**Hint:** State the properties of linear equation in order to classify the given equations as linear or non linear.

**Question.1.** Which of these is a linear equation in two variables?

(a) 5x-2y=0

(b) x+x^{2}-2y+8=0

(c) x-2y+10=x^{2}+y

(d) 3x-2y+1=0

**Question.2.** If x^{2n}-1+y^{m-4}=0 is a linear equation, which of these is also a linear equation?

(a) x^{n}+y^{m}=0

(b) x^{\frac{n}{5}}+y^{\frac{m}{5}}=0

(c) x^{n+\frac{1}{2}}+y^{m+4}=0

(d) x^{\frac{1}{n}}+y^{\frac{m}{5}}=0

**Ans.1.** (a) 5x-2y=0**Ans.2.** (d) x^{\frac{1}{n}}+y^{\frac{m}{5}}=0

**Hint:** Interpret the concepts of linear equations in order to represent any given situation algebraically and graphically.

**Question.3.** Rohit earned ₹3550 by selling some bags each for ₹500 and some baskets each for ₹150. Aarav earned ₹3400 by selling the same number of bags each for ₹400 and the same number of baskets each for ₹ 200 as Rohit sold. Which of these equations relates the number of bags x, and the number of baskets, y?

(a) 500x+150y=3550 and 400x+200y=3400

(b) 500x+150y=3400 and 400x+200y=3550

(c) 400x+150y=3550 and 500x+200y=3400

(d) 500x+200y=3550 and 400x+150y=3400

**Question.4.** The cost of production per unit for two products, A and B, are ₹100 and ₹80 respectively. In a week, the total production cost is ₹32000. In the next week, the production cost reduces by 20%, and the total cost of producing the same number of units of each product is ₹25600. Which of these are the equations that can be used to find the number of units of A, x, and the number of units of B, y?

(a) 100x+80y=25600 and 80x+64y=32000

(b) 100x+64y=32000 and 80x+100y=25600

(c) 80x+80y=32000 and 100x+64y=25600

(d) 100x+80y=32000 and 80x+64y=25600

**Ans.3.** (a) 500x+150y=3550 and 400x+200y=3400**Ans.4.** (d) 100x+80y=32000 and 80x+64y=25600

**Hint:** Demonstrate given two linear equations in order to comment on the nature / behaviour of the lines representing the linear equations.

**Question.5.** Which of these linear equations have a unique solution?

(a) (b) (c) (d)

**Question.6.** Consider the graph shown. Which of these is true about the given graph?

(a) These lines have a unique solution as they are intersecting at a point.

(b) These lines have infinitely many solutions as they lie in the same quadrant.

(c) These lines have a unique solution as the coefficient of x in both the equations is one.

(d) These lines have infinitely many solutions as they lie in the same quadrant.

**Ans.5.** (b) **Ans.6.** (a) These lines have a unique solution as they are intersecting at a point.

**Hint:** Use different algebraic methods in order to solve a pair of linear equations.

**Question.7.** Consider the equations shown.

4x+3y=41 and

x+3y=26

Which of these is the correct way of solving the given pair of equations?

(a) 4x-x+3y-3y=41-26

(b) 4(x+3y)+3y=41

(c) 4x+x+3y-3y=41-26

(d) 4x+3y+3y=41-26

**Question.8.** In the equations shown below, a and b are unknown constants.

3ax+4y=-2 and

2x+by=14

If (–3, 4) is the solution of the given equations, what are the values of a and b?

(a) a=2, b=5

(b) a=-2, b=5

(c) a=5, b=2

(d) a=5, b=-2

**Ans.7.** (a) 4x-x+3y-3y=41-26**Ans.8.** (a) a=2, b=5

**Hint:** Use the most appropriate algebraic method in order to solve the given pair of linear equations.

**Question.9.** Consider the equations shown.

p+q=5 &

p-q=2

Which of these are the values of p and q?

