Practice Questions Chapter 3

Section A — Basic Concepts

Q1. The general form of a pair of linear equations in two variables is:

A. $ax^2+by=c$
B. $ax+by+c=0$
C. $ax^2+by^2=c$
D. $ax+by^2+c=0$

Answer: B

Solution: A linear equation in two variables is of the form $ax+by+c=0$, where $a$ and $b$ are not both zero.

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Q2. Which of the following represents a pair of linear equations?

A. $x^2+y=5,\ x+y=2$
B. $2x+3y=7,\ 4x-5y=9$
C. $xy=6,\ x+y=4$
D. $x^2+y^2=25,\ x-y=3$

Answer: B

Solution: Both equations in B have variables only to the first power and no product of variables.

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Q3. The graph of a linear equation in two variables is a:

A. Circle
B. Parabola
C. Straight line
D. Hyperbola

Answer: C

Solution: Every linear equation in two variables represents a straight line.

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Q4. A pair of linear equations having exactly one solution represents:

A. Parallel lines
B. Coincident lines
C. Intersecting lines
D. Curves

Answer: C

Solution: Two intersecting lines have exactly one common point.

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Q5. A pair of linear equations having infinitely many solutions represents:

A. Intersecting lines
B. Coincident lines
C. Parallel lines
D. Curves

Answer: B

Solution: Coincident lines overlap completely, so every point on the line is a common solution.

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Q6. A pair of linear equations having no solution represents:

A. Coincident lines
B. Intersecting lines
C. Parallel distinct lines
D. The same line

Answer: C

Solution: Distinct parallel lines never meet, so there is no common solution.

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Q7. If two lines intersect at one point, the pair is:

A. Inconsistent
B. Consistent and has a unique solution
C. Consistent and has infinitely many solutions
D. Neither

Answer: B

Solution: One common point means exactly one solution; hence the pair is consistent.

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Q8. If two lines are coincident, the number of solutions is:

A. 0
B. 1
C. 2
D. Infinitely many

Answer: D

Solution: Coincident lines have infinitely many common points.

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Q9. If two lines are parallel and distinct, the number of solutions is:

A. 0
B. 1
C. 2
D. Infinitely many

Answer: A

Solution: Distinct parallel lines have no common point.

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Q10. Which method involves drawing the two equations on a coordinate plane?

A. Substitution method
B. Elimination method
C. Graphical method
D. Factorisation method

Answer: C

Solution: In the graphical method, both equations are represented as lines and their intersection gives the solution.

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Section B — Conditions for Solutions

For
$$a_1x+b_1y+c_1=0$$
and
$$a_2x+b_2y+c_2=0,$$

Q11. The pair has a unique solution when:

A. $\frac{a_1}{a_2}=\frac{b_1}{b_2}$
B. $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$
C. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
D. $a_1=a_2=b_1=b_2$

Answer: B

Solution: If
$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2},$$
the lines intersect at exactly one point.

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Q12. The pair has no solution when:

A. $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$
B. $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$
C. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
D. $c_1=c_2=0$

Answer: B

Solution: Equal ratios of $a$ and $b$, but unequal $c$ ratios, represent distinct parallel lines.

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Q13. The pair has infinitely many solutions when:

A. $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$
B. $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$
C. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
D. $a_1=a_2$

Answer: C

Solution: Equal ratios of all three coefficients mean the two equations represent the same line.

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Q14. The equations $2x+3y=7$ and $4x+6y=14$ have:

A. No solution
B. Unique solution
C. Infinitely many solutions
D. Exactly two solutions

Answer: C

Solution:
$$\frac24=\frac36=\frac7{14}=\frac12.$$
Therefore, the lines are coincident.

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Q15. The equations $2x+3y=7$ and $4x+6y=15$ have:

A. Unique solution
B. No solution
C. Infinitely many solutions
D. Two solutions

Answer: B

Solution:
$$\frac24=\frac36=\frac12,\qquad \frac7{15}\ne\frac12.$$
Therefore, the lines are parallel and distinct.

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Q16. The equations $2x+3y=7$ and $4x+5y=9$ have:

A. No solution
B. Unique solution
C. Infinitely many solutions
D. Two solutions

Answer: B

Solution:
$$\frac24=\frac12,\qquad \frac35\ne\frac12.$$
Therefore, the lines intersect at one point.

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Q17. The pair $3x+2y=5,\ 6x+4y=10$ represents:

A. Parallel lines
B. Intersecting lines
C. Coincident lines
D. Perpendicular lines

Answer: C

Solution:
$$\frac36=\frac24=\frac5{10}=\frac12.$$

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Q18. The pair $3x+2y=5,\ 6x+4y=12$ represents:

A. Coincident lines
B. Parallel lines
C. Intersecting lines
D. Same x-axis

Answer: B

Solution:
$$\frac36=\frac24=\frac12,\qquad \frac5{12}\ne\frac12.$$
Hence parallel distinct lines.

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Q19. For $kx+3y=7$ and $2x+6y=9$ to have no solution, $k$ must be:

A. 1
B. 2
C. 3
D. 6

Answer: A

Solution:
For no solution,
$$\frac{k}{2}=\frac36=\frac12.$$
Therefore $k=1.$

Q20. For $2x+3y=7$ and $4x+ky=14$ to have infinitely many solutions, $k$ is:

A. 3
B. 4
C. 6
D. 7

Answer: C

Solution:
$$\frac24=\frac3k=\frac7{14}=\frac12.$$
Thus
$$\frac3k=\frac12\Rightarrow k=6.$$

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Section C — Solving Linear Equations

Q21. The solution of 

$$x+y=7,\quad x-y=3$$

is:

A. $(5,2)$
B. $(2,5)$
C. $(4,3)$
D. $(3,4)$

Answer: A

Solution:
Adding:
$$2x=10\Rightarrow x=5.$$
Then $y=7-5=2$.

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Q22. The solution of

$$2x+y=9,\quad x-y=3$$
is:

A. $(4,1)$
B. $(3,4)$
C. $(2,5)$
D. $(5,-1)$

Answer: A

Solution:
Adding the equations:
$$3x=12\Rightarrow x=4.$$
Then $4-y=3\Rightarrow y=1$.

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Q23. The solution of

$$3x+2y=12,\quad x-y=1$$
is:

A. $(2,1)$
B. $(14/5,9/5)$
C. $(9/5,14/5)$
D. $(3,2)$

Answer: B

Solution:
From $x-y=1$,
$$x=y+1.$$
So
$$3(y+1)+2y=12$$
$$5y=9\Rightarrow y=\frac95.$$
Thus
$$x=\frac95+1=\frac{14}{5}.$$

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Q24. The solution of

$$x+2y=9,\quad 2x-y=3$$
is:

A. $(3,3)$
B. $(2,4)$
C. $(4,2)$
D. $(1,4)$

Answer: A

Solution:
From $2x-y=3$,
$$y=2x-3.$$
Then
$$x+2(2x-3)=9$$
$$5x=15\Rightarrow x=3,\quad y=3.$$

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Q25. The solution of

$$5x+2y=19,\quad 3x+2y=13$$
is:

A. $(3,2)$
B. $(2,3)$
C. $(4,-1)$
D. $(1,7)$

Answer: A

Solution:
Subtract:
$$2x=6\Rightarrow x=3.$$
Then
$$15+2y=19\Rightarrow y=2.$$

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Q26. [PYQ — CBSE 2024]

If
$$2x+y=13,\quad4x-y=17,$$
then $x-y$ equals:

A. 1
B. 2
C. 3
D. 4

Answer: B

Solution:
Adding:
$$6x=30\Rightarrow x=5.$$
Then
$$2(5)+y=13\Rightarrow y=3.$$
Therefore
$$x-y=5-3=2.$$

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Q27. [PYQ — CBSE 2024]

Solve:
$$7x-2y=5,\quad8x+7y=15.$$

A. $(95/81,130/81)$
B. $(1,1)$
C. $(2,1)$
D. $(1,2)$

Answer: B

Solution:
Multiply the first equation by 7:
$$49x-14y=35.$$
Multiply the second by 2:
$$16x+14y=30.$$
Adding:
$$65x=65\Rightarrow x=1.$$

Then from $7x-2y=5$:
$$7-2y=5\Rightarrow y=1.$$

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Q28. If $x+y=10$ and $x-y=4$, then $x$ is:

A. 3
B. 5
C. 7
D. 14

Answer: C

Solution:
Adding:
$$2x=14\Rightarrow x=7.$$

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Q29. If $x+y=10$ and $x-y=4$, then $y$ is:

A. 2
B. 3
C. 6
D. 7

Answer: B

Solution:
$$x=7,\quad y=10-7=3.$$

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Q30. If

$$3x+4y=10,\quad2x-y=5,$$
then $x+y$ equals:

A. $35/11$
B. $29/11$
C. $39/11$
D. $49/11$

Answer: A

Solution:
From
$$2x-y=5,$$
$$y=2x-5.$$
Substitute:
$$3x+4(2x-5)=10$$
$$11x=30\Rightarrow x=\frac{30}{11}.$$
Then
$$y=\frac{60}{11}-\frac{55}{11}=\frac5{11}.$$
Hence
$$x+y=\frac{35}{11}.$$

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Section D — Parameter-Based Questions

Q31. For what value of $k$ will

$$x+ky=5,\quad2x+4y=10$$
have infinitely many solutions?

A. 1
B. 2
C. 3
D. 4

Answer: B

Solution:
For coincident lines:
$$\frac12=\frac{k}{4}=\frac5{10}.$$
Thus
$$k=2.$$

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Q32. For what value of $k$ will

$$2x+3y=7,\quad4x+ky=14$$
have infinitely many solutions?

A. 3
B. 4
C. 6
D. 8

Answer: C

Solution:
$$\frac24=\frac3k=\frac7{14}.$$
Hence
$$\frac3k=\frac12\Rightarrow k=6.$$

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Q33. For what value of $k$ will

$$3x+2y=5,\quad6x+ky=12$$
have no solution?

A. 2
B. 3
C. 4
D. 6

Answer: C

Solution:
For parallel lines:
$$\frac36=\frac2k.$$
Thus
$$\frac12=\frac2k\Rightarrow k=4.$$
Also
$$\frac5{12}\ne\frac12,$$
so there is no solution.

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Q34. For what value of $k$ will

$$kx+2y=5,\quad3x+6y=10$$
have no solution?

A. 1
B. 2
C. 3
D. 4

Answer: A

Solution:
For parallel lines:
$$\frac{k}{3}=\frac26=\frac13.$$
Therefore
$$k=1.$$
Also $5/10=1/2\ne1/3$, so there is no solution.

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Q35. [PYQ — CBSE 2023]

If
$$x-y=1,\quad x+ky=5$$
has solution $x=2,y=1$, find $k$.

A. $-3$
B. $-2$
C. 3
D. 4

Answer: C

Solution:
Substitute $x=2,y=1$:
$$2+k=5$$
$$k=3.$$

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Q36. For what value of $k$ do

$$3x-y+8=0,\quad6x-ky=-16$$
represent coincident lines?

A. 1
B. 2
C. 3
D. 4

Answer: B

Solution:
Multiply the first equation by 2:
$$6x-2y+16=0$$
or
$$6x-2y=-16.$$
Therefore $k=2$.

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Q37. For what value of $k$ do

$$2x+3y=7$$
and
$$(k+1)x+(2k-1)y=4k+1$$
have infinitely many solutions?

A. 3
B. 4
C. 5
D. 6

Answer: C

Solution:
For infinitely many solutions:
$$\frac{k+1}{2} = \frac{2k-1}{3} = \frac{4k+1}{7}.$$

Using the first two:
$$3(k+1)=2(2k-1)$$
$$3k+3=4k-2$$
$$k=5.$$

Check:
$$\frac{6}{2}=\frac9{3}=\frac{21}{7}=3.$$

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Q38. [PYQ — CBSE 2024]

The pair
$$5x+2y-7=0,\quad2x+ky+1=0$$
has no solution. Find $k$.

A. $4/5$
B. $5/4$
C. $5/2$
D. 4

Answer: A

Solution:
For no solution:
$$\frac52=\frac2k.$$
Therefore
$$5k=4\Rightarrow k=\frac45.$$

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Q39. For what value of $k$ do

$$x+2y=3,\quad2x+ky=6$$
have infinitely many solutions?

