CBSE Class 9 Maths Ganita Manjari Part 2 Ch 13 Two Variables One Line MCQs with Answers Set 01

Answer: (c) 2x − 5y + 1 = 0

Explanation:
1. In a linear equation each variable appears on its own, to the power 1.
2. x² has power 2; xy multiplies the variables; 3/x puts x in a denominator.
3. Only 2x − 5y + 1 = 0 fits, with a = 2, b = −5, c = 1.

Teacher's Note:
1. Check powers, products and denominators.
2. Linear means straight-line graph.

Answer: (b) −7

Explanation:
1. Move all terms to the left: 2x + 3y − 7 = 0.
2. So a = 2, b = 3, c = −7.
3. The rule 'coefficient of x positive' fixes the sign; otherwise −2x − 3y + 7 = 0 would also be correct.

Teacher's Note:
1. Sign changes when a term crosses the = sign.
2. Read the condition on the sign of a.

Answer: (a) 4x − 3y − 12 = 0

Explanation:
1. LCM of 3 and 4 is 12.
2. 12 × x/3 = 4x and 12 × y/4 = 3y, and 12 × 1 = 12.
3. So 4x − 3y = 12, i.e. 4x − 3y − 12 = 0.

Teacher's Note:
1. Multiply every term, including the right side.
2. (b) swaps the coefficients.

Answer: (a) a and b are not both zero

Explanation:
1. If a = b = 0, the equation becomes c = 0 with no variable at all.
2. So at least one of a, b must be non-zero.
3. 3y − 5 = 0 (a = 0) is still allowed, and c can be 0 as in 2x + 5y = 0.

Teacher's Note:
1. 'Not both zero' is weaker than 'both non-zero'.
2. Give 3y − 5 = 0 as an example.

Answer: (c) infinitely many

Explanation:
1. Pick any value for x; the equation then gives a matching y.
2. Since x can be chosen in infinitely many ways, there are infinitely many solutions.
3. On a graph, the solutions form a line, which has infinitely many points.

Teacher's Note:
1. One equation, two unknowns: always infinitely many.
2. Connect with the graph.

Answer: (d) (2, −1)

Explanation:
1. (1, 2): 3 − 10 = −7. (5, 1): 15 − 5 = 10.
2. (−2, 1): −6 − 5 = −11. (2, −1): 6 + 5 = 11 ✓.
3. Only (2, −1) works; (−2, 1) gives −11, a sign trap.

Teacher's Note:
1. Substitute carefully with signs.
2. Test every option.

Answer: (a) a straight line

Explanation:
1. Written as y = mx + d, equal steps in x give equal steps in y.
2. So the points lie along a straight line.
3. Every solution is on the line, and every point of the line is a solution.

Teacher's Note:
1. That is why it is called 'linear'.
2. Two points are enough to draw it.

Answer: (d) (4, 0)

Explanation:
1. On the x-axis, y = 0.
2. 5x = 20, so x = 4. Point: (4, 0).
3. (b) is the y-axis crossing (put x = 0).

Teacher's Note:
1. x-axis: put y = 0.
2. y-axis: put x = 0.

Answer: (b) 6

Explanation:
1. The y-intercept is found by putting x = 0.
2. 4y = 24, so y = 6.
3. 8 is the x-intercept.

Teacher's Note:
1. Do not mix up the two intercepts.
2. The line crosses the y-axis at (0, 6).

Answer: (a) 7x − 2y = 0

Explanation:
1. A line passes through (0, 0) exactly when its constant term is 0.
2. 7(0) − 2(0) = 0 ✓.
3. The others give 0 = 3, 0 = 5 and 0 = 1, which are false.

Teacher's Note:
1. Look for c = 0.
2. Substituting (0, 0) is the quick test.

Answer: (d) −2

Explanation:
1. Slope = \( \frac{y_2 - y_1}{x_2 - x_1} \) = \( \frac{-7 - 5}{4 - (-2)} \) = \( \frac{-12}{6} \) = −2.
2. It is negative because y falls as x rises.
3. Reversing the order of points gives the same result.

Teacher's Note:
1. Keep the same order in top and bottom.
2. Watch the double negative in 4 − (−2).

Answer: (d) not defined

Explanation:
1. x = −3 is a vertical line: every point has x = −3.
2. So x₂ − x₁ = 0 and the slope formula divides by zero.
3. Hence the slope is undefined. A horizontal line has slope 0.

Teacher's Note:
1. Vertical → undefined; horizontal → 0.
2. Division by zero has no meaning.

Answer: (a) \( -\frac{2}{7} \)

Explanation:
1. 7y = −2x + 9, so y = \( -\frac{2}{7} \)x + \( \frac{9}{7} \).
2. Slope m = \( -\frac{2}{7} \).
3. Shortcut: m = \( -\frac{a}{b} \) = \( -\frac{2}{7} \).

Teacher's Note:
1. m = −a/b from standard form.
2. Do not drop the minus sign.

Answer: (b) decreases

Explanation:
1. Increasing x by 1 changes y by exactly m.
2. If m is negative, that change is a decrease.
3. So the line falls from left to right; d plays no part.

Teacher's Note:
1. Sign of slope = direction of line.
2. Negative slope: downhill left to right.

Answer: (d) \( \frac{3}{8} \)

Explanation:
1. Slope = rise ÷ run.
2. = 3 ÷ 8 = \( \frac{3}{8} \).
3. (a) inverts it; (b) wrongly uses 3 + 8 as the run.

