ANDHERI/ VILE PARLE / DADAR / CHEMBUR / THANE / CHURCHGATE / NERUL / POWAI
MAX.MARKS: 240
PAPER - II SUB.:MATHEMATICS
MAX. TIME: 03 HR.
SECTION – I Only one correct answer type Instructions : In this section (Total Marks : 60), for each question you will be awarded 3 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other cases, minus one (-1) mark will be awarded.
1.
Infinite number of triangles are formed as shown in figure. If total area of these triangles is A then 8A is equal to (A) 3 (B) 4 (C) 1 (D) 2
y
3 1 O
1 9
1 3
2
x 1 27
..........
2.
If roots of the cubic 64x3 – 144x2 + 92x – 15 = 0 are in arithmetic progression, then the difference between the largest and smallest root is equal to (A) 2 (B) 1 (C) 1/2 (d) 1/4
3.
Let a, b and c be three distinct real roots of the cubic x3 + 2x2 – 4x – 4 = 0. If the equation x3 + qx2 + rx + s = 0 has roots 1/a , 1/b and 1/c, then the value of (q + r + s) is equal to (A) 3/4 (B) 1/2 (C) 1/4 (D) 1/6
4.
A sequence is such that the sum of its any number of terms, beginning from the first, is four times as large as the square of the number of terms. If the nth term of such a sequence is 996, then value of n is equal to (A) 100 (B) 112 (C) 125 (D) 132
5.
The orthocentre of the triangle formed by the lines x – 7y + 6 = 0, 2x – 5y – 6 = 0 and 7x + y – 8 = 0 is (A) (8, 2) (B) (0, 0) (C) (1, 1) (D) (2, 8)
6.
1 (1 – x ) (1 – x ) 2 – For x > 0, the sum of the series – ............ is equal to 1 x (1 x ) 2 (1 x )3 (A) 1/4
(B) 1/2
(C) 3/4 Page [1]
(D) 1
7.
8.
If P( + 1, – 3) be any point in xy plane, then the number of integral values of for which the point P lies between the lines x + 2y = 1 and 2x + 4y = 14, is (A) 0 (B) 1 (C) 2 (D) 3 A line with positive rational slope, passes through the point M(6, 0) and is at a distance of 5 from N(1, 3). The slope of line equals (A) 15/8 (B) 8/15 (C) 5/8 (D) 8/5 n
9.
If P =
(tan(3n+1)
– tan) and Q =
r 0
10.
11.
(A) P = 2Q (B) P = 3Q (C) 2P = Q (D) 3P = Q The equations of perpendicular bisectors of two sides AB and AC of a triangle ABC are x + y + 1 = 0 and x – y + 1 = 0 respectively. If circumradius of ABC is 2 units, then the locus of vertex A is (A) x2 + y2 + 2x – 3 = 0 (B) x2 + y2 + 2x + 3 = 0 (C) x2 + y2 – 2x + 3 = 0 (D) x2 + y2 – 2x – 3 = 0 If the points of intersection of lines L1 : y – m1x – k = 0 and L2 : y – m2x – k = 0 (m1 m2) lies inside a triangle formed by the lines 2x + 3y = 1, x + 2y = 3 and 5x – 6y – 1 = 0, then true set of values of k are 1 3 (A) , 3 2
12.
13. 14. 15.
16.
1 (B) , 1 2
3 (C) 0, 2
4
(B)
5 3
1
(C)
5 3
Locus of a point P(x, y) satisfying the equation is (A) a finite line segment (C) part of a circle with finite radius
18.
The value of
n n 3
5
2 5 3
(D) None of these
x 2 y 2 24 y 144 13 x 2 y 2 10 x 25 ,
(B) an infinite ray (D) a pair of straight lines
1 is equal to 5n 3 4n
1 1 1 1 (B) (C) (D) 120 96 24 144 A non constant arithmetic progression has common differene d and first term is (1 – ad). If the sum of the first 20 terms is 20, then the value of a is equal to (A) 2/19 (B) 2/9 (C) 10 (D) 19/2
(A) 19.