A. 2
B. 3
C. 4
D. 5

Answer: C

Solution:
The second equation must be twice the first:
$$2x+4y=6.$$
Hence $k=4$.

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Q40. For

$$2x+3y=5,\quad4x+6y=k$$
to have infinitely many solutions, $k$ must be:

A. 5
B. 10
C. 15
D. 20

Answer: B

Solution: The second equation must be twice the first:
$$4x+6y=10.$$
Therefore $k=10$.

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Section E — Graphical Interpretation

Q41. The equations $x=3$ and $y=4$ represent:

A. Two parallel lines
B. Two coincident lines
C. A vertical and a horizontal line intersecting at $(3,4)$
D. Two curves

Answer: C

Solution: $x=3$ is vertical and $y=4$ is horizontal. They intersect at $(3,4)$.

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Q42. [PYQ — CBSE 2023]

The equations $x=5$ and $y=7$ have solution:

A. $(7,5)$
B. $(5,7)$
C. $(5,5)$
D. $(7,7)$

Answer: B

Solution: $x=5$ fixes the x-coordinate and $y=7$ fixes the y-coordinate. Hence $(5,7)$.

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Q43. [PYQ — CBSE 2023]

The equations $x=0$ and $y=-3$ intersect at:

A. $(0,3)$
B. $(3,0)$
C. $(0,-3)$
D. $(-3,0)$

Answer: C

Solution: $x=0$ is the y-axis. Therefore, with $y=-3$, the intersection is $(0,-3)$.

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Q44. The graph of $x=2$ is a:

A. Horizontal line
B. Vertical line
C. Sloping line
D. Curve

Answer: B

Solution: $x$ has a fixed value, so the line is parallel to the y-axis.

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Q45. The graph of $y=-3$ is a:

A. Vertical line
B. Horizontal line
C. Circle
D. Parabola

Answer: B

Solution: $y$ has a fixed value, so the line is parallel to the x-axis.

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Q46. Two intersecting lines have:

A. No common point
B. Exactly one common point
C. Infinitely many common points
D. Two common points

Answer: B

Solution: Two distinct straight lines can intersect at only one point.

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Q47. Two coincident lines have:

A. One common point
B. Two common points
C. No common point
D. Infinitely many common points

Answer: D

Solution: Coincident lines overlap completely.

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Q48. Two distinct parallel lines have:

A. One common point
B. No common point
C. Infinitely many common points
D. Two common points

Answer: B

Solution: Parallel distinct lines never intersect.

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Q49. The equations $2x+y=4$ and $4x+2y=8$ represent:

A. Intersecting lines
B. Parallel lines
C. Coincident lines
D. Perpendicular lines

Answer: C

Solution:
$$\frac24=\frac12,\quad\frac12=\frac12,\quad\frac48=\frac12.$$
All ratios are equal.

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Q50. The equations $2x+y=4$ and $4x+2y=10$ represent:

A. Coincident lines
B. Parallel lines
C. Intersecting lines
D. Same line

Answer: B

Solution:
$$\frac24=\frac12,\qquad\frac12=\frac12,$$
but
$$\frac4{10}\ne\frac12.$$
Therefore, parallel distinct lines.

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Section F — Word Problems

Q51. The sum of two numbers is 50 and their difference is 10. The numbers are:

A. 20, 30
B. 30, 20
C. 25, 25
D. 40, 10

Answer: B

Solution:
$$x+y=50,\quad x-y=10.$$
Adding:
$$2x=60\Rightarrow x=30.$$
Thus $y=20$.

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Q52. [PYQ — CBSE 2024]

The sum of two numbers is 105 and their difference is 45. The numbers are:

A. 60, 45
B. 75, 30
C. 70, 35
D. 80, 25

Answer: B

Solution:
$$x+y=105,\quad x-y=45.$$
Adding:
$$2x=150\Rightarrow x=75.$$
Thus $y=30$.

