List of top Questions asked in KEAM

Let \( f:\mathbb{N}\to\mathbb{N} \) be such that \( f(1)=2 \) and \( f(x+y)=f(x)f(y) \). If \( \sum_{k=1}^{n} f(a+k)=16(2^n-1) \), then \( a \) is

Point equidistant from \( (2,0,3), (0,3,2), (0,0,1) \)

If \( ^nC_{r-1}=36, ^nC_r=84, ^nC_{r+1}=126 \), then \( n= \)

Let \( f:(-1,1)\to(-1,1) \) be continuous, \( f(x)=f(x^2) \), and \( f(0)=\frac{1}{2} \). Find \( 4f\left(\frac{1}{4}\right) \)

Coefficient of \( x \) in \( (1 - 3x + 7x^2)(1 - x)^{16} \)

If \( \vec{a} \) and \( \vec{b}=3\hat{i}+6\hat{j}+6\hat{k} \) are collinear and \( \vec{a}\cdot\vec{b}=27 \), then \( \vec{a} \) is

Distance between two parallel lines \( y = 2x + 4 \) and \( y = 2x - 1 \) is

If \( ^{56}P_{r+6} : \, ^{54}P_{r+3} = 30800 : 1 \), then \( r \) is equal to

\( \binom{7}{0}+\binom{7}{1} + \binom{7}{2}+\binom{7}{3} + \cdots + \binom{7}{7} \)

If \( |\vec{a}|=13, |\vec{b}|=5 \) and \( \vec{a}\cdot\vec{b}=30 \), then \( |\vec{a}\times\vec{b}| \) is

If \( x^2 + y^2 + 2gx + 2fy + 1 = 0 \) represents a pair of straight lines, then \( f^2 + g^2 \) is equal to

\( \sin 15^\circ = \)

If \( \theta \) is the angle between the pair of straight lines \( x^2 - 5xy + 4y^2 + 3x - 4 = 0 \), then \( \tan^2\theta \) is equal to

The area of a triangle is 5 sq. units. Two of its vertices are (2,1) and (3,-2). The third vertex lies on \( y = x + 3 \). The coordinates of the third vertex can be

If \( 3\hat{i} + 2\hat{j} - 5\hat{k} = x(2\hat{i} - \hat{j} + \hat{k}) + y(\hat{i} + 3\hat{j} - 2\hat{k}) + z(-2\hat{i} + \hat{j} - 3\hat{k}) \), then

Equation of line through \( (a,b) \) parallel to \( \frac{x}{a} + \frac{y}{b} = 1 \)

If \( (x,y) \) is equidistant from \( (a+b,b-a) \) and \( (a-b,a+b) \), then

The coordinate of the point dividing internally the line joining the points \( (4,-2) \) and \( (8,6) \) in the ratio \( 7:5 \) is

If the points \( (2a,a), (a,2a), (a,a) \) form a triangle of area 18, find centroid

The area of the triangle formed by the points \( (a,b+c), (b,c+a), (c,a+b) \) is

\( \int \frac{x e^x}{(1+x)^2} \, dx = \)

The remainder when \( 2^{2000} \) is divided by 17 is

The area of the triangle in the complex plane formed by \( z, iz \) and \( z + iz \) is

The maximum value of \( 5\cos \theta + 3\cos \left(\theta + \frac{\pi}{3}\right) + 3 \) is

Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function. If \( f \) is even, then \( f'(0) \) is equal to