Question 1

Graphically, the pair of equations
6x–3y+10=06x – 3y + 10 = 06x–3y+10=0
2x–y+9=02x – y + 9 = 02x–y+9=0
are represented by two lines that are:
(a) Intersecting at exactly one point
(b) Intersecting at exactly two points
(c) Coincident
(d) Parallel

Answer 1: (d) Parallel

Question 2

The pair of equations x+2y+5=0x + 2y + 5 = 0x+2y+5=0 and −3x–6y+1=0-3x – 6y + 1 = 0−3x–6y+1=0 have:
(a) Unique solution
(b) Exactly two solutions
(c) Infinitely many solutions
(d) No solution

Answer 2: (d) No solution

Question 3

When the pair of linear equations is consistent, then the lines will be:
(a) Parallel
(b) Always coincident
(c) Intersecting or coincident
(d) Always intersecting

Answer 3: (c) Intersecting or coincident

Question 4

The pair of equations y=0y = 0y=0 and y=–7y = –7y=–7 have:
(a) One solution
(b) Two solutions
(c) Infinitely many solutions
(d) No solution

Answer 4: (d) No solution

Question 5

The pair of equations x=ax = ax=a and y=by = by=b graphically represents lines that are:
(a) Parallel
(b) Intersecting at (b, a)
(c) Coincident
(d) Intersecting at (a, b)

Answer 5: (d) Intersecting at (a, b)

Question 6

Do the following pair of linear equations have no solution? Explain your answer.

(i) 2x+4y=32x + 4y = 32x+4y=3 and 12y+6x=612y + 6x = 612y+6x=6
(ii) x=2yx = 2yx=2y and y=2xy = 2xy=2x
(iii) 3x+y–3=03x + y – 3 = 03x+y–3=0 and 2x+23y=22x + \frac{2}{3}y = 22x+32​y=2

Answer 6:
(i) Yes. The equations are parallel, so they have no solution.
(ii) No. The equations intersect, so they have a unique solution.
(iii) No. The equations are coincident, so they have infinitely many solutions.

Question 7

Do the following equations represent a pair of coincident lines? Justify your answer.

(i) 3x+17y=33x + \frac{1}{7}y = 33x+71​y=3 and 7x+3y=77x + 3y = 77x+3y=7
(ii) −2x–3y=1-2x – 3y = 1−2x–3y=1 and 6y+4x=–26y + 4x = –26y+4x=–2
(iii) x2+y+25=0\frac{x}{2} + y + \frac{2}{5} = 02x​+y+52​=0 and 4x+8y+516=04x + 8y + \frac{5}{16} = 04x+8y+165​=0

Answer 7:
(i) No, the equations are not coincident.
(ii) Yes, the equations are coincident.
(iii) No, the equations are not coincident.

Question 8

For which value(s) for λ\lambdaλ, do the pair of given linear equations λx+y=λ2\lambda x + y = \lambda^2λx+y=λ2 and x+λy=1x + \lambda y = 1x+λy=1 have
(i) no solution?
(ii) infinitely many solutions?
(iii) unique solution?

Answer 8:
(i) λ=−1\lambda = -1λ=−1 for no solution.
(ii) λ=1\lambda = 1λ=1 for infinitely many solutions.
(iii) All real values of λ\lambdaλ except 111 for a unique solution.

Question 9

For which value(s) of kkk would the pair of given equations kx+3y=k−3kx + 3y = k – 3kx+3y=k−3 and 12x+ky=k12x + ky = k12x+ky=k have no solution?

Answer 9: k=−6k = -6k=−6 for no solution.

Question 10

For which values of aaa and bbb, would the following pair of given linear equations consist of infinitely many solutions?

x+2y=1x + 2y = 1x+2y=1
(a−b)x+(a+b)y=a+b−2(a – b)x + (a + b)y = a + b – 2(a−b)x+(a+b)y=a+b−2

Answer 10: a=3a = 3a=3, b=1b = 1b=1 for infinitely many solutions.

