3D LEVELI 1 The locus of the point, which moves such that its distance from (1, 2, 2) is unity, is (A) x2 + y2 + z2 2x + 4y 4z + 8 = 0 (B) x2 + y2 + z2 2x 4y 4z + 8 = 0 (C) x2 + y2 + z2 + 2x + 4y 4z + 8 = 0 (D) x2 + y2 + z2 2x + 4y + 4z + 8 = 0 *2 The angle between the lines whose direction ratios are 1, 1, 2; 3 1, 3 1, 4 is

(A) cos11

65 (B) 6

(C) 3 (D)

4

*3. The plane passing through the point (a, b, c) and parallel to the plane x + y + z = 0 is (A) x + y + z = a + b + c (B) x + y + z + (a + b + c) = 0 (C) x + y + z + abc = 0 (D) ax + by + cz = 0

4. The equation of line through the point (1, 2, 3) parallel to line x 4 y 1 z 102 3 8

are

(A) x 1 y 2 z 32 3 8

(B) x 1 y 2 z 31 2 3

(C) x 4 y 1 z 101 2 3

(D) none of these

5. The value of k, so that the lines x 1 y 2 z 33 2k 2

, x 1 y 5 z 63k 1 5

are perpendicular

to each other, is

(A) 107

(B) 87

(C) 67

(D) 1

*6. The angle between a line with direction ratios 2:2:1 and a line joining (3,1,4,) to (7,2,12)

(A) cos123

(B) cos

132

(C) tan123

(D) none of these

7. The equation of a plane which passes through (2, 3, 1) and is normal to the line joining the

points (3, 4, 1) and (2, 1, 5) is given by (A) x + 5y 6z + 19 = 0 (B) x 5y + 6z 19 = 0 (C) x + 5y + 6z +19 = 0 (D) x 5y 6z 19 = 0 8. Direction cosines of the line joining the points (0, 0, 0) and (a, a, a) are

(A) 1 1 1, ,2 2 2

(B) 1, 1, 1

(C) 1 1 1, ,3 3 3

(D) none of these

*9. The length of perpendicular from the point (1, 2, 2)) on the line x 1 y 2 z 22 3 4

is

(A) 29 (B) 6 (C) 21 (D) none of these 10. Two lines not lying in the same plane are called (A) parallel (B) coincident (C) intersecting (D) skew 11. The distance of the point (x, y, z) from the x y plane is (A) x (B) y (C) 3 (D) z 12. A point (x, y, z) moves parallel to x axis. Which of three variables x, y, z remains fixed? (A) x and y (B) y and z (C) z and x (D) None of these *13. Let P (2, 3, 5), Q (1, 2, 3), R (7, 0, 1) then Q divides PR. (A) externally in the ratio 1 : 2 (B) internally in the ratio 1 : 2 (C) externally in the ratio 3 : 5 (D) internally in the ratio 1: 3 14. The xy plane divides the line segment joining (1, 2, 3) and (3, 4, 5) internally in the ratio (A) 3 : 5 (B) 3 : 4 (C) 4 : 3 (D) None of these 15. The direction cosines of the joining (1, 1, 1) and (1, 1, 1) are

(A) 1 1, ,02 2

(B) 2, 2,0

(C) 1 1, ,02 2

(D) 2, 2,0

16. Two lines with direction cosines 1 1 1 2 2 2l ,m ,n and l ,m ,n are at right angles iff (A) l1 l2 + m1 m2 + n1 n2 = 0 (B) l1 = l2, m1 = m2, n1 = n2 (C) l1 l2 = m1 m2 = n1 n2 (D) None of these 17. The foot of perpendicular from , , on x axis is (A) (, 0, 0) (B) (0, , 0) (C) (0, 0, ) (D) (0, 0, 0) 18. The direction cosines of a line equally inclined to the positive direction of axes are

(A) < 1, 1, 1> (B) 1 1 1, ,3 3 3

(C) 1 1 1, ,2 2 2

(D) None of these

19. A plane meets the coordinate axes at P, Q and R such that the centroid of the triangle is (1, 1, 1). The equation of plane is, (A) x + y + z = 3 (B) x + y + z = 9 (C) x + y + z = 1 (D) x + y + z = 1/3 *20. A plane meets the axes in P, Q and R such that centroid of the triangle PQR is (1, 2, 3). The

equation of the plane is

(A) 6x + 3y + 2z = 6 (B) 6x +3 y + 2z = 12 (C) 6x + 3y + 2z = 1 (D) 6x + 3y + 2z = 18 21. The direction cosines of a normal to the plane 2x 3y 6z + 14 = 0 are

(A) 2 3 6, ,7 7 7

(B) 2 3 6, ,7 7 7

(C) 2 3 6, ,7 7 7

(D) None of these

*22. The equation of the plane whose intercept on the axes are thrice as long as those made by

the plane 2x 3y + 6z 11 = 0 is (A) 6x 9y + 18z 11 = 0 (B) 2x 3y + 6z + 33 = 0 (C) 2x 3y + 6z = 33 (D) None of these 23. The angle between the planes 2x y + z = 6 and x + y + 2z = 7 is (A) /4 (B) /6 (C) /3 (D) /2 *24. The angle between the lines x = 1, y = 2 and y + 1 = 0 and z = 0 is (A) 00 (B) /4 (C) /3 (D) /2 LEVELII 1. The three lines drawn from O with direction ratios [1, 1, k], [2, 3, 0] and [1, 0, 3] are

coplanar. Then k = (A) 1 (B) 0 (C) no such k exists (D) none of these 2. A plane meets the coordinates axes at A, B, C such that the centroid of the triangle is (3, 3, 3). The equation of the plane is (A) x + y + z = 3 (B) x + y + z = 9 (C) 3x + 3y + 3z = 1 (D) 9x + 9y + 9z = 1 3. The equation of the plane through the intersection of the planes x 2y + 3z 4 = 0,

2x 3y + 4z 5 = 0 and perpendicular to the plane x + y + z 1 = 0 is (A) x y + 2 = 0 (B) x z + 2 = 0 (C) y z + 2 = 0 (D) z x + 2 = 0

4. The coordinates of the point of intersection of the line x 1 y 3 z 21 3 2

with the plane

3x + 4y + 5z = 5 are (A) (5, 15, 14) (B) (3, 4, 5) (C) (1, 3, 2) (D) (3, 12, 10)

5. The angle between the line x 1 y 1 z 23 2 4

and the plane 2x + y 3z + 4 = 0 is

(A) cos14

406 (B) sin

14

406

(C) 30 (D) none of these

*6. The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l2 + m2 n2 = 0 is given by

(A) 23 (B)

6

(C) 56 (D)

3

*7. The angle between the line x 2 y 1 z 32 1 2

and the plane 3x + 6y 2z + 5 = 0 is

(A) cos1421

(B) sin

1421

(C) sin1621

(D) sin

1 421

*8. Shortest distance between lines x 6 y 2 z 21 2 2

and x 4 y z 13 2 2

is

(A) 108 (B) 9 (C) 27 (D) None of these 9. The acute angle between the plane 5x 4y + 7z 13 = 0 and the yaxis is given by

(A) sin1590

(B) sin

1490

(C) sin1790

(D) sin

1490

10. The planes x + y z = 0, y + z x = 0, z + x y = 0 meet (A) in a line (B) taken two at a time in parallel lines (C) in a unique point (D) none of these 11. The graph of the equation x2 + y2 = 0 in the three dimensional space is (A) z axis (B) (0, 0) point (C) y z plane (D) x y plane 12. A line making angles 450 and 600 with the positive directions of the x axis and y axis

respectively, makes with the positive direction of z axis an angle of (A) 600 (B) 1200 (C) both (A) and (B) (D) Neither (A) nor (B) 13. The angle between two diagonals of a cube is

(A) 1 1cos2

(B) 1 1cos3

(C) 1 1cos3

(D) 1 3cos2

14. If a line makes angles , , with the axes, then cos2 + cos2 + cos2 = (A) 1 (B) 1 (C) 2 (D) 2

15. The equation (x 1) . (x 2) = 0 in three dimensional space is represented by (A) a pair of straight line (B) a pair of parallel planes (C) a pair of intersecting planes (D) a sphere *16. The equation of the plane containing the line 2x + z 4 = 0 and 2y + z = 0 and passing

through the point (2, 1, 1) is (A) x + y z = 4 (B) x y z = 2 (C) x + y + z + 2 = 0 (D) x + y + z = 2 *17. The locus of xy + yz = 0 is, in 3 D ; (A) a pair of straight lines (B) a pair of parallel lines (C) a pair of parallel planes (D) a pair of intersecting planes

18. The lines 6x = 3y = 2z and x 1 y 2 z 32 4 6

are

(A) parallel (B) skew (D) intersecting (D) coincident

*19. The line 1 1 1x x y y z z

0 1 2

is

(A) parallel to x axis (B) perpendicular to x axis (C) perpendicular to YOZ plane (D) None of these

20. For the line y 1x 1 z 3l :3 2 1

and plane P : x 2y z = 0 ; of the following assertions,

the one/s which is/are true : (A) l lies on P (B) l is parallel to P (C) l is perpendicular to P (D) None of these

21. The coordinates of the point of intersection of the line x 6 y 1 z 31 0 4

and the plane

x y z 3 are (A) (2, 1, 0) (B) (7, 1, 7) (C) (1, 2, 6) (D) (5, 1, 1)

*22. The Cartesian equation of the plane perpendicular to the line, x 1 y 3 z 42 1 2

and

passing through the origin is (A) 2x y + 2z 7 = 0 (B) 2x + y + 2z = 0 (C) 2x y + 2z = 0 (D) 2x y z = 0

Level III *1. The length of projection of the segment joining (x1 , y1 , z1 ) and (x2 , y2 , z2 ) on the line

x y zl m n

is

(A) 2 1 2 1 2 1l x x m y y n z z (B) 2 1 2 1 2 1x x y y z z

(C) 2 1 2 1 2 1x x y y z zl m n

(D) None of these

2. The shortest distance between the lines x 1 y 2 z 3 x 2 y 3 z 5and2 3 4 3 4 5

is

(A) 16

(B) 16

(C) 13

(D) 13

3. The equation of the plane through the point (1, 2, 0) and parallel to the lines

x y 1 z 2 x 1 2y 1 z 1and3 0 1 1 2 1

is

(A) 2x + 3y + 6z 4 = 0 (B) x 2y + 3z + 5 = 0 (C) x + y 3z+ 1 = 0 (D) x + y + 3z = 1

*4. The distance of the plane through (1, 1, 1) and perpendicular to the line x 1 y 1 z 13 0 4

from the origin is

(A) 34

(B) 43

(C) 75

(D) 1

*5. The reflection of the point (2, 1, 3) in the plane 3x 2y z = 9 is

(A) 26 15 17, ,7 7 7

(B) 26 15 17, ,7 7 7

(C) 15 26 17, ,7 7 7

(D) 26 17 15, ,7 7 7

6. The coordinates of the foot of perpendicular from the point A (1, 1, 1) on the line joining the points B (1, 4, 6) and C (5, 4, 4) are

(A) (3, 4, 5) (B) (4, 5, 3) (C) (3, 4, 5) (D) (3, 4, 5) 7. The equation of the right bisecting plane of the segment joining the points (a, a, a) and (a, a, a) ; a 0 is (A) x + y + z = a (B) x + y + z = 3a (C) x + y + z = 0 (D) x + y + z + a = 0 8. The angle between the plane 3x + 4y = 0 and the line x2 + y2 = 0 is (A) 00 (B) 300

(C) 600 (D) 900 9. If the points (0, 1, 2) ; (3, 4, 5) ; (6, 7, 8) and (x, x, x) are noncoplanar then x = (A) any real number (B) 1 (C) 1 (D) 0 *10. The equation of the plane through intersection of planes x + 2y + 3z = 4 and 2x + y z = 5

and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is (A) 7x 2y + 3z + 81 (B) 23y + 14x 9z + 48 = 0 (C) 23x + 14y 9z + 48 = 0 (D) 51x + 15y 50z + 173 = 0 11. The equation of the plane passing through the intersection of planes x + 2y + 3z + 4 = 0 and

