List of practice Questions

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A x B, find A and B, where x, y and z are distinct elements.

The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.

Let A = {1, 2, 3, ... ,14}. Define a relation R from A to A by R = {(x, y): 3x - y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

Define a relation R on the set Non natural numbers by R = {(x, y): y= x+ 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈A, y ∈B}. Write R in roster form.

The given figure shows a relationship between the sets P and Q. Write this relation

1. in set-builder form
2. in roster form.

What is its domain and range?

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{x}{1+tan\,x}\)

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by 
{(a, b): a, b ∈ A, b is exactly divisible by a}.

1. Write R in roster form
2. Find the domain of R
3. Find the range of R

Determine the domain and range of the relation R defined by R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}.

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \((x+sec\,x)(x-tan\,x)\)

Write the relation R = {(x, x³ ): x is a prime number less than 10} in roster form.

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): \(\frac{x}{sin\,n^x}\)

Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.

Fill in the blanks 

1. The volume of a cube of side 1 cm is equal to________\(m^3\).
2. The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to__________\(\text(mm)^2\).
3. A vehicle moving with a speed of 18 km \(h^{-1}\) covers ________m in 1 s.
4. The relative density of lead is 11.3. Its density is ....g \(\text {cm}^{-3}\) or ________kg \(\text m^{-3}\).

The slope of a line is double of the slope of another line. If tangent of the angle between them is \(\frac{1}{3}\) , find the slopes of he lines.

If the lines y = 3x + 1 and 2y = x + 3 are equally inclined to the line y = mx + 4, find the value of m.

If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y - 5 = 0 and 3x - 2y + 7 = 0 is always 10. Show that P must move on a line.

Fill in the blanks by suitable conversion of units  

1. 1 \(\text {kg}\) \(\text m^{2}\text s^{-2}\) = _______ \(\text g\) \(\text{cm}^2\text s^{-2}\) .
2. 1 \(\text m\) = ______ly.
3. 3.0 \(\text m^{2}\text s^{-2}\) = ________ \(\text {km}\) \(\text h^{-2}\)
4. G = 6.67 × \(10^{-11}\) N \(\text m^2\) \((\text{kg})^{-2}\) =_______ \((\text{cm})^3\text s^{-2} \text g^{-1}\).

Find equation of the line which is equidistant from parallel lines 9x + 6y - 7 = 0 and 3x + 2y + 6 = 0.

A line passes through (x₁, y₁) and (h, k). If slope of the line is m, show that k-y₁=m(h-x₁)

A calorie is a unit of heat (energy in transit) and it equals about 4.2 J where 1J = 1 \(\text {kg} \;\text m^2\; \text s^{-2}.\) Suppose we employ a system of units in which the unit of mass equals \(\alpha\) kg, the unit of length equals \(\beta\) m, the unit of time is \(\gamma\) s. Show that a calorie has a magnitude 4.2 \(\alpha^{-1}\; \beta^{-2}\; \gamma^{2}\) in terms of the new units.

A ray of light passing through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

Explain this statement clearly : “To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary : 

1. atoms are very small objects 
2. a jet plane moves with great speed 
3. the mass of Jupiter is very large 
4. the air inside this room contains a large number of molecules 
5. a proton is much more massive than an electron 
6. the speed of sound is much smaller than the speed of light.

Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9.