–3 ,0 (D) 2
Let all the points on the curve x2 + y2 – 10x = 0 are reflected about the line y = x + 3. The locus of the reflected points is in the form x2 + y2 + gx + fy + c = 0. The value of (g + f + c) is equal to (A) 28 (B) –28 (C) 38 (D) –38 2 2 The minimum value of expression 25 sec + 64 cosec is equal to (A) 144 (B) 169 (C) 189 (D) 196 If the lines y = x & x + y = 0 are tangents to a variable circle, then locus of centre of the circle, is (A) x + y = 1 (B) xy = 0 (C) xy = 1 (D) 2x + 3y = 5 2 2 Let a, b, c R and satisfying (a + c) + 4b – 4ab – 4bc = 0 then the variable line ax + by + c = 0 passes through a fixed point whose co-ordinates are (A) (1, –2) (B) (–1, 2) (C) (1, 2) (D) (–1, –2) P is a point on the line y + 2x = 1 and Q and R are two points on the line 3y + 6x = 6 such that the triangle PQR is an equilatral triangle. Area of this triangle is equal to (A)
17.
sin( 3r) , then r 1 )
cos(3
Page [2]
20.
If the equation ax2 – 6xy + y2 + 2gx + 2fy + c = 0 represents a pair of lines whose slopes are m and m2, then sum of all values of a is (A) 17 (B) –19 (C) 19 (D) –17
SECTION – II One or more than one correct answer type Instructions : In this section (Total Marks : 52), for each question you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) and zero marks otherwise. There are no negative marks in this section.
21.
Let f(x) = x2 + ax + b and g(x) = x2 + cx + d be two quadratic polynomials with real coefficients and satisfy ac = 2(b + d). Then which of the following is (are) correct? (A) Exactly one of either f(x) = 0 or g(x) = 0 must have real roots. (B) Atleast one of either f(x) = 0 or g(x) = 0 must have real roots. (C) Both f(x) = 0 and g(x) = 0 must have real roots. (D) Both f(x) = 0 and g(x) = 0 must have imaginary roots.
22.
The sum of three positive numbers , , is equal to
. If ecot , ecot and ecot form a G.P., then 2
which of the following hold(s) good ? (A)
cot cot
(C) cot cot = 3 23.
(B) 2 cot = cot + cot (D) 4 cos sin 2
Which of the following statement(s) is(are) correct? (A) The smallest natural number n so that (2 – n) x2 – n < 8x + 4 x R is equal to 5 2 2 (B) Let x1, x2 be two positive real numbers, then the minimum value of 162 x 2 x12 is 2. x 1 2 x2
2 2 3 (C) Number of integer satisfying the inequality ( x 4x 5) ( x – 1) (x 1) 0 is equal to 3. x 4 (x – 2)
24. 25.
26.
27.
(D) Let sequence < un > be an arithmetic sequence. If vn = un2 and wn = vn+1 – vn, then the sequence < wn > is an A.P. Equation of straight lines parallel to the line 4y = 3x + 24 and at a distance of 4 units from it are (A) 4y = 3x + 4 (B) 4y = 3x – 4 (C) 4y = 3x + 8 (D) 4y = 3x + 44 2 The value(s) of t for which the lines 2x + 3y = 5, t x + t y – 6 = 0 and 3x – 2y – 1 = 0 are concurrent, can be (A) t = 2 (B) t = – 3 (C) t = – 2 (D) t = 3 If one of the lines given by the equation ax2 + 6xy + by2 = 0 bisects the angle between the coordinate axes then value of (a + b) can be (A) –6 (B) 3 (C) 6 (D) 12 If the sum of first 100 terms of an arithmetic progression is –1 and the sum of the terms at even positions in first 100 terms is 1, then which of the following statement(s) is(are) correct ? (A) Common difference of the arithmetic progression is –3/50. (B) First term of the arithmetic progression is 149/50. (C) 100th term of the arithmetic progression is 74/25. (D) Sum of an infinite geometric progression whose first term is 47/25 and common ratio is common difference of arithmetic sequence, equals 2. Page [3]
28.
29.
