Q.1. Find the area of the triangle formed by the vertices (4, 5), (10, 12) and (-3, 2)

A. 3

B. 4.5

C. 4

D. 3.5

Sol : Option D
The area of the triangle having its vertices as P(x_{1}, y_{1}), Q(x_{2}, y_{2}) and R(x_{3}, y_{3}) is given by
1/2 [x_{1( y2 - y3) + x2( y3 - y1) + x3( y1 - y2)]
= 1/2 [4(12+2) + 10(-2-5) + (-3)(5-12)] = 1/2[4.14 - 10.7 + (-3)(-7)] = 1/2[56 - 70 + 21] = 3.5
Hence, solution is option D }

Q.2. Find the equation of straight line passing through (2, 3) and perpendicular to the line 3x + 2y + 4 = 0

A. y=5/3x- 2

B. 3Y=2x+5

C. 3Y=5x-2

D. None of these

Sol : Option B
The given line is 3x + 2y + 4 = 0 or y = -3x / 2 - 2
Any line perpendicular to it will have slope = 2 / 3
Thus equation of line through (2, 3) and slope 2 / 3 is
(y – 3) = 2 / 3 (x – 2)
3y – 9 = 2x – 4
3y – 2x – 5 = 0.

Q.3. Find the coordinates of the point which will divide the line joining the points (3, 5) and (11, 8) externally in the ratio 5: 2.

A. (5/3 , 1/3)

B. (3/49 , 1/10)

C. (49/3 , 10)

D. None of these

Sol : Option C
The external division case will use the formula
x = (mx2 – nx1)/(m – n)
y = (my2 – ny1)/(m – n)
where m:n is 5:2 in our case.
Putting the values you will get points (49/3, 10).

Q.4. Find the coordinate of the point which will divide the line joining the point (2,4) and (7,9) internally in the ratio 1:2?

A. (5/3 , 1/3)

B. (3/8 , 3/11)

C. (8/3 , 11/3)

D. (11/3 , 17/3)

Sol : Option D
The internal division will use the formula (mx2 + nx1)/(m + n)
y = (my2 + ny1)/(m + n).
So, the point becomes (11/3, 17/3).

Q.5. What is the slope of the line passing through the points J (-2, 3) and (2, 7)?

A. 1

B. 2

C. &redic;2

D. 4.&redic;3

Sol : Option A
[(y_{2} - y_{1}) / (x_{2} - x_{1})] = [(7-3) / (2-(-2)) ] = 4/4 = 1
Hence, answer is option A

Q.6. Find the equation of the line whose slope is 3 and y intercept is – 4.

A. y = 2x – 3

B. Y=3x+4

C. Y=3x- 4

D. 4.y=√3x-2

Sol : Option C
Given m = 3 and c = - 4. Substituting values in y = mx + c, we get y = 3x – 4.
Hence, solution is option C

Q.7. Find the equation of the line passing through (2, -1) and parallel to the line 2x – y = 4.

A. y = 2x – 5

B. Y=2x+6

C. Y=√2x +7

D. 4.y=2x+5

Sol : Option A
The given line is 2x – y = 4 ⇒ y = 2x – 4 (Converting into the form of y = mx + c)
Its slope = 2. The slope of the parallel line should also be 2.
Hence for the required line
m = 2 and (x1 , y1) = (2, -1).
Equation = y - y_{1} /x - x_{1} = y_{2} - y_{1} /x_{2} - x_{1}
⇒ y - y_{1} /x - x_{1} = m
⇒ y – y1 = m (x – x1) ⇒ y – ( - 1) = 2 (x – 2)
⇒ y = 2x – 5.

Q8. Find the equation of the line parallel to the line passing through (5,7) and (2,3) and having x intercept as -4.

A. 3y = 4x -16

B. 4y=3x-16

C. 3y = 4x +16

D. 4y=3x+16

Sol : Option C
Slope of the given line = (7 – 3) / (5 – 2) = 4/3.
So the slope of the required line is also 4/3. One point on this line is (-4, 0). Hence the equation of the line is
y – 0 = 4/3 (x + 4) ⇒ 3y = 4x +16.

Q9. Find the coordinates of the circum-centre of the triangle whose vertices are (0, 0), (8,0) and (0,6). Find the Circum-radius also.

A. (4, 3),6

B. (3,4),5

C. (4, 3),5

D. (4, 3),3

Sol : Option B
Circum-centre is the point of intersection of the perpendicular bisectors of the three sides of a triangle.
Let S (x, y) be the circumcentre.
SA = SB = SC. ∴ &redic;(x^{2} + y^{2}) = &redic; (x-8)^{2} + (y-2) )^{2}.
&redic;(x^{2} + y^{2}) = &redic; (x-0)^{2} + (y-6) )^{2}.
Squaring x^{2} + y^{2} = x^{2} – 16x + 64 + y^{2}. So x = 4.
Also x^{2} + y^{2} = x^{2} + y^{2} – 12y + 36.
So y = 3. Hence coordinates of the circumcentre is (4, 3).
Circumradius = SA = &redic; x^{2} + y = &redic;16+9 = 5.

Q10. Find the equation of the line passing through (2, -1) and perpendicular to the line 2x – y = 4.

A. x+2y=0

B. Y=5x-2

C. Y=2x-5

D. None of these

Sol : Option A
Slope of the given line is 2. By comparing with y=mx+c
The perpendicular lines have the product of their slopes= (-1)
So, slope of new line will be (-1/2)
Equation of line will be y=mx+c
M=-1/2
So we get, y= (-1/2)x+k
We are given the point (2,-1) that will satisfy this line.
Putting these coordinates we get k=0 and the line becomes=x+2y=0