List of practice Questions

Examine whether the following statements are true or false:
(i) {a, b} \(⊄\) {b, c, a}
(ii) {a, e} \(⊂\) {x: x is a vowel in the English alphabet}
(iii) {1, 2, 3} \(⊂\) {1, 3, 5}
(iv) {a} \(⊂ \) {a. b, c}
(v) {a} \(∈\) (a, b, c)
(vi) {x: x is an even natural number less than 6} \(⊂\) {x: x is a natural number which divides 36}

Make correct statements by filling in the symbols \( ⊂ \) or \(⊄\) in the blank spaces:
(i) {2, 3, 4} … {1, 2, 3, 4, 5}
(ii) {a, b, c} … {b, c, d}
(iii) {x: x is a student of Class XI of your school} … {x: x student of your school}
(iv) {x: x is a circle in the plane} … {x: x is a circle in the same plane with radius 1 unit}
(v) {x: x is a triangle in a plane}…{x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane}… {x: x is a triangle in the same plane} 
(vii) {x: x is an even natural number} … {x: x is an integer}

From the sets given below, select equal sets:
A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}
E = {-1, 1}, F = {0, a}, G = {1, -1}, H = {0, 1}

Are the following pair of sets equal? Give reasons.
(i) A = {2, 3}; B = {x: x is solution of \(x^2+ 5x + 6 = 0\)}
(ii) A = {x: x is a letter in the word FOLLOW}; B = {y: y is a letter in the word WOLF}

State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0, 0)

Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x: x is a natural numbers, x < 5 and x > 7 }
(iv) {y: y is a point common to any two parallel lines}

Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3 ...}
(iii) {1, 2, 3 ... 99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99

List all the elements of the following sets: 
(i) A = {x: x is an odd natural number} 
(ii) B = {x: x is an \(\frac{-1}{2} < x <\frac{ 9}{2}\)  integer,} 
(iii) C = {x: x is an \(x^2 ≤ 4 \) integer, 
(iv) D = {x: x is a letter in the word “LOYAL“} 
(v) E = {x: x is a month of a year not having 31 days} 
(vi) F = {x: x is a consonant in the English alphabet which proceeds k}.

Write the following sets in the set-builder form:
(i) (3, 6, 9, 12)  (ii) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}  (iv) {2, 4, 6 …}
(v) {1, 4, 9 … 100}

Which of the following statements is incorrect regarding the polar satellite?

Write the following sets in roster form: 
(i) A = {x: x is an integer and \(-3 < x < 7\)}. 
(ii) B = {x: x is a natural number less than 6}. 
(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8} 
(iv) D = {x: x is a prime number which is divisor of 60}. 
(v) E = The set of all letters in the word TRIGONOMETRY. 
(vi) F = The set of all letters in the word BETTER

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol \(∈\) or \(∉\) in the blank spaces: 
(i) 5…A  (ii) 8…A  (iii) 0…A  (iv) 4…A  (v) 2…A  (vi) 10…A

Which of the following are sets? Justify our answer. 
(i) The collection of all months of a year beginning with the letter J. 
(ii) The collection of ten most talented writers of India. 
(iii) A team of eleven best-cricket batsmen of the world. 
(iv) The collection of all boys in your class. 
(v) The collection of all natural numbers less than 100. 
(vi) A collection of novels written by the writer Munshi Prem Chand. 
(vii) The collection of all even integers. 
(viii) The collection of questions in this Chapter. 
(ix) A collection of most dangerous animals of the world.

Find the distance of the line \(4x + 7y + 5 = 0\)  from the point (1, 2) along the line  \(2x - y = 0\)

The hypotenuse of a right-angled triangle has its ends at the points (1, 3) and (-4, 1). Find the equation of the legs (perpendicular sides) of the triangle.

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

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 three lines whose equations are y = m₁x + c₁, y = m₂x + c₂ and y = m₃x + c₃ are concurrent, then show that m₁ (c₂ – c₃) + m₂ (c₃ – c₁) + m₃ (c₁ – c₂) = 0.

Find the equation of the lines through the point (3, 2) which make an angle of 45° with the line  \(x -2y = 3.\)

Find the equation of the line passing through the point of intersection of the lines \(4x + 7y - 3 = 0\) and \(2x - 3y+ 1 = 0\)  that has equal intercepts on the axes.

Show that the equation of the line passing through the origin and making an angle θ with the line \(y=mx+c\) is \(\frac{y}{x}=\frac{m±tanθ}{1-+mtanθ}\)

In what ratio, the line joining (-1, 1) and (5, 7) is divided by the line x + y = 4?

Find the perpendicular distance from the origin to the line joining the points \((cosθ,sinθ) \) and \((cos\phi,sin\phi).\)

Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines  \(x - 7y+ 5 = 0\) and \(3x + y = 0.\)

Find the equation of a line drawn perpendicular to the line \(\frac{x}{4} +\frac{ y}{6} = 1 \) through the point where it meets the y-axis