An arithmetic progression has the following property : For an even number of terms, the ratio of the sum of first half of the terms to the sum of second half is always equal to a constant 'k'. Let the first term of arithmetic progression is 1,then which of the following statement(s) is(are) correct ? (A) Absolute difference of all possible values of k is 4/3 (B) The sum of all possible values of k is 4/3 (C) If the number of terms of A.P. is 20, then the sum of all terms of all possible APs is 420. (D) The number of possible non-zero values of common difference of the A.P. is 1. A variable line through M (1, k) encloses a triangle with co-ordinate axes whose area is 4. If four such distinct lines exists then k can be (A) –2 (B) 1 (C) 0 (D) –1 n
30.
Let fn(x) =
(tan nx sec 2nx ) then which of the following is(are) correct ? n 1
(A) f 2 2 3 – 1 12
(B) f 3 – 2 8
(C) f 4 0 3
5 – 4 (D) f 5 3 6
31.
32.
If 9th, 13th and 15th terms of an arithmetical progression are the first three terms of a geometric series whose sum of infinite terms is 80, then which of the following hold(s) good ? (A) Sum of the first 16 terms of the geometric progression is 860. (B) First term of the arithmetic progression is 80. (C) First term of the geometric progression is 40. (D) If d is the common difference of arithmetic progression and r is the common ratio of geometric progression then d r 5 / 2 . Which of the following statement(s) is(are) correct ? (A) Sum of the reciprocal of all the n harmonic means inserted between a and b is equal to n times the harmonic mean between two given numbers a and b. (B) Sum of the cubes of first n natural number is equal to square of the sum of the first n natural numbers. 2n
(C) If a, A1, A2, A3, ............ A2n, b are A.P. then
A
i
n (a b).
i 1
(D) If the sum of the infinite geometric progression g1, g2, g3 ....... is unity, then the value of the common ratio of the progression such that (4g2 + 5g3) is minimum equals 2/5 . 33.
If it is possible to draw a line which belongs to the following given family of lines (y – 2x + 1) + 1 (2y – x – 1) = 0, (3y – x – 6) + 2 (y – 3x + 6) = 0, (x + y – 2) + 3 (6x + y – ) = 0, then the possible values of can be (A) 4 (B) –3 (C) 1 (D) 0
Page [4]
SECTION – III Linked Comprehension Type Instructions : In this section (Total Marks : 24), for each question you will be awarded 3 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other cases, minus one (-1) mark will be awarded.
Comprehension-I In a ABC , vertex A is at (14,12). Equation of median through B is y = 6x – 30 and equation of angle-bisector through C is 2x + 3y = 36. 34. The equation of the side BC is : (A) 5x + 12y = 25 (B) 5x – 12y = 25
(C) 12x + 5y = 60
(D) 12x – 5y = 60
35. The equation of angle-bisector through B is : (A) y = 8x – 40 (B) y = 8x + 40 (C) 8x + y = 40 (D) 8x – y = 40 36. The abscissa through centroid is : (A) y = 8 (B) y + 8 = 0
(C) 3x – 19 = 0 (D) 3x + 19 = 0
Comprehension-II In a ABC, the equations of altitudes AP and BQ are 5x – 12y + 74 = 0 and x = 5 respectively. P, Q and R are feet of perpendiculars on the sides BC , CA and AB respectively. If the equation of side AB be 4x – 3y = 20, then answer the following questions : 37. The equation of the third altitude CR is : (A) 3x – 4y + 18 = 0 (B) 3x + 4y = 48 (C) 4x + 3y = 179/4
(D) 4x – 3y = –19/4
38. The altitude BQ cuts the circumcircle of ABC at : (A) (5 , 63/4) (B) (5 , 33/4) (C) (5 , 24)
(D) (5 , 93/4)
39. The orthocenter of PQR is : (A) (8 , 6) (B) (6 , 8)
(D) (3 , 4)
(C) (4 , 3)
Comprehension-III If l, m, n are three consecutive odd integers then the family of lines lx – my + n = 0 passes through a fixed point M(a, b). 40.
