List of practice Questions

The lengths of the two focal chords of the parabola \( y^2 = 16x \) is 25 units each. If these two chords cut the parabola at \( A, B, C, D \), then the area (in sq. units) of the quadrilateral formed by \( A, B, C, D \) is:

If \( \theta \) is the angle between the tangents drawn from the point \( (-1, -1) \) to the circle \( x^2+y^2-4x-6y+c=0 \) and \( \cos\theta = -\frac{7}{25} \), then the radius of the circle is:

If the tangents drawn from a point \( P \) to the ellipse \( 4x^2+9y^2-16x+54y+61=0 \) are perpendicular, then the locus of \( P \) is:

The equations of the asymptotes of a hyperbola are \( x+y+3=0 \), \( 2x-y+1=0 \). If \( (1,-2) \) is a point on this hyperbola, then the equation of its conjugate hyperbola is:

If the power of the point \( (1,6) \) with respect to the circle \( x^2+y^2+4x-6y-a=0 \) is \( -16 \), then \( a \) is:

Let \( A(5,4) \) and \( B(5,-4) \) be two points. If \( P \) is a point in the coordinate plane such that \( |APB| = \frac{\pi}{4} \), then the point \( P \) lies on the curve:

When the axes are rotated through an angle \( \theta \) about the origin in the anticlockwise direction and then translated to the new origin (2, -2), if the transformed equation of \( x^2+y^2=4 \) is \( X^2+Y^2+aX+bY+c=0 \), then \( a+b+c= \):

From a point \( P(-4, 0) \), two tangents are drawn to the circle \( x^2+y^2-4x-6y-12=0 \) touching the circle at \( A \) and \( B \). If the equation of the circle passing through \( P, A, B \) is \( x^2+y^2+2gx+2fy+c=0 \), then \( (g,f) = \):

If the perpendicular distances from the points \( (2,3) \), \( (4,a) \), and \( (\alpha, \beta) \) onto the line \( 3x + 4y - 3 = 0 \) are equal and \( 4\alpha - 3\beta + 1 = 0 \), then the sum of all possible values of \( \alpha \), \( \alpha \) and \( \beta \) is:

The triangle formed by the lines \( 2x^2 + xy - 6y^2 = 0 \) and \( x+y-1=0 \) is:

The equation of the base of an equilateral triangle is \( x + y = 2 \) and its opposite vertex is \( (2,1) \). If \( m_1, m_2 \) are the slopes of the other two sides and the length of its side is \( a \), then \( |m_1 - m_2| + a\sqrt{2} = \):

If \( \left(\frac{2}{3},0\right) \) is the centroid of the triangle formed by the lines \( 4x^2 - y^2 = 0 \) and \( lx + my + n = 0 \), then \( l+m+n= \):

The probability distribution of a discrete random variable \( X \) is given below: \[ \begin{array}{c|cccc} X = x & -1 & 0 & 1 & 2 \\ P(X = x) & \frac{1}{3} & \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \end{array} \] Then the value of \( 6 \sum x^2 P(X = x) - \text{var}(X) \) is:

The mean and variance of the observations \( x_1, x_2, x_3, \dots, x_{15} \) are respectively 2 and 4. If the mean and variance of the observations \( Y_1, Y_2, \dots, Y_{10} \) are respectively 2 and 5, then the variance of the observations \( x_1, x_2, \dots, x_{15}, Y_1, Y_2, \dots, Y_{10} \) is:

If 3 squares are chosen at random from the 64 squares of a chessboard, then the probability that all of them lie along the same diagonal line is:

Three letters are chosen at random from the letters of the word VARIABLE and all possible three-letter words (with or without meaning) are formed with them. Then the probability of getting a three-letter word having a consonant as its middle letter is:

If the average number of accidents occurring at a particular junction on a highway in a week is 5, then the probability that at most one accident occurs in a particular week is:

A rational number is selected at random from the distinct rational numbers of the form \( \frac{p}{q} \) formed with \( p \) and \( q \) belonging to the set \( \{1,2,3,4,5,6\} \). The probability that the rational number selected is a proper fraction is:

In a shoe rack there are 4 pairs of shoes and 4 shoes are drawn one after the other at random without replacement. Then the probability of getting at least one correct pair of shoes among the four shoes drawn is:

If the magnitudes of \( \vec{a} \), \( \vec{b} \), and \( \vec{a} + \vec{b} \) are respectively \( 3 \), \( 4 \), and \( 5 \), then the magnitude of \( \vec{a} - \vec{b} \) is:

A unit vector that is perpendicular to the vector \( 2\vec{i} - \vec{j} + 2\vec{k} \) and coplanar with the vectors \( \vec{i} + \vec{j} - \vec{k} \) and \( 2\vec{i} - 2\vec{j} - \vec{k} \) is

In triangle \(ABC\), if \(a = 6\), \(b = 8\), and \(c = 10\), then \(\dfrac{2r_1}{r_2} =\)

If the vectors \(2\vec{i} + 3\vec{j} + k\), \(-3\vec{i} - 2\vec{j} - 4k\), and \(l\vec{i} - \vec{j} + 3\vec{k}\) form a right-angled triangle for a positive value of \(l\), then the length of its hypotenuse is

In \(\triangle ABC\), if \(a : b : c = 4 : 5 : 6\), then \( \dfrac{\cos A + 3 \cos C}{\cos B} = \)

If \( \vec{i} + \vec{j} - \vec{k}, -\vec{i} + 2\vec{j} + \vec{k}, \vec{j} + 2\vec{k}, 2\vec{i} - \vec{j} + 2\vec{k} \) are the position vectors of four points \( A, B, C, D \) respectively, then the shortest distance between the lines \( AB \) and \( CD \) is: