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Question Bank No: 1

1. If the foot of the perpendicuar from the origin to a straight line is at the point (3, -4) then the equation of the line is

a)		3x – 4y = 25
b)		3x – 4y + 25 = 0
c)		4x + 3y – 25 = 0
d)		4x – 3y + 25 = 0

2. The equations of the lines through (-1, 1) and making an angle $45^{0}$ with the line x + y = 0 are given by

a)		$x^{2}$ – xy + x – y = 0
b)		xy – $y^{2}$ + x – y = 0
c)		xy + x + y = 0
d)		xy + x + y + 1 = 0

3. The equation of the two sides of a square whose area is 25 sq. Units are 3x – 4y = 0 and 4x + 3y = 0. The equations of the other two sides of the square are

a)		3x – 4y $\pm$ 25 = 0, 4x + 3y $\pm$ 25 = 0
b)		3x – 4y $\pm$ 5 = 0, 4x + 3y $\pm$ 5 = 0
c)		3x – 4y $\pm$ 5 = 0, 4x + 3y $\pm$ 25 = 0
d)		None of these

4. If one vertex of an equilateral triangle is at (2, -1) and the base is x + y – 2 = 0, then the length of each side is

a)		$\sqrt{\frac{3}{2}}$
b)		$\sqrt{\frac{2}{3}}$
c)		$\frac{2}{3}$
d)		$\frac{2}{3}$

5. The ratio in which the segment of the line joining ( $x_{1}$ , $y_{1}$ ) and ( $x_{2}$ , $y_{2}$ ) is cut by the line Ax + By + C = 0, is

a)		$\frac{A x_{1} + B y_{1} + C}{A x_{2} + B y_{2} + C}$
b)		$- \frac{A x_{1} + B y_{1} + C}{A x_{2} + B y_{2} + C}$
c)		$\frac{A x_{2} + B y_{2} + C}{A x_{1} + B y_{1} + C}$
d)		$- \frac{A x_{2} + B y_{2} + C}{A x_{1} + B y_{1} + C}$

6. If the axes are rotated through an angle of $30^{0}$ in the clockwise direction, the point (4, -2 $\sqrt{3}$ ) in the new system was formerly

a)		(2, $\sqrt{3}$ )
b)		( $\sqrt{3}$ , -5)
c)		( $\sqrt{3}$ , 2)
d)		(2, 3)

7. If each of the points ( $x_{1}$ , 4) and (-2, $y_{1}$ ) lies on the line joining the points (2, -1) and (5, -3) then the point P( $x_{1}$ , $y_{1}$ ) lies on the line

a)		x = 3y
b)		x = -3y
c)		y = 2x + 1
d)		2x + 6y + 1 = 0

8. The orthocentre of the triangle formed by the lines x y = 1, 2x + 3y = 6 and 4x – y + 4 = 0 lies in

a)		I quadrant
b)		II quadrant
c)		III quadrant
d)		IV quadrant

9. The number of lines that can be drawn through the point (4, $\sqrt{13}$ ) at a distance of 3 units from the point (-2, 0) is

a)		0
b)		1
c)		2
d)		infinite

10. Two sides of an isosceles triangle are given by the equation 7x – y + 3 = 0 and x + y – 3 = 0. If its third side passes through the point (1, -10) then its equations are

a)		X – 3y – 7 = 0 or 3x + y – 31 = 0
b)		x – 3y – 31 = 0 or 3x + y – 7 = 0
c)		x – 33y – 31 = 0 or 3x + y + 7 = 0
d)		None of these

11. If a straight line L perpendicular to the line 5x – y = 1 such that the area of the $Δ$ formed by the line L and the coordinate axes is 5, then the equation of the line L is

a)		X + 5y + 5 = 0
b)		x + 5y $\pm$ $\sqrt{2}$ = 0
c)		x + 5y $\pm$ $\sqrt{5}$ = 0
d)		x + 5y $\pm$ $5 \sqrt{2}$ = 0

12. If u = $a_{1}$ x+ $b_{1}$ y + $c_{1}$ = 0, $υ$ = $a_{2}$ x+ $b_{2}$ y + $c_{2}$ = 0 and $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ , then u + k $υ$ = 0 represents

a)		A family of concurrent lines
b)		a family of parallel lines
c)		u = 0 or $υ$ = 0
d)		None of these

