List of top Questions asked in KEAM- 2019

The equation of the tangent to the curve \( y = x + \frac{4}{x^2} \) that is parallel to the x-axis is

The equation of the plane containing the line \( \frac{x-\alpha}{l} = \frac{y-\beta}{m} = \frac{z-\gamma}{n} \) is \( a(x-\alpha)+b(y-\beta)+c(z-\gamma)=0 \), where \( al + bm + cn \) is equal to

Let \( f(x) \) and \( g(x) \) be two differentiable functions for \( 0 \leq x \leq 1 \) such that \( f(0)=2, g(0)=0, f(1)=6 \). If there exists a real number \( c \in (0,1) \) such that \( f'(c)=2g'(c) \), then \( g(1) \) is equal to

A bag contains 3 black and 2 white balls. A ball is drawn at random and is put back in the bag along with one ball of the same colour. A ball is again drawn at random. What is the probability that it is white?

The number \(81\) is the coefficient of \( x^k \) in the binomial expansion of \( \left(x^2 + \frac{3}{x}\right)^4 \), \( x \neq 0 \). Then the value of \( k \) equals

The possible number of arrangements starting with K of the word KALINGA is

In a chess tournament, assume that your probability of winning a game is 0.3 against level 1 players, 0.4 against level 2 players and 0.5 against level 3 players. It is further assumed that among the players 50% are at level 1, 25% are at level 2 and the remaining are at level 3. Suppose that you win the game. Then the probability that you had played with level 1 player is

The coefficient of \( x^3 \) in the expansion of \( (1 + x + 2x^2)(1 - 2x)^5 \) is

The constant term in the expansion of \( \left(x^2 - \frac{2}{x}\right)^6 \) is

A sum of Rs. 280 is to be used to award four prizes. If each prize after the first prize is Rs. 20 less than its preceding prize, then the value of the fourth prize is

If the equation of the sphere through the circle \( x^2 + y^2 + z^2 = 9; \; 2x + 3y + 4z = 5 \) and through the point \( (1,2,3) \) is \( 3(x^2 + y^2 + z^2) - 2x - 3y - 4z = C \), then the value of \( C \) is

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

The area of the region bounded by the curves \( y = |x - 2| \), \( x = 1 \), \( x = 3 \) and \( y = 0 \) is:

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is

If the mean of the first \(n\) odd numbers is \( \frac{n^2}{81} \), then \(n\) equals

A pair of fair dice are rolled together. The probability of getting a total of 8 is

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of red ball, the number of blue balls must be

The equation of the tangent to the curve given by \( x^2 + 2x - 3y + 3 = 0 \) at the point \( (1,2) \) is

The solution of the differential equation \( 5y\,dx = 2x\,dy \) passing through the point \( (1,1) \) is:

The maximum value of \( y = \left(\frac{1}{x}\right)^x \), \( x > 0 \) is

The value of the integral \( \int_{0}^{\pi} \frac{\cos x}{1+\sin^2 x} \, dx \) is

The area enclosed between the curves \( y = 2x^2 + 1 \) and \( y = x^2 + 5 \) is:

If \( x = 2\cos t - \cos 2t \) and \( y = 2\sin t - \sin 2t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{2} \) is

Let \( z_1 = 1 + i\sqrt{3} \) and \( z_2 = 1 + i \), then \( \arg\left( \frac{z_1}{z_2} \right) \) is

The complex number \( \sqrt{2}\left[ \sin \frac{\pi}{8} + i \cos \frac{\pi}{8} \right]^6 \) represents