List of top Questions asked in KEAM

If \( y = \sec(\tan^{-1} x) \), then \( \frac{dy}{dx} \) is equal to

The functions \(f, g\) and \(h\) satisfy the relations \( f'(x)=g(x+1) \) and \( g'(x)=h(x-1) \). Then \( f''(2x) \) is equal to

If \( y^2 = 100\tan^{-1}x + 45\sec^{-1}x + 100\cot^{-1}x + 45\cosec^{-1}x \), then \( \frac{dy}{dx} \) is

If the combined mean of two groups is \( \frac{40}{3} \) and if the mean of one group with 10 observations is 15, then the mean of the other group with 8 observations is equal to

\[ \lim_{x \to 0} \left(\frac{10\sin 9x}{9\sin 10x}\right) \left(\frac{8\sin 7x}{7\sin 8x}\right) \left(\frac{6\sin 5x}{5\sin 6x}\right) \left(\frac{4\sin 3x}{3\sin 4x}\right) \left(\frac{\sin x}{\sin 2x}\right) \]

A function \(f\) satisfies the relation \( f(n^2) = f(n) + 6 \) for \(n \ge 2\) and \( f(2) = 8 \). Then the value of \( f(256) \) is

The value of $\displaystyle \lim_{y \to \infty} \left[ y \sin\left(\frac{1}{y}\right) - \frac{1}{y} \right]$ is equal to}

The first term of an A.P. is $148$ and the common difference is $-2$. If the A.M. of first $n$ terms of the A.P. is $125$, then the value of $n$ is

The number of points at which the function \( f(x)=\frac{1{\log_e|x|} \) is discontinuous is}

The mean of five observations is 4 and their variance is $5.2$. If three of these observations are $2, 4$ and $6$, then the other two observations are

The foot of the perpendicular from the point \( (1,6,3) \) to the line \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) is

If a straight line makes the angles \(60^\circ,45^\circ,\alpha\) with x, y and z axes respectively, then \( \sin^2\alpha \) is

The plane \( x+3y+13=0 \) passes through the line of intersection of the planes \( 2x-8y+4z=p \) and \( 3x-5y+4z+10=0 \). If the plane is perpendicular to the plane \( 3x-y-2z-4=0 \), then the value of \( p \) is equal to

Let \(A\) and \(B\) be two events. Then \(1 + P(A \cap B) - P(B) - P(A)\) is equal to

The equation of the plane which bisects the line segment joining the points \( (3,2,6) \) and \( (5,4,8) \) and is perpendicular to the same line segment, is

The angle between the straight line \( \vec{r} = (\hat{i}+2\hat{j}+\hat{k}) + s(\hat{i}-\hat{j}+\hat{k}) \) and the plane \( \vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k}) = 4 \) is

The angle between the lines \( 2x = 3y = -z \) and \( 6x = -y = -4z \) is

If a straight line makes angles \( \alpha, \beta, \gamma \) with the coordinate axes, then evaluate \[ 1 - \frac{\tan^2\alpha}{1+\tan^2\alpha} + \frac{1}{\sec^2\beta} - 2\sin^2\gamma \]

The projection of the line segment joining \( (2,0,-3) \) and \( (5,-1,2) \) on a straight line whose direction ratios are \( 2,4,4 \) is equal to

Let the position vectors of points \(A, B, C\) be \( \vec{a}, \vec{b}, \vec{c} \) respectively. Let \(Q\) be the centroid. Then \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)

If \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \vec{c} = \hat{i} + \alpha \hat{j} + \beta \hat{k} \) are coplanar and \( |\vec{c}| = \sqrt{3} \), then

Let \( P(1,2,3) \) and \( Q(-1,-2,-3) \) be the two points and let \( O \) be the origin. Then \( |\overrightarrow{PQ} + \overrightarrow{OP}| = \)

Let \(ABCD\) be a parallelogram. If \( \vec{AB} = \hat{i} + 3\hat{j} + 7\hat{k} \), \( \vec{AD} = 2\hat{i} + 3\hat{j} - 5\hat{k} \), and \( \vec{p} \) is a unit vector parallel to \( \vec{AC} \), then \( \vec{p} \) is equal to

The angle between the two vectors \( \hat{i} + \hat{j} + \hat{k} \) and \( 2\hat{i} - 2\hat{j} + 2\hat{k} \) is equal to

If \( \vec{a} = \lambda\hat{i} + 2\hat{j} + 2\hat{k} \) and \( \vec{b} = 2\hat{i} + 2\hat{j} + \lambda\hat{k} \) are at right angle, then the value of \( |\vec{a}+\vec{b}| - |\vec{a}-\vec{b}| \) is