Practice MCQs for Class 10 Mathematics Chapter 03 Pair of Linear Equations in Two Variables
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Multiple Choice Questions
(a) Yes, because both sides are 8
(b) No, because the left side is 7
(c) No, because the left side is 8
(d) Yes, because the left side is 7
Show Answer & Explanation
Answer: (b) No, because the left side is 7
Explanation:
1. Put x = 12: 12/3 + 12/4 = 4 + 3 = 7, which is not equal to 8. So x = 12 is not a solution.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) Yes
(b) No, left side is −11 and right side is −11
(c) No, left side is −9 and right side is −11
(d) No, left side is 5 and right side is 1
Show Answer & Explanation
Answer: (a) Yes
Explanation:
1. Put x = −4: left side = (−16 + 7) − 2 = −11 and right side = −12 + 1 = −11. Both sides are equal, so x = −4 is a solution.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 1/3
(b) x = 9
(c) x = −3
(d) x = 3
Show Answer & Explanation
Answer: (d) x = 3
Explanation:
1. Collect the terms: 6/x − 3/x = 12 − 11, so 3/x = 1 and x = 3.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) 0
(b) 10
(c) 2
(d) −2
Show Answer & Explanation
Answer: (c) 2
Explanation:
1. 2x = 16 gives x = 8. Then x² + x − 70 = 64 + 8 − 70 = 2.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) first quadrant
(b) third quadrant
(c) second quadrant
(d) fourth quadrant
Show Answer & Explanation
Answer: (b) third quadrant
Explanation:
1. Both coordinates are negative, so the point lies in the third quadrant.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) positive side of the x-axis
(b) negative side of the y-axis
(c) positive side of the y-axis
(d) negative side of the x-axis
Show Answer & Explanation
Answer: (d) negative side of the x-axis
Explanation:
1. Its y-coordinate is 0, so it lies on the x-axis, and its x-coordinate is negative, so it is on the negative side.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) 7
(b) −11
(c) 11
(d) −7
Show Answer & Explanation
Answer: (c) 11
Explanation:
1. Substitute x = −1, y = −1: 2(−1) − k(−1) = 9, so −2 + k = 9 and k = 11.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) a = 5, b = 7
(b) a = 7, b = 5
(c) a = 7, b = −5
(d) a = 12, b = 7
Show Answer & Explanation
Answer: (b) a = 7, b = 5
Explanation:
1. x = 1: a + b = 12. x = 2: 2a + b = 19. Subtracting, a = 7 and then b = 5. Check with x = −3: 7(−3) + 5 = −16, which matches the table.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) A, B and C
(b) only A and B
(c) only B and C
(d) only A and C
Show Answer & Explanation
Answer: (a) A, B and C
Explanation:
1. For A: 9(1) − 7 = 2. For B: 9(−1) − 7 = −16. For C: 9(0) − 7 = −7. All three satisfy the equation.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) (0, −2)
(b) (2/7, 0)
(c) (−2, 0)
(d) (−2/7, 0)
Show Answer & Explanation
Answer: (d) (−2/7, 0)
Explanation:
1. On the x-axis y = 0, so 7x = −2 and x = −2/7. The point is (−2/7, 0).
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) Yes
(b) No
(c) Yes, only if y = 0
(d) Cannot be said
Show Answer & Explanation
Answer: (b) No
Explanation:
1. Substituting: 2(−3) − 5(−2) = −6 + 10 = 4, which is not 7. So the point is not on the line.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 2, y = 1
(b) x = 1, y = 2
(c) x = 3, y = 2
(d) x = 0, y = 1
Show Answer & Explanation
Answer: (a) x = 2, y = 1
Explanation:
1. For x = 2, y = 1: 2(2) + 1 = 5. For x = 1, y = 2: 4 ≠ 5; for x = 3, y = 2: 8 ≠ 5; for x = 0, y = 1: 1 ≠ 5.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) (1, 3)
(b) (2, 0)
(c) (5, 2)
(d) (3, 1)
Show Answer & Explanation
Answer: (d) (3, 1)
Explanation:
1. From x − 2y = 1, x = 1 + 2y. Then 2(1 + 2y) + y = 7, so 5y = 5, y = 1 and x = 3. The point is (3, 1).
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) (0, 7)
(b) (7, 0)
(c) (7/2, 0)
(d) (0, 7/2)
Show Answer & Explanation
Answer: (c) (7/2, 0)
Explanation:
