List of top Questions asked in KEAM

Let \( A \) and \( B \) be two events such that \( P(A \cup B) = P(A) + P(B) - P(A)P(B) \). If \( 0 < P(A) < 1 \) and \( 0 < P(B) < 1 \), then \( P(A \cup B)' = \)

The value of \( \lim_{x \to 3} \frac{x^5 - 3^5}{x^8 - 3^8} \) is equal to:

The standard deviation of 9, 16, 23, 30, 37, 44, 51 is:

The distance between the \( x \)-axis and the point \( (3, 12, 5) \) is:

Foot of the perpendicular drawn from the origin to the plane \( 2x - 3y + 4z = 29 \) is:

The angle between a normal to the plane \( 2x - y + 2z - 1 = 0 \) and the \( z \)-axis is:

The angle between the straight lines \( x - 1 = \frac{2y + 3}{3} = \frac{z + 5}{2} \) and \( x = 3r + 2; y = -2r - 1; z = 2 \), where \( r \) is a parameter, 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:

A unit vector parallel to the straight line \( \frac{x - 2}{3} = \frac{3 + y}{-1} = \frac{z - 2}{-4} \) is:

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 \( \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} = \)

If \( \lambda(3\hat{i} + 2\hat{j} - 6\hat{k}) \) is a unit vector, then the values of \( \lambda \) are:

Equation of the plane through the mid-point of the line segment joining the points P(4, 5, -10) and Q(-1, 2, 1) and perpendicular to PQ is:

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:

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:

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:

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:

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:

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: