List of practice Questions

If \( \sec \theta = a + \frac{1}{4a}, 0^\circ<\theta<90^\circ \), then \( \csc \theta + \cot \theta = \):
Simplify the expression: \[ \frac{\sin \theta(1 + \tan \theta) + \cos \theta (1 + \cot \theta)}{\csc \theta - \sin \theta} \cdot \frac{\sec \theta}{\cos \theta (\tan \theta + \cot \theta)} = ? \]
Calculate the variance of: 2, 4, 5, 6, 8, 17.
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?
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 \( k \)?
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 \( |x - y| \)?
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?
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:
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:
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?
If \( a + b + c = 2 \), \( a^2 + b^2 + c^2 = 36 \), then the value of \( a^3 + b^3 + c^3 - 3abc \) is:
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 graphs of the equations \( 2x + 3y = a \) and \( x + 2y = b \) intersect at a point \( P(\alpha, \beta) \). What is the value of \( (3\alpha + 2\beta) \)?
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 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 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?
Last year, the ratio of the prices of two articles A and B was 3 : 5. This year, the price of A is increased by 25% and that of B is decreased by ₹210. If the ratio of the present prices of A and B is 15 : 14, then the price of A last year was:
A person borrowed a certain sum on compound interest and agreed to return it in two years in two equal annual instalments. If the rate of interest is 10% p.a. and each annual instalment is ₹4,840, then the interest paid by him was:
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?
Water flows at the rate of 20 metre/minute through a cylindrical pipe whose radius is 1.5 cm. Using this pipe, how long (in hours) would it take to fill a conical vessel whose radius is 120 cm and depth is 72 cm?
If \( a_1, a_2, a_3, \dots \) is an arithmetic progression with the common difference of 1 and \( a_2 + a_4 + a_6 + \dots + a_{98} = 93 \), then \( \sum_{i=1}^{98} a_i \) is equal to \( k \). The sum of the digits of \( k \) is:
If \( a \), \( b \), and \( c \) are three fractions such that \( a<b<c \), and if the smallest fraction is divided by the middle fraction, the result is \( \frac{15}{16} \), which exceeds the largest fraction by \( \frac{3}{16} \). If \( a + b + c = \frac{49}{24} \), then what is the difference between \( c \) and \( b \)?
The value of the expression \[ \frac{0.1\overline{8} \times 11.0 \times 0.8\overline{3}}{2.\overline{4} \times 0.\overline{6} \times 3 \times 0.1\overline{6}} \] is:
What is the value of the expression \[ \frac{(4.8)^4 + (3.5)^4 + 282.24}{(4.8)^2 + (3.5)^2 - 16.8} \]