12.Which of the following is true for a gradient / a slope ?

a. If a straight line is parallel to the X – axis ,its gradient is zero

b. A gradient of a straight line is the ratio of horizontal distance to the vertical distance.

A straight line has a gradient with a negative value if it inclines downwards to The left.

d. A straight line has a gradient with a positive value if it inclines downwards to The right

Math Quiz on Coordinate Geometry and Linear Equations

1) A stone is thrown vertically upward with height y = 40x − 5x². What is the maximum height reached?

50

60

75

80

The quadratic function opens downward because the coefficient of x² is negative. The maximum value occurs at the vertex. Using x = −b/(2a), where a = −5 and b = 40, gives x = 4. Substituting x = 4 into y = 40x − 5x² gives y = 160 − 80 = 80. Therefore, the maximum height is 80 meters.

Explanation

The quadratic function opens downward because the coefficient of x² is negative. The maximum value occurs at the vertex. Using x = −b/(2a), where a = −5 and b = 40, gives x = 4. Substituting x = 4 into y = 40x − 5x² gives y = 160 − 80 = 80. Therefore, the maximum height is 80 meters.

2) Which statement about the gradient of a straight line is true?

Gradient is vertical over horizontal

Gradient is horizontal over vertical

A line parallel to the x-axis has zero gradient

Downward to the right gives negative gradient

Gradient is defined as change in vertical distance divided by change in horizontal distance. A line parallel to the x-axis has no vertical change, so its rise is zero. Since slope equals rise over run, a zero rise results in a gradient of zero. Other options incorrectly reverse the ratio or misinterpret slope direction.

Explanation

Gradient is defined as change in vertical distance divided by change in horizontal distance. A line parallel to the x-axis has no vertical change, so its rise is zero. Since slope equals rise over run, a zero rise results in a gradient of zero. Other options incorrectly reverse the ratio or misinterpret slope direction.

3) Which equation represents a line parallel to y = 5x − 4?

Y = 2x − 4

5x + y = 12

−5x − y = 6

5x − y + 3 = 0

Parallel lines have equal gradients. The line y = 5x − 4 has slope 5. Rearranging each option into slope-intercept form shows only 5x − y + 3 = 0 simplifies to y = 5x + 3, which has slope 5. Therefore, it is parallel to the given line.

Explanation

Parallel lines have equal gradients. The line y = 5x − 4 has slope 5. Rearranging each option into slope-intercept form shows only 5x − y + 3 = 0 simplifies to y = 5x + 3, which has slope 5. Therefore, it is parallel to the given line.

4) A line passes through (−1, −5) with gradient 3. Which is its equation?

Y = 3x + 2

5x + y = 12

−5x − y = 6

3x − y + 2 = 0

Using point-slope form y − y₁ = m(x − x₁), substitute m = 3 and point (−1, −5). This gives y + 5 = 3(x + 1). Expanding yields y + 5 = 3x + 3, which rearranges to 3x − y + 2 = 0. This matches the correct option.

Explanation

Using point-slope form y − y₁ = m(x − x₁), substitute m = 3 and point (−1, −5). This gives y + 5 = 3(x + 1). Expanding yields y + 5 = 3x + 3, which rearranges to 3x − y + 2 = 0. This matches the correct option.

5) What is the gradient of the line through points (1, −3) and (3, 7)?

−5

5

1/5

−1/5

Gradient equals change in y divided by change in x. From (1, −3) to (3, 7), change in y is 7 − (−3) = 10 and change in x is 3 − 1 = 2. Dividing gives 10 ÷ 2 = 5. Therefore, the slope of the line is 5.

Explanation

Gradient equals change in y divided by change in x. From (1, −3) to (3, 7), change in y is 7 − (−3) = 10 and change in x is 3 − 1 = 2. Dividing gives 10 ÷ 2 = 5. Therefore, the slope of the line is 5.

6) Which is the equation of the line through (−2, 4) and (3, −2)?

−6x + 5y − 8 = 0

6x + 5y − 8 = 0

6x + 5y + 8 = 0

5x − 6y − 8 = 0

First calculate slope: (−2 − 4)/(3 − (−2)) = −6/5. Using point-slope form with (−2, 4) gives y − 4 = (−6/5)(x + 2). Rearranging and multiplying through yields 6x + 5y − 8 = 0, which matches the correct equation.

Explanation

First calculate slope: (−2 − 4)/(3 − (−2)) = −6/5. Using point-slope form with (−2, 4) gives y − 4 = (−6/5)(x + 2). Rearranging and multiplying through yields 6x + 5y − 8 = 0, which matches the correct equation.

7) Which pairs of points have a slope of 2/3?

I only

II only

II and III only

I, II, and III

Calculate slope for each pair. Pair II gives (2 − 0)/(8 − 5) = 2/3. Pair III gives (−1 − 3)/(−3 − 3) = −4/−6 = 2/3. Pair I gives 4/3, which is incorrect. Therefore, only II and III satisfy the condition.

Explanation

Calculate slope for each pair. Pair II gives (2 − 0)/(8 − 5) = 2/3. Pair III gives (−1 − 3)/(−3 − 3) = −4/−6 = 2/3. Pair I gives 4/3, which is incorrect. Therefore, only II and III satisfy the condition.

8) A line passes through (−1, −1) and is parallel to 4x − 2y = 5. What is its equation?

Y = 2x − 1

Y = 2x + 1

Y = 4x − 1

Y = 4x + 1

Parallel lines have equal slopes. Rearranging 4x − 2y = 5 gives y = 2x − 5/2, so slope is 2. Using point-slope form with (−1, −1) gives y + 1 = 2(x + 1), which simplifies to y = 2x + 1.

Explanation

Parallel lines have equal slopes. Rearranging 4x − 2y = 5 gives y = 2x − 5/2, so slope is 2. Using point-slope form with (−1, −1) gives y + 1 = 2(x + 1), which simplifies to y = 2x + 1.

9) A line perpendicular to 2x + 5y − 3 = 0 passes through the intersection of two given lines. Which is correct?

2y − 5x = −1

2x − 5y = −1

5x − 2y = 4

5x − 2y = 8

The given line has slope −2/5. A perpendicular line has slope 5/2. Solving the two given equations gives the intersection point, then applying point-slope form with slope 5/2 produces the equation 2x − 5y = −1. This matches the correct option.

Explanation

The given line has slope −2/5. A perpendicular line has slope 5/2. Solving the two given equations gives the intersection point, then applying point-slope form with slope 5/2 produces the equation 2x − 5y = −1. This matches the correct option.

10) What is the solution of 2x + y = 9 and 3x + 2y = 13?

{5, 1}

{−5, 1}

{5, −1}

{−5, −1}

Solve simultaneously. From 2x + y = 9, express y = 9 − 2x. Substitute into 3x + 2y = 13 to get 3x + 2(9 − 2x) = 13. Simplifying gives x = 5 and y = −1, forming the solution set {5, −1}.

Explanation

Solve simultaneously. From 2x + y = 9, express y = 9 − 2x. Substitute into 3x + 2y = 13 to get 3x + 2(9 − 2x) = 13. Simplifying gives x = 5 and y = −1, forming the solution set {5, −1}.