1. Put y = 0: 2x = 7, so x = 7/2. The point is (7/2, 0).
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) inconsistent
(b) consistent with a unique solution
(c) consistent with infinitely many solutions
(d) dependent
Show Answer & Explanation
Answer: (b) consistent with a unique solution
Explanation:
1. a1/a2 = 1/2 and b1/b2 = 1/(−3). Since 1/2 ≠ −1/3, the lines intersect at one point, so the pair is consistent with a unique solution.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) a unique solution
(b) infinitely many solutions
(c) exactly two solutions
(d) no solution
Show Answer & Explanation
Answer: (d) no solution
Explanation:
1. The ratios of coefficients satisfy 2/2 = 3/3 = 1 but c1/c2 = 12/6 = 2. The lines are parallel, so there is no solution.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) no solution
(b) a unique solution
(c) infinitely many solutions
(d) exactly two solutions
Show Answer & Explanation
Answer: (c) infinitely many solutions
Explanation:
1. Write the second as 9x − 15y + 12 = 0. Then 3/9 = −5/−15 = 4/12 = 1/3, so the lines coincide and there are infinitely many solutions.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) inconsistent
(b) consistent and dependent
(c) consistent with a unique solution
(d) neither consistent nor inconsistent
Show Answer & Explanation
Answer: (b) consistent and dependent
Explanation:
1. 1/2 = 3/6 = 1/2, so the lines coincide. The system is consistent with infinitely many solutions (dependent).
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 49/29, y = 19/29
(b) x = 19/29, y = 49/29
(c) x = 1, y = 1
(d) x = 29/49, y = 29/19
Show Answer & Explanation
Answer: (a) x = 49/29, y = 19/29
Explanation:
1. From the second, x = 3 − 2y. Then 7(3 − 2y) − 15y = 2, so 21 − 29y = 2 and y = 19/29. Hence x = 3 − 38/29 = 49/29.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 2, y = 1
(b) x = 3, y = 1
(c) x = −1, y = 3
(d) x = 1, y = 2
Show Answer & Explanation
Answer: (d) x = 1, y = 2
Explanation:
1. x = 5 − 2y. Then 2(5 − 2y) + 3y = 8 gives 10 − y = 8, so y = 2 and x = 1.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 3, y = 2
(b) x = 2, y = 3
(c) x = 1, y = 4
(d) x = 2, y = 2
Show Answer & Explanation
Answer: (b) x = 2, y = 3
Explanation:
1. Multiply by 10: 2x + 3y = 13 and 4x + 5y = 23. Doubling the first gives 4x + 6y = 26. Subtracting: y = 3, so 2x = 13 − 9 = 4 and x = 2.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = −2, y = −1
(b) x = 2, y = 1
(c) x = −1, y = −2
(d) x = −3, y = −2
Show Answer & Explanation
Answer: (a) x = −2, y = −1
Explanation:
1. From the second, x = y − 1. Then 3(y − 1) − 5y + 1 = 0 gives −2y − 2 = 0, so y = −1 and x = −2.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 2, y = −1
(b) x = 1, y = 2
(c) x = 0, y = 17/5
(d) x = 1, y = 1
Show Answer & Explanation
Answer: (d) x = 1, y = 1
Explanation:
1. Multiply the second equation by 5: 35x − 5y = 30. Adding to the first: 47x = 47, so x = 1 and y = 7 − 6 = 1.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = −2, y = 3
(b) x = 3, y = −2
(c) x = 2, y = −3
(d) x = 1, y = −1
Show Answer & Explanation
Answer: (c) x = 2, y = −3
Explanation:
1. Multiply the first by 2 and the second by 3: 34x + 24y = −4 and 45x + 24y = 18. Subtracting: 11x = 22, so x = 2 and 12y = −2 − 34 = −36, y = −3.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 1, y = 2
(b) x = −1, y = −2
(c) x = −2, y = −1
(d) x = −1, y = 2
Show Answer & Explanation
Answer: (b) x = −1, y = −2
Explanation:
1. Adding the equations: 56x − 14y = −28, so 4x − y = −2 ...(i). Subtracting the second from the first: 30x − 60y = 90, so x − 2y = 3 ...(ii). From (i) y = 4x + 2; then x − 8x − 4 = 3 gives x = −1 and y = −2.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) x = 1, y = −2
(b) x = −2, y = 1
(c) x = 3, y = −1/4
(d) x = 2, y = −1
Show Answer & Explanation
Answer: (d) x = 2, y = −1
Explanation:
1. Multiply the first by 3 and the second by 4: 9x − 12y = 30 and 16x + 12y = 20. Adding: 25x = 50, so x = 2 and y = (6 − 10)/4 = −1.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
(a) no solution
(b) a unique solution
(c) infinitely many solutions
(d) exactly two solutions
Show Answer & Explanation
Answer: (c) infinitely many solutions
Explanation:
1. 2/3 = −6/−9 = 10/15 = 2/3, so the two equations represent the same line.
Teacher's Note:
1. Form the equations from the data, solve them by substitution, elimination or cross-multiplication, and verify the answer in both equations.
2. For the conditions on k, compare a1/a2, b1/b2 and c1/c2.
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