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Q53. A notebook and a pen together cost ₹50. Two notebooks and three pens cost ₹120. The cost of one notebook is:

A. ₹20
B. ₹30
C. ₹40
D. ₹50

Answer: B

Solution:
$$x+y=50$$
$$2x+3y=120.$$
From first equation, $x=50-y$.

$$2(50-y)+3y=120$$
$$100+y=120\Rightarrow y=20.$$
Thus
$$x=30.$$

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Q54. The sum of the ages of a father and son is 50 years. The father is 4 times the age of the son. The son's age is:

A. 8 years
B. 10 years
C. 12 years
D. 15 years

Answer: B

Solution:
Let son's age $=x$. Father's age $=4x$.

$$x+4x=50$$
$$5x=50\Rightarrow x=10.$$

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Q55. The sum of two angles is $90^\circ$. One angle is twice the other. The smaller angle is:

A. $20^\circ$
B. $30^\circ$
C. $40^\circ$
D. $60^\circ$

Answer: B

Solution:
Let smaller angle $=x$. Larger $=2x$.

$$x+2x=90$$
$$3x=90\Rightarrow x=30^\circ.$$

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Q56. A two-digit number has digits whose sum is 9. The number obtained by reversing its digits is 27 less than the original number. The original number is:

A. 36
B. 45
C. 63
D. 72

Answer: C

Solution:
Let tens digit $=x$, units digit $=y$.

$$x+y=9.$$

Original number $=10x+y$, reversed $=10y+x$.

$$10y+x=10x+y-27$$
$$9(y-x)=-27$$
$$x-y=3.$$

Adding with $x+y=9$:
$$2x=12\Rightarrow x=6,\quad y=3.$$

Original number $=63$.

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Q57. The cost of 2 pens and 3 pencils is ₹24. The cost of 3 pens and 2 pencils is ₹21. The cost of one pen is:

A. ₹3
B. ₹4
C. ₹5
D. ₹6

Answer: A

Solution:
$$2p+3q=24$$
$$3p+2q=21.$$

Multiply first by 3:
$$6p+9q=72.$$
Second by 2:
$$6p+4q=42.$$
Subtract:
$$5q=30\Rightarrow q=6.$$
Then
$$3p+12=21\Rightarrow p=3.$$

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Q58. A boat travels 30 km downstream and 18 km upstream in the same time. If the speed of the stream is 3 km/h, the speed of the boat in still water is:

A. 9 km/h
B. 10 km/h
C. 12 km/h
D. 15 km/h

Answer: C

Solution:
Let boat speed in still water $=v$.

Downstream speed $=v+3$.
Upstream speed $=v-3$.

$$\frac{30}{v+3}=\frac{18}{v-3}.$$

Cross-multiply:
$$30(v-3)=18(v+3)$$
$$30v-90=18v+54$$
$$12v=144$$
$$v=12.$$

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Q59. The perimeter of a rectangle is 50 cm. Its length is 5 cm more than its breadth. Its length is:

A. 10 cm
B. 12.5 cm
C. 15 cm
D. 20 cm

Answer: C

Solution:
Let breadth $=x$, length $=x+5$.

$$2(l+b)=50$$
$$l+b=25.$$
Therefore
$$x+5+x=25$$
$$2x=20\Rightarrow x=10.$$
Length $=15$ cm.

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Q60. The sum of two numbers is 20. One number is 4 more than the other. The larger number is:

A. 8
B. 10
C. 12
D. 14

Answer: C

Solution:
Let smaller number $=x$. Larger $=x+4$.

$$x+x+4=20$$
$$2x=16\Rightarrow x=8.$$
Larger number $=12$.

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Section G — Higher-Order Thinking

Q61. If $x+y=8$ and $2x+2y=16$, the pair has:

A. No solution
B. One solution
C. Infinitely many solutions
D. Two solutions

Answer: C

Solution: The second equation is twice the first, so both equations represent the same line.

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Q62. If $x+y=8$ and $2x+2y=18$, the pair has:

A. No solution
B. One solution
C. Infinitely many solutions
D. Exactly two solutions

Answer: A

Solution: Dividing the second equation by 2 gives
$$x+y=9.$$
Thus $x+y$ cannot simultaneously be 8 and 9.