A variable line passes through M(a, b) and meets the co-ordinate axes at A and B, then locus of mid point of AB, is equal to (A) 2x – y – 2xy = 0 (B) 2x – y + 2xy = 0 (C) 2x + y – 2xy = 0 (D) 2x + y + 2xy = 0
41.
Number of straight lines passes through M(a, b) and making an area of 2 square unit with coordinate axes in the first quadrant, is equal to (A) 1 (B) 0 (C) 2 (D) 4
SECTION – III Single / Double Digit Answer Type Instructions : In this section (Total Marks : 80), for each question you will be awarded 4 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks otherwise. There are no negative marks in this section.
42.
A cricketer has to score 4500 runs. Let an denotes the number of runs he scores in the nth match. If a1 = a2 = ................ = a10 = 150 and a10, a11, a12, ............... are in A.P. with common difference –2, then find the total number of matches played by him to score 4500 runs. Page [5]
43.
Let Sk, k = 1, 2, 3, ....... denote the sum of infinite geometric series whose first term is (k2 – 1) and the common ratio is
44.
45.
46.
47.
1 . Find the value of k
Sk
2
.
k –1
k 1
The sum of the first 24 terms of a G.P. is 2 and the sum of the reciprocals of the first 24 terms of that G.P. in unity. If the product of the first 24 terms of the series is 2n , where n N then find the value of n. If 2a2 – 7ab – ac + 3b2 – 2bc – c2 = 0 then the family of lines ax + by + c = 0 are either concurrent at the point P(x1, y1) or at the point Q(x2, y1) or at the point Q(x2, y2). Find the distance of the origin from the line passing through the points P and Q, the distance being measured parallel to the line 3x – 4y = 2. A variable line passes through P(2, 3) and cuts the co-ordinates axes at A and B. If the parallelogram OACB (where O is the origin) is completed then find number of ordered pairs (x, y) of integers which lie on the locus of point C. Let the point M(2, 1) be shifted through a distance 3 2 units measured parallel to the line L : x + y – 1 = 0 in the direction of decreasing ordinates, to reach at N. If the image of N in the line L is R, then find the distance of R from the line 3x – 4y + 25 = 0. 1099
48.
Let a n be an arithmetic sequence. If
a
1099 100
2r
10
r 1
and
a
2 r –1
1099 , then find the common
r 1
difference of arithmetic sequence. 49. 50. 51.
52.
Let a and b be positive integers. The value of xyz is 55 or 343/55 according as a, x, y, z, b are in arithmetic progression or harmonic progression. Find the value of (a2 + b2). If 5a + 4b + 20c = t, then find the value of t for which the line ax + by + c –1 = 0 always passes through a fixed point. Two parallel lines l1 and l2 having non-zero slope, are passing through the points (0, 1) and (–1, 0) respectively. Two other lines l3 and l4 are drawn through (0, 0) and (1, 0), which are perpendicular to l1 and l2 respectively. The two sets of lines intersect in four points which are vertices of a square. If the area of this square can be expressed in the form p/q where p, q N, then the least value of (p + q). If three non-zero distinct real numbers form an arithmetic progression and the squares of these numbers taken in the same order constitute a geometric progression, then find the sum of all possible common ratios of the geometric progression.
53.
r
s
p 1 1 0 if r s If the value of rs where rs and p & q are coprime natural q 2 3 r 0 s 0 1 if r s
numbers, then find the value of p 2 q 2 .
p m2 n If 3m (n 3m m 3n ) 2q where p and q are coprime natural numbers, then find the m 1 n 1
54.
value of p q . 55.
Let a, b, c, d be four distinct real numbers in A.P. Find the smallest positive value of k satisfying
56.
2(a – b) + k (b – c)2 + (c – a)3 = 2(a – d) + (b – d)2 + (c – d)3. For a, b > 0, let 5a – b, 2a + b, a + 2b be in A.P. and (b + 1)2, ab + 1, (a – 1)2 are in G.P. then compute (a–1 + b–1). Page [6]
3 4 5 6 .............. . Compute the value of S–1. 23 2 4 ·3 2 6 ·3 27 ·5
57.
Let S denote sum of the series
58.
59.