NCERT Exemplar Class 10 Maths Chapter 3 Question 11 to 19

Question 11

Find out the value(s) of ppp in (i) to (iv) and ppp and qqq in (v) for the following pair of equations:

(i) 3x−y−5=03x – y – 5 = 03x−y−5=0 and 6x−2y−p=06x – 2y – p = 06x−2y−p=0, if the lines are parallel.
(ii) −x+py=1-x + py = 1−x+py=1 and px−y=1px – y = 1px−y=1, if the pair has no solution.
(iii) −3x+5y=7-3x + 5y = 7−3x+5y=7 and 2px−3y=12px – 3y = 12px−3y=1, if the lines intersect at a unique point.
(iv) 2x+3y−5=02x + 3y – 5 = 02x+3y−5=0 and px−6y−8=0px – 6y – 8 = 0px−6y−8=0, if the pair has a unique solution.
(v) 2x+3y=72x + 3y = 72x+3y=7 and 2px+py=28−qy2px + py = 28 – qy2px+py=28−qy, if the pair has infinitely many solutions.

Answer 11:
(i) p≠10p \neq 10p=10
(ii) p=1p = 1p=1
(iii) p≠910p \neq \frac{9}{10}p=109​
(iv) p≠−4p \neq -4p=−4
(v) p=4p = 4p=4, q=8q = 8q=8

Question 12

Two given straight paths are shown by the equations x−3y=2x – 3y = 2x−3y=2 and −2x+6y=5-2x + 6y = 5−2x+6y=5. Check if the paths cross each other or not.

Answer 12: No, the paths are parallel and do not cross each other.

Question 13

Write the pair of linear equations that have the unique solution x=−1x = -1x=−1, y=3y = 3y=3. How many pairs of this kind can you write?

Answer 13:
Example: 2x+6y=162x + 6y = 162x+6y=16 and 4x+12y=324x + 12y = 324x+12y=32.
There are infinitely many pairs of this kind.

Question 14

When 2x+y=232x + y = 232x+y=23 and 4x−y=194x – y = 194x−y=19, find the values of 5y−2x5y – 2x5y−2x and yx−2\frac{y}{x} – 2xy​−2.

Answer 14:
5y−2x=315y – 2x = 315y−2x=31
yx−2=−57\frac{y}{x} – 2 = -\frac{5}{7}xy​−2=−75​

Question 15

Find out the values for xxx and yyy in the following rectangle:

Answer 15:
x=1x = 1x=1
y=4y = 4y=4

Question 16

Aftab explains to his daughter, “Seven years ago, I was seven times as old as you were then. Moreover, three years from now, I shall be three times as old as you will be.” Represent the given situation algebraically and graphically.

Answer 16:
Let xxx be Aftab’s current age, and yyy be his daughter’s current age.
Algebraically: x−7=7(y−7)x – 7 = 7(y – 7)x−7=7(y−7) and x+3=3(y+3)x + 3 = 3(y + 3)x+3=3(y+3).
Graphically, plot the equations and find their intersection.

Question 17

The coach of the cricket team buys 3 bats and 6 balls for Rs.3900. Later, she buys another bat and 3 more balls for Rs.1300. Represent the given situation algebraically and geometrically.

Answer 17:
Let the cost of a bat be xxx and the cost of a ball be yyy.
Algebraically: 3x+6y=39003x + 6y = 39003x+6y=3900 and x+3y=1300x + 3y = 1300x+3y=1300.
Graphically, plot these equations and find their intersection.

Question 18

The cost of 2 kg of apples and 1 kg of grapes was Rs.160. A month later, the cost of 4 kg of apples and 2 kg of grapes is Rs.300. Represent the situation algebraically and geometrically.

Answer 18:
Let the cost of 1 kg of apples be xxx and the cost of 1 kg of grapes be yyy.
Algebraically: 2x+y=1602x + y = 1602x+y=160 and 4x+2y=3004x + 2y = 3004x+2y=300.
Graphically, plot these equations and find their intersection.

Question 19

Are the following pairs of given linear equations consistent? Explain your answer.

(i) −3x–4y=12-3x – 4y = 12−3x–4y=12 and 4y+3x=124y + 3x = 124y+3x=12
(ii) 35x–y=12\frac{3}{5}x – y = \frac{1}{2}53​x–y=21​ and 15x–3y=16\frac{1}{5}x – 3y = \frac{1}{6}51​x–3y=61​
(iii) 2ax+by=a2ax + by = a2ax+by=a and ax+2by−2a=0ax + 2by – 2a = 0ax+2by−2a=0, a,b≠0a, b \neq 0a,b=0
(iv) x+3y=11x + 3y = 11x+3y=11 and 2(2x+6y)=222(2x + 6y) = 222(2x+6y)=22

Answer 19:
(i) No, inconsistent (parallel lines).
(ii) Yes, consistent (unique solution).
(iii) Yes, consistent (infinitely many solutions).
(iv) No, inconsistent (parallel lines).