(a) p=3.5, q=1.5

(b) p=1.5, q=3.5

(c) p=4, q=2

(d) p=2, q=4

**Question.10.** Consider the equations shown:

ax+by=ab &

2ax+3by=3b

Which of these is the value of y in terms of a?

(a) y=2a-35

(b) y=2ab-3b

(c) y=9a-35

(d) y=3-2a

**Ans.9.** (a) p=3.5, q=1.5**Ans.10.** (d) y=3-2a

**Hint:** Use the concepts of pair of linear equations in two variables in order to represent any given situation algebraically and find its solution.

**Question.11.** Shivi gave a note of ₹2,000 for a pair of jeans worth ₹500. She was returned 11 notes in denominations of ₹200 and ₹100. Which pair of equations can be used to find the number of ₹200 notes, x, and the number of ₹100 notes x? How many notes of ₹200 did she get?

(a) x+y=11 and 200x+100y=1500; 8

(b) y=x+11 and 200x+100y=2000; 4

(c) x+y=11 and 200x+100y=1500; 4

(d) x+y=11 and 100x+200y=2000; 10

**Question.12.** Arnav wants to plant some saplings in columns. If he increases the number of saplings in a column by 4, the number of columns decreases by 1. If he decreases the number of saplings by 5 in a column, the number of columns increased by 2.

Which of these graphs relates the number, x, of columns and the number, y, of plants in a column?

(a) (b) (c) (d)

**Ans.11.** (c) x+y=11 and 200x+100y=1500; 4**Ans.12.** (d)

**Hint:** Calculate the ratio of coefficients of linear equations in order to discuss the nature of pair of linear equations.

**Question.13.** Consider the equations as shown:

9x+6y=5 and

3x+2y=7

Which of these is true about the given equations?

(a) This is a pair of coincident lines as \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

(b) This is a pair of Intersecting lines as \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}

(c) This is a pair of coincident lines as \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}

(d) This is a pair of parallel lines as \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}

**Question.14.** Consider the equations as shown:

(x-a)(y-b)=(x-2a)(y-\frac{b}{2}) and

x(x+\frac{1}{2b})+y(y+\frac{a}{2})-2xy=5+(x-y)^{2}

On comparing the coefficients, a student says these pairs of equations is consistent. Is he/she correct? Which of these explains why?

(a) Yes; because they are intersecting lines.

(b) Yes; because they are parallel lines.

(c) No; because they are parallel lines.

(d) No; because they are intersecting lines.

**Ans.13.** (d) This is a pair of parallel lines as \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}} **Ans.14.** (a) Yes; because they are intersecting lines.

**Hint:** Rewrite the given equations (using substitution method) which are reducible to a pair of linear equations in order to find the solution of those equations.

**Question.15.** Consider a pair of equations as shown.

\frac{7}{x+2}+\frac{2}{y-2}=\frac{33}{20} and

\frac{2}{x+2}+\frac{6}{y-2}=\frac{23}{20}

Which of these pair of equations is equivalent to the given pair of equations?

(a) 7u+2v=\frac{20}{33} and 2u+6v=\frac{20}{23}

(b) 7u+6v=\frac{33}{20} and 2u+2v=\frac{23}{20}

(c) 7u+2v=\frac{33}{20} and 2u+6v=\frac{23}{20}

(d) 2u+6v=\frac{33}{20} and 7u+2v=\frac{23}{20}

**Question.16.** Consider a pair of equations as shown.

\frac{5}{x+2}+\frac{7}{y+2}=\frac{31}{12} and

\frac{4}{x+2}+\frac{3}{y+2}=\frac{17}{12}

What is the value of x and y ?

(a) x=2 and y=4

(b) x=4 and y=2

(c) x=14 and y=16

(d) x=16 and y=14

**Ans.15.** (c) 7u+2v=\frac{33}{20} and 2u+6v=\frac{23}{20}**Ans.16.** (b) x=4 and y=2