Pair of Linear Equations — Chapter Test | Class 10 IMO

Q1

5 pens and 3 pencils cost ₹45, while 3 pens and 5 pencils cost ₹35. The cost of one pen is:

Adding gives 8(pen + pencil) = 80, subtracting gives 2(pen − pencil) = 10; solving, pen = ₹7.50.

Q2

A system that has at least one solution is called:

Having a solution makes the system consistent.

Q3

A consistent pair of equations with infinitely many solutions is called:

Coincident lines give a dependent (and consistent) system.

Q4

A train covers a distance at a uniform speed. If the speed were 10 km/h more it would take 2 h less, and 10 km/h less it would take 3 h more. The distance is:

With distance D, speed s and time t: D = st, D = (s + 10)(t − 2), D = (s − 10)(t + 3). Solving gives s = 50, t = 12, D = 600 km.

Q5

For what value of k do the lines 3x + ky = 7 and 6x − 10y = 3 have a unique solution for every k except:

Unique solution fails when 3/6 = k/(−10), giving k = −5.

Q6

Solving x − 2y = 0 and 3x + 4y = 20 gives:

x = 2y; then 3(2y) + 4y = 20 gives 10y = 20, so y = 2 and x = 4.

Q7

A pair of linear equations has a unique solution when:

The lines intersect at one point exactly when a₁/a₂ ≠ b₁/b₂.

Q8

Solving 2/x + 3/y = 13 and 5/x − 4/y = −2 (with u = 1/x, v = 1/y) gives:

The pair becomes 2u + 3v = 13, 5u − 4v = −2, giving u = 2, v = 3, so x = 1/2 and y = 1/3.

Q9

Two cars start from the same point and travel in the same direction. After 2 hours they are 40 km apart. The difference in their speeds is:

Relative speed × 2 h = 40 km, so the difference in speeds is 40 ÷ 2 = 20 km/h.

Q10

Using substitution with y = 2x in x + y = 9 gives:

x + 2x = 9 gives 3x = 9, so x = 3 and y = 6.

Q11

If 2x + 3y = 12 and x = 3, then y equals:

6 + 3y = 12 gives 3y = 6, so y = 2.

Q12

If a₁/a₂ = b₁/b₂ but c₁/c₂ is different, the lines are:

Equal coefficient ratios with a different constant ratio means parallel lines.

Q13

If a pair of equations is inconsistent, their graphs:

Inconsistent means no common point, i.e. parallel lines.

Q14

A pair of linear equations has no solution when:

Parallel lines (no solution) satisfy a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Q15

A father is three times as old as his son. In 12 years he will be twice as old. The son's present age is:

f = 3s and f + 12 = 2(s + 12) give 3s + 12 = 2s + 24, so s = 12.