CBSE Class 10 Pair of Linear Equations in Two Variables Sure Shot Questions Set 13

Question. When \( \frac{1}{2x} + \frac{1}{3y} = 2 \) and \( \frac{1}{3x} + \frac{1}{2y} = \frac{13}{6} \) are solved by reducing them to a pair of linear equations, we get
(a) \( x = \frac{1}{2} , y = \frac{1}{3} \)
(b) \( x = \frac{1}{4} , y = \frac{1}{5} \)
(c) \( x = \frac{1}{5} , y = \frac{1}{4} \)
(d) \( x = \frac{1}{5} , y = \frac{1}{7} \)
Answer: (a) \( x = \frac{1}{2} , y = \frac{1}{3} \)

Question. The value of \( x \) and \( y \) for the following system of equations: \( \frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2 \) and \( \frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1 \), respectively are
(a) \( x = 4, y = 9 \)
(b) \( x = 5, y = 10 \)
(c) \( x = 8, y = -9 \)
(d) None of the options
Answer: (a) \( x = 4, y = 9 \)

Question. Given the following system of equations: \( \frac{10}{x + y} + \frac{2}{x - y} = 4 \) and \( \frac{15}{x + y} - \frac{5}{x - y} = -2 \). Then which of the following is true?
(a) \( x = 2, y = 5 \)
(b) \( x = 1, y = 2 \)
(c) \( x = 3, y = 2 \)
(d) None of the options
Answer: (c) \( x = 3, y = 2 \)

Question. If \( \frac{5}{x - 1} + \frac{1}{y - 2} = 2 \) and \( \frac{6}{x - 1} - \frac{3}{y - 2} = 1 \)
(a) \( x = 1, y = 2 \)
(b) \( x = 2, y = 3 \)
(c) \( x = 4, y = 5 \)
(d) \( x = 5, y = 7 \)
Answer: (c) \( x = 4, y = 5 \)

Question. If \( \frac{2}{x} + \frac{3}{y} = 13 \) and \( \frac{5}{x} - \frac{4}{y} = -2 \), then \( x + y \) equals
(a) \( \frac{1}{6} \)
(b) \( \frac{-1}{6} \)
(c) \( \frac{5}{6} \)
(d) \( \frac{-5}{6} \)
Answer: (c) \( \frac{5}{6} \)

Question. If \( \frac{2}{x} + 2y = 15 \) and \( \frac{2}{x} - 4y = 3 \), then the values of \( x \) and \( y \), respectively are
(a) \( \frac{2}{11}, 2 \)
(b) \( 3, \frac{1}{3} \)
(c) \( 4, \frac{1}{4} \)
(d) \( \frac{1}{4}, 4 \)
Answer: (a) \( \frac{2}{11}, 2 \)

Question. If \( \frac{1}{x} = u \) and \( \frac{1}{y} = v \), then \( \frac{2}{x} + \frac{3}{y} = 13 \) becomes
(a) \( 3u + 2v = 13 \)
(b) \( 2u + 3v = 13 \)
(c) \( \frac{2}{u} + \frac{3}{v} = 13 \)
(d) \( \frac{2}{v} + \frac{3}{u} = 13 \)
Answer: (b) \( 2u + 3v = 13 \)

Question. Equation \( \frac{5}{x - 1} + \frac{1}{y - 2} = 2 \) is reduced in linear equation as \( 5p + q = 2 \). Then the values of \( p \) and \( q \) respectively are
(a) \( \frac{1}{x - 1}, \frac{1}{y - 2} \)
(b) \( \frac{1}{y - 1}, \frac{1}{x - 2} \)
(c) \( x - 1, y - 2 \)
(d) \( x + 1, y + 2 \)
Answer: (a) \( \frac{1}{x - 1}, \frac{1}{y - 2} \)

