heat-5 progression+binomial by abhijit kumar jha · pdf file · 2012-10-07heat-5...

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HEAT-5 PROGRESSION+BINOMIAL BY ABHIJIT KUMAR JHA

TIME-2HRS 30 MINTS MAX MARKS(68(3)=204)

Q.1 If a, b, c be in A.P., b, c, d in G.P. & c, d, e in H.P., then a, c, e will be in :(A) A.P. (B) G.P. (C) H.P. (D) none of these

Q.2 If a, b, c are in H.P., then a, a c, a b are in :(A) A.P. (B) G.P. (C) H.P. (D) none of these

Q.3 If three positive numbers a , b, c are in H.P. then e n a c n a b c ( ) ( ) 2 simplifies to(A) (a � c)2 (B) zero (C) ( a � c) (D) 1

Q.4 The sum 1

122 rr

is equal to :

(A) 1 (B) 3/4 (C) 4/3 (D) none

Q.5 In a potato race , 8 potatoes are placed 6 metres apart on a straight line, the first being 6 metres from thebasket which is also placed in the same line. A contestant starts from the basket and puts one potato ata time into the basket. Find the total distance he must run in order to finish the race.(A) 420 (B) 210 (C) 432 (D) none

Q.6 If the roots of the cubic x3 � px2 + qx � r = 0 are in G.P. then(A) q3 = p3r (B) p3 = q3r (C) pq = r (D) pr = q

Q.7 Along a road lies an odd number of stones placed at intervals of 10 m. These stones have to be assembledaround the middle stone. A person can carry only one stone at a time. A man carried out the job startingwith the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Thenthe number of stones is(A) 15 (B) 29 (C) 31 (D) 35

Q.8 If 2log)12.5( x

; 4log)12( x1 and 1 are in Harmonical Progression then

(A) x is a positive real (B) x is a negative real(C) x is rational which is not integral (D) x is an integer

Q.9 If a, b, c are in G.P., then the equations, ax2 + 2bx + c = 0 & dx2 + 2ex + f = 0 have a common root, if

d

a,

e

b,

f

c are in :

(A) A.P. (B) G.P. (C) H.P. (D) none

Q.10 If the sum of the roots of the quadratic equation, ax2 + bx + c = 0 is equal to sum of the squares of their

reciprocals, then a

c,

b

a,

c

b are in :

(A) A.P. (B) G.P. (C) H.P. (D) noneQ.11 If for an A.P. a1 , a2 , a3 ,.... , an ,....

a1 + a3 + a5 = � 12 and a1 a2 a3 = 8then the value of a2 + a4 + a6 equals(A) � 12 (B) � 16 (C) � 18 (D) � 21

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Q.12 Given four positive number in A.P. If 5 , 6 , 9 and 15 are added respectively to these numbers , we geta G.P. , then which of the following holds?(A) the common ratio of G.P. is 3/2(B) common ratio of G.P. is 2/3(C) common difference of the A.P. is 3/2(D) common difference of the A.P. is 2/3

Q.13 Consider an A.P. with first term 'a' and the common difference d. Let Sk denote the sum of the first K

terms. LetS

Skx

x is independent of x, then

(A) a = d/2 (B) a = d (C) a = 2d (D) none

Q.14 Concentric circles of radii 1, 2, 3......100 cms are drawn. The interior of the smallest circle is colouredred and the angular regions are coloured alternately green and red, so that no two adjacent regions areof the same colour. The total area of the green regions in sq. cm is equal to(A) 1000 (B) 5050 (C) 4950 (D) 5151

Q.15 Consider the A.P. a1 , a2 ,..... , an ,....the G.P. b1 , b2 ,....., bn ,.....

such that a1 = b1 = 1 ; a9 = b9 and 369a9

1rr

then

(A) b6 = 27 (B) b7 = 27 (C) b8 = 81 (D) b9 = 18[ Apex : Q.68 of Test - 1 Scr. 2004 ]

