List of practice Questions

The power of a point (2,-1) with respect to a circle C of radius 4 is 9. The centre of the circle C lies on the line x+y=0 and in the 2nd quadrant. If \((\alpha, \beta)\) is the centre of the circle C, then \(\beta - \alpha = \)
The angle between the tangents drawn from the point P(k, 6k) to the circle \(x^2+y^2+6x-6y+2=0\) is \(2\tan^{-1}(\frac{4}{3})\). If the coordinates of P are integers, then k =
The tangents drawn from a point (2,-1) touch the circle \(x^2+y^2+4x-2y+1=0\) at the points A and B. If C is the centre of the circle, then the area (in sq. units) of the triangle ABC is
If \( \theta \) is the angle between the circles \( x^2+y^2-4x+2y-4=0 \) and \( x^2+y^2-2x+4y-11=0 \), then \( \sin\theta = \)
If the line x+y=2 cuts the circle \(x^2+y^2+2x-4y+4=0\) at two points A and B then the radius of the circle passing through A, B and orthogonal to \(x^2+y^2-2x-4y-4=0\) is
A normal chord PQ drawn at a point P on the parabola \(y^2=5x\) subtends a right angle at the vertex. If P lies in the first quadrant, then the other end Q of the normal chord is
(a,b) are the new coordinates of the point (2,3) after shifting the origin to the point (3,2) by translation of axes. If (c,d) are the new coordinates of the point (a,b) after rotating the axes through an angle \( \frac{\pi}{4} \) about the origin in the anti-clockwise direction, then d-c =
The lines x+y+4=0, x-2y-4=0 and 3x+4y-2=0
The area of the triangle formed by the line L with the coordinate axes is 12 sq. units. If L passes through the point (12,4) and the product P of X-intercept of L and square of the Y-intercept of L is negative, then P =
The area of the quadrilateral formed by the lines x+2y+3=0, 2x+4y+9=0, x-2y+3=0 and 3x-6y+11=0 is
If (-1,-1) is the point of intersection of the pair of lines \(2x^2+5xy-3y^2+2gx+2fy+c=0\) then g+f=
If the length of the chord 2x+3y+k=0 of the circle \(x^2+y^2-2x+4y-11=0\) is \(2\sqrt{3}\), then the sum of all possible values of k is
An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ball drawn is not red is
Among every 8 units of a product, one is likely to be defective. If a consumer has ordered 5 units of that product, then the probability that atmost one unit is defective among them is \
If A = (0,1), B = (1,2), C = (-2,1) then the equation of the locus of a point P such that area of triangle PAB = area of triangle PAC is
If \( \vec{a} = \vec{i}-2\vec{j}+2\vec{k} \), \( \vec{b} = 6\vec{i}+3\vec{j}-2\vec{k} \), \( \vec{c} = -4\vec{i}+3\vec{j}+12\vec{k} \) are three vectors then the value of the expression is...
Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = |\vec{b}| \) and \( |\vec{a}+2\vec{b}| = |2\vec{a}-\vec{b}| \). If \( \vec{c} \) is a vector parallel to \( \vec{a} \) then the angle between \( \vec{b} \) and \( \vec{c} \) is
If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}|=|\vec{b}|=\sqrt{6} \) and \( \vec{a} \cdot \vec{b} = -1 \), then \( |\vec{a} \times \vec{b}| \sin(\vec{a}, \vec{b}) = \)
If the volume of a tetrahedron having \( \vec{i}+2\vec{j}-3\vec{k} \), \( 2\vec{i}+\vec{j}-3\vec{k} \) and \( 3\vec{i}-\vec{j}+p\vec{k} \) as its coterminous edges is 2, then the values of p are the roots of the equation
The coefficient of variation for the following data is
If two smallest squares are chosen at random on a chess board then the probability of getting these squares such that they do not have a side in common is
Number of solutions of the equation \( \tan^2 x + 3\cot^2 x = 2\sec^2 x \) lying in the interval \( [0, 2\pi] \) is
\( \sin^{-1}(-\cos 2) + \cos^{-1}(\sin 3) + \tan^{-1}(\cot 5) = \)
If \( x = \log_e 3 \), then \( \tanh(2x) + \operatorname{sech}(2x) = \)
If a=3, b=5, c=7 are the sides of a triangle ABC, then \( \cot A + \cot B + \cot C = \)