List of top Questions asked in KEAM

The shortest distance between the point \( (2,3,4) \) and the line \( \dfrac{x-4}{-2}=\dfrac{y-4}{2}=\dfrac{z-6}{1} \) is

The projections of a line segment on the coordinate axes are \(5,6,8\). Then the length of the line segment is

The equation of the straight line joining the points \((1,2,3)\) and \((3,4,k)\) is \(\frac{x-3}{1}=\frac{y-4}{1}=\frac{z-k}{5}\). Then the value of \(k\) is

Let \( \vec{a}, \vec{b} \) and \( \vec{c} \) be the sides of a triangle \( ABC \) such that \( \overrightarrow{BC}=\vec{a}, \overrightarrow{CA}=\vec{b} \) and \( \overrightarrow{AB}=\vec{c} \). If \( BC=AC=3 \) and \( \vec{b}\cdot\vec{c}=-9 \), then \( \vec{a}\cdot\vec{b} \) is equal to

If \( \vec{a}=\hat{i}+2\hat{j}-2\hat{k} \) and \( \vec{b}=2\hat{i}+\hat{j}+2\hat{k} \), then a unit vector perpendicular to \( \vec{a}+\vec{b} \) and \( \vec{a}-\vec{b} \) is

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors, and \(\theta\) be the angle between them. If \(\vec{a}-\vec{b}\) is a unit vector, then \(\theta\) is equal to

The unit vector that bisects the angle between two vectors \( 2\hat{i}+\hat{j}+2\hat{k} \) and \( \hat{i}+2\hat{j}-2\hat{k} \) is

The distance between the foci of the hyperbola \( x^2-4y^2=16 \), is

If the length of the latus rectum of an ellipse is one-fourth of the major axis, then the eccentricity of the ellipse is

If one end of the latus rectum of the parabola \(y^2=16x\) is \((4,8)\), then the coordinates of the other end of the latus rectum are

If the two circles \( (x-2)^2+(y-3)^2=9 \) and \( (x-2)^2+(y+3)^2=a^2 \) intersect in two distinct points, then

If the normal form of the equation of a straight line \( x+\sqrt{3}y=2\sqrt{3} \) is \( x\cos\alpha+y\sin\alpha=p \), then the values of \( \alpha \) and \( p \) are respectively

The point with integral coordinates on the line \(x+y=1\), that lie at a distance \(2\) units from the line \(5x+12y=0\), is

If the line joining of two points \((1,0)\) and \((4,3)\) is rotated about the point \((1,0)\) in counter clockwise direction through an angle \(15^\circ\), then the equation of the line in the new position is

If \( \sin^{-1}x+\sin^{-1}y=\dfrac{2\pi}{3} \), then \( \cos^{-1}x+\cos^{-1}y \) is equal to

If \( \cos^{-1}x+\cos^{-1}y+\cos^{-1}z=3\pi \), then \( (x+y+z) \) is equal to

If \( \theta \in \left[\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right] \), then \( \sin^{-1}(\sin\theta) \) is equal to

\(\frac{1-\cos 2x}{1+\cos 2x}-\sec^2 x=\)

The period of the function \(f(x)=2\sin 4x+3\cos 2x\) is

The value of \( \tan \dfrac{\pi}{12}+\tan \dfrac{\pi}{6}+\left(\tan \dfrac{\pi}{12}\tan \dfrac{\pi}{6}\right) \) is equal to

If $\tan(x-y)=\frac{4}{5}$, $\tan(x+y)=\frac{6}{5}$ and $0<x,y<\frac{\pi}{4}$, then $\tan 2x$ is:

The solution set of inequality \( |x+2|<3 \) is

Let \( A = \{x : x \text{ is a positive multiple of } 2 \text{ less than } 36\},\)
\(B = \{x : x \text{ is a positive multiple of } 3 \text{ greater than } 16\}, \text{ and }\)
\(C = \{x : x \text{ is a positive multiple of } 4 \text{ less than } 42\}. \text{ Then } (A \cap B) \cap C =\)

If \(\frac{x+1}{x-1}<2\), then \(x\) lies in the interval

If the following system of linear equations
\[ x-2y+z=5 \] \[ 2x-y+2z=7 \] \[ x+2y+\lambda z=5 \] has a unique solution, then \(\lambda \ne\)