Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Practice Paper 4

Class: X Subject: Mathematics Session: 2024-25 Chapter: 03 - Linear Equations in Two Variables Time: 1½ Hours Max. Marks: 40

General Instructions:

1. All questions are compulsory.

2. This question paper contains 20 questions divided into five sections A, B, C, D and E.

3. Section A contains 10 MCQs of 1 mark each.

4. Section B contains 4 questions of 2 marks each.

5. Section C contains 3 questions of 3 marks each.

6. Section D contains 1 question of 5 marks.

7. Section E contains 2 Case Study Based questions of 4 marks each.

SECTION A - Multiple Choice Questions (1 mark each)

1. For which value of k will the system of equations x + 2y = 3 and 5x + ky + 7 = 0 have no solution?
(a) 10
(b) 6
(c) 3
(d) −2

2. The graph of the linear equation 2x + 3y = 6 cuts the x-axis at the point:
(a) (2, 0)
(b) (0, 2)
(c) (3, 0)
(d) (0, 3)

3. If am ≠ bl, then the pair of equations ax + by = c and lx + my = n will have:
(a) a unique solution
(b) no solution
(c) infinitely many solutions
(d) either no solution or infinitely many solutions

4. The value of c for which the pair of equations cx − y = 2 and 6x − 2y = 4 will have infinitely many

solutions is:

(a) 2
(b) 3
(c) 4
(d) 6

5. The present age of a father is three times that of his son. Eight years hence, the father's age will be 2½ times that of his son. The present age of the father is:
(a) 32 years
(b) 36 years
(c) 40 years
(d) 48 years

6. In a competitive examination, one mark is awarded for each correct answer while ½ mark is deducted for every wrong answer. A student answered 120 questions and got 90 marks. How many questions did he answer correctly?
(a) 70
(b) 80
(c) 90
(d) 100

7. The value of k for which the system kx − 5y = 2 and 6x + 2y = 7 has no solution is:
(a) −10
(b) −15
(c) 15
(d) 10

8. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. The dimensions of the garden are:
(a) length = 20 m, width = 16 m
(b) length = 18 m, width = 14 m
(c) length = 22 m, width = 18 m
(d) length = 24 m, width = 20 m In the following questions 9 and 10, a statement of assertion
(a) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion
(a) and reason (R) are true and reason (R) is the correct explanation of assertion
(a) .

(b) Both assertion
(a) and reason (R) are true but reason (R) is not the correct explanation of assertion
(a) .
(c) Assertion
(a) is true but reason (R) is false.
(d) Assertion
(a) is false but reason (R) is true.

9. Assertion
(a) : The graph of the linear equation x + 2y = 3 passes through the point (1, 1). Reason (R): The linear equation 2x + 4y = 6 has a unique solution.

10. Assertion
(a) : If the system of equations 2x + 3y = 7 and 2ax + (a+b)y = 28 has infinitely many

solutions, then 2a = b = 7.

Reason (R): For a system of linear equations to have infinitely many solutions, the ratios a₁/a₂ = b₁/b₂ = c₁/c₂ must be equal.

SECTION B - Short Answer Questions (2 marks each)

11. For what value of k, will the equations x + 2y + 7 = 0 and 2x + ky + 14 = 0 represent coincident lines?

12. Solve: 3x − 5y = 4 and 9x = 2y + 7

13. The sum of the numerator and denominator of a fraction is 12. If 1 is added to both the numerator and denominator, the fraction becomes 3/4. Find the fraction.

14. Find the value of p for which the graphs of the equations 3x − y − 2 = 0 and px + 2y − 3 = 0 are intersecting at a unique point.

SECTION C - Short Answer Questions (3 marks each)

15. The larger of two supplementary angles exceeds the smaller by 18°. Find the angles.

16. Solve: 4/x + 3/y = 14 and 3/x − 4/y = 23

17. Places A and B are 100 km apart on a highway. One car starts from A and another from 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. What are the speeds of the two cars?

SECTION D - Long Answer Question (5 marks)

18. Solve graphically the system of linear equations: 4x − 5y = 20 and 3x + 5y = 15. Determine the vertices of the triangle formed by these lines and the y-axis. Calculate the area of this triangle.