Question. If \( \frac{2}{x} + \frac{3}{y} = 13 \) and \( \frac{5}{x} - \frac{4}{y} = -2 \), then the values of \( x \) and \( y \) respectively are
(a) \( \frac{1}{2}, \frac{1}{3} \)
(b) \( \frac{1}{3}, \frac{1}{4} \)
(c) \( \frac{1}{2}, \frac{1}{5} \)
(d) \( 2, 3 \)
Answer: (a) \( \frac{1}{2}, \frac{1}{3} \)

Question. \( \frac{1}{3x + y} + \frac{1}{3x - y} = \frac{3}{4} \) and \( \frac{1}{2(3x + y)} - \frac{1}{2(3x - y)} = \frac{-1}{8} \), where \( 3x + y \neq 0, 3x - y \neq 0 \), then
(a) \( 1, 2 \)
(b) \( 1, 1 \)
(c) \( 1, 3 \)
(d) \( 1, 4 \)
Answer: (b) \( 1, 1 \)

Question. If \( \frac{3a}{x} - \frac{2b}{y} + 5 = 0 \) and \( \frac{a}{x} + \frac{3b}{y} - 2 = 0 \), then
(a) \( x = \frac{1}{a}, y = \frac{1}{b} \)
(b) \( x = \frac{1}{b}, y = \frac{1}{a} \)
(c) \( x - a, y = b \)
(d) \( x = a, y = -b \)
Answer: (a) \( x = \frac{1}{a}, y = \frac{1}{b} \)

Question. If \( \frac{1}{2x} - \frac{1}{y} = -1 \) and \( \frac{1}{x} + \frac{1}{2y} = 8 \), (\( x \neq 0, y \neq 0 \)), then
(a) \( x = \frac{1}{4}, y = \frac{1}{2} \)
(b) \( x = \frac{1}{3}, y = \frac{1}{5} \)
(c) \( x = \frac{1}{6}, y = \frac{1}{8} \)
(d) \( x = \frac{1}{6}, y = \frac{1}{4} \)
Answer: (d) \( x = \frac{1}{6}, y = \frac{1}{4} \)

Solving Word Problems on Pair of Linear Equations in Two Variables

Question. Ritu can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. The pair of linear equations representing the above situation is
(a) \( x + y = 10, x + y = 2 \)
(b) \( x + y = 10, x - y = 2 \)
(c) \( x - y = 10, x + y = 2 \)
(d) \( x - y = 10, x - y = 2 \)
Answer: (b) \( x + y = 10, x - y = 2 \)

Question. A part of monthly hostel charges in a college is fixed and the remaining depends on the number of days one has taken food in the mess. When a student ‘A’ takes food for 22 days, he has to pay ₹ 1380 as hostel charges; whereas a student ‘B’, who takes food for 28 days, pays ₹ 1680 as hostel charges. The fixed charges and the cost of food per day respectively are
(a) ₹ 280, ₹ 50
(b) ₹ 240, ₹ 60
(c) ₹ 290, ₹ 100
(d) ₹ 170, ₹ 210
Answer: (a) ₹ 280, ₹ 50

Question. Atul sold a television set and a mobile phone for ₹ 10500, thereby making a profit of 10% on the television set and 25% on the mobile phone. If he had taken a profit of 25% on the television set and 10% on the mobile phone, he would have got ₹ 10650. The cost of each item is
(a) TV : ₹ 7000, Mobile : ₹ 5000
(b) TV : ₹ 6000, Mobile : ₹ 4000
(c) TV : ₹ 4000, Mobile : ₹ 5000
(d) TV : ₹ 5000, Mobile : ₹ 4000
Answer: (d) TV : ₹ 5000, Mobile : ₹ 4000

Question. Base of an isosceles triangle is \( \frac{2}{3} \) times its congruent sides. Perimeter of the triangle is 32 cm. The length of each side of that triangle is
(a) 8 cm, 8 cm, 6 cm
(b) 12 cm, 12 cm, 8 cm
(c) 10 cm, 10 cm, 12 cm
(d) None of the options
Answer: (b) 12 cm, 12 cm, 8 cm

