CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables MCQs Set 21

Answer: (a) 45 years

Explanation:
1. Let the father be x years and the son y years. x + y = 65 and 2(x − y) = 50, so x − y = 25.
2. Adding: 2x = 90, so x = 45.

Teacher's Note:
1. Write the two conditions as equations.
2. Add the equations to remove y.

Answer: (a) 40 km/h, 30 km/h

Explanation:
1. Let the speeds be x and y (x > y). Same direction: 9(x − y) = 90, so x − y = 10.
2. Opposite directions: \( \frac{9}{7}(x+y)=90 \), so x + y = 70.
3. Solving gives x = 40 and y = 30.

Teacher's Note:
1. Same direction: subtract speeds. Opposite direction: add speeds.
2. Distance = speed × time.

Answer: (a) k = 15

Explanation:
1. The lines are consistent if they meet or coincide. Here \( \frac{7}{21}=\frac{-1}{-3}=\frac{1}{3} \), so they are parallel or coincident.
2. They coincide when \( \frac{5}{k}=\frac{1}{3} \), that is k = 15.

Teacher's Note:
1. Equal a- and b-ratios mean no unique solution.
2. Consistent then requires the c-ratio to match too.

Answer: (d) lies between −1 and \( \frac{3}{2} \)

Explanation:
1. Adding the equations: \( x=\frac{5}{k+1} \), and then \( y=x-2=\frac{3-2k}{k+1} \).
2. x > 0 needs k > −1. y > 0 needs \( \frac{3-2k}{k+1}>0 \), so −1 < k < 3/2.

Teacher's Note:
1. First quadrant means x > 0 and y > 0.
2. Solve both inequalities together.

Answer: (a) b = 2a

Explanation:
1. \( \frac{2}{2a}=\frac{3}{a+b}=\frac{7}{28}=\frac{1}{4} \).
2. So a = 4 and a + b = 12, giving b = 8 = 2a.

Teacher's Note:
1. Coincident lines have all three ratios equal.
2. Use 7/28 = 1/4 first.

Answer: (b) x = −1/2, y = 2

Explanation:
1. Put x = −1/2, y = 2 in the first equation: 2(−a/2 − 2b) + a + 4b = −a − 4b + a + 4b = 0.
2. In the second: 2(−b/2 + 2a) + b − 4a = −b + 4a + b − 4a = 0. Both hold.

Teacher's Note:
1. Substituting each option is quick here.
2. Solving also works by eliminating y.

Answer: (a) a = 8, b = 5

Explanation:
1. \( \frac{3}{5}=\frac{-(a+1)}{1-2a}=\frac{2b-1}{3b} \).
2. From the first two: −5(a + 1) = 3(1 − 2a), so a = 8.
3. From the first and third: 9b = 10b − 5, so b = 5.

Teacher's Note:
1. Infinitely many solutions means all three ratios are equal.
2. Solve a and b from separate pairs.

Answer: (d) \( x=\frac{c(a+b)}{a^2+b^2},\ y=-\frac{c(a-b)}{a^2+b^2} \)

Explanation:
1. Multiply the first by a and the second by b, then add: \( (a^2+b^2)x=c(a+b) \).
2. Then \( y=\frac{c-ax}{b}=\frac{c(b-a)}{a^2+b^2} \).

Teacher's Note:
1. Eliminate y by multiplying with a and b.
2. Substitute back for y.

Answer: (a) \( \frac{12x+10y}{100} \)

Explanation:
1. Interest = \( \frac{12x}{100}+\frac{10y}{100}=\frac{12x+10y}{100} \).

Teacher's Note:
1. Simple interest for 1 year = principal × rate / 100.
2. Add the two interests.

Answer: (b) 12x + 10y = 13000

Explanation:
1. \( \frac{12x+10y}{100}=130 \), so 12x + 10y = 13000.

Teacher's Note:
1. Multiply both sides by 100.
2. Do not divide by 2 unless asked.

Answer: (a) 5x + 6y = 6700

Explanation:
1. Interchanged: 10% of x and 12% of y gives interest 130 + 4 = 134.
2. 10x + 12y = 13400, so 5x + 6y = 6700.

Teacher's Note:
1. The new interest is Rs. 134.
2. Divide by 2 to simplify.

Answer: (b) x + y = 1200

Explanation:
1. Adding 6x + 5y = 6500 and 5x + 6y = 6700 gives 11x + 11y = 13200.
2. So x + y = 1200.