Teacher's Note:
1. Rise on top, run below.
2. Run is horizontal distance only.

Answer: (c) \( \frac{1}{2} \)

Explanation:
1. A bigger slope means a steeper line.
2. As decimals: 0.4, about 0.429, 0.5, about 0.444.
3. The largest is \( \frac{1}{2} \).

Teacher's Note:
1. Convert to decimals to compare.
2. Or use a common denominator (630).

Answer: (c) infinitely many solutions

Explanation:
1. Doubling the first equation gives the second exactly.
2. So it is really one equation, which has infinitely many solutions.
3. Ratios: 3/6 = 2/4 = 5/10, all equal.

Teacher's Note:
1. All three ratios equal → coincident lines.
2. Check by multiplying.

Answer: (b) 2x − 3y = 7 and 4x − 6y = 9

Explanation:
1. (b): 2/4 = −3/−6 = 1/2, but 7/9 ≠ 1/2.
2. First two ratios equal, third different → parallel lines → no solution.
3. (a) is coincident; (c) and (d) intersect once.

Teacher's Note:
1. Write the three ratios for each pair.
2. Only the third ratio separates 'none' from 'infinite'.

Answer: (d) (4, 2)

Explanation:
1. Adding: 2x = 8, so x = 4.
2. Then 4 + y = 6, so y = 2.
3. Check: 4 − 2 = 2 ✓. (a) reverses the coordinates.

Teacher's Note:
1. Sum and difference: add to find x.
2. Check in both equations.

Answer: (b) no solution

Explanation:
1. A solution is a point that lies on both lines.
2. The lines are parallel and separate, so they never meet.
3. No point lies on both, so there is no solution (inconsistent pair).

Teacher's Note:
1. Parallel = no solution.
2. Such a pair is called inconsistent.

Answer: (c) infinitely many solutions

Explanation:
1. Coincident lines lie exactly on top of each other.
2. Every point of one is a point of the other.
3. So every one of the infinitely many points is a solution.

Teacher's Note:
1. One equation is a multiple of the other.
2. Coincident = infinitely many.

Answer: (a) a unique solution

Explanation:
1. 'No solution' and 'infinitely many' both need a₁/a₂ = b₁/b₂.
2. If those ratios differ, neither case is possible.
3. So the lines cross once: a unique solution, whatever c₁ and c₂ are.

Teacher's Note:
1. Unequal first two ratios → intersecting lines.
2. c₁/c₂ does not matter here.

Answer: (b) (2, 3)

Explanation:
1. The solution is the point where the two lines cross.
2. Reading the crossing point from the grid: x = 2, y = 3.
3. (c) and (d) are where only ℓ₁ meets the axes; (a) reverses the coordinates.

Teacher's Note:
1. Read x first, then y.
2. Only the crossing point lies on both lines.

Answer: (c) infinitely many solutions

Explanation:
1. 0 = 0 is always true and puts no restriction on x or y.
2. It means one equation is a multiple of the other: the lines coincide.
3. A false result like 0 = 5 would mean no solution.

Teacher's Note:
1. True leftover → infinitely many.
2. False leftover → none.

Answer: (d) 2

Explanation:
1. All three ratios must be equal: 3/9 = 7/21 = 1/3.
2. So k/6 = 1/3, giving k = 2.
3. Check: 3 × (2x + 3y = 7) gives 6x + 9y = 21 ✓.

Teacher's Note:
1. Use the known ratio to find k.
2. Always check by multiplying.

Answer: (b) 2

Explanation:
1. Unique solution needs 3/6 ≠ p/4, i.e. 1/2 ≠ p/4.
2. This fails only when p = 2.
3. At p = 2, doubling the first gives 6x + 4y = 16, and 16 ≠ 5, so the lines are parallel: no solution.

Teacher's Note:
1. Set the first two ratios equal to find the exception.
2. Then check what happens at that value.

Answer: (d) 4x + 3y = 250, 3x + 4y = 270

Explanation:
1. 4 pens + 3 notebooks: 4x + 3y = 250.
2. 3 pens + 4 notebooks: 3x + 4y = 270.
3. (a) swaps the totals; (b) contradicts itself; (c) uses the combined total.

Teacher's Note:
1. Match each total to its own purchase.
2. Read the words in order.

Answer: (c) 10y + x

Explanation:
1. The original number is 10x + y.
2. Reversed: y is in the tens place, x in the units place: 10y + x.
3. Example: x = 7, y = 3 gives 73 and 37.

Teacher's Note:
1. Place value: tens digit × 10.
2. 'yx' is not an algebraic expression for the number.

Answer: (a) \( \frac{3}{4} \)

Explanation:
1. Points: (−4, −1) and (4, 5).
2. Slope = \( \frac{5 - (-1)}{4 - (-4)} \) = \( \frac{6}{8} \) = \( \frac{3}{4} \).
3. The line rises left to right, so the slope is positive.

Teacher's Note:
1. Read coordinates carefully from the figure.
2. Rise over run, not run over rise.

Answer: (a) none

Explanation:
1. A common solution would need x = 4 and x = 7 together, which is impossible.
2. The graphs are two parallel vertical lines, 3 units apart.
3. The ratio test cannot be used here since b₁ = b₂ = 0.

Teacher's Note:
1. Reason directly when the ratio test does not apply.
2. Vertical parallel lines never meet.