13. The point P(1, 1) is translated parallel to 2x = y in the first quadrant through a unit distance. The coordinates of the new position of P are

a)		$(1 \pm \frac{2}{\sqrt{5}}, 1 \pm \frac{1}{\sqrt{5}})$
b)		$(1 \pm \frac{1}{\sqrt{5}}, 1 \pm \frac{2}{\sqrt{5}})$
c)		$(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}})$
d)		$(\frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}})$

14. The number of lines that can be drawn through the point (5, 2) at a distance of 5units from the point (2, -2) is

a)		0
b)		1
c)		2
d)		infinite

15. The straight lines 2x + 3y = 5 and 6x – 4y + $λ$ = 0, $λ$ $\in$ R, are sides of (if the third line is not parallel to any of these two lines)

a)		An equilateral triangle
b)		right angled triangle
c)		obtuse angled triangle
d)		cannot be the sides of a triangle

16. The line segment joining the points (1, 2) and (k, 1) is divided by the line 3x + 4y – 7 = 0 in the ratio 4 : 9, then k is

a)		-2
b)		2
c)		-3
d)		3

17. If A and B are two fixed points, then the locus of a point which moves in such a way that the angle APB is a right angle is

a)		a circle
b)		an ellipse
c)		a parabola
d)		None of these

18. The angle between the lines x cos $α$ + y sin $α$ = $p_{1}$ and x cos $β$ + y sin $β$ = $p_{2}$ where $α$ > $β$ .

a)		$α$ - $β$
b)		$α$ + $β$
c)		$α$ $β$
d)		2 $α$ - $β$

19. One vertex and a diagonal of a square are (-4, 5) and 7x - y + 8 = 0. Then, the equation of the second diagonal is

a)		x + 7y + 31 = 0
b)		x + 7y - 31 = 0
c)		x - 7y + 31 = 0
d)		x - 7y - 31 = 0

20. The points (0, -1), (-2, 3), (6, 7) and (8, 3) are

a)		Colliner
b)		vertices of a parallelogram which is not a rectangle
c)		vertices of a rectangle which is not a square
d)		None of these

21. The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P(h, k) with the lines y = x and x+y =2 is 4 $h^{2}$ . The locus of P is

a)		$2 x - y - 1 = 0$
b)		$2 x + y + 1 = 0$
c)		$2 x - y + 1 = 0$
d)		$2 x + y - 3 = 0$

22. If the pair of lines ${ax}^{2} + 2 (a + b) xy + b y^{2} = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one sector is thrice that of another, then

a)		$3 a^{2} - 2 ab + 3 b^{2} = 0$
b)		$3 a^{2} - 10 ab + 3 b^{2} = 0$
c)		$3 a^{2} + 2 ab + 3 b^{2} = 0$
d)		$3 a^{2} + 10 ab + 3 b^{2} = 0$

23. If the two rays in the first quadrant, x +y = $| a |$ and ax-y = 1 intersect, then a $\in$ ( $a_{0}, \infty$ ) where $a_{0}$ =

a)		0
b)		-1
c)		1
d)		2

24. A straight line through the point A (3, 4) is such that its intercept between the axes is bisected at A. Its equation is

a)		4x +3y =24
b)		3x +4y = 25
c)		x +y = 7
d)		3x - 4y + 7 = 0

25. If a vertex of a triangle is (1, 1) and the midpoints of two sides of a triangle through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is

a)		$(- \frac{1}{3}, \frac{7}{3})$
b)		$(- 1, \frac{7}{3})$
c)		$(\frac{1}{3}, \frac{7}{3})$
d)		$(1, \frac{7}{3})$

26. If non- zero numbers a, b, c are in H.P., then the line $\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$ passes through the point

a)		(-1, -2)
b)		(-1, 2)
c)		(1, - $\frac{1}{2}$ )
d)		(1, -2)

27. The centre of the circle which circumscribes the square formed by $x^{2} - 8 x + 12$ = 0 and $y^{2} - 14 y + 45 = 0$ is

a)		(3, 7)
b)		(4, 7)
c)		(2, 5)
d)		(6, 9)