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Q63. If $2x+3y=12$ and $4x+6y=24$, then:

A. No solution
B. Unique solution
C. Infinitely many solutions
D. Exactly two solutions

Answer: C

Solution: The second equation is twice the first.

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Q64. If $2x+3y=12$ and $4x+6y=25$, then:

A. Unique solution
B. No solution
C. Infinitely many solutions
D. Two solutions

Answer: B

Solution: Dividing the second by 2:
$$2x+3y=\frac{25}{2},$$
which contradicts the first equation.

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Q65. The pair

$$3x+4y=7,\quad6x+8y=14$$
has:

A. Unique solution
B. No solution
C. Infinitely many solutions
D. Exactly two solutions

Answer: C

Solution:
$$\frac36=\frac48=\frac7{14}=\frac12.$$

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Q66. The pair

$$3x+4y=7,\quad6x+8y=15$$
has:

A. Unique solution
B. No solution
C. Infinitely many solutions
D. Two solutions

Answer: B

Solution:
$$\frac36=\frac48=\frac12,$$
but
$$\frac7{15}\ne\frac12.$$
Hence no solution.

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Q67. If

$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2},$$
then the two lines:

A. Are parallel
B. Are coincident
C. Intersect at exactly one point
D. Never meet

Answer: C

Solution: Unequal ratios of the x- and y-coefficients indicate intersecting lines.

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Q68. A consistent pair of equations can have:

A. Only zero solutions
B. Only one solution
C. One or infinitely many solutions
D. Exactly two solutions

Answer: C

Solution: A consistent system has at least one solution. It may have a unique solution or infinitely many solutions.

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Q69. An inconsistent pair of equations has:

A. One solution
B. No solution
C. Infinite solutions
D. Two solutions

Answer: B

Solution: Inconsistent means there is no common solution.

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Q70. If the graphs of two equations are the same straight line, then:

A. $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$
B. $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$
C. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
D. $a_1=a_2=0$

Answer: C

Solution: Equal ratios of all corresponding coefficients indicate coincident lines.

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Section H — Board-Level MCQs

Q71. [PYQ — CBSE 2024]

The pair
$$x+2y+5=0,\quad -3x=6y-1$$
has:

A. Unique solution
B. Exactly two solutions
C. Infinitely many solutions
D. No solution

Answer: D

Solution:
Rewrite:
$$x+2y+5=0$$
and
$$3x+6y-1=0.$$

Thus
$$\frac13=\frac26=\frac13,$$
but
$$\frac5{-1}=-5.$$

Therefore,
$$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}.$$
Hence no solution.

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Q72. [PYQ — CBSE 2023]

The pair
$$2x=5y+6,\quad15y=6x-18$$
represents:

A. Intersecting lines
B. Parallel lines
C. Coincident lines
D. Perpendicular lines

Answer: C

Solution:
Rewrite:
$$2x-5y-6=0$$
$$6x-15y-18=0.$$

Now
$$\frac26=\frac{-5}{-15}=\frac{-6}{-18}=\frac13.$$
Therefore, the lines are coincident.

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Q73. [PYQ — CBSE 2023]

If
$$x-y=1,\quad x+ky=5$$
has solution $x=2,y=1$, then $k$ is:

A. $-2$
B. $-3$
C. 3
D. 4

Answer: C

Solution:
$$2+k(1)=5$$
$$k=3.$$

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Q74. [PYQ — CBSE 2023]

The pair
$$x+2y+5=0,\quad-3x-6y+1=0$$
has:

A. Unique solution
B. Exactly two solutions
C. Infinitely many solutions
D. No solution

Answer: D

Solution:
The equations can be written as:
$$x+2y=-5$$
and
$$x+2y=\frac13.$$
The same left-hand side has different constants, so the lines are parallel and distinct.

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Q75. [PYQ — CBSE 2024]

If
$$x+2y=9,\quad y-2x=2,$$
then $x$ and $y$ are:

A. $(1,4)$
B. $(4,1)$
C. $(2,3)$
D. $(3,2)$

Answer: A

Solution:
$$y-2x=2\Rightarrow y=2x+2.$$

Substitute:
$$x+2(2x+2)=9$$
$$5x+4=9$$
$$x=1.$$
Thus
$$y=4.$$

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Q76. [PYQ — CBSE 2024]

The point $(-4,3)$ lies on
$$x+y+1=0$$
because:

A. $-4+3+1=0$
B. $-4-3+1=0$
C. $4+3+1=0$
D. $4-3+1=0$

Answer: A

Solution:
Substitute:
$$-4+3+1=0.$$
Hence the point lies on the line.

