List of top Mathematics Questions

The foot of the perpendicular from the point \( (1,6,3) \) to the line \( \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) 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 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 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

The projection of the line segment joining \( (2,0,-3) \) and \( (5,-1,2) \) on a straight line whose direction ratios are \( 2,4,4 \) is equal to

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

The angle between the straight line \( \vec{r} = (\hat{i}+2\hat{j}+\hat{k}) + s(\hat{i}-\hat{j}+\hat{k}) \) and the plane \( \vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k}) = 4 \) is

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

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

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

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

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

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

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

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

\(ABCD\) 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

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 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

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 parametric form of equation of the circle \( x^2 + y^2 - 6x + 2y - 28 = 0 \) 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 \( p_1 \) 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 \( 4p_1^2 + p_2^2 = \)