LEVEL – I

1. Let P and Q be the points on the line joining A(–2 , 5) and B(3 , 1) such that AP = PQ = QB , then the mid point of PQ is

(A) (1/2 , 3)

(B) (-1/4 ,4)

(C) (2, 3)

(D) (– 1, 4)

2. P is a point lying on line y = x then maximum value of |PA – PB| , ( where A ≡ (1 , 3) , B ≡(5 , 2)) is

(A) √5

(B) 2√2

(C) √17

(D) 3/ √2

3. Locus of point of intersection of the perpendicular lines one belonging to (x + y – 2) + λ(2x + 3y – 5) = 0 and other to (2x + y – 11) + λ(x + 2y – 13) = 0 is a

(A) circle

(B) straight line

(C) pair of lines

(D) None of these

4. A(-3, 4), B(5, 4), C and D form a rectangle. If x – 4y + 7 = 0 is a diameter of the circumcircle of the rectangle ABCD then area of rectangle ABCD is

(A) 8

(B) 16

(C) 32

(D) 64

5. The triangle ABC has medians AD, BE, CF. AD lies along the line y = x + 3 , BE lies along the line y = 2x + 4, AB has length 60 and angle C = 90°, then the area of ΔABC is

(A) 400

(B) 200

(C) 100

(D) none of these

6. A line passes through (1, 0). The slope of the line, for which its intercept between y = x – 2 and y = -x + 2 subtends a right angle at the origin, is

(A) ± 2/3,

(B) ± 3/2

(C) ± 1

(D) none of these

7. The locus of the image of origin in line rotating about the point (1 , 1) is

(A) x^{2} + y^{2} = 2(x + y)

(B) x^{2} + y^{2} = (x + y)

(C) x^{2} + y^{2} = 2(x – y)

(D) x^{2} + y^{2} = (x – y)

8. A line through the point (–a, 0) cuts from the second quadrant a triangular region with area T. The equation for the line is

(A) 2Tx + a^{2} y + 2aT = 0

(B) 2Tx – a^{2}y + 2aT = 0

(C) 2Tx + a^{2}y – 2aT = 0

(D) 2Tx – a^{2}y – 2aT = 0

9. If A_{1} , A_{2} , A_{3} , … , A_{n} are n points in a plane whose coordinates are (x_{i} , y_{i}), i = 1 , 2 , … , n respectively. A_{1}A_{2} is bisected by the point G_{1} ; G_{1}A_{3} is divided by G_{2} in ratio 1 : 2 and G_{2}A_{4} is divided by G_{3} in the ratio 1 : 3 , G_{3}A_{5} at G_{4} in the ratio 1 : 4 and so on until all the points are exhausted, then the coordinates of fixed point G_{n} – 1 so obtained will be

(A) $ \displaystyle ( \sum_{i=1}^{i=n}\frac{i x_i}{n} , \sum_{i=1}^{i=n}\frac{i y_i}{n} ) $

(B) $ \displaystyle ( \sum_{i=1}^{i=n}\frac{ x_i}{n} , \sum_{i=1}^{i=n}\frac{ y_i}{n} ) $

(C) $ \displaystyle ( \sum_{i=1}^{i=n}\frac{(i-1) x_i}{(n-1)} , \sum_{i=1}^{i=n}\frac{(i-1) y_i}{(n-1)} ) $

(D) $ \displaystyle ( n\sum_{i=1}^{i=n}x_i , n\sum_{i=1}^{i=n}y_i ) $

10. The vertex A of a triangle ABC is the point (-2, 3) whereas the line perpendicular to the sides AB and AC are x – y – 4 = 0 and 2x – y – 5 = 0 respectively. The right bisectors of sides meet at P(3/2 , 5/2) . Then the equation of the median of side BC is

(A) 5x + 2y = 10

(B) 5x – 2y = 16

(C) 2x – 5y = 10

(D) none of these

__ANSWER:__

1. A 2. A 3. A 4. C 5. A 6. D 7. A 8. B 9. B 10. D

LEVEL – I

11. The orthocentre of a triangle whose vertices are (0, 0), (√3 , 0) and (0, √6) is

(A) (2, 1)

(B) (3, 2)

(C) (4, 1)

(D) none of these

12. The locus of a point P which divides the line joining (1, 0) and (2 cosθ, 2 sinθ) internally in the ratio 2 : 3 for all θ ∈ R is

(A) a straight line

(B) a circle

(C) a pair of straight line

(D) a parabola

13. Let ax + by + c = 0 be a variable straight line, where a, b and c are 1st, 3rd and 7th terms of some increasing A.P. Then the variable straight line always passes through a fixed point which lies on

(A) x^{2} + y^{2} = 13

B) x^{2} + y^{2} = 5

(C) y^{2} = 9x

(D) 3x+4y=9

14. The image of the point (3, 8) with respect to the line x + 3y = 7 is

(A) (–1, –4)

(B) (–1, 4)

(C) (1, 4)

(D) none of these

15. Number of points lying on the line 7x + 4y + 2 = 0 which is equidistant from the lines 15x^{2} + 56xy + 48y^{2} = 0 is

(A) 0

(B) 1

(C) 2

(D) 4

16. The point (4, 1) undergoes the following three transformations successively

(a) Reflection about the line y = x

(b) Transformation through a distance 2 units along the positive direction of the x-axis.

(c) Rotation through an angle p/4 about the origin in the anti clockwise direction.

The final position of the point is given by the co-ordinates

(A) (-4/√2 , 1/√2)

(B) (-1/√2 , 7/√2)

(C) (-1/√2 , 4/√2)

(D) (-3/√2 , 4/√2 )

17. The three lines 4x – 7y + 10 = 0, x + y = 5 and 7x + 4y = 15 form the sides of a triangle. Then the point (1, 2) is its

(A) centroid

(B) incentre

(C) orthocentre

(D) none of these

18. If (-6, -4) and (3, 5) are the extremities of the diagonal of a parallelogram and (-2, 1) is its third vertex, then its fourth vertex is

(A) (-1, 0)

(B) (0, -1)

(C) (-1, 1)

(D) none of these

19. Area of the triangle formed by the line x + y = 3 and angle bisector of the pair of straight lines x^{2} – y^{2} + 2y – 1 = 0 is

(A) 2 sq. units

(B) 4 sq. units

(C) 6 sq. units

(D) 8 sq. uints

20. If P1, P2, P3 be the perpendicular from the points (m^{2}, 2m), (mm’, m + m’) and (m’^{2}, 2m’) respectively on the line x cosα + y sinα + sin^{2}α/cos^{2}α = 0, then P1, P2, P3 are in

(A) A.P.

(B) G.P.

(C) H.P.

(D) none of these

__ANSWER:__

11. D 12. B 13. A 14. A 15. B 16. B 17. C 18. A 19. A 20. B

LEVEL – I

21. If P = (1, 0), Q = (-1, 0), R = (2, 0) are 3 given points, then the locus of the point S satisfying the relation SQ^{2} + SR^{2} = 2SP^{2} is

(A) a straight line parallel to the x-axis

(B) circle through the origin

(C) circle with centre at the origin

(D) a straight line parallel to the y-axis.

22. If the quadratic equation ax^{2} + bx + c = 0 has –2 as one of its roots then ax + by + c = 0 represents

(A) A family of concurrent lines

(B) A family of parallel lines

(C) A single line

(D) A line perpendicular to x-axis

23. The line 3x + 2y = 24 meets y-axis at A and x-axis at B. The perpendicular bisector of AB meets the x-axis at C, then area of DABC is

(A) 78

(B) 92

(C)

(D) none of these

24. Two vertices of a triangle are (5, -1) and (-2, 3). If orthocenter of the triangle is origin, then the co-ordinates of third vertex is

(A) (4, 7)

(B) (3, 7)

(C) (-4, -7)

(D) None of these

25. The number of integral values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is

(A) 2

(B) 0

(C) 4

(D) 1

26. If A(cosα, sinα), B(sinα, -cosα), C(2, 1) are the vertices of a ΔABC, then as a varies the locus of its centroid is

(A) x^{2} + y^{2} – 2x – 4y + 1 = 0

(B) 3(x^{2} + y^{2}) – 2x – 4y + 1 = 0

(C) x^{2} + y^{2} – 2x – 4y + 3 = 0

(D) none of these

27. The straight line y = x–2 rotates about a point where it cuts the x-axis and becomes perpendicular to the straight line ax + by + c = 0. Then its equation is

(A) ax + by + 2a = 0

(B) ax – by – 2a = 0

(C) bx + ay – 2b = 0

(D) ay – bx + 2b = 0

28. It is desired to construct a right angled triangle ABC (∠C = π/2) in xy plane so that it’s sides are parallel to coordinates axes and the medians through A and B lie on the lines y = 3x + 1 and y = mx + 2 respectively. The values of ‘m’ for which such a triangle is possible is /are

(A) –12

(B)3/4

(C) 4/3

(D) 1/12

29. If 3a + 2b + 6c = 0, the family of lines ax + by + c = 0 passes through a fixed point whose coordinates are given by

(A) (1/2 , 1/3)

(B) (2, 3)

(C) (3, 2)

(D) (1/3 , 1/2)

30. Area of the parallelogram whose sides are x cosα + y sinα = p, x cosα + y sinα = q, xcosβ + y sinβ = r, x cosβ + y sinβ = s is

(A) pq + rs

(B) |pq tanα + rs tanβ|

(C) |(p – q)(r – s)cosec(α – β)|

(D) |(p – q)(r – s) tan(α + β)|

31. Consider the equation y – y_{1} = m(x-x_{1}). If m and x_{1} are fixed and different lines are drawn for different values of y_{1}, then

(A) the line will pass through a fixed point

(B) there will be a set of parallel lines.

(C) all the lines will be parallel to the line y=y1.

(D) none of these

32. The medians AD and BE of a triangle ABC with vertices A(0, b), B(0, 0) and C(a, 0) are perpendicular to each other if

(A) b = √2a

(B) a = √2b

(C) b = -√2a

(D) none of these

33. If ΔOAB is an equilateral triangle (O is the origin and A is a point on the x-axis), then centroid of the triangle will be

(A) always rational

(B) rational if B is rational

(C) rational if A is rational

(D) never rational

(a point P(x, y) is said to be rational if both x and y are rational)

34. Let 2x–3y =0 be a given line and P (sinθ, 0) and Q (0, cosθ) be the two points. Then P and Q lie on the same side of the given line, if q lies in the

(A) 1st quadrant

(B) 2nd quadrant

(C) 3rd quadrant

(D) none of these

35. Two sides of a rhombus ABCD are parallel to the lines y = x + 2 and y = 7x + 3. If the diagonals of the rhombus intersect at the point (1, 2) and the vertex A is on the y-axis, then possible co-ordinates of A is

(A) (0, 0)

(B) (0, 1)

(C) (0, 3)

(D) (0, -1)

__ANSWER:__

21. (D) 22. (A) 23. (A) 24. (C) 25. (A )

26. (D) 27. (D) 28. (B) 29. (A) 30. (C)

31. (B) 32. (B) 33. (D) 34. (B) 35. (A)