List of top Mathematics Questions

If a certain variable \(x\) divides a statistical data arranged in order into two equal parts; then the value of \(x\) is called the:

For what value of \(\theta\), \(\sin^2\theta + \sin\theta + \cos^2\theta\) is equal to 2?

The distance between the points \((2, -3)\) and \((-2, 3)\) is:

If probability of winning a game is \(p\), then probability of losing the game is:

If the roots of quadratic equation \(4x^2 - 5x + k = 0\) are real and equal, then value of \(k\) is:

HCF \((132, 77)\) is:

If \(\sin \theta = \frac{1}{3}\), then \(\sec \theta\) is equal to:

The mid-point of the line segment joining the points \((-1, 3)\) and \(\left(8, \frac{3}{2}\right)\) is:

In an A.P., if \(a = 8\) and \(a_{10} = -19\), then value of \(d\) is:

A road roller is a compactor-type engineering vehicle, used to compact soil, gravel, concrete, etc, in the construction of roads and foundations. They are also used at landfills or in agriculture. A company started making road rollers 10 years ago and increased its production uniformly by a fixed number every year. The company produces 800 rollers in the 6^th year and 1130 rollers in the 9^th year.
Based on the above information, answer the following questions :

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure.
On a similar concept, a radio station tower was built in two stations A and B (B vertically below A). The tower is supported by wires AO and BO from a point O on the ground. Distance between the base C of the tower and the point O is 36 m. From O, the angles of elevation of the tops of station B and station A are 30º and 45º respectively.
Based on the above, answer the following questions :

Student-teacher ratio expresses the relationship between the number of students enrolled in a school and the number of teachers employed by the school. This ratio is important for a number of reasons. It can be used as a tool to measure teachers’ workload as well as the allocation of resources. A survey was conducted in 100 secondary schools of a state and the following frequency distribution table was prepared :

Number of students per Teacher	Number of Schools
20 - 25	5
25 - 30	15
30 - 35	25
35 - 40	30
40 - 45	15
45 - 50	10

Based on the above, answer the following questions :

The picture given below shows a circular mirror hanging on the wall with a cord. The diagram represents the mirror as a circle with centre \(O\). \(AP\) and \(AQ\) are tangents to the circle at \(P\) and \(Q\) respectively such that \(AP = 30 \, \text{cm}\) and \(\angle PAQ = 60^\circ\).

Based on the above information, answer the following questions:

Gurpreet is very fond of doing research on plants. She collected some leaves from different plants and measured their lengths in mm.
The length of the leaves from different plants are recorded in the following table.

\(\text{Length (in mm)}\)	70-80	80-90	90-100	100-110	110-120	120-130	130-140
\(\text{Number of leaves}\)	3	5	9	12	5	4	2

Based on the above information, answer the following questions :

To keep the lawn green and cool, Sadhna uses water sprinklers which rotate in circular shape and cover a particular area. The diagram below shows the circular areas covered by two sprinklers :

Using integration, find the area of the region enclosed between the curve \( y = \sqrt{4 - x^2} \), and the lines \( x = -1 \), \( x = 1 \), and the \( x \)-axis.

It is given that the function \( f(x) = x^4 - 62x^2 + ax + 9 \) attains a local maximum value at \( x = 1 \). Find the value of \( a \), and hence obtain all other points where the given function \( f(x) \) attains local maximum or local minimum values.

Let \( A = \mathbb{R} - \{3\} \) and \( B = \mathbb{R} - \{a\} \). Find the value of \( a \) such that the function \( f: A \to B \) defined by \( f(x) = \frac{x-2}{x-3} \) is onto. Also, check whether the given function is one-one or not.

Two vertices of the parallelogram ABCD are given as \( A(-1, 2, 1) \) and \( B(1, -2, 5) \). If the equation of the line passing through \( C \) and \( D \) is: \[ \frac{x - 4}{1} = \frac{y + 7}{-2} = \frac{z - 8}{2}, \] then find the distance between sides \( AB \) and \( CD \). Hence, find the area of parallelogram ABCD.

Find the equation of the line passing through the point of intersection of the lines \[ \frac{x - 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \] and \[ \frac{x - 1}{0} = \frac{y}{-3} = \frac{z - 7}{2}, \] and perpendicular to these given lines.

Evaluate: \[ \int \frac{2 + \sin 2x}{1 + \cos 2x} e^x \, dx. \]

Find: \[ I = \int \frac{\sqrt{x}}{(x+1)(x-1)} \, dx \]

Find the absolute maximum and absolute minimum values of the function \( f(x) = x^2 + \frac{2}{x} \) on the interval \([1, 2]\).

Find the intervals in which the function \( f(x) = \frac{\log x}{x} \) is strictly increasing or strictly decreasing.

Find the value of \( a \) and \( b \) so that the function \( f(x) \), defined as: \[ f(x) = \begin{cases} \frac{x - 2}{|x - 2|} + a, & x < 2, \\ a + b, & x = 2, \\ \frac{x - 2}{|x - 2|} + b, & x > 2, \end{cases} \] is a continuous function.