Q.16 For an increasing A.P. a1, a2, ...... an if a1 + a3 + a5 = � 12 : a1a3a5 = 80 then which of the following doesnot hold?(A) a1 = � 10 (B) a2 = � 1 (C) a3 = � 4 (D) a5 = 2

Q.17 Consider a decreasing G.P. : g1, g2, g3, ...... gn ....... such that g1 + g2 + g3 = 13 and 23

22

21

ggg =91

then which of the following does not hold?(A) The greatest term of the G.P. is 9. (B) 3g4 = g3(C) g1 = 1 (D) g2 = 3

Q.18 If p , q, r in H.P. and p & r be different having same sign then the roots of the equation px2 + qx + r = 0are(A) real & equal (B) real & distinct (C) irrational (D) imaginary

Q.19 The point A(x1, y1) ; B(x2, y2) and C(x3, y3) lie on the parabola y = 3x2. If x1, x2, x3 are in A.P. and y1,y2, y3 are in G.P. then the common ratio of the G.P. is

(A) 3 + 22 (B) 3 + 2 (C) 3 � 2 (D) 3 � 22

Q.20 If a, b, c are in A.P., then a2 (b + c) + b2 (c + a) + c2 (a + b) is equal to :

(A) ( )a b c 3

8(B)

2

9 (a + b + c)3 (C)

3

10(a + b + c)3 (D)

1

9(a + b + c)3




Q.21 If Sn =1

1

1 2

1 23 3 3

+...... + 1 2 3

1 2 33 3 3 3

......

......

n

n, n = 1, 2, 3,...... Then Sn is not greater than:

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

Q.22 If Sn denotes the sum of the first n terms of a G.P. , with the first term and the common ratio both positive,then(A) Sn , S2n , S3n form a G.P. (B) Sn , S2n , � Sn , S3n , �S2n form a G.P.(C) S2n � Sn , S3n � S2n , S3n � Sn form a G.P. (D) S2n�Sn , S3n�S2n , S3n�Sn form a G.P.

Q.23 toequalis................10.8.6.4.2

7.5.3.1

8.6.4.2

5.3.1

6.4.2

3.1

4.2

1

(A) 4

1(B)

3

1(C)

2

1(D) 1

Q.24 Consider an A.P. a1 , a2 , a3 ,......... such that a3 + a5 + a8 = 11 and a4 + a2 = �2, then the value ofa1 + a6 + a7 is(A) �8 (B) 5 (C) 7 (D) 9

Q.25 A circle of radius r is inscribed in a square. The mid points of sides of the square have been connected byline segment and a new square resulted. The sides of the resulting square were also connected bysegments so that a new square was obtained and so on, then the radius of the circle inscribed in the nth

square is

(A) r2 2

n1

(B) r2 2

n33

(C) r2 2

n

(D) r2 2

n35

Q.26 The sum

1kk

2k

3

2 equal to

(A) 12 (B) 8 (C) 6 (D) 4

Q.27 The sum

1n2n

2n

4

25 is equal to

(A) 1372 (B) 440 (C) 320 (D) 388

Q.28 Given am+n = A ; am�n = B as the terms of the G.P. a1 , a2 , a3 ,............. then for A 0 which of thefollowing holds?

(A) ABam (B) n2 nnn BAa

(C) nm

mnnmm

1m

2

B

Aaa

(D)

nm

nnmm

1n

22

B

Aaa

Q.29 The sum of the infinite series, 12 2

5

3

5

4

5

5

5

6

5

2 2

2

2

3

2

4

2

5 +........ is :

(A) 1

2(B)

25

24(C)

25

54(D)

125

252




Q.30 Given that the term of the expansion (x1/3 x1/2)15 which does not contain x is 5 m where m N ,then m =(A) 1100 (B) 1010 (C) 1001 (D) none

Q.31 In the binomial (21/3 + 31/3)n, if the ratio of the seventh term from the beginning of the expansion tothe seventh term from its end is 1/6 , then n =(A) 6 (B) 9 (C) 12 (D) 15

Q.32 If the coefficients of x7 & x8 in the expansion of 23

xn

are equal , then the value of n is :

(A) 15 (B) 45 (C) 55 (D) 56

Q.33 The coefficient of x49 in the expansion of (x � 1)

2

1x

22

1x .....

