List of top Questions asked in KEAM

The value of \(\lim_{x \to 0} \frac{\sqrt{1 - \cos 2x}}{|x|}\) is equal to

The value of \(\lim_{x \to 5} \left( \frac{25 - x^2}{4 - \sqrt{x^2 - 9}} \right)\) is:

If \(P(A)=\frac{1}{4}, P(B)=\frac{1}{5}\) and \(P(A \cap B)=\frac{1}{8}\), then \(P(A' \cup B')\) is:

We have two data sets each of size 5. The variances are 4 and 5 and the corresponding means are 2 and 4 respectively. Then the variance of the combined data set is:

If the variance of $1,2,3,\ldots,n$ is 10, then the value of $n$ is:

If the lines \(\frac{2x-1}{2} = \frac{3-y}{1} = \frac{z-1}{3}\) and \(\frac{x+3}{2} = \frac{y+2}{5} = \frac{z+1}{a}\) are perpendicular to each other, then the value of \(a\) is:

If \(O\) is the origin and \(C\) is the midpoint of \(A(-2,1)\) and \(B(4,-3)\), then \(\vec{OC}\) is:

The equation of straight line passing through $(a,b,c)$ and parallel to x-axis is:

The equation of a line passing through the point $(1,-2,3)$ and equally inclined to the axes are:

If \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\), then the projection of \(\vec{a}\) on \(\vec{b}\) is:

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

Let \(\vec{a}, \vec{b}, \vec{c}\) be such that \(\vec{a} + \vec{b} + \vec{c} = 0\). If \(|\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5\) then \(|\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}|\) is:

The line $x - 1 = 0$ is the directrix of the parabola $y^2 - kx + 8 = 0$. Then, the values of $k$ are:

The length of the latus rectum of $x^2 = -9y$ is equal to:

The centre and radius of the circle $x^2 + y^2 - 2x + 4y = 8$ respectively are:

If \(\theta\) is the angle between two vectors \(\vec{a}\) and \(\vec{b}\) such that \(|\vec{a}| = 7, |\vec{b}| = 1\) and \(|\vec{a}\times\vec{b}|^2 = k^2 - (\vec{a}\cdot\vec{b})^2\), then the value(s) of \(k\) is/are:

If the length of the major axis of an ellipse is thrice the length of the minor axis, then its eccentricity is equal to:

The distance of the point $P(1,-3)$ from the line $2y - 3x = 4$ is:

If the points $(3,-2)$, $(a,2)$, $(8,8)$ are collinear, then the value of $a$ is:

If the slope of the line joining the points \((3,4)\) and \((-2,a)\) is equal to \(-\frac{2}{5}\), then the value of \(a\) is:

If \(\alpha\) and \(\beta\) are respectively the minimum and maximum values of \(\frac{\pi^2}{8} + 2\left(\sin^{-1}x - \frac{\pi}{4}\right)^2\), then \(\frac{\beta}{\alpha}\) is:

The value of \(2\tan^{-1}\left(\frac{1}{3}\right) + \cot^{-1}\left(\frac{3}{4}\right)\) is

The value of \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \sin^{-1}\left(\frac{1}{2}\right)\) is equal to

Let $L$ be an arc of a circle which subtends $45^\circ$ at the centre. If the radius of circle is $4$ cm, then the length of $L$ in centimeter is