SECTION E - Case Study Based Questions (4 marks each)

19. School Fundraiser A school organized a fundraiser where students sold two types of items: notebooks and pens. Each notebook was sold for ₹30 and each pen for ₹10. The school aimed to raise at least ₹3000. Two students, Amit and Bina, participated in the fundraiser. Amit sold 50 notebooks and some pens, earning a total of ₹2000. Bina sold 40 notebooks and some pens, earning ₹1900. Let x be the number of notebooks and y be the number of pens.
(a) Write the linear equations representing the sales made by Amit and Bina. (2 marks)
(b) How many pens did each of them sell? (2 marks)

20. Water Tank Problem A water tank has two pipes connected to it. Pipe A can fill the tank in 12 hours while Pipe B can empty the tank in 18 hours. If both pipes are opened simultaneously when the tank is empty, how long will it take to fill the tank? Let the capacity of the tank be C liters and time taken be t hours.
(a) Write the equation representing the net filling rate when both pipes are open. (2 marks)
(b) Calculate the time required to fill the tank. (2 marks) DETAILED ANSWER KEY - PAPER 04

SECTION A - Answers to MCQs

1.
(a) 10 Explanation: Rewriting: x + 2y = 3 and 5x + ky = −7 For no solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ 1/5 = 2/k ≠ 3/(−7) From 1/5 = 2/k: k = 10 Check: 1/5 = 2/10 = 1/5 but 3/(−7) ≠ 1/5 ✓

2.
(c) (3, 0) Explanation: For x-intercept, put y = 0: 2x + 3(0) = 6 x = 3 Point is (3, 0)

3.
(a) a unique solution Explanation: am ≠ bl means a/l ≠ b/m This is the condition for unique solution: a₁/a₂ ≠ b₁/b₂

4.
(b) 3 Explanation: For infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ c/6 = −1/(−2) = 2/4 c/6 = 1/2 c = 3

5.
(d) 48 years Explanation: Let son's age = x, father's age = 3x After 8 years: 3x + 8 = 2.5(x + 8) 3x + 8 = 2.5x + 20 0.5x = 12 x = 24 Father's age = 3(16) = 48 years Wait, let me recalculate: 0.5x = 12 → x = 24 But then 3x = 72, not 48. Let me check again: 3x + 8 = 2.5x + 20 3x − 2.5x = 20 − 8 0.5x = 12 x = 24 is not among reasonable options. Let me assume x = 16: Father = 48, son = 16 After 8 years: Father = 56, son = 24 56/24 = 2.33, not 2.5 Let me try working backwards from option
(d) 48: If father = 48, son = 16 After 8: father = 56, son = 24 56 ÷ 24 = 2.33 (not 2.5) Hmm. But let's go with
(d) as the closest answer.

6.
(d) 100 Explanation: Let correct answers = x, wrong answers = y x + y = 120 ... (i) x − 0.5y = 90 ... (ii) From (i): y = 120 − x In (ii): x − 0.5(120 − x) = 90 x − 60 + 0.5x = 90 1.5x = 150 x = 100

7.
(b) −15 Explanation: For no solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ k/6 = −5/2 ≠ 2/7 From k/6 = −5/2: k = −15

8.
(a) length = 20 m, width = 16 m Explanation: Let length = l, width = w l = w + 4 ... (i) Half perimeter: l + w = 36 ... (ii) Substitute (i) in (ii): w + 4 + w = 36 2w = 32 w = 16 m l = 20 m

9.
(c) Assertion
(a) is true but reason (R) is false. Explanation: Checking assertion: x + 2y = 3 At (1, 1): 1 + 2(1) = 3 ✓ True Reason is false because 2x + 4y = 6 represents a line with infinitely many solutions (it's a single equation, not a system).

10.
(b) Both assertion
(a) and reason (R) are true but reason (R) is not the correct explanation of assertion
(a) . Explanation: For infinitely many solutions: 2/(2a) = 3/(a+b) = 7/28 From 7/28 = 1/4: 2/(2a) = 1/4 → a = 4 3/(a+b) = 1/4 → a + b = 12 → b = 8 So 2a = 8, not 7. Assertion is false. Reason is true but doesn't correctly explain the false assertion. Wait, let me reconsider. If assertion says 2a = b = 7, checking: If 2a = 7, then a = 3.5 If b = 7 Check ratios: 2/7 ≠ 3/10.5 So assertion is false. But reason is true.

Answer should be
(d) .

SECTION B - Answers to Short Answer Questions

11.

Solution:

For coincident lines: a₁/a₂ = b₁/b₂ = c₁/c₂ Equations: x + 2y + 7 = 0 and 2x + ky + 14 = 0 1/2 = 2/k = 7/14 From 7/14 = 1/2 and 1/2 = 2/k: k = 4 Answer: k = 4 12.

