List of top Questions asked in KEAM- 2015

Let the position vectors of points \(A, B, C\) be \( \vec{a}, \vec{b}, \vec{c} \) respectively. Let \(Q\) be the centroid. Then \( \overrightarrow{QA} + \overrightarrow{QB} + \overrightarrow{QC} = \)

The angle between the lines \( 2x = 3y = -z \) and \( 6x = -y = -4z \) is

The equation of the plane which bisects the line segment joining the points \( (3,2,6) \) and \( (5,4,8) \) and is perpendicular to the same line segment, is

If a straight line makes angles \( \alpha, \beta, \gamma \) with the coordinate axes, then evaluate \[ 1 - \frac{\tan^2\alpha}{1+\tan^2\alpha} + \frac{1}{\sec^2\beta} - 2\sin^2\gamma \]

Let \( \overrightarrow{OB} = \hat{i} + 2\hat{j} + 2\hat{k} \) and \( \overrightarrow{OA} = 4\hat{i} + 2\hat{j} + 2\hat{k} \). The distance of the point \( B \) from the straight line passing through \( A \) and parallel to the vector \( 2\hat{i} + 3\hat{j} + 6\hat{k} \) is

Let \( P(1,2,3) \) and \( Q(-1,-2,-3) \) be the two points and let \( O \) be the origin. Then \( |\overrightarrow{PQ} + \overrightarrow{OP}| = \)

If \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \vec{c} = \hat{i} + \alpha \hat{j} + \beta \hat{k} \) are coplanar and \( |\vec{c}| = \sqrt{3} \), then

Let \(ABCD\) be a parallelogram. If \( \vec{AB} = \hat{i} + 3\hat{j} + 7\hat{k} \), \( \vec{AD} = 2\hat{i} + 3\hat{j} - 5\hat{k} \), and \( \vec{p} \) is a unit vector parallel to \( \vec{AC} \), then \( \vec{p} \) is equal to

The angle between the two vectors \( \hat{i} + \hat{j} + \hat{k} \) and \( 2\hat{i} - 2\hat{j} + 2\hat{k} \) is equal to

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

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 distance between the directrices of the hyperbola \( x^2 - y^2 = 9 \) 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 one end of the latus rectum of the parabola \( y^2 - 4x - 2y - 3 = 0 \) is at

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

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

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

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 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

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

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:

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 \( p_1 \) 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