4x + 3y + 2z + 1 = 0 and the origin is (A) 3x + 2y + z + 1 = 0 (B) 3x + 2y + z = 0 (C) 2x + 3y + z = 0 (D) x + y + z = 0 12. If the plane x + y z = 4 is rotated through 900 about the line of intersection with the plane x + y + 2z = 4 then equation of the plane in its new position is (A) 5x + y + 4z + 20 = 0 (B) 5x + y + 4z = 20 (C) x + 5y + 4z = 20 (D) None of these 13. The equation of the plane passing through the line of intersection of the planes 4x 5y 4z = 1 and 2x + y + 2z = 8 and the point (2, 1, 3) is (A) 32x 5y + 8z = 83 (B) 32x + 5y 8z = 83 (C) 32x 5y + 8z + 83 = 0 (D) None of these 14. The equation of the plane passing through the points (2, 1, 2) and (1, 3, 2) and parallel to x axis is (A) x + 2y = 4 (B) 2y + x + z = 4 (C) x + y + z = 4 (D) 2y + z = 4 15. The equation of the plane passing through the point (3, 3, 1) and is normal to the line

joining the points (2, 6, 1) and (1, 3, 0) is (A) x + 3y + z + 11 = 0 (B) x + y + 3z + 11 = 0 (C) 3x + y + z = 11 (D) None of these *16. If a point moves so that the sum of the squares of its distances from the six faces of a cube

having length of each edge 2 units is 46 units, then the distance of the point from (1,1, 1) is (A) a variable . (B) a constant equal to 7 units. (C) a constant equal to 4 units. (D) a constant equal to 49 units. 17. Planes are drawn parallel to the coordinate planes through the points (1, 2, 3) and (3, 4, 5). The length of the edges of the parallelepiped so found, are (A) 4, 6, 8 (B) 3, 4, 5 (C) 2, 4, 5 (D) 2, 6, 8 18. The length of a line segment whose projections on the coordinate axes are 6, 3, 2, is (A) 7 (B) 6 (C) 5 (D) 4

19. The direction cosines of a line segment whose projections on the coordinate axes are 6, 3, 2, are

(A) 6 3 2, ,7 7 7

(B) 6 3 2, ,7 7 7

(C) 6 3 2, ,7 7 7

(D) None of these

20. If P, Q, R, S are (3, 6, 4), (2, 5, 2), (6, 4, 4) , (0, 2, 1) respectively then the projection of PQ

on RS is (A) 2 units (B) 4 uints (C) 6 uints (D) 8 uints 21. Let f be a oneone function with domain (2, 1, 0) and range (1, 2, 3) such that exactly one

of the following statements is true. f (2) = 1, f (1) 1, f (0) 2 and the remaining two are false. The distance between points (2, 1, 0) and ( f (2), f (1), f (0) ) is

(A) 2 (B) 3 (C) 4 (D) 5

ANSWERS

LEVEL I 1. A 2. C 3 A 4. A 5. A 6. A 7. A 8. C 9. D 10. (D) 11. (D) 12. (B) 13. (B) 14. (A) 15. A 16. (A) 17. (A) 18. (B) 19. (A) 20. (D) 21. (A) 22. (C) 23. (C) 24. (D) LEVEL II 1. A 2. B 3. B 4. A 5. B 6. D 7. B 8. B 9. D 10. C 11. (D) 12. (C) 13. (B) 14. (A) 15. (B) 16. (D) 17. (D) 18. (D) 19. (B) 20. 21. (D) 22. (C)

Level III 1. (A) 2. (B) 3. (D) 4. (C) 5. (B) 6. A 7. (C) 8. (A) 9. (A) 10. (D) 11. (B) 12. (B)

13. (A) 14. (D) 15. (A) 16. (B) 17. (D) 18. (A) 19. (A) 20. (A) 21. (D)

AD

LEVELI

1. Number of critical points of f (x) = 1x

|4x|2

2

are

(A) 1 (B) 2 (C) 3 (D) none of these 2. If the function f (x) = cos |x| 2ax + b increases for all x R, then (A) a b (B) a = b/2 (C) a < 1/2 (D) a 3/2 3. Area of the triangle formed by the positive x-axis and the normal and the tangent to

x2 + y2 = 4 at (1, 3 ) is (A) 2 3 sq. units (B) 3 sq. units (C) 4 3 sq. units (D) none of these

4. A tangent to the curve y = 2x

2 which is parallel to the line y = x cuts off an intercept from the

y-axis is (A) 1 (B) 1/3 (C) 1/2 (D) 1/2 5. A particle moves on a co-ordinate line so that its velocity at time t is v (t) = t2 2t m/sec.

Then distance travelled by the particle during the time interval 0 t 4 is (A) 4/3 (B) 3/4 (C) 16/3 (D) 8/3 6. The derivative of f (x) = |x| at x = 0 is (A) 1 (B) 0 (C) 1 (D) does not exist 7. f (x) = [x2 + 3x4 + 5x6 + 5] have only ------------- value in (,) at x = ------------ 8. If y = a log x + bx2 + x has its extremum values at x = -1 and x =2 then a= ------- b = -------------- 9. The value of b for which the function f (x) = sin x bx + c is decreasing in the interval (,)

is given by (A) b < 1 (B) b 1 (C) b > 1 (D) b 1

10. Equation of the tangent to the curve y = e|x| at the point where it cuts the line x=1 (A) is ey + x =2 (B) is x + y = e

(C) is ex + y = 1 (D) does not exist 11. The greatest and least values of the function f(x) = ax + b x + c, when a > 0, b > 0, c > 0 in

the interval [0,1] are (A) a+b+c and c (B) a/2 b2+c, c

(C) 2

cba , c (D) None of these

12. The absolute minimum value of x4 x2 2x+ 5

(A) is equal to 5 (B) is equal to 3 (C) is equal to 7 (D) does not exist

13. Through the point P (, ) where >0 the straight line 1by

ax

is drawn so as to form with

co-ordinates axes a triangle of area S. If ab >0, then the least value of S is (A) 2 (B) 1/2 (C) (D) None of these

14. If f(x) = A ln |x| + B x2 + x has its extreme values at x = 2 and x = 1 then

(A) A = 2, B = 1/2 (B) A = 2 , B = 1/2 (C) A = 2, B =1 (D) None of these

15. The function 2tan3x-3tan2x+12tanx + 3, x

2

,0 is

(A) increasing (B) decreasing (C) increasing in (0, /4) and decreasing in (/4, /2) (D) none of these 16. The tangent to the curve y = 2x at the point whose ordinate is 1, meets the x axis at the

point (A) (0, ln2) (B) (ln 2, 0) (C) (-ln2, 0) (D) (-1/ln2, 0) 17. The minimum value of ax + by, where xy = r2, is (r, ab >0) (A) 2r ab (B) 2ab r (C) 2r ab (D) None of these

18. The range of the function f(x) = sin-1

21x2 + cos-1

21x2 , where [.] is the greatest

integer function, is

(A)

,2

(B)

2,0 (C) {} (D)

2

,0

19. The domain of f(x) =

4xx5log

2

41 +

10Cx is

(A) (0, 1]U [4, 5) (B) (0, 5) (C) {1, 4} (D) None of these 20. A function whose graph is symmetrical about the origin is given by (A) f (x) = ex + e-x (B) f (x) = loge x (C) f (x + y) = f (x) + f (y) (D) none of these 21. Let f (x) be a function whose domain is [-5, 7]. Let g (x) = |2x + 5|, then the domain of fog (x)

is (A) [-5, 1] (B) [-4, 0] (C) [-6, 1] (D) none of these

22. 1xcosxcos1xsinxsinlim 24

24

x

is equal to,

(A) 0 (B) 1

(C) 1 (D) does not exist 23. Pick up the correct statement of the following where [ ] is the greatest integer function,

(A) If f (x) is continuous at x = a then [f (x)] is also continuous at x = a. (B) If f (x) is continuous at x = a then [f (x)] is differentiable at x = a. (C) If f (x) is continuous at x = a then f (x) is also continuous at x = a. (D) None of these

24. The greatest value of f (x) = cos (xe[x] + 7x2 3x), x [-1, ) is

(A) 1 (B) 1 (C) 0 (D) none of these.

25. The equation of the tangent to the curve f (x) = 1 + e2x where it cuts the line y = 2 is

(A) x + 2y = 2 (B) 2x + y = 2 (C) x 2y = 1 (D) x 2y + 2 = 0

26. The angle of intersection of curves y = 4 x2 and y = x2 is.

27. The greatest value of the function f (x) =

4xsin

x2sin on the interval

2

,0 is.

28. Let f(x) = x sinx and g(x) = x tanx, where x

2

,0 . Then for these value of x.

(A) f(x). g(x) > 0 (B) f(x) . g(x) < 0

(C) 0xgxf

(D) none of these

29. Suppose that f(x) 0 for all x [0, 1] and f is continuous in [0, 1] and 0dx)x(f1

0 , then

x [0, 1], f is (A) entirely increasing (B) entirely decreasing (C) constant (D) None of these

LEVELII 1. Let h (x) = f (x) + ln{f(x)} + {f (x)}2 for every real number x, then (A) h (x) is increasing whenever f (x) is increasing (B) h (x) is increasing whenever f (x) is decreasing (C) h (x) is decreasing whenever f (x) is increasing (D) nothing can be said in general 2. Let f (x) = a0 + a1x2 + a2x4 + + anx2n, where 0 < a0 < a1 < a2 < < an, then f (x) has (A) no minimum (B) only one minimum (C) no maximum (D) neither a maximum nor a minimum

3. The maximum value of xcosxsin

xcosxsin

in the interval

2

,0 is

(A) 1/2 (B) 1/4

(C) 22

1 (D) 1/3

4. If y = .......xsinxsinxsin , then the value of dxdy

is

(A) 1yxsin

(B)

1yxsin

(C) 1y2xcos

(D)

1y2xcos

5. The curve y exy + x = 0 has a vertical tangent at the point

(A) (1, 1) (B) at no point (C) (0, 1) (D) (1, 0)

6. A differentiable function f (x) has a relative minimum at x = 0 then the function

y = f(x) + ax + b has a relative minimum at x = 0 for (A) all a and all b (B) all b if a = 0 (C) all b > 0 (D) all a 0

7. Let f(x) =

Then.

0x,1xx

0x,xsin12

(A) f has a local maximum at x = 0 (B) f has a local minimum at x = 0 (C) f is increasing every where (D) f is decreasing everywhere 8. Let f(x) = xn+1 + a. xn, where a is a positive real number, n I+ . Then x = 0 is a point of (A) local minimum for any integer n (B) local maximum for any integer n (C) local minimum if n is an even integer (D) local minimum if n is an odd integer 9. f(x) = max ( sinx, cosx) x R. Then number of critical points [ -2, 2] is /are ; (A) 5 (B) 7 (C) 9 (D) none of these 10. Let (x) = (f(x))3 3(f(x))2 + 4f(x) + 5x + 3 sinx + 4 cos x x R, then (A) is increasing whenever f is increasing

(B) is increasing when ever f is decreasing (C) is decreasing whenever f is decreasing

(D) Nothing can be said

11. A function f(x) = 3x2x2x3x

2

2

is:

(A) Maximum at x = -3 (B) Minimum at x = -3 and maximum at x = 1 (C) No point of maxima or minima (D) Function is decreasing in its domain.

12. Let f(x) =

0xx6x50x)x3xsin(

2

2. Then f(x) has

(A) local maxima at x = 0 (B) Local minima at x = 0 (C) Global maxima at x = 0 (D) Global minima at x = 0 13. If a, b, c, d are four positive real numbers such that abcd =1, then minimum value of (1+a)

(1+b) (1+c) (1+d) is (A) 8 (B) 12 (C) 16 (D) 20 14. If f(x) + 2f(1- x) = x2 + 2 xR, then f(x) is given as

(A) 32x 2 (B) x2 2

(C) 1 (D) None of these 15. xcosxsinlim

4/5x

, where [ . ] denotes the Integral part of x.

(A) is equal to 1 (B) is equal to 2 (C) is equal to 3 (D) Does not exist

16. If f (x) = x1

xx1ln2

x1

, then the value of f (0) so that f (x) is continuous at x = 0, is;

(A) 2 (B) 1 (C)1/2 (D) None of these

17. If f (x) = x1

x

, then

(A) f (x) is differentiable x R (B) f (x) is no where differentiable (C) f (x) is not differentiable at finite no. of point (D) None of these

18. If f1 (x) = sin x + tan x, f2 (x) = 2x then

(A) f1 (x) > f2 (x) x ( 0, /2) (B) f1 (x) < f2 (x) x ( 0, /2) (C) f1 (x) f2 (x) = 0 has exactly one root x ( 0, /2) (D) None of these

19. .Let f (x) =

1x1x

,3x2,a1x

. If f (x) has a local minima at x = 1. Then exhaustive set of

values of a is; (A) a 4 (B) a 5 (C) a 6 (D) a 7

20. A differentiable function f (x) has a relative minimum at x = 0 then the function y = f (x) + ax +

b has a relative minimum at x = 0 for

(B) all a and all b (B) all b if a = 0 (D) all b > 0 (D) all a 0

21. The maximum value of f(x) = |x ln x| in x(0,1) is;

(A) 1/e (B) e (C) 1 (D) none of these

22. If f (x) = x

0

)1t( (et 1) (t 2) (t + 4) dt then f (x) would assume the local minima at;

(A) x = 4 (B) x = 0 (C) x = 1 (D) x = 2.