492

1x is equal to

(A) � 2

502

11 (B) + ve coefficient of x

(C) � ve coefficient of x (D) � 2

492

11

Q.34 The last digit of (3P + 2) is :(A) 1 (B) 2 (C) 4 (D) 5where P = 34n and n N

Q.35 The sum of the binomial coefficients of 21

xx

n

is equal to 256 . The constant term in the

expansion is(A) 1120 (B) 2110 (C) 1210 (D) none




Q.36 The coefficient of x4 in x

x2

32

10

is :

(A) 405

256(B)

504

259(C)

450

263 (D)

405

512

Q.37 The remainder, when (1523 + 2323) is divided by 19, is(A) 4 (B) 15 (C) 0 (D) 18

Q.38 Let n)347( = p + when n and p are positive integers and (0, 1) then (1 � ) (p + ) is

(A) rational which is not an integer (B) a prime(C) a composite (D) none of these

Q.39 If (11)27 + (21)27 when divided by 16 leaves the remainder(A) 0 (B) 1 (C) 2 (D) 14

Q.40 Last three digits of the number N = 7100 � 3100 are(A) 100 (B) 300 (C) 500 (D) 000

Q.41 The last two digits of the number 3400 are :(A) 81 (B) 43 (C) 29 (D) 01

Q.42 If (1 + x + x²)25 = a0 + a1x + a2x² + ..... + a50 .

x50 then a0 + a2 + a4 + ..... + a50 is :(A) even (B) odd & of the form 3n(C) odd & of the form (3n 1) (D) odd & of the form (3n + 1)

Q.43 The sum of the series (1² + 1).1! + (2² + 1).2! + (3² + 1). 3! + ..... + (n² + 1). n! is :(A) (n + 1). (n+2)! (B) n.(n+1)! (C) (n + 1). (n+1)! (D) none of these

Q.44 Let Pm stand for nPm . Then the expression 1 . P1 + 2 . P2 + 3 . P3 + ..... + n . Pn =

(A) (n + 1) ! 1 (B) (n + 1) ! + 1 (C) (n + 1) ! (D) none of these

Q.45 The expression 1

4 1

1 4 1

2

1 4 1

2

7 7

x

x x

is a polynomial in x of degree

(A) 7 (B) 5 (C) 4 (D) 3

Q.46 If the second term of the expansion aa

a

n

1 13

1

/

is 14a5/2 then the value of n

n

C

C3

2

is :

(A) 4 (B) 3 (C) 12 (D) 6

Q.47 If (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + x3 + ...... + xn)

a0 + a1x + a2x2 + a3x

3 + ...... + amxm then a rr

m

0 has the value equal to

(A) n! (B) (n + 1) ! (C) (n � 1)! (D) none




Q.48 The value of 4 {nC1 + 4 . nC2 + 42 . nC3 + ...... + 4n 1} is :(A) 0 (B) 5n + 1 (C) 5n (D) 5n 1

Q.49 If n be a positive integer such that n 3, then the value of the sum to n terms of the series

1 . n n1

1! (n 1) +

n n 1 2

2 ! (n 2) �

n n n 1 2 3

3! (n 3) + ...... is :

(A) 0 (B) 1 (C) � 1 (D) none of these

Q.50 In the expansion of (1 + x)43 if the coefficients of the (2r + 1)th and the (r + 2)th terms are equal, thevalue of r is :(A) 12 (B) 13 (C) 14 (D) 15

Q.51 The positive value of a so that the coefficient of x5 is equal to that of x15 in the expansion of xa

x2

3

10

is

(A) 1

2 3(B)

1

3(C) 1 (D) 2 3

Q.52 In the expansion of x

x x

x

x x

1

1

12 3 1 3 1 2

10

/ / / , the term which does not contain x is :

