List of top Questions asked in KEAM- 2015

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

If \( f(x) = 3x^2 - 7x + 5 \), then \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} \) 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 \( f(x) = \sin^{-1} \left( \frac{1 - \cos 2x}{2 \sin x} \right) \), then \( |f'(x)| \) 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

\[ \lim_{x \to \infty} \left(\frac{x^2}{3x-2} - \frac{x}{3}\right) \]

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

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

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

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

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

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 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

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

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

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

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

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 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 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

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

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 \( \vec{a} = \lambda\hat{i} + 2\hat{j} + 2\hat{k} \) 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

Let \( \overrightarrow{OB} = \hat{i} + 2\hat{j} + 2\hat{k} \) and \( \overrightarrow{OA} = 4\hat{i} + 2\hat{j} + 2\hat{k} \). The distance of the point \( B \) from the straight line passing through \( A \) and parallel to the vector \( 2\hat{i} + 3\hat{j} + 6\hat{k} \) is