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Q77. If

$$x+y=10,\quad x-y=2,$$
then $x/y$ is:

A. 2
B. 3
C. 4
D. 6

Answer: B

Solution:
Adding:
$$2x=12\Rightarrow x=6.$$
Then $y=4$.

Therefore:
$$\frac{x}{y}=\frac64=\frac32.$$

Correction: None of the listed options is correct. The correct answer is
$$\boxed{\frac32}.$$

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Q78. If

$$3x+2y=12,\quad3x-2y=4,$$
then $x+y$ equals:

A. $14/3$
B. 5
C. 6
D. 7

Answer: A

Solution:
Adding:
$$6x=16\Rightarrow x=\frac83.$$
Subtracting:
$$4y=8\Rightarrow y=2.$$
Hence
$$x+y=\frac83+2=\frac{14}{3}.$$

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Q79. If

$$5x+3y=21,\quad5x-3y=9,$$
then $xy$ equals:

A. 6
B. 9
C. 12
D. 18

Answer: A

Solution:
Adding:
$$10x=30\Rightarrow x=3.$$
Then
$$15+3y=21\Rightarrow y=2.$$
Thus
$$xy=3\times2=6.$$

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Q80. If $x=2,y=3$, which pair is satisfied?

A. $x+y=6,\ x-y=1$
B. $x+y=5,\ x-y=-1$
C. $2x+y=8,\ x+2y=7$
D. $3x+y=8,\ x+3y=12$

Answer: B

Solution:
$$x+y=2+3=5$$
and
$$x-y=2-3=-1.$$

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Section I — Assertion and Reason

For Q81–90 choose:

A. Both A and R are true, and R correctly explains A.
B. Both A and R are true, but R does not correctly explain A.
C. A is true, but R is false.
D. A is false, but R is true.

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Q81.

Assertion: $2x+3y=5$ and $4x+6y=10$ have infinitely many solutions.

Reason: The second equation is twice the first.

Answer: A

Solution: The equations represent the same line. Hence infinitely many solutions, and the reason correctly explains this.

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Q82.

Assertion: $2x+3y=5$ and $4x+6y=11$ have no solution.

Reason: Their corresponding lines are parallel and distinct.

Answer: A

Solution:
$$\frac24=\frac36=\frac12$$
but
$$\frac5{11}\ne\frac12.$$
Hence parallel distinct lines and no solution.

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Q83.

Assertion: A pair of intersecting straight lines has a unique solution.

Reason: Two intersecting straight lines have exactly one common point.

Answer: A

Solution: The common point is the solution of both equations.

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Q84.

Assertion: Coincident lines have no solution.

Reason: Coincident lines have infinitely many common points.

Answer: D

Solution: The assertion is false. Coincident lines have infinitely many solutions. The reason is true.

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Q85.

Assertion: If
$$\frac{a_1}{a_2}\ne\frac{b_1}{b_2},$$
the pair has a unique solution.

Reason: The corresponding lines intersect at exactly one point.

Answer: A

Solution: Unequal coefficient ratios indicate intersecting lines.

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Q86.

Assertion: The pair $x=2,\ y=3$ has a unique solution.

Reason: $x=2$ is vertical and $y=3$ is horizontal.

Answer: B

Solution: Both statements are true. Their intersection at $(2,3)$ is what directly explains the unique solution; merely identifying their orientations does not fully state the intersection argument.

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Q87.

Assertion: The pair $x=0,\ y=-3$ has a unique solution.

Reason: The y-axis and $y=-3$ intersect at $(0,-3)$.

Answer: A

Solution: $x=0$ is the y-axis. It intersects $y=-3$ at exactly $(0,-3)$.

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Q88.

Assertion: A pair of parallel lines is inconsistent.

Reason: Distinct parallel lines do not have a common point.

Answer: A

Solution: No common point means no common solution, so the system is inconsistent.