(A) 10C0 (B) 10C7 (C) 10C4 (D) none

Q.53 If the 6th term in the expansion of the binomial 18 3

210

8

xx x

/log

is 5600, then x equals to

(A) 5 (B) 8 (C) 10 (D) 100

Q.54 Co-efficient of t in the expansion of,( + p)m 1 + ( + p)m 2 ( + q) + ( + p)m 3 ( + q)2 + ...... ( + q)m 1

where q and p q is :

(A) m

tt tC p q

p q

(B)

mt

m t m tC p q

p q

(C) m

tt tC p q

p q

(D)

mt

m t m tC p q

p q

Q.55 (1 + x) (1 + x + x2) (1 + x + x2 + x3) ...... (1 + x + x2 + ...... + x100) when written in the ascending powerof x then the highest exponent of x is ______ .(A) 4950 (B) 5050 (C) 5150 (D) none

Q.56 Let 5 2 6n = p + f where n N and p N and 0 < f < 1 then the value of, f2 f + pf p is

(A) a natural number (B) a negative integer(C) a prime number (D) are irrational number




Q.57 Number of rational terms in the expansion of 2 34 100 is :

(A) 25 (B) 26 (C) 27 (D) 28

Q.58 The greatest value of the term independent of x in the expansion of 10

x

cossinx

is

(A) 10C5 (B) 25 (C) 25 · 10C5 (D) 55

10

2

C

Q.59 If (1 + x � 3x2)2145 = a0 + a1x + a2x2 + ......... then a0 � a1 + a2 � a3 + ..... ends with

(A) 1 (B) 3 (C) 7 (D) 9

Q.60 Coefficient of x6 in the binomial expansion

92

x2

3

3

x4

is

(A) 2438 (B) 2688 (C) 2868 (D) none

Q.61 The term independent of ' x ' in the expansion of 91

3

18

xx

, x > 0 , is times the corresponding

binomial co-efficient . Then ' ' is :

(A) 3 (B) 1

3(C)

1

3(D) 1

Q.62 The expression [x + (x31)1/2]5 + [x (x31)1/2]5 is a polynomial of degree :(A) 5 (B) 6 (C) 7 (D) 8

Q.63 Given (1 � 2x + 5x2 � 10x3) (1 + x)n = 1 + a1x + a2x2 + .... and that 2

1a = 2a2 then the value of n is

(A) 6 (B) 2 (C) 5 (D) 3

Q.64 The sum of the series aC0 + (a + b)C1 + (a + 2b)C2 + ..... + (a + nb)Cn iswhere Cr's denotes combinatorial coefficient in the expansion of (1 + x)n, n N(A) (a + 2nb)2n (B) (2a + nb)2n (C) (a +nb)2n � 1 (D) (2a + nb)2n � 1

Q.65 The coefficient of the middle term in the binomial expansion in powers of x of (1 + x)4 and of(1 � x)6 is the same if equals

(A) � 3

5(B)

3

10(C) �

10

3(D)

5

3

Q.66 (2n + 1) (2n + 3) (2n + 5) ....... (4n 1) is equal to :

(A) ( ) !

. ( ) ! ( ) !

4

2 2 2

n

n nn (B) ( ) ! !

. ( ) ! ( ) !

4

2 2 2

n n

n nn (C) ( ) ! !

( ) ! ( ) !

4

2 2

n n

n n (D)

( ) ! !

! ( ) !

4

2 2

n n

nn




Q.67 If Sn =

n

0r rn C

1 and Tn =

n

0r rn C

r then

n

n

S

T is equal to

(A) 2

n(B) 1

2

n (C) n � 1 (D)

2

1n2

Q.68 The coefficient of xr (0 r n 1) in the expression :(x + 2)n1 + (x + 2)n2. (x + 1) + (x + 2)n3 . (x + 1)² + ...... + (x + 1)n1 is :(A) nCr (2

r 1) (B) nCr (2nr 1) (C) nCr (2

r + 1) (D) nCr (2nr + 1)


heat-5 progression+binomial by abhijit kumar jha · pdf file · 2012-10-07heat-5...

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