Question. Places A and B are 80 km apart from each other on a highway. A car starts from A and another from B at the same time. If they move in same direction they meet in 8 hrs and if they move in opposite directions they meet in 1 hr 20 minutes. The speeds of cars started from places A and B respectively are
(a) 40 km/hr, 55 km/hr
(b) 20 km/hr, 30 km/hr
(c) 35 km/hr, 25 km/hr
(d) 30 km/hr, 28 km/hr
Answer: (c) 35 km/hr, 25 km/hr

Question. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. The speed of the stream and that of the boat in still water, respectively are
(a) 10 km/hr, 12 km/hr
(b) 3 km/hr, 8 km/hr
(c) 5 km/hr, 4 km/hr
(d) 4 km/hr, 6 km/hr
Answer: (b) 3 km/hr, 8 km/hr

Question. Aruna has only ₹ 1 and ₹ 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹ 75, then the number of ₹ 1 and ₹ 2 coins are, respectively
(a) 35 and 15
(b) 35 and 20
(c) 15 and 35
(d) 25 and 25
Answer: (d) 25 and 25

Question. The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages, (in years) of the son and the father are, respectively
(a) 4 and 24
(b) 5 and 30
(c) 6 and 36
(d) 3 and 24
Answer: (c) 6 and 36

Question. A purse contains 25 paise and 10 paise coins. The total amount in the purse is ₹ 8.25. If the number of 25 paise coins is one-third of the number of 10 paise coins in the purse, then the total number of coins in the purse is
(a) 60
(b) 40
(c) 80
(d) 72
Answer: (a) 60

Question. A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, the number of hens will be
(a) 18
(b) 26
(c) 32
(d) 40
Answer: (b) 26

Question. If 3 chairs and 1 table costs ₹ 1500 and 6 chairs and 1 table costs ₹ 2400, the pair of linear equations to represent this situation is
(a) \( 6x + y = 1500, 3x + y = 2400 \)
(b) \( \frac{x}{3} + y = 1500, \frac{x}{6} + y = 2400 \)
(c) \( 3x + y = 1500, 6x + y = 2400 \)
(d) None of the options
Answer: (c) \( 3x + y = 1500, 6x + y = 2400 \)

Question. A fraction becomes \( \frac{1}{3} \) when 2 is subtracted from the numerator and it becomes \( \frac{1}{2} \) when 1 is subtracted from the denominator. The fraction is
(a) \( \frac{2}{5} \)
(b) \( \frac{5}{18} \)
(c) \( \frac{4}{13} \)
(d) \( \frac{7}{15} \)
Answer: (d) \( \frac{7}{15} \)

Question. The difference between two numbers is 26 and the larger number exceeds thrice of the smaller number by 4. The numbers are
(a) 39, 13
(b) 12, 38
(c) 37, 11
(d) None of the options
Answer: (c) 37, 11

Question. Meena went to a bank to withdraw ₹ 2,000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Meena got 25 notes in all. How many notes of ₹ 50 and ₹ 100 she received?
(a) ₹ 50 : 10, ₹ 100 : 15
(b) ₹ 50 : 12, ₹ 100 : 10
(c) ₹ 50 : 15, ₹ 100 : 10
(d) None of the options
Answer: (a) ₹ 50 : 10, ₹ 100 : 15

Question. A motor boat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. The speed of the stream is
(a) 60 km/hr
(b) 6 km/min
(c) 6 km/s
(d) 6 km/hr
Answer: (d) 6 km/hr

Question. The two consecutive odd positive integers, sum of whose squares is 290 are
(a) 5, 13
(b) 11, 13
(c) 13, 17
(d) None of the options
Answer: (b) 11, 13

Question. A two-digit number is obtained by either multiplying the sum of digits by 8 and then subtracting 5 or by multiplying the difference of digits by 16 and adding 3. The number is
(a) 23
(b) 34
(c) 83
(d) 119
Answer: (c) 83

Question. The sum of a two-digit number and the number obtained by interchanging the digits is 132. If the two digits differ by 2, the number is
(a) 45
(b) 75
(c) 85
(d) 115
Answer: (b) 75

Question. Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. The present age of father and son, respectively are
(a) 40 years, 12 years
(b) 30 years, 6 years
(c) 32 years, 8 years
(d) 42 years, 10 years
Answer: (d) 42 years, 10 years

Case Study Based Questions

Rupesh purchased 3 chairs and one table for ₹ 1600 and his friend purchased 5 chairs and 2 tables for ₹ 2900. If in both the cases the price of chairs and tables are the same, Rupesh wants to know some answers of his question what are in his mind. Help him to solve his problems.

Question. Using the cost price of one chair as ‘x’ and the cost price of one table as ‘y’, the linear pair of equations of the above statement are as follows
(a) \( 3x + y = 1600, 5x + 2y = 2900 \)
(b) \( 3x - y = 1600, 5x - 2y = 2900 \)
(c) \( 3x + y = 1600, 5x - 2y = 2900 \)
(d) \( 3x - y = 1600, 5x + 2y = 2900 \)
Answer: (a) \( 3x + y = 1600, 5x + 2y = 2900 \)

Question. The linear pair of equations formed above by the given information are
(a) Consistent
(b) Inconsistent
(c) Dependent
(d) Both (a) and (c)
Answer: (a) Consistent

Question. If lines are drawn for the linear pair of equations formed by the above information, they will
(a) intersect at one point
(b) be parallel to each other
(c) be coincident
(d) not exist
Answer: (a) intersect at one point

Question. The solutions of the linear pair of equations formed by the above information are
(a) ₹ 300, ₹ 700
(b) ₹ 500, ₹ 600
(c) ₹ 450, ₹ 700
(d) ₹ 200, ₹ 1100
Answer: (a) ₹ 300, ₹ 700

Question. If Rupesh had purchased 2 chairs and 2 tables the sum of total cost will be
(a) ₹ 2200
(b) ₹ 1800
(c) ₹ 2000
(d) ₹ 2400
Answer: (c) ₹ 2000

Travelling by a car on a highway gives lots of fun and thrills. People drive and enjoy the moment of thrills. On morning two friends who are living at a distance of 150 km apart decided to meet at another place. So, they drive their cars from point A and point B at the same time and meet after 15 hours on a highway. In an another incident they decide to meet in a hurry. So, they drive their cars in opposite directions and meet in one hour.

Question. Using the speed of car at A, \( x \) km/h and the speed of car at B, \( y \) km/h, the pair of linear equations representing the situation is
(a) \( 15x + 15y = 150, x + y = 150 \)
(b) \( 15x - 15y = 150, x + y = 150 \)
(c) \( 15x - 15y = 150, x - y = 150 \)
(d) \( 15x + 15y = 150, x - y = 150 \)
Answer: (b) \( 15x - 15y = 150, x + y = 150 \)

Question. On comparing the coefficients of the pair of linear equations formed by the above situations, following conditions can arise:
(a) \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
(b) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
(c) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
(d) None of the options
Answer: (a) \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)

Question. The lines drawn on the graph paper for the pair of linear equations formed due to above situations can have one of following possibilities.
(a) The two lines intersect at one point
(b) The two lines do not intersect
(c) The two lines coincide
(d) Can not be said anything
Answer: (a) The two lines intersect at one point

Question. The solutions of the pair of linear equations formed by above situation is
(a) a unique solution
(b) no solution
(c) infinitely many solution
(d) not to be determined
Answer: (a) a unique solution

Question. The speeds of car at A and car at B are
(a) 70 km/h, 80 km/h
(b) 90 km/h, 60 km/h
(c) 80 km/h, 70 km/h
(d) 60 km/h, 90 km/h
Answer: (c) 80 km/h, 70 km/h