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Q89.

Assertion: A pair with infinitely many solutions is consistent.

Reason: A consistent pair has at least one solution.

Answer: A

Solution: Infinitely many solutions certainly means at least one solution.

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Q90.

Assertion: $2x+y=4$ and $4x+2y=8$ have a unique solution.

Reason: The second equation is twice the first.

Answer: D

Solution: The assertion is false. Since the second equation is twice the first, both equations represent the same line and have infinitely many solutions. The reason is true.

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Section J — Case-Based Questions

Case Study 1

A school canteen sells sandwiches and juice. Let the cost of one sandwich be $x$ rupees and one juice be $y$ rupees.

Suppose:
$$2x+3y=130$$
and
$$3x+2y=140.$$

Q91. The equations represent:

A. A pair of quadratic equations
B. A pair of linear equations
C. A pair of exponential equations
D. A pair of trigonometric equations

Answer: B

Solution: Both variables have degree 1, so both are linear equations.

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Q92. The value of $x$ is:

A. ₹22
B. ₹30
C. ₹32
D. ₹40

Answer: C

Solution:
$$2x+3y=130$$
$$3x+2y=140.$$

Multiply first by 2:
$$4x+6y=260.$$

Multiply second by 3:
$$9x+6y=420.$$

Subtract:
$$5x=160$$
$$x=32.$$

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Q93. The value of $y$ is:

A. ₹20
B. ₹22
C. ₹25
D. ₹30

Answer: B

Solution:
Substitute $x=32$:
$$3(32)+2y=140$$
$$96+2y=140$$
$$2y=44$$
$$y=22.$$

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Q94. The cost of 1 sandwich and 1 juice is:

A. ₹44
B. ₹50
C. ₹54
D. ₹60

Answer: C

Solution:
$$x+y=32+22=54.$$

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Q95. The pair has:

A. No solution
B. Unique solution
C. Infinitely many solutions
D. Two solutions

Answer: B

Solution:
$$\frac23\ne\frac32.$$
Therefore, the lines intersect at exactly one point.

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Case Study 2

Two straight lines are represented by
$$3x+4y=12$$
and
$$6x+8y=24.$$

Q96. The second equation is:

A. Half of the first
B. Twice the first
C. Three times the first
D. Unrelated to the first

Answer: B

Solution:
$$2(3x+4y=12)$$
gives
$$6x+8y=24.$$

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Q97. The two lines are:

A. Intersecting
B. Parallel
C. Coincident
D. Perpendicular

Answer: C

Solution:
$$\frac36=\frac48=\frac{12}{24}=\frac12.$$
Therefore, they are coincident.

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Q98. The number of solutions is:

A. 0
B. 1
C. 2
D. Infinitely many

Answer: D

Solution: Coincident lines have infinitely many common points.

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Q99. The correct condition for this pair is:

A. $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$
B. $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$
C. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
D. $a_1=a_2$

Answer: C

Solution:
$$\frac36=\frac48=\frac{12}{24}=\frac12.$$

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Q100. Which statement is correct?

A. The pair is inconsistent.
B. The pair is consistent with a unique solution.
C. The pair is consistent with infinitely many solutions.
D. The pair has exactly two solutions.

Answer: C

Solution: The equations represent coincident lines, so there are infinitely many common solutions. Therefore, the pair is consistent with infinitely many solutions.

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Most Important Formulas for CBSE Exam

For
$$a_1x+b_1y+c_1=0$$
and
$$a_2x+b_2y+c_2=0:$$

1. Unique solution

$$\boxed{\frac{a_1}{a_2}\ne\frac{b_1}{b_2}}$$

The lines intersect at one point.

2. No solution

$$\boxed{\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}}$$

The lines are parallel and distinct.

3. Infinitely many solutions

$$\boxed{\frac{a_1}{a_2}= \frac{b_1}{b_2}= \frac{c_1}{c_2}}$$

The lines are coincident.

4. Graphical interpretation

• Intersecting lines → one solution

• Parallel lines → no solution

• Coincident lines → infinitely many solutions

5. Methods of solving

• Graphical method

• Substitution method

• Elimination method

• Cross-multiplication method