23. f(x) = tan-1 (sinx + cosx) is an increasing function in

(A) (0,/4) (B) (0, /2) (C) (-/4, /4) (D) none of these.

24. Let f: RR, where f(x) = x3 - ax, aR. Then set of values of a so that f(x) is increasing in its entire domain is; (A) (-, 0) (B) (0, ) (C) (-, ) (D) none of these

25. The curves y = 4x2 + 2x 8 and y = x3 x + 10 touch each other at the point.. 26. Let f be differentiable for all x. if f (1) = -2 and f (x) 2 for all x [1, 6], then (A) f (6) < 8 (B) f (6) 8 (C) f (6) 5 (D) f (6) 5

27. The function f (x) = 42

x1x2 decreases in the interval..

28. The function f (x) = (x + 2) e x increases in ------------------- and decreases in -------------------------------- 29. The function y = x cot-1 x log (x + 1x 2 ) is increasing on (A) (-, 0) (B) (-,) (C) (0, ) (D) R {0}

30. Let f : (0, ) R defined by f(x) = x + xcosx

9 2

. Then minimum value of f(x) is

(A) 10 1 (B) 6 1 (C) 3 1 (D) none of these 31. Let a, n N such that a n3 then 33 a1a is always

(A) less than 2n31 (B) less than 3n2

1

(C) more than 3n1 (D) more than 2n4

1

32. The global minimum value of function f(x) = x3 + 3x2 + 10x + cosx in [-2,3] is (A) 0 (B) 3-2

(C) 16-2 (D) -15 33. The minimum value of the function defined by f(x) = Maximum {x, x+1, 2-x} is (A) 0 (B) 1/2 (C) 1 (D) 3/2 LEVELIII 1. If the parabola y = ax2 + bx + c has vertex at (4, 2) and a [1, 3], then difference between

the extreme values of abc is equal to, (A) 3600 (B) 144 (C) 3456 (D) None of these 2. Let , and be the roots of f(x) = x3 + x2 5x 1 = 0. Then [] +[] +[], where [.] denotes

the greatest integer function, is equal to (A) 1 (B) 2 (C) 4 (D) 3 3. The number of solutions of the equation x3 +2x2 +5x + 2cosx = 0 in [0, 2] is (A) 0 (B) 1 (C) 2 (D) 3 4. Let S be the set of real values of parameter for which the equation

f(x) = 2x3 3( 2+)x2 + 12x has exactly one local maximum and exactly one local minimum. Then S is a subset of

(A) (-4, ) (B) (-3, 3) (C) (3, ) (D) (-, 3) 5. Consider a function y = f (x) defined parametrecally as x = 2t + t , y = t t , t R. then function is

(A) Differentiable at x = 0 (B) non-differentiable at x = 0 (C) nothing can be said about differentiablity at x = 0 (D) None of these

6. If the line ax + by + c = 0 is normal to the curve x y + 5 = 0 then

(A) a > 0 , b > 0 (B) b > 0 , a < 0

(C) a < 0 , b < 0 (D) b < 0 , a > 0

7. The number of roots of x3-3x+1 = 0 in [1,2] is/are; (A) One (B) Two (C) Three (D)none of these

8. A cubic f(x) vanishes at x = -2 and has extrema at x = -1 and x = 31 such that

1

1 314dxxf

then f (x) = 9. If g(x) = f(x) + f(1x) and f(x) < 0, 0 x 1, then

(A) g(x) is decreasing in (0, 1) (B) g(x) is decreasing in

21,0

(C) g(x) is decreasing in

1,21 (D) g(x) is increasing in (0, 1)

10. Let g(x) > 0 and f(x) < 0 x R then (A) g(f(x + 1)) > g(f(x 1)) (B) f(g(x 1)) < f(g(x + 1)) (C) g(f(x + 1) < g(f(x 1)) (D) g(g(x + 1)) < g(g(x + 1))

11. The function

( )1 4ax bf x

x x

has a local maxima at (2, 1) then

(A) b = 1, a = 0 (B) a = 1, b = 0 (C) b = 1, a = 0 (D) None of these 12. 1 2( ) 2 , ( ) 3sin cosf x x f x x x x , then for x (0, /2): (A) 1 2( ) ( )f x f x (B) 1 2f x f x

(C) 1 2( ) ( )f x f x (D) 1 2f x f x 13. y = f(x) is a parabola, having its axis parallel to y axis. If the line y = x touches this parabola

at x = 1 then (A) (1) (0) 1f f (B) (0) (1) 1f f (C) (1) (0) 1f f (D) (0) (1) 1f f 14. If f(x) = 2 (2 1) 3x xe ae a x is increasing for all values of x then (A) a (, ) {0} (B) a (, 0] (C) a (0, ) (D) a [0, ) 15. If 2a + 3b + 6c = 0, then equation 2 0ax bx c has roots in the interval (A) (0, 1) (B) (2, 3) (C) (1, 2) (D) (0, 2) 16. The equation 23 4 0x ax b has at least one root in (0, 1) if (A) 4a + b + 3 = 0 (B) 2a + b + 1 = 0 (C) b = 0, a = -4/3 (D) None of these

17. If f(x) satisfies the conditions of Rolles theorem in [1, 2] then 2

1

( )f x dx is equal to (A) 3 (B) 0 (C) 1 (D) 1 18. If f(x) =

2 22 /x ax e is a non-decreasing function then for a > 0; (A) x [a, 2a) (B) x (, a] [0, a] (C) x (a, 0) (D) None of these

19. The function ( )1 tan

xf xx x

has

(A) One point of minimum in the interval (0, /2) (B) One point of maximum in the interval (0, /2) (C) No point of maximum, no point of minimum in (0, /2) (D) Two points of maximum in (0, /2) 20. The number of solutions of the equation ( ) ( ) 0,f xa g x where a > 0, g(x) 0 and has

minimum value of is (A) 1 (B) 2 (C) 4 (D) 0

ANSWERS

LEVEL I 1. A 2. C 3. A 4. D 5. C 6. D 7. 0 8. 2, 1/2 9. C 10. A 11. A 12. B 13. C 14. D 15. A 16. D 17. A 18. C 19. C 20. D 21. C 22. C 23. C 24. B 25. B 26. 2 2 27. 2 28. B 29. C LEVEL II 1. A 2. B 3. C 4. D 5. D 6. B 7. A 8. C 9. B 10. A 11. C 12. B 13. C 14. A 15. B 16. C 17. C 18. A 19. B 20. B 21. A 22. D 23. C 24. A

25. 3, 34; 1 74,3 9

26. B 27. 1 1, 0 ,2 2

28. (0, 1); R (0, 1) 29. B 30. B 31. A 32. D 33. C LEVEL III 1. C 2. 3. A 4. D 5. A 6. A, C 7. A 8. x3 x2 + x 2

9. C 10. C 11. B 12. C 13. C 14. D 15. A 16. B 17. B 18. B 19. B 20. D

1

Area

LEVELI 1. Area common to the curves y = x3 and y = x is

(A) 125 (B)

65

(C) 85 (D) none of these

2. The area bounded by the parabola y2 = x, straight line y = 4 and y-axis is

(A) 3

64 (B) 3

16

(C) 7 2 (D) none of these 3. The area bounded by the curves y = |x| 1 and y = |x| + 1 is (A) 1 (B) 2 (C) 2 2 (D) 4 4. The area bounded by the curve y = sin x and the x-axis , for 0 x 2 is (A) 2 sq. units (B) 1 sq. units (C) 6 sq units (D) 4 sq. units 5. The area enclosed by y = ln x, its normal at (1, 0) and y-axis is (A) 1/2 (B) 3/2 (C) Not defined (D) none of these 6. The area bounded by y 1 = |x|, y =0 and |x| = 1/2 will be (A) 3/4 (B) 3/ 2 (C) 5/4 (D) none of these

7. The area bounded by the parabola y2 = 4 x and its latus rectum is

(A) 1 (B) (C) 8/3 (D) none of these

8. The area of the region bounded by y = |x-1| and y = 1 is

(A) 1/ 2 (B) 1 (B) 2 (D) none of these

9. The area of the region bounded by the parabola y = x2-3x with y 0 is (A) 3 (B) 3 (C) 9/2 (D) 9/2 10. The area of the smaller region bounded by the circle x2+y2 = 1 and |y| = x+1 is

(A) 21

4

(B) 12

(C) 2 (D) 1

2

11. The area bounded by the curves |x| + |y| 1 and x2 + y2 1 is (A) 2 sq. units (B) sq. units (C) - 2 sq. units (D) + 2 sq. units

2

12. Area bounded by f(x) = max.(sinx, cosx); 0 x /2 x = /2 and the coordinate axes is equal

to (A) 2 sq. units (B) 2 sq. units

(C) 2

1sq. units (D) None of these

13. If the area bounded by the curve , y =f(x), the lines x=1, x = b and the x-axis is (b-1)

cos (3b + 4), b > 1, then f(x) is (A) (x-5) sin (3x+4) (B) (x-1) sin (x+1)+ (x+1) cos (x-1) (C) cos (3x+4) 3(x-1) sin (3x+4) (D) (x-5) cos (3x+4)

14. The area of region that is completely bounded by the graph of f(x) = 2x 1 and g(x) = 2 4x

is

(A) 3 (B) 203

(C) 323

(D) None of these

15. The area bounded by the curves 2 4y x and x + 2y = 4, is (A) 9 (B) 18 (C) 72 (D) 36 16. The area of the region bounded by the curve 2 2y x x and y x is

(A) 92

(B) 72

(C) 112

(D) None of these

17. The total area enclosed by , 1y x x and y = 0, is (A) 1 (B) 2 (C) 3 (D) 4 18. The area of the region bounded by the function 3( )f x x , the x-axis and the lines x = 1

and x = 1 is

(A) 14

(B) 13

(C) 18

(D) 12

19. The area of the region bounded by the curve y = x and 22 2y x is

(A) 13

(B) 16

(C) 19

(D) None of these

20. The area bounded by the axes and the curve 2y x is (A) 1 (B) 2 (C) 4 (D) None

3

LEVELII 1. Area bounded by the curves y = x2 + 2, y = x, x = 0 and x = 1 is

(A) 2

17 (B) 6

17

(C) 6

19 (D) 6

13

2. The area bounded between the curves x = y2 and x = 3 2y2 is (A) 2 (B) 3 (C) 4 (D) 1 3. Area bounded by the curve ay = 3(a2 x2) and the x-axis is (A) a2 (B) 2a2 (C) 3a2 (D) 4a2 4. Area bounded by the curves x2 = y and y = x + 2 and x-axis is

(A) 92

(B) 35

(C) 65 (D)

67

5. If Am represents the area bounded by the curve y = ln xm, the x-axis and the lines x= 1

and x= e, then Am+ m Am-1 is (A) m (B) m2 (C) m2/2 (D) m2-1 6. The area bounded by the curves y = ln x, y = | ln x| and the y-axis is (A) 3 (B) 2 (C) 4 (D) 8 7. If area bounded by y = f(x), the coordinate axes and the line x = a is given by aea, then

f(x) is (A) ex(x+1) (B) ex (C) x ex (D) xex+1 8. The area common to y2 = x and x2 = y is

(A) 1 (B) 2/3 (C) 1/3 (D) none of these

9. The area bounded by y = |x-1| and y = 3 -|x| is

(A) 2 (B) 3 (C) 4 (D) 1

10. The area cut off from the parabola 4y=3x2 by the straight line 2y=3x+12 is (A) 25 sq.units (B) 27 sq.units (C) 36 sq.units (D) 16 sq.units 11. The area bounded by the curve y = x2+ 2x+1, the tangent at (1, 4) and the y-axis is (A) 1 (B) 1/2 (C) 1/3 (D) 1/4

