List of top Questions asked in KEAM

If the direction cosines of a vector of magnitude 3 are \( \frac{2}{3}, \frac{-a}{3}, \frac{2}{3} \), \( a > 0 \), then the vector is:

Equation of the line through the point (2, 3, 1) and parallel to the line of intersection of the planes \( x - 2y - z + 5 = 0 \) and \( x + y + 3z = 6 \) is:

If \( \vec{a} = \hat{i} + 2\hat{j} + 2\hat{k} \), \( |\vec{b}| = 5 \) and the angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{6} \), then the area of the triangle formed by these two vectors as two sides is:

If \( \vec{a} \cdot \vec{b} = 0 \) and \( \vec{a} + \vec{b} \) makes an angle of \( 60^\circ \) with \( \vec{a} \), then:

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are three non-zero vectors such that each one of them being perpendicular to the sum of the other two vectors, then the value of \( |\vec{a} + \vec{b} + \vec{c}|^2 \) is:

Let \( \vec{u}, \vec{v} \) and \( \vec{w} \) be vectors such that \( \vec{u} + \vec{v} + \vec{w} = \vec{0} \). If \( |\vec{u}| = 3, |\vec{v}| = 4 \) and \( |\vec{w}| = 5 \) then \( \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = \)

Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is a vector such that \( \vec{a} \cdot \vec{b} = |\vec{b}|^2 \) and \( |\vec{a} - \vec{b}| = \sqrt{7} \), then \( |\vec{b}| = \)

If \( \hat{i} + \hat{j}, \, \hat{j} + \hat{k}, \, \hat{i} + \hat{k} \) are the position vectors of the vertices of a triangle ABC taken in order, then \( \angle A \) is equal to:

If the eccentricity of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is \( \frac{5}{4} \) and \( 2x + 3y - 6 = 0 \) is a focal chord of the hyperbola, then the length of transverse axis is equal to:

The length of the transverse axis of a hyperbola is \( 2\cos \alpha \). The foci of the hyperbola are the same as that of the ellipse \( 9x^2 + 16y^2 = 144 \). The equation of the hyperbola is:

A circle of radius \( \sqrt{8} \) is passing through origin and the point \( (4, 0) \). If the centre lies on the line \( y = x \), then the equation of the circle is:

The parametric form of the ellipse \( 4(x + 1)^2 + (y - 1)^2 = 4 \) is:

A point P on an ellipse is at a distance 6 units from a focus. If the eccentricity of the ellipse is \( \frac{3}{5} \), then the distance of P from the corresponding directrix is:

If the length of the latus rectum and the length of transverse axis of a hyperbola are \( 4\sqrt{3} \) and \( 2\sqrt{3} \) respectively, then the equation of the hyperbola is:

The centre of the circle whose radius is \( 5 \) and which touches the circle \( x^2 + y^2 - 2x - 4y - 20 = 0 \) at \( (5, 5) \) is:

A circle passes through the points \( (0, 0) \) and \( (0, 1) \) and also touches the circle \( x^2 + y^2 = 16 \). The radius of the circle is:

The slope of the straight line joining the centre of the circle \( x^2 + y^2 - 8x + 2y = 0 \) and the vertex of the parabola \( y = x^2 - 4x + 10 \) is:

The shortest distance between the circles \( (x-1)^2 + (y+2)^2 = 1 \) and \( (x+2)^2 + (y-2)^2 = 4 \) is:

A straight line perpendicular to the line \( 2x + y = 3 \) is passing through \( (1,1) \). Its y-intercept is:

If \( p \) and \( q \) are respectively the perpendiculars from the origin upon the straight lines whose equations are \( x\sec\theta + y\csc\theta = a \) and \( x\cos\theta - y\sin\theta = a\cos 2\theta \), then \( 4p^2 + q^2 \) is equal to:

The number of points \( (a, b) \), where \( a \) and \( b \) are positive integers, lying on the hyperbola \( x^2 - y^2 = 512 \) is:

If \( 0 < x < \pi \), then \( \frac{\sin 8x + 7\sin 6x + 18\sin 4x + 12\sin 2x}{\sin 7x + 6\sin 5x + 12\sin 3x} = \)

The circumcentre of the triangle with vertices \( (8, 6), (8, -2) \) and \( (2, -2) \) is at the point:

The ratio by which the line \( 2x + 5y - 7 = 0 \) divides the straight line joining the points \( (-4, 7) \) and \( (6, -5) \) is:

The points \( (2, 5) \) and \( (5, 1) \) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line \( y = 2x + k \), then the value of \( k \) is: