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

Question. Graphically, the pair of equation
6x-3y+10=0
2x-y+9=0
Represents two lines which are
(a) Intersecting at exactly one point
(b) Intersecting at exactly two point
(c) Coincident
(d) Parallel
Answer: (d) Parallel

Question. The pair of equation x+2y+5=0 and -3x-6y+1=0 have :
(a) A unique solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No Solution
Answer: (d) No Solution

Question. If a pair of linear equations is consistent, then the lines will be:
(a) Parallel
(b) Always coincident
(c) Intersecting or coincident
(d) Always interesting
Answer: (c) Intersecting or coincident

Question. The pair of equation x=a and y=b graphically represents lines which are:
(a) Parallel
(b) Intersecting at (b,a)
(c) Coincident
(d) Intersecting at (a,b)
Answer: (d) Intersecting at (a,b)

Question. The pair of equation y=0 and y=-7 has:
(a) One solution
(b) Two solution
(c) Infinitely many solution
(d) No solution
Answer: (d) No solution

Question. One equation of a pair of dependent linear equations is -5x+7y=2. The second equation can be:
(a) 10x+14y+4=0
(b) -10x-14y+4=0
(c) -10x+14y+4=0
(d) 10x-14y = -4
Answer: (d) 10x-14y = -4

Question. For what value of k, do the equations 3x-y+8=0 and 6x-ky=-16 represents coincident lines?
(a) \( \frac{1}{2} \)
(b) \( -\frac{1}{2} \)
(c) 2
(d) -2
Answer: (c) 2

Question. If the lines given by 3x+2y=2 and 2x+5y+1=0 are parallel, then the value of k is
(a) \( -\frac{5}{4} \)
(b) \( \frac{2}{5} \)
(c) \( \frac{15}{4} \)
(d) \( \frac{3}{2} \)
Answer: (c) \( \frac{15}{4} \)

Question. A pair of linear equation which has a unique solution x=2, y=-3 is
(a) x+y=-1 and 2x-3y=-5
(b) 2x+5y=-11 and 4x+10y=-22
(c) 2x-y=1 and 3x+2y=0
(d) X-4y-14=0 and x-y-13=0
Answer: (b) and (d) 2x+5y=-11 and 4x+10y=-22 and X-4y-14=0 and x-y-13=0

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. If x=a, y=b is the solution of the equations x-y=2 and x+y=4, then the values of a and b are, respectively
(a) 3 and 5
(b) 5 and 3
(c) 3 and 1
(d) -1 and 3
Answer: (c) 3 and 1

Question. The larger of the two supplementary angles exceed the smaller by 18°, then the angles are:
(a) 99°, 81°
(b) 98°, 82°
(c) 97°, 83°
(d) None of the options
Answer: (a) 99°, 81°

Question. x and y are 2 different digits. If the sum of the two digit numbers formed by using both the digits is a perfect square, then value of x +y is
(a) 10
(b) 11
(c) 12
(d) 13
Answer: (b) 11

Question. In a number of two digits, unit’s digit is twice the tens digit. If 36 be added to the number, the digits are reversed. The number is
(a) 36
(b) 63
(c) 48
(d) 84
Answer: (c) 48

Assertion and Reason Based MCQ

Direction: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is the NOT the correct explanation of A
(c) A is true but R is false
(d) A is false and R is True

Question. Assertion (A): If the pair of linear equations 3x+y=3 and 6x+ky =8 does not have a solution, then the value of k=2.
Reason (R): If the pair of linear equations x+y-4=0 and 2x+ky =3 does not have a solution, then the value of k=2.
(a) A
(b) B
(c) C
(d) D
Answer: (b) Both A and R are true but R is the NOT the correct explanation of A

Question. Assertion (A): For all real values of c, the pair of equation x-2y=8 and 5x-10y=c have a unique solution.
Reason (R): Two lines are given to be parallel. The equation of one of the lines is 4x+3y=14, 12x+9y=5
(a) A
(b) B
(c) C
(d) D
Answer: (d) A is false and R is True

Question. Assertion (A): If the equation 3x-y+8=0 and 6x-ky =-16 represents coincident lines, then the value of k=2.
Reason (R): If the lines given by 3x+2ky=2 and 2x+5y+1 =0 are parallel, then the value of k=12.
(a) A
(b) B
(c) C
(d) D
Answer: (c) A is true but R is false

Question. Assertion (A): If 4 chairs and 3 tables cost ₹2100 and 5 chairs and 2 tables cost ₹1750 , then the cost of 1 chair is ₹150.
Reason (R): Sum of the ages of a father and the son is 40 years. If father’s age is 3 times that of his son, then the son’s age is 12 years.
(a) A
(b) B
(c) C
(d) D
Answer: (c) A is true but R is false

Question. Assertion (A): The solution of the pair of linear equations x+y=5 and 2x-3y =4 is \( x=\frac{19}{5} \) and \( y=\frac{6}{5} \)
Reason (R): The solution of the pair of linear equations 3x+4y=10 and 2x-2y=2 is x=2 and y=1.
(a) A
(b) B
(c) C
(d) D
Answer: (b) Both A and R are true but R is the NOT the correct explanation of A

2 Mark Question

Question. Given the linear equation 3x+4y = 9. Write another linear equation in these two variables such that the geometrical representation of the pair so formed is: (1) intersecting lines (2) coincident lines.
Answer: (1) One of the possible equation 3x-5y=10 (2) One of the possible equation 6x+8y=18

Question. For what value of p does the pair of linear equations given below has unique solution?
4 x+ py +8 = 0 and 2 x+2 y + 2 =0 .
Answer: \( p \neq 4 \)

Question. Is the system of linear equations 2 x+3 y − 9 =0 and 4 x+6 y − 18 =0 consistent? Justify your answer.
Answer: Consistent

Question. Two lines are given to be parallel. The equation of one of the lines is 4x + 3y = 14, then find the equation of the second line.
Answer: One of the possible solution 12x+9y=5

Question. Find the value(s) of k for which the pair of linear equations kx + y = k² and x + ky = 1 have infinitely many solutions.
Answer: K=1 or k= -1

3 Mark Question

Question. Solve graphically: 2x − 3y + 13 = 0; 3x − 2y + 12 = 0
Answer: X= -2 and y=3

Question. Find the value of k for which the following pair of equations has no solution :
x + 2y = 3, (k-1)x+(k+1)y=(k+2)
Answer: K=3

Question. Solve x + y = 5 and 2x − 3y = 4 by elimination method and the substitution method.
Answer: \( X = \frac{19}{5} \) and \( y = \frac{6}{5} \)

Question. Half the perimeter of a rectangular garden, whose length is 4 m more then its width, is 36 m. Find the dimensions of garden.
Answer: Length = 20m and width = 16m

Question. Determine graphically whether the following pair of linear equations :
3x - y = 7
2x + 5y + 1 = 0 has : unique solution infinitely many solutions or no solution.
Answer: Unique solution

Question. Solve : 99x + 101y = 499, 101x + 99y = 501
Answer: X=2, y=3

5 Mark Question

Question. 2 man and 7 boys can do a piece of work in 4 days. It is done by 4 men and 4 boys in 3 days. How long would it take for one man or one boy to do it ?
Answer: 15 days

Question. A fraction become \( \frac{9}{11} \) if 2 is added to both numerator and denominator. If 3 is added to both numerator and denominator it becomes \( \frac{5}{6} \). find the fraction.
Answer: \( \frac{7}{9} \)

Question. The ratio of incomes of two persons is 11:7 and the ratio of their expenditures is 9:5. If each of them manages to save Rs 400 per month, find their monthly incomes.
Answer: 2200 and 1400

Question. Solve the following pair of equations graphically: 2x + 3y = 12, x − y − 1 = 0. Shade the region between the two lines represented by the above equations and the X -axis.
Answer: [Draw the graph]

Question. A chemist has one solution which is 50% acid and a second which is 25% acid. How much of each should be mixed to make 10 litre of 40% acid solution.
Answer: x=6, y=4

Case Based Questions

Question. Read the following text and answer the following questions on the basis of the same:
It is common that governments revise travel fares from time to time based on various factors such as inflation (a general increase in prices and fall in the purchasing value of money) on different types of vehicles like auto, Rickshaws, taxis, Radio cab etc. The auto charges in a city comprise of a fixed charge together with the charge for the distance covered. Study the following situations.
City A: 10 km journey, ₹75 paid; 15 km journey, ₹110 paid.
City B: 8 km journey, ₹91 paid; 14 km journey, ₹145 paid.
SITUATION 1: In a city A, for a journey of 10 km, the charge paid is ₹75 and for a journey of 15 km, the charge paid is ₹110.
SITUATION 2: In a city B, for a journey of 8 km, the charge paid is ₹91 and for a journey of 14 km, the charge paid is ₹145.
REFER SITUATION 1
(i) If the fixed charges of auto rickshaw be ₹ x and the running charges be ₹ y km/hr. then write the pair of linear equation representing the situation is …………………….
(ii) A person travels a distance of 50 km. find the amount paid by Him……………………………….
REFER SITUATION 2
(iii) What will a person have to pay for travelling a distance of 30 km?
(iv) The graph of lines representing the conditions are …..
Answer:
(i) X+10y=75, x+15y=75
(ii) Rs. 355
(iii) Rs. 289
(iv) Draw graph

Question. Read the following text and answer the following questions on the basis of the same:
Place A and B are 100 km apart on a highway. One car starts from A and another form B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour.
(i) Assuming that the speed of first car and second car be u km/h and v km/h respectively. What is the relative speed of both cars while they are travelling in the same direction?
(ii) What is the relative speed of both cars while they are travelling towards each other?
(iii) What is the actual speed of the car?
(iv) What is the actual speed of the other car?
Answer:
(i) (u-v)km/h
(ii) (u+v)km/h
(iii) 60km/h
(iv) 40km/h

Question. Read the following text and answer the following questions on the basis of the same:
John and Jivanti are playing with the marbles in the playground. They together have 45 marbles and John has 15 marbles more than Jivanti.
(i) The number of marbles Jivanti had……………
(ii) The number of marbles John had………………
(iii) If 45 is replaced by 55 in the above case discussed in the question, then the number of marbles Jivanti have………………..
(iv) According to Question 3, the number of marbles John have:
Answer:
(i) 15
(ii) 30
(iii) 20
(iv) 35

Question. Read the following text and answer the following questions on the basis of the same:
TOWER OF PISA : To prove that objects of different weights fall at the same rate, Galileo dropped two objects with different weights from the Leaning Tower of Pisa in Italy. The objects hit the ground at the same time. An object dropped off the top of Leaning Tower of Pisa falls vertically with constant acceleration. If s is the distance of the object above the ground (in feet) t seconds after its release, then s and t are related by an equation of the form \( s = a + bt^2 \) where a and b are constants. Suppose the object is 180 feet above the ground 1 second after its release and 132 feet above the ground 2 seconds after its release.
(i) Find the constants a and b .
(ii) How high is the Leaning Tower of Pisa?
(iii) How long does the object fall?
Answer:
(i) a=196, b=-16
(ii) 196 feet
(iii) 3.5 sec

Question. Read the following text and answer the following questions on the basis of the same:
Varsha is a licensed architect and design very innovative house. She has made a house layout for her client which is given below. In the layout, the design and measurements has been made such that area of two bedrooms and kitchen together is 95 sq. m.
(i)Which pair of linear equations does describe this situation?
(ii) What is the length of the outer boundary of the layout?
(iii) What is the area of bedroom 1 ?
(iv) What is the area of living room in the layout?
(v) What is the cost of laying tiles in Kitchen at the rate of Rs. 50 per sq.
Answer:
(i) 2x+y=19 and x+y=13
(ii) 54m
(iii) Area of bedroom=30sq.m
(iv) 75 sq.m
(v) Rs. 1750