List of top Mathematics Questions asked in KEAM

If $y = f(x)$ is continuous on $[0, 6]$, differentiable on $(0, 6)$, $f(0) = -2$ and $f(6) = 16$, then at some point between $x = 0$ and $x = 6$, $f'(x)$ must be equal to:

Two sides of a triangle are $8$ m and $5$ m in length. The angle between them is increasing at the rate $0.08$ rad/sec. When the angle between the sides is $\frac{\pi}{3}$, the rate at which the area of the triangle is increasing is:

The equation of the tangent to the curve $y = x^3 - 6x + 5$ at $(2, 1)$ is:

Let $f(x) = 2x^3 - 5x^2 - 4x + 3,\ \frac{1}{2} \leq x \leq 3$. The point at which the tangent to the curve is parallel to the x-axis is:

If $y = 8x^3 - 60x^2 + 144x + 27$ is a strictly decreasing function in the interval:

If $f'(4) = 5$, $g'(4) = 12$, $f(4)g(4) = 2$ and $g(4) = 6$, then $\left( \frac{f}{g} \right)'(4) =$

If the derivative of $(ax - 5)e^{3x}$ at $x = 0$ is $-13$, then the value of $a$ is equal to:

Let $y = \tan^{-1}(\sec x + \tan x)$. Then $\frac{dy}{dx} =$

The minimum value of $2x^3 - 9x^2 + 12x + 4$ is:

The slope of the curve $y = e^x \cos x$, $x \in (-\pi, \pi)$ is maximum at:

If $s = \sec^{-1} \left( \frac{1}{2x^2 - 1} \right)$ and $t = \sqrt{1 - x^2}$, then $\frac{ds}{dt}$ at $x = \frac{1}{2}$ is:

If $f(x) = 3x + 5$ and $g(x) = x^2 - 1$, then $(f \circ g)(x^2 - 1)$ is equal to:

The period of the function $f(x) = \tan(4x - 1)$ is:

If $2^x + 2^y = 2^{x+y}$, then the value of $\frac{dy}{dx}$ at $(1, 1)$ is equal to:

If $f(x) = \frac{\sin^{-1} x}{\sqrt{1 - x^2}}$, then the value of $(1 - x^2) f'(x) - x f(x)$ is:

If $f(x) = \left( \frac{x}{2} \right)^{10}$, then $f(1) + \frac{f'(1)}{1!} + \frac{f''(1)}{2!} + \frac{f'''(1)}{3!} + \dots + \frac{f^{(10)}(1)}{10!}$ is equal to:

Let $f(x) = \begin{cases} \cos x & \text{if } x \geq 0 \\ -\cos x & \text{if } x<0 \end{cases}$. Which one of the following statements is not true?

The value of $\lim_{x \to 0} \frac{\cot 4x}{\csc 3x}$ is equal to:

The value of $\lim_{n \to \infty} \frac{{}^nC_3 - {}^nP_3}{n^3}$ is equal to:

In a class, in an examination in Mathematics, 10 students scored 100 marks each, 2 students scored zero and the average of the remaining students is 72 marks. If the class average is 76, then the number of students in the class is:

The straight line $\vec{r} = (\hat{i} + \hat{j} + 2\hat{k}) + t(2\hat{i} + 5\hat{j} + 3\hat{k})$ is parallel to the plane $\vec{r} \cdot (2\hat{i} + \hat{j} - 3\hat{k}) = 5$. Then the distance between the straight line and the plane is:

If the mean of the numbers $a, b, 8, 5, 10$ is $6$ and their variance is $6.8$, then $ab$ is equal to:

The point of intersection of the straight lines $\vec{r} = (3\hat{i} - 4\hat{j} + 5\hat{k}) + \lambda(-\hat{i} - 2\hat{j} + 2\hat{k})$ and $\frac{3-x}{-1} = \frac{y+4}{2} = \frac{z-5}{7}$ is:

The vector equation of the straight line $\frac{x-2}{1} = \frac{y}{-3} = \frac{1-z}{2}$ is:

If the two lines $\frac{x-1}{2} = \frac{1-y}{-a} = \frac{z}{4}$ and $\frac{x-3}{1} = \frac{2y-3}{4} = \frac{z-2}{2}$ are perpendicular, then the value of $a$ is equal to: