List of top Mathematics Questions asked in KEAM

The slope of the normal to the curve \( y = x^3 - 3x^2 + 2x + 1 \) at the point where the tangent is horizontal is

A spherical balloon is being inflated such that its radius increases at a constant rate of \( 2 \,\text{cm/sec} \). The rate at which the volume is increasing when the radius is \( 5 \,\text{cm} \) is

Let \( f(x) = (3\sin^2(10x+11) - 7)^2 \) for \( x \in \mathbb{R} \). Then the maximum value of the function is

The slope of the tangent to the curve \( y = 3x^2 - 5x + 6 \) at \( (1,4) \) is

If \( |t|<1 \), \( \sin x = \frac{2t}{1+t^2} \), \( \tan y = \frac{2t}{1-t^2} \), then \( \frac{dy}{dx} \) is

If \( y = \frac{x}{x+1} + \frac{x+1}{x} \), then \( \frac{d^2y}{dx^2} \) at \( x=1 \) is equal to

If a circular plate is heated uniformly, its area expands $3c$ times as fast as its radius, then the value of $c$ when the radius is $6$ units, is

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

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

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

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

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}

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 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

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

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

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 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

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

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

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