If the varibale line 3x – 4y + k = 0 lies between the circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 16x – 2y + 61 = 0 without intersecting or touching either circle, then the range of k is (a, b) where a, b I. Find the value of (b – a). The coefficient of the quadratic equation ax2 + (a + d)x + (a + 2d) = 0 are consecutive terms of a d positively valued, increasing arithmetic sequences. Determine the least integral value of such a that the equation has real solutions.
60.
....... 9 If the sum of the series 0.9 99 999 9999 ....... 999 can be expressed in the string of 101 digits all equal to 9
61.
form of (10a – b) where a, b N, then evaluate a – b. The first term of a geometric progression is equal to b – 2, the third term is b + 6, and the arithmetic mean of the first and third term to the second term is in the ratio 5 : 3. Find the positive integral value of b.
SECTION – IV Matrix Match Type Instructions : This section contains 3 questions. For each part of this question you will be awarded 2 marks ONLY if you have darken ALL the bubble(s) corresponding to the correct answer(s) and zero marks otherwise. Thus this section carries a maximum of 24 Marks. There are no negative marks.
62.
Column-I
Column-II
(A) Minimum value of y = sec2x + 4cos2x for all permissible real values of x, is equal to (B) Consider an arithmetic sequence of positive integers. If the sum of the first ten terms is equal to the 58th term, then the least possible value of the first term is equal to (C) The number of distinct lines representing the altitudes, medians and internal angle bisectors of a triangle which is isosceles but not equilateral, is equal to (D) The equations y = mx + 3 and y = (2m – 1) x + 4 are satisfied by atleast one pair of real numbers (x, y), then m can be equal to 63.
(P) 3 (Q) 4
(R) 5
(S) 7 (T) 9
Column-I Column-II (A) In an A.P. the series containing 99 terms, the sum of all the (P) odd numbered terms is 2550. The sum of all the 99 terms of the A.P. is (B) f is a function for which f(1) = 1 and f(n) = n + f(n – 1) for each (Q) natural number n 2. The value of f(100) is (C) Suppose, f(n) = log2(3) · log3(4) · log4(5) .........logn – 1(n) (R)
5010 5049 5050
100
then the sum
f (2
k
) equals
k 2
(D) Concentric circles of radii 1, 2, 3.......100 cms are drawn. The interior of the smallest circle is coloured red and the outer regions (starting from adjacent) are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green region in Page [7]
(S)
5100
sq. cm is k then 'k' equals 64.
(T)
Column-I
Column-II
(A) If first term of an infinite geometric sequence is the value of d (sin2 2x) when x = and common ratio is the minimum value of dx 8 P(x) = x2 – x + 1,then the sum of geometric sequence is equal to (B) Line L is perpendicular to the line x + 3y – 3 = 0 and it passes through (1, 7), then y-intercept of L is equal to (C) If the straight line y = 6x intersect the curve y = x2 + a at two distinct points then the possible value(s) of a can be (D) The family of lines cx – y = 3 + 2c, where c is a parameter are concurrent at a point P, whose distance from the line 3x + 4y – 9 = 0 is equal to ANSWER-KEY
1. A 8. B 15. A 22. ABCD 29. BD 36. A
2. 9. 16. 23. 30. 37.
B A A AD A B
43. 49. 56. 62. 63. 64.
44. 50. 57. Q Q R
12, 2n 212 4096 20 51. 6 2 58. 6 (B) Q (C) (B) R (C) (B) P (C)
14 50 6 (A) (A) (A)
4950
3. 10. 17. 24. 31. 38.
C A A AD BCD A
4. 11. 18. 25. 32. 39.
C A B AB BC B
5. 12. 19. 26. 33. 40.
45. 52. 59. S Q PQR
1 6 7 (D) (D) (D)
46. 8 53. 11 60. 1 Q,R,S,T R T
***
Page [8]
C C D AC AB C
(P)
4
(Q)
6
(R)
8
(S)
9
(T)
3
6. 13. 20. 27. 34. 41.
B B B AD C C
7. B 14. B 21. B 28. ABCD 35. A 42. 34
47. 54. 61.
10 7 3
48. 55.
9 1