4

12. The area bounded by y = lnx, the xaxis and the ordinates x = 0 and x = 1 is (A) 1 (B) 3/2 (C) 1 (D) none of these 13. The area bounded by the straight lines y = 0, x + y 2 = 0 and the straight line which equally

divides the common area included between the curves y = x2 and y = x is equal to (A) 1 sq. unit (B) 2sq, units (C) 3 sq. units (D) None of these 14. The area of the smaller region bounded by the circle 2 2 1x y and the lines 1y x is:

(A) 12 2

(B) 12

(C) 2

(D) 12

15. The area of the region bounded by 21 1y x and x y is

(A) 13

(B) 43

(C) 23

(D) 83

16. Area enclosed by the curve 2 1 1x y is

(A) 215

sq. units (B) 415

sq. units

(C) 2 sq. units (D) 4 sq. units 17. If the area bounded by a continuous function y = f(x), co-ordinate axes and the line x = a,

where a R+, is equal to a ea , then one such function can be (A) 1xe x (B) ( 1)xe x (C) xe (D) None 18. Value of the parameter a such that the area bounded by 2 2 1,y a x ax co-ordinate axes

and the line x = 1, attains the least value, is

(A) 14

(B) 34

(C) 12

(D) None of these

19. The area bounded by . xy x e and lines 1, 0x y is, (A) 4 (B) 6 (C) 1 (D) 2 20. The slope of the tangent to a curve y = f(x) at (x, f(x)) is 2x + 1. If the curve passes through

the point (1, 2), then the area of the region bounded by the curve, the x-axis and the line x = 1 is:

(A) 16

(B) 6

(C) 56

(D) 65

5

LEVELIII

1. The area enclosed in the region 1by

ax

2

2

2

2

and 1by

ax

is

(A) ab21

4ab

(B)

4ab

(C) ab (D) none of these 2. The area of the loop of the curve x2 = y2(1-y) is (A) 2/15 (B) 15/14 (C) 4/15 (D) 8/15 3. The area common to the region determined by y x , and x2+y2 < 2 has the value (A) -2 (B) 2-1 (C) 3 - 3/2 (D) none of these 4. The area of the region for which 0 < y< 3 2x-x2 and x> 0 is

(A) dxxx233

1

2 (B) dxxx223

0

2

(C) dxxx231

0

2 (D) dxxx223

1

2

5. The area enclosed between the curves y = sin2x and y = cos2 x in the interval 0 x is (A) 2 (B) (C) 1 (D) None of these 6. The area between the curves y = xex and y = x ex and the line x = 1 is (A) 2e (B) e (C) 2/e (D) 1/e 7. If An is the area bounded by y = (1-x2)n and coordinate axes, n N, then (A) An = An-1 (B) An < An-1 (B) An > An-1 (D) An = 2 An-1

8. Let ( ) min 1 , 1f x x x , then area bounded by f(x) and x-axis is: (A)

16

(B) 56

(C) 76

(D) 116

9. Let 2 ; 0

( ); 0

x xf x

x x

Area bounded by the curve y = f(x), y = 0 and x = 3a is 92a

, then a =

(A) 1 or 12

(B) 1 or 12

6

(C) 1 or 12

(D) None

10. The interval [a, b] such that the value of 22b

a

x x dx is maximum, is

(A) [2, 1] (B) [2, 1] (C) [1, 2] (D) [1, 2] 11. If A(n) represents the area bounded by the curve y = n lnx, where n N and n > 1, the x-axis

and the lines x = 1 and x= e, then the value of A(n) + n A(n 1) is equal to

(A) 2

1n

e (B)

2

1n

e

(C) 2n (D) 2e x 12. Area of the region which consists of all the points satisfying the conditions 8x y x y

and xy 2, is equal to: (A) 2 (9 ln8) sq. units (B) 4 (7 ln2) sq. units (C) 4 (9 ln8) sq. units (D) 4 (7 ln8) sq. units 13. A point P moves in xy plane in such a way that 1x y , where [ ] denotes the

G.I .F. Area of the region representing all possible positions of the point P is equal to (A) 8 sq. units (B) 4 sq. units (C) 16 sq. units (D) 2 2 sq. units 14. Area of the region bounded by 2 2 2 2x y and the axes is

(A) 38

sq. units (B) 32

sq. units

(C) 34

sq. units (D) None

15. The area of the smaller region in which the curve 3

100 50x xy

, where [ ] denotes G.I.F.,

divides the circle 2 22 1 4,x y is equal to

(A) 2 3 3

3

sq. units (B) 3 3

3

sq. units

(C) 5 3 3

3

sq. units (D) 4 3 3

3

sq. units

16. Area bounded by the curve

2xy e , x-axis and the lines x = 1, x = 2 is given to be equal to a sq. units. Area bounded by the curve y = ln( )x , y-axis and the lines y = e and 4y e is equal to:

(A) 42e e a (B) 4e e a (C) 42 2e e a (D) 42 2e e a 17. Area bounded by the curves 2, 2xy e y x x and the line x = 0, x = 1 is equal to

7

(A)3 2

3e

sq. units (B) 4 5

4e

sq. units

(C) 4 74

e sq. units (D) 3 53

e sq. units

18. Value of the parameter a such that area bounded by 2 3y x and the line y = ax + 2,

attain its minimum value is, (A) 1 (B) 0 (C) 1 (D) 1 19. Consider a triangle OAB formed by the points O (0, 0), A (2, 0), B 1, 3 . P(x, y) is an

arbitrary interior point of the triangle, moving in such a way that ( , ) ( , ) ( , ) 3d P OA d P AB d P OB , where d(P, OA), d(P, AB) and d(P, OB) represent the

distance of P from the sides OA, AB and OB respectively. Area of the region representing all possible positions of the point P is equal to

(A) 2 3 sq. units (B) 6 sq. units (C) 3 sq. units (D) None 20. Let f(x) = 2 ,ax bx c where a R and 2 4 0.b ac Area bounded by ( )y f x , x-axis

and the lines x = 0, x = 1 is equal to

(A) 1 3 (1) ( 1) 2 (0)6

f f f (B) 1 5 (1) ( 1) 8 (0)12

f f f

(C) 1 3 (1) ( 1) 2 (0)6

f f f (D) 1 5 (1) ( 1) 8 (0)12

f f f

ANSWERS

LEVEL I 1. A 2. A 3. B 4. D 5. B 6. C 7. C 8. B 9. D 10. A 11. C 12. A 13. C 14. C 15. D 16. A 17. A 18. D 19. D 20. B LEVEL II 1. B 2. A 3. D 4. 5. B 6. B 7. A 8. C 9. C 10. B 11. C 12. A 13. A 14. B 15. C 16. C 17. A 18. B 19. D 20. C LEVEL III 1. A 2. C 3. D 4. C 5. B 6. C 7. B 8. C 9. A 10. D 11. C 12. D

8

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

1

BT LEVELI 1. The co-efficient of x in the expansion of (1-2x3+3x5)[1+(1/x)]8 is (A) 56 (B) 65 (C) 154 (D) 62 2. If the fourth term in the expansion of (px+1/x)n is 5/2 then the value of p is (A) 1 (B) 1/ 2 (C) 6 (D) 2 3. If x = 1/3, Then the greatest term in the expansion of (1+4x)8 is

(A) 564

43

(B) 565

34

(C) 565

43

(D) 564

52

4. The two consecutive terms in the expansion of (3+2x)74 whose coefficients are equal is (A) 30th and 31st term terms (B) 29th and 30th terms (C) 31st and 32nd terms (D) 28th and 29th terms

5. If z=55

2i

23

2i

23

, then

(A) Re(z) =0 (B) Im(Z) =0 (C) Re(z) >0, Im(z) >0 (D) Re(z) >0, Im(z)

2

(C) a = 2, n = 6 (D) none of these 10. In the coefficients of the (m + 1)th term and the (m + 3) th term in the expansion of (1 +x)20

are equal then the value of m is (A) 10 (B) 8 (C) 9 (D) none of these 11. The number of distinct terms in the expansion of (2x + 3y z + -7)n is (A) n + 1 (B) (n + 4)C4 (C) (n + 5)C5 (D) nC5 12. The coefficient of x5 in the expansion of (1 x + 2x2)4 is 13. The two successive terms in the expansion of (1+x)24 whose coefficients are in the ratio 4 :1

are (A) 3rd and 4th (B) 4th and 5th (C) 5th and 6th (D) 6th and 7th

14. The expression nnnnnn CCCC 4..........4.4 2

210 , equals

(A) n22 (B) n32 (C) n5 (D) None of these

15. 602 when divided by 7 leaves the remainder (A) 1 (B) 6 (C) 5 (D) 2

16. The sum of the coefficients in the expansion of 21632 )31( xx is (A) 1 (B) 1 (C) 0 (D) None of these

17. The value of

0

11CC

n

n

1

21CC

n

n

.

1

1n

nn

n

CC

is equal to

(A) !)1( 1

nn n

(B)!

)1(n

n n (C)

)!1(

1

nnn

(D) )!1(

)1( 1

nn n

18. The sum of the rational terms in the expansion of 10

51

32

is

19. If in the expansion of (1 + x)m (1 x)n, the co-efficient of x and x2 are 3 and 6 respectively,

then m is 20. For 2 r n, nCr + 2 nCr1 + nCr2 is equal to 21. If (1 + x + 2x2)20 = a0 + a1x + a2x2 + + a40x40 then a1 + a3 + a5 + .+a37 equals

to

22. The largest term in the expansion of (3 + 2x)50 where x = 51 is

23. Let R = 1n21155 and f = R [R] where [.] denotes the greatest integer function, then Rf =

24. 23n 7n 1 is divisible by

3

25. If (1 x + x2)n = a0 + a1x + a2x2 + +a2nx2n, then a0 + a2 + a4 + + a2n equals to

26. If the rth term in the expansion of 10

2x2

3x

contains x4, then r is equal to

27. 1.nC1 + 2.nC2 + 3.nC3 + ..+ n.nCn is equal to

(A) n2.4

1nn (B) 2n+1 3

(C) n.2n 1 (D) none of these 28. If the coefficient of (2r + 2)th and (r + 1)th terms of the expansion (1 +x)37 are equal then r = (A) 12 (B) 13 (C) 14 (D) 18

29. The value of 1n

C2...........

4C

23

C2

2C

2C2 n1n3423120 is equal to

30. If the co-efficient of rth, (r+1)th and (r+2)th terms in the expansion of (1+x)14 are in A.P., then the value of r is

(A) 5 (B) 6 (C) 7 (D) 9 31. If (1+ax)n = 1+8x +24x2+------- then (A) a= 3 (B) n= 5 (C) a= 2 (D) n =4 32. If ab 0 and the co-efficient of x7 in [ax2+(1/bx)]11 is equal to the co-efficient of x-7 in

11

2bx1ax

, then a and b are connected by the relation

(A) a = 1/b (B) a = 2/b (C) ab = 1 (D) ab = 2 LEVELII 1. Co-efficient of x5 in the expansion of (1+x2)5 (1+x)4 is (A) 40 (B) 50 (C) 30 (D) 60 2. The term independent of x in the expansion of (x+1/x)2n is

(A) !n

2).1n2(.5.3.1 n (B) !n!n

2).1n2(.5.3.1 n

(C) !n

)1n2(.5.3.1 (D) !n!n

)1n2(.5.3.1

3. If 6th term in the expansion of 8

102

3/8xlogx

x1

is 5600, then x is equal to

(A) 5 (B) 4 (C) 8 (D) 10

4

4. If coefficient of x2 y3 z4 in (x + y +z)n is A, then coefficient of x4y4z is

(A) 2A (B) 2

nA

(C) 2A (D) none of these

5. The coefficient of x6 in {(1 + x)6 + (1 + x)7 + .+ (1 + x)15} is (A) 16C9 (B) 16C5 6C5 (C) 16C6 1 (D) none of these 6. If (1 +x)10 = a0 +a1x +a2x2 + +a10x10 then (a0 a2 +a4 a6 +a8 a10)2 + (a1 a3 +a5 a7 +a9)2

is equal to (A) 310 (B) 210 (C) 29 (D) none of these 7. The remainder of 7103 when divided by 25 is

8. The term independent of x in the expansion of 3

x2x21

is

9. The number of irrational terms in the expansion of 55

101

21

32

is;

(A) 47 (B) 56 (C) 50 (D) 48 10. If ab 0 and the co-efficient of x7 in (ax2+(1/bx))11 is equal to the co-efficient of x-7 in

11

2bx1ax

, then a and b are connected by the relation

(A) a= 1/b (B) a =2/b (C) ab= 1 (D) ab=2

11. If (1 + 2x + 3x2)10 =

20

0r

rr xa then a2 is equal to;

(A) 210 (B) 620 (C) 220 (D) none of these

12. If Pn denotes the product of all the co-efficients in the expansion of (1+x)n, then n

1n

PP is equal

to

(A) !n1n n (B) !1n

1n 1n

(C) !n1n 1n (D) !1n

1n n

13. Value of

n

0r

2r

n

2rsinC , is equal to;

(A) 2n (B) 2n 1 (C) 2n + 1 (D) 2n 1 1

5

14. If 1 ba , then

n

0r

rnrr

n baC equals

(A) 1 (B) n (C) na (D) nb

15. If { x } denotes the fractional part of x , then

832n

, Nn is

(A) 3/8 (B) 7/8 (C) 1/8 (D) None of these. 16. The coefficient of mx in : ,)1.......()1()1( 1 nmm xxx nm is

(A) 11

mn C (B) 1

1

m

n C (C) mn C (D) 1m

n C

17. The expansion 5

21

35

21

3 1xx1xx

is a polynomial of degree

18. In the expansion of n

23

x1x

, n N, if the sum of the coefficients of x5 and x10 is 0 then n

is (A) 25 (B) 20 (C) 15 (D) none of these

19. The sum 21 10C0 10C1 + 2. 10C2 22 . 10C3 + + 29. 10C10 is equal to

(A) 21 (B) 0

(C) 103.21 (D) none of these

20. If the second, third and fourth terms in the expansion of (a+b) n are 135, 30 and 10/3 respectively, then (A) a = 3 (B) b = 1/3 (C) n = 5 (D) all the above LEVELIII

1. The co-efficient of x53 in the expansion

100

0m

mm100m

100 2)3x(C is

(A) 100C53 (B) - 100C53 (C) 65C53 (D) 100C65

2. If n is an even natural number and coefficient of xr in the expansion of x1x1 n

is 2n, (|x| < 1),

then

(A) r n/2 (B) r 2

2n

(C) r 2

2n (D) r n

6

3. Let n be an odd natural number and A =

21n

1r rn C1 . Then value of

n

1r rn Cr is equal to

(A) n( A-1) (B) n( A+1)

(C) 2

nA (D) nA

4. ..........!5n!5

1!3n!3

1!1n!1

1

is equal to

(A) !n

2 1n for even values of n only (B) !n

12 1n for odd values of n only

(C) !n

2 1n for all n N (D) none of these

5. The greater of two numbers 300! and 300300 is 6. The co-efficient of x4 in the expansion of (1+x+x2+x3)11 is (A) 1001 (B) 990 (C) 900 (D) 895

7. Value of

n

1r

r

0mm

rr

n CC is equal to;

(A) 2n 1 (B) 3n -1 (C) 3n 2n (D) none of these

8. Value of

n

0r

2r

n Cr is equal to

(A) n . 2nCn (B) 2

Cn nn2

(C) n2 . 2nCn (C) 2

Cn nn22

9. If

n

1r rn C

r = , then value of

n

0r rn C1 is equal to;

(A) 2

n (B) n2

(C) 2

n (D) none of these

10. Value of

n

0rr

n xrnsinrxcosC is;

(A)2n 1 sin nx (B) 2n 1 cos nx (C) 2n cos nx (D) 2n sin nx 11. Value of

nji0j

nCi is;

(A) n.2n 3 (B) (n 1) . 2n 3 (C) n(n 1) . 2n 3 (D) none of these

7

12. The coefficient of xn in the polynomial ( x+ nC0) ( x+3 nC1) ( x+5 nC2) ..( x+(2n + 1) nCn) is (A) n2n (B) n2n + 1 (C) (n +1)2n (D) n2n + 1

13. Value of 2

2

0

1.2

nn

rr

r Cr

is equal to

(A)

1 22 2 1 22 1 2 2

n n nn n

(B)

2 1 22 2 1 22 1 2 2

n n nn n

(C)

2 1 22 2 2 12 1 2 2

n n nn n

(D) None of these

14. If R = 2 15 3 8 n and f = R [R]; where [ ] denotes G. I. F., then R f is equal to (A) 211 n (B) 2 111 n (C) 2 111 n (D) 11

15. Value of 20

n ni j

i j nC C

is

(A) 2 2. 2n nnn C (B) 2 21 2n nnn C (C) 2 21 2n nnn C (D) 2 21 2n nnn C

16. The remainder when 1037 is divided by 25 is (A) 0 (B) 18 (C) 16 (D) 9 17. The number 100101 1 is divisible by (A) 10 (B) 210 (C) 310 (D) 410

18. Integral part of 2 15 5 11 n is (A) Even (B) Odd (C) Neither (D) Cant Say 19. Let 2( ) 10 3 4 5;n nf n n N . The greatest value of the integer which divides f(n) for all

n is (A) 27 (B) 9 (C) 3 (D) None

20. If 8

0

2 2 1,1 6

nn

rr

r Cr

then n is (A) 8 (B) 4 (C) 6 (D) 5

8

ANSWERS

LEVEL I 1. C 2. B 3. B 4. A 5. B 6. B 7. D 8. A 9. A 10. C 11. B 12. 56 13. C 14. C 15. A 16. B 17. B 18. 41 19. 12 20. n+2Cr 21. 239 219 22. 50C6 344 (2x)6 23. 42n+1 24. 49

25. n3 12 26. 3 27. C 28. A

29. n 13 1n 1

30. D 31. C 32. C

LEVEL II 1. D 2. A 3. D 4. C 5. A 6. B 7. 7 8. 3 30 12C C 9. B 10. C 11. A 12. A 13. B 14. A 15. C 16. A 17. 7 18. C 19. A 20. D LEVEL III 1. B 2. D 3. B 4. C 5. 300! 6. B 7. B 8. B 9. B 10. A 11. C 12. C 13. A 14. C. 15. D 16. B 17. A, B, C, D 18. A 19. B 20. D

Quiz Bank-Circle-1

CIRCLE

LEVEL-I 1. The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a circle, the condition will be (A) a = b and c = 0 (B) f = g and h = 0 (C) a = b and h = 0 (D) f = g and c = 0 2. The equation x2 + y2 + 2gx + 2fy + c = 0 represents a real circle if (A) g2 + f2 c < 0 (B) g2 + f2 c 0 (C) always (D) none of these

3. Equation of a circle with centre (4,3) touching the circle x2 + y2 = 1 is

(A) x2 +y2 8x 6y 9 = 0 (B) x2 + y2 8x 6y + 11 = 0 (C) x2 + y2 8x 6y 11 = 0 (D) x2 + y2 8x 6y + 9 = 0

4. A square is inscribed in the circle x2 + y2 2x + 4y + 3 = 0. Its sides are parallel to the axes. Then

the one vertex of the square is (A) (1 + 2 , -2) (B) (1 - 2 , -2) (C) (1, -2 + 2 ) (D) none of these 5. The number of common real tangents that can be drawn to the circle x2 + y2 2x 2y = 0 and x2 +

y2 8x 8y + 14 = 0 is____________________________ 6. The lines 3x 4y + 4 = 0 and 6x 8y 7 = 0 are tangents to the same circle. The radius of the

circle is____________________________________________ 7. The straight line y = mx + c cuts the circle x2+y2 = a2 at real points if

(A) c)m1(a 22 (B) c)m1(a 22

(C) c)m1(a 22 (D) c)m1(a 22 8. A line is drawn through a fixed point P (,) to cut the circle x2+y2 = r2 at A and B. Then PA.PB is

equal to (A) (+)2-r2 (B) 2+2-r2 (C) (-)2+r2 (D) None of these 9. The locus of the centre of a circle of radius 2 units which rolls on the outside of the circle

x2 + y2 +3x 6y 9 = 0 is . 10. The values of a and b for which the two circles : x2 + y2 + 2(1 a)x + 2(1 + b)y + (2 c) = 0 and x2 + y2 + 2(1 + a)x + 2(1 b)y + (2 + c) = 0 cut

orthogonally are .. 11. A circle of radius 2 lies in the first quadrant and touches both the axes of co-ordinates. Then the

equation of the circle with centre (6, 5) and touching the above circle externally is (A) (x 6)2 + (y 5)2 = 4 (B) (x 6)2 + (y 5)2 = 9 (C) (x 6)2 + (y 5)2 = 36 (D) none of these

Quiz Bank-Circle-2 12. Two circles x2 + y2 2x 3 = 0 and x2 + y2 4x 6y 8 = 0 are such that (A) they touch each other (B) they intersect each other (C) one lies inside the other (D) each lies outside the other 13. The least distance of point (10, 7) from the circle x2 + y2 4x 2y 20 = 0 is (A) 10 (B) 15 (C) 5 (D) none of these 14. The number of common tangents to the circles x2 + y2 x = 0 and x2 + y2 + x=0 is (A) 2 (B) 1 (C) 4 (D) 3 15. The radius of the circle passing through the point (2, 6) two of whose diameters are x + y = 6 and

x + 2y = 4 is (A) 10 (B) 2 5 (C) 6 (D) 4 16. The intercept on the line y = x by the circle x2 + y2 2x = 0 is AB. The equation of the circle with

AB as diameter is (A) x2 + y2 + x + y = 0 (B) x2 + y2 = x + y (C) x2 + y2 3x + y = 0 (D) none of these 17. Equation of tangent to the circle x2 + y2 + 2x 2y + 1 = 0 at (0, 1) (A) x = 0 (B) y = 0 (C) xy = 0 (D) none of these 18. The equation x2 + y2 2x + 4y + 5 = 0 represents (A) a point (B) a pair of straight lines (C) a circle (D) none of these 19. The equation of the chord of the circle x2 + y2 4x = 0 which is bisected at the point (1, 1) is (A) x + y = 2 (B) 3x y = 2 (C) x 2y + 1 = 0 (D) x y = 0 20. The line x + y = 1 is a normal to the circle 2x2 + 2y2 5x + 6y 1 = 0 if (A) 5 6 = 4 (B) 4 + 5 = 6 (C) 4 + 6 = 5 (D) none of these 21. The locus of the point (3h+2, k), where (h, k) lies on the circle x2+y2 = 1 is (A) a hyperbola (B) a circle (C) a parabola (D) an ellipse

Quiz Bank-Circle-3

LEVEL-II

1. The centre of the circle passing through the points (0, 0), (1, 0) and touching the circle x2+y2= 9 is

(A)

21,

23 (B)

23,

21

(C)

21,

21 (D)

2,

21

2. The coordinates of mid point of the chord cut off by 2x 5y + 18 = 0 by the circle

x2 + y2 6x + 2y 54 = 0 are (A) (1, 4) (B) (2, 4) (C) (4, 1) (D) (1, 1) 3. Equation of tangent drawn from origin to the circle x2 + y2 2rx + 2hy + h2 = 0 are (A) x = 0 (B) y = 0 (C) (h2 r2)x 2rhy = 0 (D) (h2 r2)x + 2rhy = 0 4. If 2 circles (x 1)2 + (y 3)2 = r2 and x2 + y2 8x + 2y + 8 = 0 intersect at 2 distinct points, then (A) 2 < r < 8 (B) r > 2 (C) r = 2 (D) r < 2 5. The equation of circle passing through (1, 3) and the points common to the two circles

x2 + y2 6x + 8y 16 = 0, x2 + y2 + 4x 2y 8 = 0 is (A) x2 + y2 4x + 6y + 24 = 0 (B) 2x2 + 2y2 + 3x + y 20 = 0 (C) 3x2 + 3y2 5x + 7y 19 = 0 (D) none of these 6. The common chord of x2+ y2 4x 4y = 0 and x2 + y2 = 16 subtends at the origin an angle equal to

(A) 6 (B)

4

(C) 3 (D)

2

7. The locus of the centre of the circle which touches externally the circle x2+y26x6y+14=0 and

also touches the y-axis is given by the equations (A) x2 6x 10y + 14 = 0 (B) x2 10x 6y + 14 = 0 (C) y2 6x 10y + 14 = 0 (D) y2 10x 6y + 14 = 0 8. If the tangent at the P on the circle x2 + y2 + 2x + 2y = 7 meets the straight line 3x 4y = 15 at a

point Q on the x-axis, then length of PQ is (A) 3 7 (B) 4 7 (C) 2 7 (D) 7 9. A straight line is drawn through the centre of the circle x2 + y2 2ax = 0, parallel to the straight

line x + 2y = 0 and intersecting the circle at A and B. Then the area of AOB is

(A) 5

a2 (B) 5

a3

(C) 3

a2 (D) 3

a3

Quiz Bank-Circle-4

10. The equation of the circle of radius 2 which touches the line x + y = 1 at (2, 1) is (A) x2 + y2 4x +2y+ 3= 0 (B) x2 + y2 + 6x +7= 0 (C) x2 + y2 2x +4y+ 3= 0 (D) none of these 11. If the coordinates of one end of a diameters of the circle x2 + y2 8x 4y + c = 0 are (3, 2),

then the coordinates of the other end are (A) (5, 3) (B) (6, 3) (C) (1, 8) (D) (11, 2) 12. The equation of the locus of the centre of circles touching the yaxis and circle x2 + y2 2x= 0 is (A) x2 = 4y (B) x2 = 4y (C) y2 = 4x (S) y2 = 4x 13. The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x 6y + 9 sin2 + 13 cos2 = 0 is 2. The equation of the locus of P is (A) x2 + y2 + 4x 6y + 4 = 0 (B) x2 + y2 + 4x 6y 9 = 0 (C) x2 + y2 + 4x 6y 4 = 0 (D) x2 + y2 + 4x 6y + 9 = 0 14. The number of common tangents to the circles x2 + y2 6x 14y + 48 = 0 and x2 + y2 6x = 0 is (A) 1 (B) 2 (C) 3 (D) 4 15. The equation of the smallest circle passing through the intersection of the line x + y = 1 and the

circle x2 + y2 = 9 is (A) x2 + y2 + x + y 8 = 0 (B) x2 + y2 x y 8 = 0 (C) x2 + y2 x + y 8 = 0 (D) none of these

16. A, B, C, D are the points of intersection with the co-ordinate axes of the lines ax + by = ab and bx + ay = ab then (A) A, B, C, D are concyclic (B) A,B,C,D forms a parallelogram (C) A, B, C, D forms a rhombus (D) None of these

17. If the lines 2x 3y 5 = 0 and 3x-4y = 7 are diameters of a circle of area 154 square units, then the equation of the circle is

(A) x2+y2+2x-2y-62 = 0 (B) x2+y2+2ax 2y 47 = 0 (C) x2+y2-2x+2y-47 = 0 (D) x2+y2-2x+2y-62 = 0

18. The equation of the circle whose diameter is the common chord of the circle x2+y2+3x+2y+1= 0 and x2+y2+3x+4y+2 = 0 is

(A) x2+y2+8x+10y+2 = 0 (B) x2+y2-5x+4y+7 = 0 (C) 2x2+2y2+6x-2y-1 = 0 (D) None of these

19. The length of the tangent from any point on the circle 15x2 +15y2 48x + 64y = 0 to the two circles 5x2 + 5y2 24x + 32y + 75 = 0 and 5x2 + 5y248x + 64y + 300 = 0 are in the ratio of

(A) 1 : 2 (B) 2 : 3 (C) 3 : 4 (D) None of these

20. The tangents drawn from the origin to the circle x2+y2-2rx-2hy+h2 = 0 are perpendicular if (A) h = r (B) h = r (C) r2+ h2 = 1 (D) r2 = h25.

Quiz Bank-Circle-5 21. If a variable circle of radius 4 cuts the circle x2 + y2 = 1 orthogonally then locus of its centre

will be (A) x2 + y2 = 16 (B) x2 + y2 =17 (C) x2 + y2 - 2x - 4y = 1 (D) 2x - 4y + 5 = 0

22. If four points

ii t

1,t ( i = 1, 2, 3, 4) are concyclic then t1t2 t3t4 =

(A) 1 (B) -1 (C) 4 (D) 1/4

23. The number of common tangents that can be drawn to the circle x2+y24x 6y 3 = 0 and x2 + y2 + 2x + 2y + 1 = 0 is

(A) 1 (B) 2 (C) 3 (D) 4

24. The circle x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touch if

(A) c1

b1

a1

22 (B) 222 c1

b1

a1

(C) 0c1

b1

a1

(D) none of these

25. The equation (x2 a2)2 + (y2 b2)2 = 0 represents points (A) which are collinear (B) which lie on a circle centred (0, 0) (C) which lie on a circle centre (a, b) (D) none of these 26. The equations of the circle which touch both the axes and the line x = a are

(A) x2+y2 ax ay+4a2 =0 (B) x2+y2 + ax ay+

4a2 =0

(C) x2+y2 -ax ay+4a2 =0 (D) None of these

27. If the abscissae and ordinates of two points P and Q are the roots of the equation x2+2ax-b2 = 0

and x2+2px-q2 = 0 respectively, then the equation of the circle with PQ as diameter is (A) x2+y2+2ax+2py-b2-q2 = 0 (B) x2+y2-2ax-2py+b2+q2 = 0 (C) x2+y2-2ax-2py-b2-q2 = 0 (D) x2+y2+2ax+2py+b2+q2 = 0 28. If the distances from the origin of the centre of three circles x2+y2 +2ix c2=0 (i= 1, 2, 3) are in

G.P. then the length of the tangent drawn to them from any point on the circle x2+y2 = c2 are in (A) A.P. (B) G.P. (C) H.P. (D) None of these 29. If the chord of contact of tangents drawn from a point on the circle x2 + y2 =a2 to the circle

x2 + y2 = b2 touches the circle x2 + y2 =c2, a, b, c> 0, then a, b, c are related as 30. The triangle PQR is inscribed in the circle x2 + y2 = 25. If Q and R have co-ordinates (3, 4) and

(4, 3) respectively, then QPR is equal to (A) /2 (B) /3 (C) /4 (D) /6

Quiz Bank-Circle-6 31. If the circle x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect ortohgonally, then k is (A) 2 or 3/2 (B) 2 or 3/2 (C) 2 or 3/2 (D) none of these 32. If the tangent to the circle x2 + y2 = 5 at the point (1, -2) also touches the circle x2 + y2 - 8x + 6y + 20 = 0, then its point of contact is (A) (3, 1) (B) (-3, 1) (C) (3, -1) (D) (-3, -1) 33. The equation of the circle having its centre on the line x + 2y 3 = 0 and passing through the point

of intersection of the circles x2 + y2 2x 4y + 1= 0 and x2 + y2 4x 2y + 1 = 0 is (A) x2 + y2 6x + 1 = 0 (B) x2 + y2 3x + 4 = 0 (C) x2 + y2 2x 2y + 1= 0 (D) x2 + y2 + 2x 4y + 4 = 0 34. Two circles x2 + y2 = 6 and x2 + y2 6x + 8 = 0 are given. Then the equation of the circle through

their point of intersection and the point (1, 1) is (A) x2 + y2 + x y = 0 (B) x2 + y2 3x + 1 = 0 (C) x2 + y2 4y + 2 = 0 (D) none of these 35. Given that the circles x2 + y2 2x + 6y + 6 = 0 and x2 + y2 5x + 6y + 15 = 0 touch, the equation

to their common tangents is (A) x = 3 (B) y = 6 (C) 7x 12y 21 = 0 (D) 7x + 12y + 21 = 0 36. If an equilateral triangle is inscribed in the circle x2 + y2 = k2, the length of each side is equal to (A) k/3 (B) k3 (C) k (D) 2k 37. The equation of the circle through the origin and cutting intercepts of length 2 and 3 from the

positive sides of x and y is ____________________________ 38. If the circle x2+y2+4x+22y + c = 0 bisects the circumference of the circle x2 + y2 2x + 8y + d = 0

then c d is equal to (A) 60 (B) 50 (C) 40 (D) 56 39. If an equilateral triangle is inscribed in the circle x2 + y2 = 25 then length of its each side is

(A) 5 2 (B) 235

(C) 5 3 (D) none of these 40. If the coordinates at one end of a diameter of the circle x2 + y2 8x 4y + c = 0 are (11, 2) then

the coordinates at the other end are (A) (3, 2) (B) (3, 2) (C) (3, 2) (D) (3, 2) 41. S1 = x2 + y2 = 9, S2 = x2 + y2 8x 6y + n2 = 0 , n Z. If the two circle have exactly two common

tangent then the number of possible value of n is (A) 7 (B) 8 (C) 9 (D) 10

Quiz Bank-Circle-7

42. If the common chord of x2 + (y )2 = 16 and x2 + y2 = 16 subtends a right angle at the origin then is equal to

(A) 4 (B) 4 2 (C) 4 2 (D) 8 43. The locus of the middle point of chord of length 4 of the circle x2 + y2 = 16 is (A) a straight line (B) a circle of radius 2 (C) a circle of radius of radius 2 3 (D) an ellipse 44. The number of points with integral coordinates that are interior to the circle x2 + y2 = 16 is (A) 43 (B) 49 (C) 45 (D) 51 45. If equation of circle is ax2 + (2a 3)y2 4x 1 = 0, then its centre is (A) (2, 0) (B) (2/3, 0) (C) (2/3, 0) (D) none of these 46. The shortest distance between the circles x2 +y2 = 1 and x2 +y2 10x 10y+ 41 = 0 is

(A) 41 -1 (B) 0 (C) 41 (D) 5 42

47. Two circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by = 0 touch if

(A) a2 + b2 = c2 (B) c1 =

1 12 2a b

(C) 222 b1

a1

c1

(D) c2 = 4b2(a2 - c)

48. If y = 2x be the equation of a chord of the circle x2 + y2 = 2ax, then the equation of the circle, of

which this chord is a diameter, is (A) 2( x2+y2) 5a( x + 2y ) = 0 (B) x2+y2 2a( x + 2y ) = 0 (C) 5(x2+y2) 2a( x + 2y ) =0 (D) none of these.

49. PA is tangent to x2 + y2 = a2 and PB is tangent to x2 + y2 = b2 (b > a) . If APB = 2 , then locus

of point P is (A) x2 y2 = a2 + b2 (B) x2 + y2 = b2 a2 (C) x2 + y2 = a2 + b2 (D) none of these 50. f(x, y) = x2 + y2 + 2ax + 2by + c = 0 represents a circle. If f(x, 0) = 0 has equal roots, each being 2

and f(0, y) = 0 has 2 and 3 as its roots, then centre of circle is

(A)

25,2 (B)

25,2

(C) data are not sufficient (D) data are inconsistent 51. Tangents PA and PB are drawn to x2+y2=4 from the point P(3, 0). Area of triangle PAB is equal to (A) 5

95 sq. units (B) 5

31 sq. units

(C) 59

10 sq. units (D) 53

20 sq. units

Quiz Bank-Circle-8 52. Radius of bigger circle touching the circle x2+y2 4x 4y + 4 = 0 and both the co-ordinate axes is (A) 3 + 2 2 (B) 2 223 (C) 6 + 2 2 (D) 2 226 53. The lines 3x 4y + = 0 and 6x 8y + = 0 are tangents to the same circle. The radius of the

circle is

(A) 20

2 (B) 20

2

(C) 20

2 (D) none of these.

Quiz Bank-Circle-9

LEVEL-III 1. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes

one complete roll on x-axis along the positive direction of x-axis, then its equation in the new position is

(A) x2 + y2 + 20x 10y + 1002 = 0 (B) x2 + y2 + 20x + 10y + 1002 = 0 (C) x2 + y2 20x 10y + 1002 = 0 (D) none of these 2. Let AB be a chord of circle x2 + y2 = 3 which subtends 450 angle at P where P is any moving point

on the circle. The locus of centroid of PAB is

(A) 31

31y

31x

22

(B)

31

31y

31x

22

(C) 31

31y

31x

22

(D) none of these

3. Two circles, each radius 5, have a common tangent at (1, 1) whose equation is 3x +4y 7=0 then

their centre are (A) (4, 5), (2,3) (B) (4, 3), (2, 5) (C) (4, 5), (2, 3) (D) none of these 4. The equation of the circle of radius 2 2 whose centre lies on the line x y = 0 and which

touches the line x + y = 4 and whose centres coordinates satisfy the inequality x + y > 4 is (A) x2 + y2 8x 8y + 24 = 0 (B) x2 + y2 = 8 (C) x2 + y2 8x + 8y + 24 = 0 (D) x2 + y2 + 8x + 8y + 24 = 0 5. The circle passing through distinct point (1, t), (t, 1) and ( t, t) for all values of t , passes

through the point (A) (-1, -1) (B) (1, 1) (C) ( 1, -1) (D) (-1, 1) 6. The equation of the locus of the midpoints of the chords of the circle 4x2 + 4y2 12x + 4y + 1 = 0

that subtends an angle 32 at its centre is ______

7. The area of the triangle formed by the positive x-axis and the normal and tangent to the circle

x2 + y2 = 4 at the point (1, 3 ) is _____________________ 8. A circle is inscribed in an equilateral triangle of side a. the area of any square inscribed in this

circle is _______________________________________ 9. Tangents OP and OQ are drawn from the origin O to the circle x2+y2+2gx+2fy+c=0. Then the

equation of the circumcircle of the triangle OPQ is (A) x2+y2+2gx+2fy = 0 (B) x2+y2+gx+fy = 0 (C) x2+y2-gx-fy=0 (D) x2+y2-2gx-2fy = 0 10. The locus of the mid points of the chords of the circle x2+y2+4x-6y-12 = 0 which subtends an

angle of 3 radians at its centre is

(A) (x+2)2+(y-3)2 = 6.25 (B) (x-2)2+(y+3)2 = 6.25 (C) (x+2)2+(y-3)2 = 18.75 (D) (x+2)2+(y+3)2 = 18.75

Quiz Bank-Circle-1011. The locus of the mid-points of a chord of the circle x2 + y2 = 4, which subtends a right angle at the

origin is (A) x + y = 2 (B) x2 + y2 =1

(C) x2 + y2 = 2 (D) x + y = 1 12. If two distinct chords, drawn from the point (p, q) on the circle x2 +y2 = px + qy (where p, q 0 )

are bisected by the x-axis, then (A) p2 = q2 (B) p2 = 8q2 (C) p2 < 8q2 (D) p2 > 8q2 13. The locus of the centre of a circle which touches a given line and passes through a given

point, not lying on the given line, is (A) a parabola (B) a circle (C) a pair of straight line (D) none of these .

14. The tangents drawn from the origin to the circle x2+y2 + 2gx + 2fy + f2 =0 are perpendicular if (A) g = f (B) g = -f (C) g = 2f (D) 2g = f 15. Two circles with radii r1 and r2, r1 > r2 2 , touch each other externally. If be the angle

between the direct common tangents, then

(A)

21

211

rrrrsin (B)

21

211

rrrrsin2

(C) = sin-1

21

21

rrrr (D) none of these.

16. Tangents are drawn to the circle x2 + y2 = 50 from a point P lying on the x-axis. These tangents

meet the y-axis at points P1 and P2. Possible coordinates of P so that area of triangle PP1P2 is minimum, are

(A) (10, 0) (B) (10 2 , 0) (C) (-10, 0) (D) (-10 2 , 0) 17. Two distinct chords of the circle x2 + y2 2x 4y = 0 drawn from the point P(a, b) gets bisected

by the y-axis, then (A) (b + 2)2 > 4a (B) (b 2)2 > 4a (C) (b 2)2 > 2a (D) none of these 18. A circle S of radius a is the director circle of another circle S1. S1 is the director circle of circle S2

and so on. If the sum of the radii of all these circles is 2, then the value of a is (A) 2 + 2 (B) 2 2

(C) 2 2

1 (D) 2 +2

1

19. Circles are drawn having the sides of triangle ABC as their diameters. Radical centre of these

circles is the (A) circumcentre of triangle (B) Incentre of triangle ABC (C) orthcentre of triangle ABC (D) centroid of ABC 20. The circle x2 + y2 + 2a1x + c = 0 lies completely inside the circle x2 + y2 + 2a2x + c =0, then (A) a1a2 > 0, c < 0 (B) a1a2 > 0, c > 0 (C) a1a2 < 0, c < 0 (D) a1a2 < 0, c > 0

Quiz Bank-Circle-11

ANSWERS

LEVEL I 1. C 2. B 3. D 4. D 5. 3 6. 3/4 7. A 8. B

9. 2

23 169x y 32 4

10. a = b = 0 11. B

12. B 13. C 14. D 15. A 16. B 17. A 18. C 19. D 20. A 21. D LEVEL II 1. D 2. A 3. A 4. A 5. D 6. D 7. D 8. C 9. A 10. C 11. D 12. C 13. D 14. D 15. B 16. A 17. C 18. C 19. A 20. A 21. B 22. A 23. C 24. A 25. B 26. C 27. A 28. B 29. G.P. 30. C 31. D 32. B 33. A 34. B 35. A 36. B

37. 2

2 3 13x 1 y2 4

38. B 39. C

40. C 41. C 42. C 43. C 44. C 45. B 46. D 47. D 48. C 49. C 50. D 51. C 52. B 53. A LEVEL III 1. D 2. C 3. C 4. A

5. B 6.

2 23 1 9x y2 2 4

7. 2 3 units

8.

21 a6

9. B 10. c 11. C 12. D 13. A 14. A 15. B 16. A, C 17. B 18. B 19. C 20. B

COMPLEX NUMBER

LEVEL-I 1. If z1 , z2 are two complex numbers such that arg(z1+z2) = 0 and

Im(z1z2) = 0, then (A) z1 = - z2 (B) z1 = z2 (C) z 1= 2z (D) none of these 2. Roots of the equation xn 1 = 0, n I, (A) form a regular polygon of unit circum-radius . (B) lie on a circle. (C) are non-collinear. (D) A & B 3. Which of the following is correct (A) 6 + i > 8 i (B) 6 + i > 4 - i (C) 6 + i > 4 + 2i (D) None of these 4. If (1+i3)1999 = a+ib, then (A) a = 21998, b = 219983 (B) a = 21999, b = 219993 (C) a=-21998, b = -219983 (D) None of these 5. If z = 1 + i 3 , then | arg ( z) | + | arg ( z ) | equals (A) /3 (B) 2/3 (C) 0 (D) /2

6. The equation 3i1zz3iizz

= 0 represents a circle with

(A) centre

23,

21 and radius 1 (B) centre

23,

21 and radius 1

(C) centre

23,

21 and radius 2 (D) centre

23,

21 and radius 2

7. Number of solutions to the equation (1 i)x = 2x is (A) 1 (B) 2 (C) 3 (D) no solution 8. If ,0)arg( z then )arg()arg( zz

(A) (B)4 (C)

2

(D) 2

9. The number of solutions of the equation ,022 zz where Cz is

(A) one (B) two (C) three (D) infinitely many

10. If is an imaginary cube root of unity, then (1 + 2)7 equals (A) 128 (B) 128 (C) 128 2 (D) 128 2

11. If z1 and z2 be the nth roots of unity which subtend right angle at the origin. Then n must

be of the form (A) 4k + 1 (B) 4k + 2 (C) 4k + 3 (D) 4k 12. For any two complex numbers z1 and z2 | 7 z1 + 3z2|2 + |3z1 7 z2|2 is always equal to (A) 16(|z1|2 + |z2|2) (B) 4(|z1|2 + |z2|2) (C) 8(|z1|2 + |z2|2) (D) none of these

13. If is an nth root of unity other than unity itself, then the value of 1 + + 2 + + n 1 is

14. Locus of z in the Argand plane is 2,z then the locus of z + 1 is -

(A) a straight line (B) a circle with centre (1, 0)

(C) a circle with centre (0, 0) (D) a straight line passing through (0, 0) 15. Value of 1999 299 1 is (A) 1 (B) 2 (C) 0 (D) -1 16. Square root(s) of 1 is/ are -

(A) 1 12

i (B) 1 13

i

(C) 1 12

i (D) 1 12

i

17. The real value of for which 3 2 sin1 2 sin

ii

is real is

(A) ,n n I (B) ,3

n n I

(C) ,2

n n I (D) ,2

n n I

18. Principal argument of 3z i is

(A) 56

(B) 6

(C) 56

(D) None

19. Which one is not a root of the fourth root of unity (A) i (B) 1

(C) 2i

(D) i

20. If 3 22 4 8 0z z z then (A) 1z (B) 2z

(C) 3z (D) None

LEVEL-II

1. If a,b, c are three complex numbers such that c =(1 ) a + b, for some non-zero real number , then points corresponding to a,b, c are (A) vertices of a triangle (B) collinear (C) lying on a circle (D) none of these

2. If z be any complex number such that |3z 2| + |3z +2| = 4, then locus of z is (A) an ellipse (B) a circle (C) a line-segment (D) None of these 3. If arg 1z = arg(z2), then (A) z2 = k z1-1 (k > 0) (B) z2 = kz1 (k > 0) (C) |z2| = | z 1| (D) None of these.

4. The value of the expression 2

21111 +3

21212 + 4

21313 + .

. . + (n+1)

21n1n , where is an imaginary cube root of unity, is

(A) 3

2nn 2 (B) 3

2nn 2 (C) 4

n41nn 22 (D) none of these

5. For a complex number z , | z-1| + |z +1| =2. Then z lies on a

(A) parabola (B) line segment (C) circle (D) none of these

6. If z1 and z2 are two complex numbers such that |z1| = |z2| + |z1 z2|, then

(A) Im

2

1

zz = 0 (B) Re

2

1

zz = 0

(C)

2

1

2

1

zzIm

zzRe (D) none of these.

7. If 2

1

zz =1 and arg (z1 z2) = 0, then

(A) z1 = z2 (B) |z2|2 = z1z2 (C) z1z2 = 1 (D) none of these. 8. Number of non-zero integral solutions to (3+ 4i)n = 25n is (A) 1 (B) 2 (C) finitely many (D) none of these. 9. If |z| < 4, then | iz +3 4i| is less than (A) 4 (B) 5 (C) 6 (D) 9 10. If z is a complex number, then z2 + 2z = 2 represents

(A) a circle (B) a straight line (C) a hyperbola (D) an ellipse

11. If i1i1 = A + iB, then A2 +B2 equals to

(A) 1 (B) 2 (B) -1 (D) - 2 12. A,B and C are points represented by complex numbers z1, z2 and z3. If the circumcentre

of the triangle ABC is at the origin and the altitude AD of the triangle meets the circumcircle again at P, then P represents the complex number

(A) 3

21

zzz (B)

1

32

zzz

(C) 2

13

zzz (D)

3

21

zzz

13. If |z1| = |z2| and arg(z1) +arg(z2) = /2 , then (A) arg(z1-1) + arg(z2-1) = -/2 (B) z1z2 is purely imaginary (C) (z1+z2)2 is purely imaginary (D) All the above.

14. If z1 and z2 are two complex numbers satisfying the equation 1izzizz

21

21

, then 2

1

zz

is a

(A) purely real (B) of unit modulus (C) purely imaginary (D) none of these 15. If the complex numbers z1, z2, z3, z4, taken in that order, represent the vertices of a

rhombus, then (A) z1 + z3 = z2 + z4 (B) |z1 z2| = |z2 z3|

(C) 42

31

zzzz

is purely imaginary (D) none of these

16. If 0z,z,kzzzzzz

2121

21

then

(A) for k = 1 locus of z is a straight line (B) for k {1, 0} z lies on a circle (C) for k = 0 z represents a point (D) for k = 1,z lies on the perpendicular bisector of the line segment joining

1

2

1

2zzand

zz

17. If the equation |z z1|2 + | z z2|2 = k represents the equation of a circle, where z1 2+

3i, z2 4 + 3i are the extremities of a diameter, then the value of k is

(A) 41 (B) 4

(C) 2 (D) None of these

18. If z be a complex number and ai , bi , ( i= 1,2,3) are real numbers, then the value of the

determinant

332211

332211

332211

azbazbazbzazbzazbzazbzbzazbzazbza

is equal to

(A) (a1 a2 a3 + b1 b2 b3 ) |z|2 (B) |z|2 (C) 0 (D) None of these

19. If z = x + iy satisfies the equation arg (z-2) = arg(2z+3i), then 3x-4y is equal to (A) 5 (B) -3

(C) 7 (D) 6

20. If a complex number x satisfies

1|z|2|z|26|z|2|z|log 2

2

2/1

27. If nn 2i3 , where n is an integer, then (A) n is a multiple of 5 (B) n is a multiple of 6 (C) n is a multiple of 10 (D) none of these 28. If points corresponding to the complex numbers z1, z2 and z3 in the Argand plane are A,B

and C respectively and if ABC is isosceles, and right angled at B then a possible value

of 23

21zzzz

is

(A) 1 (B) -1 (C) i (D) none of these 29. If z1 and z2 are two complex numbers satisfying the equation

1zzzz

21

21

, then

2

1zz

is a number which is

(A) Real (B) Imaginary

(C) Zero (D) None of these 30. If |z| = 1, then |z-1| is

(A) < |arg z| (B) >|arg z| (C) = |arg z| (D) None of these 31. If z1, z2 and z3, z4 are two pairs of conjugate complex numbers then

arg

4

1zz

+ arg

3

2zz

equals

(A) 2 (B)

(C) 2

3 (D) 0

32. If ||z + 2| |z 2|| = a2, z C is representing a hyperbola for a S, then S contains (A) [1, 0] (B) (, 0] (C) (0, ) (D) none of these

33. If |z| = 1 and z i, then iziz

is

(A) purely real (B) purely imaginary (C) a complex number with equal real and imaginary parts (D) none of these 34. The locus of z which satisfied the inequality log0.5|z 2| > log0.5|z i| is given by (A) x+ 2y > 1 (B) x y < 0 (C) 4x 2y > 3 (D) none of these 35. Let Z1 and Z2 be the complex roots of ax2 + bx + c = 0, where a b c > 0. Then

(A) | Z1 + Z2 | 1 (B) |Z1 + Z2 | > 2 (C) |Z1 | = |Z2| = 1 (D) none of these 36. If the roots of z3 + az2 + bz + c = 0, a, b, c C(set of complex numbers) acts as the

vertices of a equilateral triangle in the argand plane, then (A) a2 + b = c (B) a2 = b (C) a2 + b = 0 (D) none of these 37. If |z1| = 4, |z2| = 4, then |z1 + z2 + 3 + 4i| is less than (A) 2 (B) 5 (C) 10 (D) 13 38. If z = x + iy satisfies Re{z -|z 1| + 2i} = 0, then locus of z is

(A) parabola with focus

21,

21 and directrix x + y =

21

(B) parabola with focus

21,

21 and directrix x + y =

21

(C) parabola with focus

21,0 and directrix y =

21

(D) parabola with focus

0,21 and directrix x =

21

39. If |z +1| = z + 1 , where z is a complex number, then the locus of z is (A) a straight line (B) a ray (C) a circle (D) an arc of a circle 40. Length of the curved line traced by the point represented by z, when

arg41z

1z

, is

(A) 22 (B) 2

(C) 2 (D) none of these

41. If 02718128 23 izziz then

(A) 23z (B) 1z (C) 32z (D) 43z

42. If 2 iz and iz 351 then the maximum value of 1ziz is

(A) 312 (B) 231 (C) 231 (D) 7

43. ,)1z(i1sin 1

where z is not real, can be the angle of the triangle if

(A) 2)(,1)Re( zIz m (B) 1)(1,1)Re( zIz m (C) 0)()Re( zIz m (C) None of these

44. The value of )1ln( (A) does not exist (B) iln2 (C) i (D) 0

45. If 21 ,nn are positive integers then 2121 )1()1()1()1(753 nnnn iiii is a real Number

if and only if (A) 121 nn (B) 21 1 nn (C) 21 nn (D) 21 ,nn be +ve integers

46. Let 21 , zz be two nonreal complex cube roots of unity and 2

12

1 zzzz be the equation of a circle with 21 , zz as ends of a diameter then the value of is (A) 4 (B) 3 (C) 2 (D) 2

47. The center of the arc 4682

363arg

iziz

is

(A) (4,1) (B) (1,4) (C) (2,5) (D) (3,1)

48. The value of

6

1 72cos

72sin

k

kik

(A) i (B) i (C) 1 (D) 1

49. The complex numbers z1, z2 and z3 satisfying 23i1

zzzz

32

31

are the vertices of a

triangle which is (A) of area zero (B) right angled isosceles (C) equilateral (D) obtuse angled isosceles

50. If |z| = 3 then the number 3z3z

is

(A) purely real (B) purely imaginary (C) a mixed number (D) none of these

51. If iz3 + z2 z + i = 0, then |z| is equal to

52. If and are different complex numbers with || = 1, then

1 is equal to

53. If the complex numbers z1, z2, z3 are in A.P., then they lie on a

(A) circle (B) parabola

(C) line (D) ellipse

54. If z1 and z2 are two nth roots of unity, then arg

2

1

zz is a multiple of .

55. The maximum value of |z| when z satisfies the condition z2z = 2 is

56. All non-zero complex numbers z satisfying z = iz2 are.

57. Common roots of the equation z3 + 2z2 + 2z +1 = 0 and z1985 + z100 + 1 = 0 is

LEVEL-III 1. If points corresponding to the complex numbers z1, z2, z3 and z4 are the vertices of a

rhombus, taken in order, then for a non-zero real number k (A) z1 z3 = i k( z2 z4) (B) z1 z2 = i k( z3 z4) (C) z1 + z3 = k( z2 +z4) (D) z1 + z2 = k( z3 +z4) 2. If z1 and z2 are two complex numbers such that | z1 z2| = | |z1| - |z2| |, then

argz1 argz2 is equal to (A) - /4 (B) - /2 (C) /2 (D) 0

3. If f(x) and g(x) are two polynomials such that the polynomial h(x) = x f(x3) + x2 g(x6) is divisible by x2 +x +1 , then

(A) f(1) = g(1) (B) f(1) - g( 1) (C) f(1) = g(1) 0 (D) f(1) = -g(1) 0 4. Consider a square OABC in the argand plane, where O is origin and A A(z0).

Then the equation of the circle that can be inscribed in this square is; ( vertices of square are given in anticlockwise order)

(A) | z z0(1+ i)| =|z0| (B) 2

00 z

2i1z

z

(C)

00 z

2i1z

z

(D) none of these .

5. For a complex number z, the minimum value of |z| + | z - cos - isin| is (A) 0 (B) 1 (C) 2 (D) none of these 6. The roots of equation zn = (z +1)n (A) are vertices of regular polygon (B) lie on a circle (C) are collinear (D) none of these 7. The vertices of a triangle in the argand plane are 3 + 4i, 4+ 3i and 2 6 + i, then

distance between orthocentre and circumcentre of the triangle is equal to,

(A) 628137 (B) 628137

(C) 62813721

(D) 62813731

.

8. One vertex of the triangle of maximum area that can be inscribed in the curve

|z 2 i| =2,is 2 +2i , remaining vertices is / are (A) -1+ i( 2 + 3 ) (B) 1 i( 2 + 3 ) (C) 1+ i( 2 3 ) (D) 1 i( 2 3 )

9. If

2

2

1

1

z3z2

z2z3 = k, then points A(z1) , B(z2), C(3, 0) and D(2, 0) (taken in clockwise

sense) will (A) lie on a circle only for k > 0 (B) lie on a circle only for k < 0 (C) lie on a circle k R (D) be vertices of a square k( 0, 1) 10. Let z be a complex number and a be a real parameter such that

z2 + az + a2 = 0, then (A) locus of z is a pair of straight lines

(B) arg(z) = 32

(C) |z| =|a| . (D) All

11. If z1, z2, z3 . . .. zn-1 are the roots of the equation zn-1 + zn-2 + zn-3 + . . .+z +1= 0,

where n N, n > 2, then (A) n, 2n are also the roots of the same equation. (B) 1/n, 2/n are also the roots of the same equation. (C) z1, z2, . . . , zn-1 form a geometric series. (D) none of these. Where is the complex cube root of unity. 12. The value of i log(x i) + i2 +i3 log(x +i) + i4( 2 tan-1x), x> 0 ( where i = 1 ) is

(A) 0 (B) 1 (C) 2 (D) 3 13. If z = -2 + i32 , then z2n + 22n zn + 24n may be equal to (A) 22n (B) 0 (C) 3. 24n (D) none of these

14. The value of

135cos

1312sini 11

e169 is (A) 119 120i (B) -i(120 +119i) (C) 119 + 120i (D) none of these 15. Let z1 and z2 be the complex roots of the equation 3z2 + 3z+ b = 0. If the origin, together

with the points represented by z1 and z2 form an equilateral triangle then the value of b is (A) 1 (B) 2 (C) 3 (D) None of these

16. If|z-2| = min {|z-1|,| z-3|}, where z is a complex number, then

(A) Re(z) = 23 (B) Re(z) =

25

(C) Re (z)

25,

23 (D) None of these

17. If x = 1 + i, then the value of the expression x4 4x3 + 7x2 6x + 3 is (A) -1 (B) 1 (C) 2 (D) None of these 18. If z lies on the circle centred at origin. If area of the triangle whose vertices are z, z and

z + z, where is the cube root of unity, is 4 3 sq. unit. Then radius of the circle is (A) 1 unit (B) 2 units (C) 3 units (D) 4 units 19. If i [0, /6], i = 1, 2, 3, 4, 5 and sin 1z4 + sin2 z3 + sin3 z2 + sin 4 z + sin5 = 2, then

z satisfies.

(A) 43|z| (B)

21|z|

(C) 43|z|

21

(D) None of these

20. If is the angle which each side of a regular polygon of n sides subtends at its centre,

then 1 + cos + cos2 + cos3 + cos(n-1) is equal to (A) n (B) 0 (C)1 (D) None of these 21. Triangle ABC, A(z1), B(z2), C(z3) is inscribed in the circle |z| = 2. If internal bisector of the

angle A meets its circumcircle again at D(zd) then (A) 32

2d zzz (B) 31

2d zzz

(C) 122d zzz (D) none of these

ANSWERS LEVEL I 1. C 2. D 3. D 4. A 5. B 6. B 7. A 8. A 9. D 10. D 11. D 12. A 13. 0 14. B 15. C 16. A 17. A 18. A 19. C 20. B LEVEL II 1. B 2. C 3. A 4. C 5. B 6. A 7. B 8. D 9. D 10. C 11. A 12. B 13. D 14. A 15. A, B, C 16. A, B, C, D 17. B 18. C 19. D 20. B 21. D 22. A 23. 20 24. 25. 0 26. A 27. D 28. C 29. B 30. A 31. D 32. A 33. B 34. C 35. A 36. D 37. D 38. D 39. B 40. D 41. A 42. D 43. B 44. C 45. C 46. B 47. A 48. A 49. C 50. B 51. 1

52. 1 53. C 54. 2n

55. 1 + 3

56. 3 1,2 2

57. , 2

LEVEL III 1. A 2. D 3. A 4. B 5. B 6. C 7. B 8. A 9. C 10. D 11. C 12. A 13. B, C 14. A, B 15. A 16. C 17. B 18. D 19. A 20. B 21. A

Determinants

LEVELI

1. Let f (x) = x(x 1), then = )4(f)3(f)2(f)3(f)2(f)1(f)2(f)1(f)0(f

is equal to

(A) 2! (B) 3! 2! (C) 0 (D) none of these

2. If f (x) = )1x(x)1x()2x)(1x(x)1x(x3

x)1x()1x(xx21xx1

, then f (100) is equal to

(A) 0 (B) 1 (C) 100 (D) 100

3. The determinant (x) = )x1(cbcac

bc)x1(babacab)x1(a

2

2

2

(abc 0) is divisible by

(A) 1 + x (B) (1 + x)2 (C) x2 (D) none of these

4. The value of the determinant pqrprqqrp

rqp111

222

is

(A) pqr (B) p + q + r (C) p + q + r pqr (D) 0

5. If a, b, c > 0 and x, y, z R, then the determinant

1cccc

1bbbb

1aaaa

2zz2zz

2yy2yy

2xx2xx

is equal

to (A) ax + by + cz (B) a-x by c-z (C) a2x b2y c2z (D) 0 6. Given a system of equations in x, y, z: x + y + z = 6; x + 2y + 3z = 10 and x + 2y + az = b. If

this system has infinite number of solutions, then (A) a = 3, b = 10 (B) a = 3, b 10 (C) a 3, b = 10 (D) a 3, b 10 7. If each element of a determinant of 3rd order with value A is multiplied by 3, then the value of

the newly formed determinant is (A) 3A (B) 9A (C) 27A (D) none of these 8. If the value of 3rd order determinant is 11, then the value of the determinant formed by the

cofactors will be (A) 11 (B) 121 (C) 1331 (D) 14641

9. If a1 + b1 + c1 = 0 such that c111

1b1111a1

= , then the value of is

(A) 0 (B) abc (C) abc (D) none of these

10. If a, b, c are real numbers, then 1cc1c1bb1b1aa1a

is

(A) 0 (B) 6 (C) 9 (D) None of these 11. Let D be the determinant of order 3 3 with the entry Ii + k in lth row and kth column

(I = 1 . Then value of D is (A) imaginary (B) Zero (C) real and positive (D) real and negative

12. The value of the determinant abcccabbbcaa

2

2

2

111

is

(A) a3+b3+c3-3abc (B) a2+b2+c2-bc-ca-ab (C) a2b2+b2c2+c2a2 (D) None of these

13. Let =

1111

nxmlx

. Then, the roots of the equation are

(A) , , (B) l, m ,n (C) +, +, + (D) l+m, m+n, n+l

14. Let = bacacbcba

; a>0 , b>0, c >0. Then,

(A) 0 (B) a+b+c = 0 (C) >0 (D) R

15. The value of =

2

2

11

111 is