List of top Questions asked in KEAM

If \( \frac{dy}{dx} = \frac{2}{x+y} \) and \( y(1)=0 \), then \( x+y+2 \) equals

The length of the latus rectum of the parabola \( (x+2)^2 = -14(y-5) \) is

One of the foci of the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \) is

The number of solutions of \( \frac{dy}{dx} = y^{1/3} \) passing through origin is

If the circles \( x^2+y^2-8x-6y+c=0 \) and \( x^2+y^2-2y+d=0 \) cut orthogonally, then \( c+d \) equals

The differential equation which represents the family \( y^2 = 2c(x+\sqrt{c}) \) is

The values of \(k\) for which the system \[ (k+1)x + 8y = 0 \] \[ kx + (k+3)y = 0 \] has unique solution, are

A plane passes through the point \( (1,-2,1) \) and is perpendicular to planes \(2x-2y+z=0\) and \(x-y+2z=4\). Then the equation of the plane is

Let \( A = \begin{pmatrix} \alpha & 0 \\ 1 & 1 \end{pmatrix} \) and \( B = \begin{pmatrix} 1 & 0\\ 5 & 1 \end{pmatrix} \) be two matrices where \( \alpha \) is a real number. Then

If \( M \) and \( N \) are square matrices of order 3 where \( \det(M)=2 \) and \( \det(N)=3 \), then \( \det(3MN) \) is

If the lines \( \frac{x+3}{-3} = \frac{y-1}{k} = \frac{z-5}{5} \) and \( \frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{5} \) are coplanar, then the value of \( k \) is

Let \( f(x)=|x-2| \) and \( g(x)=f(f(x)) \). Then derivative of \( g \) at the point \( x=5 \) is

The value of the definite integral \( \int_0^{2\pi} \sqrt{1 + \sin \frac{x}{2}}\, dx \) is

The value of \( \lim_{n\to\infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{6n} \right] \) is

The area of the triangle whose vertices are $A(1,-1,2)$, $B(2,1,-1)$ and $C(3,-1,2)$ is

A unit vector $\vec{b}$ is coplanar with $\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} + \hat{k}$ and is perpendicular to $\hat{i} + \hat{j} + \hat{k}$. Then $\vec{b} \cdot \hat{i}$ equals

Let $R = \{(a,b): a \leq b^2\}$ be a relation on the set of all real numbers. Then $R$ is

Suppose $\alpha \hat{i} + \alpha \hat{j} + \gamma \hat{k}$, $\hat{i} + \hat{k}$ and $\gamma \hat{i} + \gamma \hat{j} + \beta \hat{k}$ are coplanar where $\alpha, \beta, \gamma$ are positive constants. Then the product $\alpha\beta$ is

The domain of definition of the function $f(x) = \frac{\log_3(x+7)}{x^2 - 5x + 6}$ is

Let $f(x) = 3x - 5$. The inverse of $f$ is given by

If $^{n}C_{2017} = {}^{n}C_{2016}$, then $^{n}C_{4033}$ equals

The image of the point $P(2,1)$ on the straight line $2x - 3y + 1 = 0$ is

If the centre of the circle inscribed in a square formed by the lines $x^2 - 8x + 12 = 0$ and $y^2 - 14y + 45 = 0$ is $(a,b)$, then $a + b$ is

If $^{5}P_r = {}^{6}P_{r-1}$, then the value of $r$ is

The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is