Teacher's Note:
1. Add the two simplified equations.
2. Divide by 11.

Answer: (b) x = Rs. 500, y = Rs. 700

Explanation:
1. Subtracting 5x + 6y = 6700 from 6x + 5y = 6500 gives x − y = −200.
2. With x + y = 1200: x = 500 and y = 700. Check: 0.12 × 500 + 0.10 × 700 = 60 + 70 = 130.

Teacher's Note:
1. Use sum and difference.
2. Verify with the original interest.

Answer: (b) Rs. 2100, Rs. 25

Explanation:
1. Let the full fare be x and the reservation charge y. Then x + y = 2125.
2. Also (x + y) + (x/2 + y) = 3200, so x + 2y = 2150.
3. Subtracting: y = 25 and x = 2100.

Teacher's Note:
1. Write the half ticket as x/2 + y.
2. Subtract the equations to find y.

Answer: (a) all real values except 10

Explanation:
1. Here a₁/a₂ = b₁/b₂ = 1/2. Lines are parallel when c₁/c₂ is different: \( \frac{5}{p} \) ≠ \( \frac{1}{2} \).
2. So p ≠ 10.

Teacher's Note:
1. Parallel lines have equal a- and b-ratios and an unequal c-ratio.
2. If p = 10 the lines coincide.

Answer: (a) 2x − 7y = 1

Explanation:
1. 10x + y = 8(x + y) + 1, so 2x − 7y = 1.

Teacher's Note:
1. The number is 10x + y.
2. Collect the terms on one side.

Answer: (a) 14y − 3x = 2

Explanation:
1. 10x + y = 13(x − y) + 2, so 3x − 14y = −2, that is 14y − 3x = 2.

Teacher's Note:
1. The number is 10x + y.
2. Move all terms to one side.

Answer: (c) 4

Explanation:
1. Double 2x − 7y = 1: 4x − 14y = 2. Subtract 3x − 14y = −2: x = 4.

Teacher's Note:
1. Make the y-coefficients equal.
2. Subtract the equations.

Answer: (b) 1

Explanation:
1. From 2x − 7y = 1 with x = 4: 7y = 7, so y = 1.

Teacher's Note:
1. Substitute x = 4.
2. Check with the second equation.

Answer: (c) 41

Explanation:
1. The number is 10 × 4 + 1 = 41. Check: 8 × 5 + 1 = 41 and 13 × 3 + 2 = 41.

Teacher's Note:
1. Number = 10x + y.
2. Check with both conditions.

Answer: (d) 15

Explanation:
1. Let the fraction be n/d. Then n + 1 = 4(d + 1) and n − 1 = 7(d − 1).
2. So n = 4d + 3 and n = 7d − 6, giving d = 3 and n = 15.

Teacher's Note:
1. Form two equations.
2. Eliminate n.

Answer: (c) 70 years

Explanation:
1. Let the ages be 7x and 3x: 7x + 10 = 2(3x + 10), so x = 10.
2. The father is 70 years old.

Teacher's Note:
1. Use a common multiplier x.
2. Add 10 to both ages.

Answer: (d) x = ±2, y = ±3

Explanation:
1. Put a = x² and b = y²: 2a + 3b = 35 and 3a + 2b = 30.
2. Solving: a = 4 and b = 9, so x = ±2 and y = ±3.

Teacher's Note:
1. Treat x² and y² as the unknowns.
2. Eliminate b.

Answer: (d) 10

Explanation:
1. For inconsistency: \( \frac{1}{5} \) = \( \frac{2}{k} \) ≠ \( \frac{-3}{7} \).
2. So k = 10.

Teacher's Note:
1. Inconsistent means parallel lines.
2. Equal a- and b-ratios, different c-ratio.

Answer: (c) 48

Explanation:
1. Let the tens digit be x. The number is 10x + 2x = 12x and the reversed number is 20x + x = 21x.
2. 12x + 36 = 21x, so x = 4 and the number is 48.

Teacher's Note:
1. Write the number as 10 × tens + units.
2. Equate the reversed number to number + 36.

Answer: (c) 7 : 1

Explanation:
1. 4(3x + 4y) = 9(x + 2y), so 3x = 2y.
2. Let x = 2k and y = 3k: (6k + 15k) : (6k − 3k) = 21k : 3k = 7 : 1.

Teacher's Note:
1. Cross-multiply to get a relation between x and y.
2. Substitute x = 2k, y = 3k.