List of practice Questions

The equation of the pair of asymptotes of the hyperbola \(\frac{(x - 3)^2}{3} - \frac{(y - 2)^2}{2} = 1\) is:
\( \lim_{x \to -9} \frac{(2.5)^{81 - x^2} - (0.4)^{x + 9}}{x + 9} = \)
If the points \( (k, 1, 5), (1, 0, 3), (7, -2, m) \) are collinear, then \( (k, m) = \)
If a line \( L \) makes angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) with the positive X-axis and positive Y-axis respectively, then the angle made by \( L \) with the positive direction of the Z-axis is:
The equation of a plane passing through \( (-1, 2, 3) \) and whose normal makes equal angles with the coordinate axes is:
Let \( S = 0 \) be the circle passing through the points \((2, 0)\), \((1, -2)\), and \((-1, 1)\). Then the point \((1, 2)\):
If the acute angle between the pair of tangents drawn from the origin to the circle \( x^2 + y^2 - 4x - 8y + 4 = 0 \) is \( \alpha \), then \( \tan \alpha = \)
Let \( C \) be the centre and \( A \) be one end of a diameter of the circle \( x^2 + y^2 - 2x - 4y - 20 = 0 \). If \( P \) is a point on \( AC \) such that \( CP : PA = 2 : 3 \), then the locus of \( P \) is:
If a cubical die is thrown, then the mean and variance of the random variable \( X \), giving the number on the face that shows up, are respectively:
The Cartesian form of the curve given by \( x = \frac{a}{2}\left(t + \frac{1}{t}\right) \), \( y = \frac{a}{2}\left(t - \frac{1}{t}\right) \), where \( t \) is a parameter, is:
If a straight line \( L \) perpendicular to the line \( 3x - 4y = 6 \) forms a triangle of area 6 square units with coordinate axes, then the minimum perpendicular distance from the point \( (1,1) \) to the line \( L \) is:
Let \( a, b, c, d \) be non-zero numbers. If the point of intersection of the lines \( 4ax + 2ay + c = 0 \) and \( 5bx + 2by + d = 0 \) lies in the fourth quadrant and is equidistant from the two coordinate axes, then the relationship between \( a, b, c, d \) is:
If the image of the point \( (3, 8) \) in the line \( x + 3y = 7 \) is \( (\alpha, \beta) \), then \( \alpha + \beta = \)
The condition that the lines joining the origin to the points of intersection of the line \( \frac{x}{a} + \frac{y}{b} = 2 \) and the circle \( (x - a)^2 + (y - b)^2 = r^2 \) are at right angles is:
The square of the distance from the origin to the point of intersection of the pair of lines \( ax^2 + 2hxy - ay^2 + 2gx + 2fy + c = 0 \) is:
If the vectors \( \vec{a} = 2\vec{i} + 3\vec{j} - \vec{k} \), \( \vec{b} = 4\vec{i} - \vec{j} + 3\vec{k} \), and \( \vec{c} = p\vec{i} + \vec{j} - \vec{k} \) are coplanar, then: \[ |\vec{a} \times \vec{c}| = ? \]
Students of two sections A and B of a class show the following results in a test conducted for 100 marks. Then, the comparison of variability between Section A and Section B is:
Students of two sections A and B of
A bag contains 12 two rupee coins, 7 one rupee coins and 4 fifty paise coins. If three coins are selected at random, then the probability that the sum of the values of the three coins is not an integral multiple of a rupee is
If \( A \) and \( B \) are any two events of a random experiment and \( P(B) \neq 1 \), then: \[ P(A|B^C) = ? \]
A bag contains 10 similar balls, of which 4 are blue and 6 are red. Three balls are taken at random from the bag one after the other without replacement. The probability that all the three balls drawn are red is:
If A and B are among 20 persons who sit at random along a round table, then the probability that there are exactly six persons between A and B is:
It is observed that there will be 25 blood specimens of normal persons, if 100 blood samples are tested. If 10 specimens are sent to a laboratory for testing, then the probability of having at least two specimens of normal persons is:
In \( \triangle ABC \), if \( (\sin A + \sin B)(\sin A - \sin B) = \sin C(\sin B + \sin C) \), then \( \angle A = \)
In \( \triangle ABC \), if \( A, B, C \) are in arithmetic progression, then \( \frac{c}{a} \sin 2A + \frac{a}{c} \sin 2C = \):
Let \( x \) be a real number and \( -2<x<2 \). When \( \frac{x+1}{(x+3)(x-2)} \) is expanded in powers of \( x \), then the coefficient of \( x^3 \) is: