) for a semester. Double the number of notebooks plus \(5\) must strictly exceed \(15\), but the number of notebooks plus \(10\) must be no more than \(22\). What is the range of notebooks they can buy?

Let $A=[a_{ij}]$ be a square matrix of order $3\times3$, where \[ a_{ij}= \begin{cases} i-2j, & i=j \\ 0, & i>j \\ 1, & i<j \end{cases} \] Then the value of $|A^T|$ is:

If \(f(x)=x^3-3x^2+1\), then the interval in which the function is decreasing is:

A coffee roaster has 12 rare coffee beans with intensity scores ranked from 1 (mildest) to 12 (strongest). You choose 7 beans at random and line them up from mildest to strongest: \(C_1<C_2<C_3<C_4<C_5<C_6<C_7\). What is the probability that the third bean \(C_3\) has an intensity score of exactly 4?

The function \[ f(x)=e^{ax}+e^{-ax}, \quad x\in R \] and \(a<0\), is strictly decreasing for all values of \(x\), where:

If \( f(x) = \int_0^x t(\sin x - \sin t)dt \) then :

Let $A(1,0)$, $B(2,-1)$ and $C\left(\dfrac{7}{3},\dfrac{4}{3}\right)$ be three points. If the equation of the bisector of the angle $ABC$ is $\alpha x+\beta y=5$, then the value of $\alpha^2+\beta^2$ is

If $\sin x \cos x = \frac{1}{4}$, then the general solution is:

Consider an A.P. $a_1,a_2,\ldots,a_n$ with $a_1>0$, $a_2-a_1=-\dfrac{3}{4}$ and $a_n=\dfrac{a_1}{4}$. If \[ \sum_{i=1}^{n} a_i=\frac{525}{2}, \] then find $\sum_{i=1}^{17} a_i$.

What is the nature of the roots of the quadratic equation $x^2 - 7x + 10 = 0$?