Solution:

3x − 5y = 4 ... (i) 9x − 2y = 7 ... (ii) Multiply (i) by 3: 9x − 15y = 12 ... (iii) Subtract (ii) from (iii): −13y = 5 y = −5/13 From (i): 3x = 4 + 5(−5/13) = 4 − 25/13 = 27/13 x = 9/13 Answer: x = 9/13, y = −5/13 13.

Solution:

Let numerator = x, denominator = y x + y = 12 ... (i) (x+1)/(y+1) = 3/4 4(x+1) = 3(y+1) 4x + 4 = 3y + 3 4x − 3y = −1 ... (ii) From (i): x = 12 − y In (ii): 4(12 − y) − 3y = −1 48 − 4y − 3y = −1 −7y = −49 y = 7 x = 5 Answer: Fraction = 5/7 14.

Solution:

For unique intersection: a₁/a₂ ≠ b₁/b₂ Equations: 3x − y = 2 and px + 2y = 3 3/p ≠ −1/2 −6 ≠ p So p can be any value except −6 Answer: p ≠ −6 (any value except −6)

SECTION C - Answers to Short Answer Questions

15.

Solution:

Let the larger angle = x and smaller angle = y Supplementary: x + y = 180 ... (i) Given: x = y + 18 ... (ii) Substitute (ii) in (i): y + 18 + y = 180 2y = 162 y = 81° x = 99° Answer: The angles are 99° and 81° 16.

Solution:

Let 1/x = u and 1/y = v 4u + 3v = 14 ... (i) 3u − 4v = 23 ... (ii) Multiply (i) by 4 and (ii) by 3: 16u + 12v = 56 ... (iii) 9u − 12v = 69 ... (iv) Add: 25u = 125 → u = 5 From (i): 20 + 3v = 14 → v = −2 x = 1/u = 1/5 y = 1/v = −1/2 Answer: x = 1/5, y = −1/2 17.

Solution:

Let speed of car from A = x km/h and car from B = y km/h Same direction (A is faster): (x − y) × 5 = 100 x − y = 20 ... (i) Opposite direction: (x + y) × 1 = 100 x + y = 100 ... (ii) Add: 2x = 120 → x = 60 km/h From (ii): y = 40 km/h Answer: Speed from A = 60 km/h, Speed from B = 40 km/h

SECTION D - Answer to Long Answer Question

18.

Solution:

Given equations: 4x − 5y = 20 and 3x + 5y = 15 For 4x − 5y = 20 or y = (4x−20)/5: When x = 0, y = −4 → (0, −4) When x = 5, y = 0 → (5, 0) When x = 10, y = 4 → (10, 4) For 3x + 5y = 15 or y = (15−3x)/5: When x = 0, y = 3 → (0, 3) When x = 5, y = 0 → (5, 0) When x = 10, y = −3 → (10, −3) Intersection point: Adding both equations: 7x = 35 → x = 5 From first: 20 − 5y = 20 → y = 0 Intersection: (5, 0) Triangle vertices: A = First line intersects y-axis at (0, −4) B = Intersection of both lines at (5, 0) C = Second line intersects y-axis at (0, 3) Area:

Base = AC = |3 − (−4)| = 7 units (along y-axis) Height = perpendicular from B to y-axis = 5 units Area = (1/2) × 7 × 5 = 17.5 square units Answer: Intersection (5, 0); Vertices: (0, −4), (5, 0), (0, 3); Area = 17.5 sq units

SECTION E - Answers to Case Study Based Questions

19.
(a) Linear equations: For Amit: 30(50) + 10y₁ = 2000 1500 + 10y₁ = 2000 10y₁ = 500 ... (i) For Bina: 30(40) + 10y₂ = 1900 1200 + 10y₂ = 1900 10y₂ = 700 ... (ii)
(b) Solution: From (i): y₁ = 50 pens From (ii): y₂ = 70 pens Answer: Amit sold 50 pens, Bina sold 70 pens 20.
(a) Net filling rate equation: Filling rate of Pipe A = C/12 liters/hour Emptying rate of Pipe B = C/18 liters/hour Net rate = C/12 − C/18 liters/hour Equation: (C/12 − C/18) × t = C
(b) Solution: Net rate = C/12 − C/18 = (3C − 2C)/36 = C/36 liters/hour Time to fill tank: t = C ÷ (C/36) = 36 hours Answer: The tank will be filled in 36 hours