List of top Questions asked in KEAM

If \( \vec{a} = \lambda\hat{i} + 2\hat{j} + 2\hat{k} \) and \( \vec{b} = 2\hat{i} + 2\hat{j} + \lambda\hat{k} \) are at right angle, then the value of \( |\vec{a}+\vec{b}| - |\vec{a}-\vec{b}| \) is

If the line \( y = kx \) touches the parabola \( y = (x-1)^2 \), then the values of \( k \) are

If a point \( P(x,y) \) moves along the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) and \( C \) is the centre of the ellipse, then \( 4\max\{CP\} + 5\min\{CP\} \) is

The one end of the latus rectum of the parabola \( y^2 - 4x - 2y - 3 = 0 \) is at

The distance between the directrices of the hyperbola \( x^2 - y^2 = 9 \) is

If the semi-major axis of an ellipse is \( 3 \) and the latus rectum is \( \frac{16}{9} \), then the standard equation of the ellipse is

\(ABCD\) is a square with side \(a\). If \(AB\) and \(AD\) are along the coordinate axes, then the equation of the circle passing through the vertices \( A, B \) and \( D \) is

The parametric form of equation of the circle \( x^2 + y^2 - 6x + 2y - 28 = 0 \) is

The line segment joining \( (5,0) \) and \( (10\cos\theta, 10\sin\theta) \) is divided internally in the ratio \(2:3\) at \(P\). If \(\theta\) varies, then the locus of \(P\) is

If the straight line \( 5x + y = k \) forms a triangle with the coordinate axes of area \(10\) sq. units, then the values of \(k\) are

A circle passes through the point \( (6,2) \). If segments of the straight lines \( x+y=6 \) and \( x+2y=4 \) are two diameters of the circle, then its radius is

The point on the circle \( (x-1)^2 + (y-1)^2 = 1 \) which is nearest to the circle \( (x-5)^2 + (y-5)^2 = 4 \) is

Let \( A(0,0) \) and \( B(8,0) \) be two vertices of a right angled triangle whose hypotenuse is \( BC \). If the circumcentre is \( (4,2) \), then the point \( C \) is:

If the points \( A(3,4) \), \( B(x_1,y_1) \) and \( C(x_2,y_2) \) are such that both \( 3,x_1,x_2 \) and \( 4,y_1,y_2 \) are in A.P., then

If coordinates of the circumcentre and the orthocentre of a triangle are respectively \( (5,5) \) and \( (2,2) \), then the coordinates of the centroid are:

If \( p_1 \) and \( p_2 \) are respectively length of perpendiculars from the origin to the straight lines \( x\sec\theta + y\cosec\theta = a \) and \( x\cos\theta - y\sin\theta = a\cos 2\theta \), then \( 4p_1^2 + p_2^2 = \)

The distance between the point \( (1,2) \) and the point of intersection of the lines \( 2x + y = 2 \) and \( x + 2y = 2 \) is

If a straight line is perpendicular to \( 2x + 8y = 10 \) and meets the x-axis at \( (5,0) \), then it meets the y-axis at

\( 2\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1\left(\frac{1}{4}\right) = \)}

If \( \sin(\theta+\phi) = n \sin(\theta-\phi), \; n \neq 1 \), then the value of \( \frac{\tan\theta}{\tan\phi} \) is

If \( 0 \le x \le 2\pi \), then the number of solutions of the equation \( \sin^8 x + \cos^8 x = 1 \) is

If \( \cos^{-1}x + \cos^{-1}y = \frac{2\pi}{7} \), then the value of \( \sin^{-1}x + \sin^{-1}y \) is equal to

If \( \cos^{-1}x > \sin^{-1}x \), then \(x\) lies in the interval

Let \(p, q, r\) be three statements. Then \( \sim (p \vee (q \wedge r)) \) is equal to

Consider the two statements \(P\): He is intelligent and \(Q\): He is strong. Then the symbolic form of the statement “It is not true that he is either intelligent or strong” is