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) x2 + y2 = 2(x + y)
(B) x2 + y2 = (x + y)
(C) x2 + y2 = 2(x – y)
(D) x2 + y2 = (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 + a2 y + 2aT = 0
(B) 2Tx – a2y + 2aT = 0
(C) 2Tx + a2y – 2aT = 0
(D) 2Tx – a2y – 2aT = 0
9. If A1 , A2 , A3 , … , An are n points in a plane whose coordinates are (xi , yi), i = 1 , 2 , … , n respectively. A1A2 is bisected by the point G1 ; G1A3 is divided by G2 in ratio 1 : 2 and G2A4 is divided by G3 in the ratio 1 : 3 , G3A5 at G4 in the ratio 1 : 4 and so on until all the points are exhausted, then the coordinates of fixed point Gn – 1 so obtained will be
(A) 
(B) 
(C) 
(D) 
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) x2 + y2 = 13
B) x2 + y2 = 5
(C) y2 = 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 15x2 + 56xy + 48y2 = 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 x2 – y2 + 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 (m2, 2m), (mm’, m + m’) and (m’2, 2m’) respectively on the line x cosα + y sinα + sin2α/cos2α = 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 SQ2 + SR2 = 2SP2 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 ax2 + 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) x2 + y2 – 2x – 4y + 1 = 0
(B) 3(x2 + y2) – 2x – 4y + 1 = 0
(C) x2 + y2 – 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 – y1 = m(x-x1). If m and x1 are fixed and different lines are drawn for different values of y1, 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 DOAB 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