# Mathematics Questions and Answers – Slope of a Line

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Slope of a Line”.

1. What is the distance between (1, 3) and (5, 6)?
a) 3 units
b) 4 units
c) 5 units
d) 25 units

Explanation: We know, distance between two points (x1, y1) and (x2, y2) is $$\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$.
So, distance between (1, 3) and (5, 6) is $$\sqrt{(1-5)^2+(3-6)^2}=\sqrt{(4)^2+(3)^2}$$ = 5 units.

2. What is the distance of (5, 12) from origin?
a) 6 units
b) 8 units
c) 10 units
d) 13 units

Explanation: We know, distance between two points (x1, y1) and (x2, y2) is $$\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$.
So, distance between (5, 12) from origin (0, 0) is $$\sqrt{(5-0)^2+(12-0)^2} = \sqrt{(5)^2+(12)^2}$$ = 13 units.

3. The coordinates of a point dividing the line segment joining (1, 2) and (4, 5) internally in the ratio 2:1 is ____________________
a) (3, 4)
b) (4, 3)
c) (5, 4)
d) (5, 3)

Explanation: The coordinates of a point dividing the line segment joining (x1, y1) internally in the ratio m: n is $$(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})$$.
So, the coordinates of a point dividing the line segment joining (1, 2) and (4, 5) internally in the ratio 2:1 is $$(\frac{2*4+1*1}{2+1}, \frac{2*5+1*2}{2+1})$$ = (3, 4).

4. In which ratio (3, 4) divides the line segment joining (1, 2) and (4, 5) internally?
a) 1:2
b) 2:1
c) 3:4
d) 4:3

Explanation: The coordinates of a point dividing the line segment joining (x1, y1) and (x2, y2) internally in the ratio m: n is $$(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})$$.
Let the ratio be k: 1.So, the coordinates of a point dividing the line segment joining (1, 2) and (4, 5) internally in the ratio k: 1 is $$(\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1})$$
=> $$(\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1})$$ is same as (3, 4).
=> (4k+1)/(k+1) = 3
=> 4k+1 = 3k+3
=> k = 2
So, ratio is 2:1.

5. The coordinates of a point dividing the line segment joining (1, 2) and (4, 5) externally in the ratio 2:1 is ____________________
a) (4, 5)
b) (6, 8)
c) (7, 8)
d) (8, 6)

Explanation: The coordinates of a point dividing the line segment joining (x1, y1) and (x2, y2) externally in the ratio m: n is $$(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n})$$.
So, the coordinates of a point dividing the line segment joining (1, 2) and (4, 5) externally in the ratio 2:1 is $$(\frac{2*4-1*1}{2-1},\frac{2*5-1*2}{2-1})$$ = (7, 8).

6. _____________ is the midpoint of (1, 2) and (5, 8).
a) (2, 5)
b) (3, 5)
c) (5, 2)
d) (5, 3)

Explanation: We know, midpoint of (x1, y1) and (x2, y2) is $$(\frac{x1+x2}{ 2}, \frac{y1+y2}{2})$$.
So, midpoint of (1, 2) and (5, 8) is ((1+5)/2, (2+8)/2) is (3, 5).

7. What is the area of triangle whose vertices are (-4, -4), (-3, 2), (3, -16)?
a) 24 sq. units
b) 27 sq. units
c) 32 sq. units
d) 37 sq. units

Explanation: We know, area of triangle joining vertices (x1, y1), (x2, y2) and (x3, y3) is (1/2)* determinant $$\begin{pmatrix}-4 & -4 &1 \\-3& 2 & 1\\ 3& -16 & 1\end{pmatrix}$$ is $$\frac{1}{2}$${(-4)(2+16) – (-4)(-3-3) + (1)(48-6)} = $$\frac{1}{2}$$|(-72)+(-24)+42| = 27 square units.

8. If area of triangle formed by three points is zero then the three points must be collinear.
a) True
b) False

Explanation: Area of triangle formed by three points is zero then the three points must be collinear i.e. they must lie on the same line.

9. Angle made by line with ____________ measured anticlockwise is called inclination of the line.
a) positive x-axis
b) negative x-axis
c) positive y-axis
d) negative y-axis

Explanation: We know, inclination of line is always measured with positive x-axis in anticlockwise direction.

10. Slope of a line is given by _________ if inclination of line is α.
a) sinα
b) cosα
c) tanα
d) cotα

Explanation: Slope of a line is given by tanα if inclination of line is α. Slope is denoted by tangent of the inclination angle.

11. Find slope of line if inclination made by the line is 60°.
a) 1/2
b) 1/√3
c) √3
d) 1

Explanation: Slope of a line is given by tanα if inclination of line is α. If inclination is 60° the slope is tan 60° = √3.

12. What is the inclination of a line which is parallel to x-axis?
a) 0°
b) 180°
c) 45°
d) 90°

Explanation: If a line is parallel to x-axis then angle formed by it with x-axis is zero. So, its inclination is zero.

13. What is the inclination of a line which is parallel to y-axis?
a) 0°
b) 180°
c) 45°
d) 90°

Explanation: If a line is parallel to y-axis then angle formed by it with x-axis is 90°. So, its inclination is 90°.

14. What is the slope of a line which is parallel to x-axis?
a) -1
b) 0
c) 1
d) Not defined

Explanation: If a line is parallel to x-axis then angle formed by it with x-axis is zero. So, its inclination is zero. Hence slope = tan 0° = 0.

15. What is the slope of a line which is parallel to y-axis?
a) -1
b) 0
c) 1
d) Not defined 