List of practice Questions

Let \( f(x) = \frac{3x - 5}{2x + 1} \). If \( f^{-1}(x) = \frac{x + a}{bx + c} \), then what is the value of \( (a - b + c) \)?
Given \( f(x) = \frac{4x + 1}{4} \) and \( g(x) = \sqrt{x^3} \), then \( (g \circ f^{-1}) \left( \frac{3}{8} \right) = \)?
Study the given symbols.

$A@ B$ means A is not smaller than B.
$A # B$ means A is not greater than B.
$A & B$ means A is neither smaller than nor equal to B.
$A % B$ means A is neither greater than nor equal to B.
$A $ B$ means A is neither smaller than nor greater than B.

Based on the symbols, read the given statements and conclusions carefully and decide which of the conclusions are true.

Statements:
A) $L @ M$
B) $N & K$
C) $M @ L$

Conclusions:
I. $N & M$
II. $M % K$
III. $K & L$
Study the given symbols.
A @ B means A is not smaller than B.
A # B means A is not greater than B.
A & B means A is neither smaller than nor equal to B.
A % B means A is neither greater than nor equal to B.
A $ B means A is neither smaller than nor greater than B. Based on the symbols, read the given statements and conclusions carefully and decide which of the conclusions is/are true. Statements:
A) R @ P
B) S # Q
C) P $ S Conclusions:
I. S % R
II. Q & R
III. Q @ P
If the signs '$' and '+' are interchanged in each of the equations given as options, which equation would be correct?
If \( 3 \sin^2 x + 10 \cos x - 6 = 0, 0^\circ<\theta<90^\circ \), then the value of \( \sec x + \csc x + \cot x \) is:
If \( \sec \theta = a + \frac{1}{4a}, 0^\circ<\theta<90^\circ \), then \( \csc \theta + \cot \theta = \):
Simplify the expression: \[ \frac{\sin \theta(1 + \tan \theta) + \cos \theta (1 + \cot \theta)}{\csc \theta - \sin \theta} \cdot \frac{\sec \theta}{\cos \theta (\tan \theta + \cot \theta)} = ? \]
The value of \( \frac{(1 + \cot \theta - \csc \theta)(1 + \tan \theta + \sec \theta)}{\tan^2 \theta + \cot^2 \theta - \sec^2 \theta \csc^2 \theta} \) is:
A die is constructed so that when it is thrown, each of the three even numbers 2, 4 and 6 is twice as likely to come up as each of the odd outcomes 1, 3 and 5. What is the probability that 4 comes up when the die is thrown once?
Calculate the variance of: 2, 4, 5, 6, 8, 17.
If the standard deviation of the series \( x_1, x_2, \dots, x_n \) is \( \sigma \), then the standard deviation of the series \( \frac{6x_1 - 7}{3}, \frac{6x_2 - 7}{3}, \dots, \frac{6x_n - 7}{3} \) is:
Let \( x \) be the median of the data: 23, 17, 19, 11, 7, 3, 13, 2, 5, 29. Let \( y \) be the median of the same data set obtained by replacing 2 by 21 and 13 by 31. What is the value of \( |x - y| \)?
The mean of the following distribution is 23.8 .

\[ \begin{array}{|c|c|c|c|c|c|} \hline \textbf{Class} & 0\text{–}10 & 10\text{–}20 & 20\text{–}30 & 30\text{–}40 & 40\text{–}50 \\ \hline \textbf{Frequency} & 7 & 5 & 3 & 4 & k \\ \hline \end{array} \]

What is the value of \( k \)?
The scores of a batsman in 10 different test matches were 42, 38, 48, 70, 46, 63, 55, 34, 54, and 44. What is the mean deviation about the median of these scores?
X and Y are the two points that are 135 m apart on the ground on either side of a pole and in the same line. The angles of elevation of a bird sitting on the top of the pole from X and Y are 30° and 60° respectively. The distance of Y from the foot of the pole (in m) is:
If \( a + b + c = 2 \), \( a^2 + b^2 + c^2 = 36 \), then the value of \( a^3 + b^3 + c^3 - 3abc \) is:
If the equations \( x^2 + px + 12 = 0 \), \( x^2 + qx + 15 = 0 \), and \( x^2 + (p+q)x + 36 = 0 \) have a common positive root, then what is the value of \( (2p - q) \)?
The graphs of the equations \( 2x + 3y = a \) and \( x + 2y = b \) intersect at a point \( P(\alpha, \beta) \). What is the value of \( (3\alpha + 2\beta) \)?
The ratio of the sum of the first \( n \) terms to the sum of the first \( s \) terms of an arithmetic progression is \( r^2 : s^2 \). What is the ratio of its 8th term to the 23rd term of this same progression?
The sum of the first \( n \) terms of a geometric progression is 255, the \( k \)-th term is 128, and the common ratio is 2. The value of \( k \) satisfies the equation:
The sum of the roots of the equation \( |x - 7|^2 + 2|x| - 7| = 24 \) is:
When 5 is subtracted from each of given \( n \) numbers, the sum of numbers so obtained is 210. When 8 is subtracted from each of the given \( n \) numbers, then the sum of numbers so obtained is 156. What is the mean of the given \( n \) numbers?
A boat can go \( 1 \frac{3}{5} \) km upstream and \( 4 \frac{4}{5} \) km downstream in 48 minutes, while it can go 2 km upstream and 600 m downstream in 33 minutes. How much time (in hours) will it take to go 28.8 km downstream?
A dealer allows 32% discount on the marked price of an article and still gains 28%. If the cost price of the article is reduced by 10%, how much discount percent should the dealer allow now to get the same percentage of profit as before?