List of top Questions asked in KEAM

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

Given that the equation $x^2 - (2a + b)x + \left(2a^2 + b^2 - b + \frac{1}{2}\right) = 0$ has two real roots. The value of $b$ is

In a G.P., $1, \frac{1}{2}, \frac{1}{4}, \ldots$, when the first $n$ number of terms are added, the sum is $\frac{1023}{512}$. Then the value of $n$ is

The value of \( \tan \frac{\pi}{8} \) is

If A.M. and G.M. of the roots of a quadratic equation are 8 and 5 respectively, then the quadratic equation is

Let \( w = \frac{1-iz}{z-i} \). If \( |w| = 1 \), which of the following must be true?

Let \( \cot \theta = -5/12 \) where \( \frac{\pi}{2} < \theta < \pi \). Then the value of \( \sin \theta \) is

Let \( \omega \ne 1 \) be a cube root of unity and \( (1+\omega)^7 = a + \omega \). Then the value of \( a \) is

If \( A \) and \( B \) are two events associated with an experiment such that \( P(A \cup B) = P(A \cap B) \), and \( P(A) = 1/3 \), then \( P(B) \) equals

Three identical fair dice are rolled. The probability that the same number appears on each of them is

For \( |z| \ge 2 \), if \( \left|z + \frac{1}{2}\right| \ge k \), the minimum possible value of \(k\) is

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

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

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

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

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

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

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

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

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

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