# Binomial+Theorem+[Practice+Question].pdf

April 21, 2018 | Author: Aditya SanjayBoob | Category: Abstract Algebra, Number Theory, Mathematical Analysis, Numbers, Mathematical Objects

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Binomial+Theorem+[Practice+Question].pdf...

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Practice Question Question based on

LEVEL –1 Q.8

Binomial Theorem for positive integral Index

equal then (where r > 1, n > 2), positive

Q.2

 a    9 b  is 3  

integer)-

Fourth term in the expansion of  (A) 40 a 7 b3

3 7 (B) 40a  b

6 4 (C) 1890 a  b

(D) 1890a4 b6

(A) r = n/2 (C) r =

Second term in the expansion of (2x + 3y) 5 will

Q.9

be (A) 46 x2y3

(B) 30 x 3y2

(C) 240 x 4 y

(D) 810 xy4

The coefficient of (3r) th term and coefficient of (r + 2)th term in the expansion of (1 + x) 2n  are

10

Q.1

Binomial Theorem

Q.10

n 1 2

(B) r = n/3 n 1

(D) r =

2

The coefficient coefficient of a 2 b3 in (a + b) 5 is(A) 10

(B) 20

(C) 30

(D) 40

The coefficient coefficient of x7 and x8 in the expansion of n

Q.3

Q.4

  x   2   are equal, then n is equal to  3 

The 5  term of the expansion of (x – 2)  is th

8

(A) 8C5x3( –2)5

(B) 8C5x3 25

(C) 8C4x4 (–2)4

(D) 8C6x2 (–2)6

The number of terms in expansion of (x – 3x 2 +

Q.11

3x3)20 is(A) 60 Q.5

(B) 61

(C) 40

(D) 41

(C) 55

(D) None of these

The coefficient of x5  in the expansion of (A) 12C525, 37

(B) 12C626.36

(C) 12C527.35

(D) None of these

The term with coefficient coefficient 6C2  in the expansion

n

Q.12

(A) T1 and T3

(B) T2 and T4

(C) T3 and T 5

(D) None of these

1    If the expansion of  x 2   , the coefficient 4    of third term is 31, then the value of n i s-

If n is a positive integer, then r th  term in the expansion of (1–x)n is-

Q.7

(B) 45

(2 + 3x)12 is-

of (1+ x)6 is-

Q.6

(A) 35

Q.13

(A) 30

(B) 31

(C) 29

(D) 32

If A and B are coefficients coefficients of x r   and xn–r

(A) nCr (–x)r

(B) nCr xr

respectively in the expansion of (1+ x) n, then-

(C) nCr-1(–x) r–1

(D) nCr–1xr–1

(A) A = B (B) A  B

  

If the 4th term in the expansion of  ax  5 2

1 

(C) A =  , B for some 

n

 is x 

, then the values of a and n are-

(D) None of these Q.14

If (1 + by)n  = (1 + 8y +24y2  + …..) then the value of b and n are r espectivelyespectively-

(A) 1/2, 6

(B) 1, 3

(A) 4, 2

(B) 2, –4

(C) 1/2, 3

(D) can not be found

(C) 2, 4

(D) –2, 4

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1

Q.15

Q.23

The number of terms in the expansion of 9

10

 x 3     2  is  2 x  

9

(1 + 5 2 x) + (1 – 5 2 x)  is-

Q.16

(A) 5

(B) 7

(C) 9

(D) 10

(A)

The number of non zero terms in the expansion

(C)

of [(1 + 3 2 x )9 – (1 – 3 2 x)9] is -

Q.17

(A) 9

(B) 10

(C) 5

(D) 15

Q.24

(B) 5

(C) 4

(D) 3

256

(B)

450

504 259

(D) None of these

263

The coefficient of x –26  in the expansion of 11

in the expansion of (x + 2 )  + (x –  2  ) is(A) 10

405

  2 2    x  4  is x   

After simplification, the total number of terms 4

The coefficient of x 4 in the expansion of

(A) 330 × 2 6

(B) – 330 × 26

(C) 330 × 2 7

(D) – 330 × 27

4

Q.25

The term independent of x in the expansion of 10

Q.18

  x 3    will be   3 2x 2     

The number of terms in the expansion of [(x – 3y)2 (x + 3y) 2]3 is-

Q.19

(A) 6

(B) 7

(A) 3/2

(B) 5/4

(C) 8

(D) None of these

(C) 5/2

(D) None of these

The number of terms in (x + a) 100 + (x – a) 100

Q.26

8

(A) 202

(B) 51

 1   expansion of  y1/ 3  y 1/ 5   is  2  

(C) 101

(D) None of these

(A) sixth

(B) seventh

(C) fifth

(D) None of these

after solving the expansion is -

Q.20

The term independent of y in the binomial

The coefficient of x 5  in the expansion of (1+ x2)5 (1+ x)4 is (A) 30

(B) 60

(C) 40

(D) None of these

Q.27

If x4  occurs in the r th  term in the expansion of 15

  4 1    x  3  , then r equalsx     (A) 7

Q. 21

(B) 8

Q.28

(A) 6

(B) 22

(C) – 6

(D) 8

The term containing x in the expansion of 5

  2 1   x   is x    (A) 2nd

(B) 3 rd

(C) 4 th

(D) 5

th

The coefficient of x 4  in the expansion of 6

(1+ x + x 2 + x3)11 is-

Question based on

(D) 10

The coefficient of x   in the expansion of (1 + x)3.(1 – x)6 is -

Q.22

(C) 9

5

(A) 990

(B) 495

(C) 330

(D) None of these

Q.29

1     The term independent of x in  2x   is 3x    (A) 160/9

(B) 80/9

(C) 160/27

(D) 80/3

Particular term in the expansion

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2

Q.30

The term independent of x in the expansion of

8

Q.37

10

  x 3     is 2   3 2 x     (A) 10C1

(B) 5/12

(C) 1

(D) None of these

Q.38

 x a  The middle term of the expansion     is a x  (A) 56a2/x2

(B) – 56a 2/x2

(C) 70

(D) –70

The

middle

term

in

the

expansion

of

10

Q.31

  3 1    x  3  isx   

If 9  term in the expansion of (x  + x )  does th

1/3

-1/3 n

not depend on x, then n is equal to(A) 10 Q.32

The

constant

  1  x    x  (A)

Q.33

(B) 13

2n ! n!

(C) 16 term

in

(D) 18

the

expansion

of

2n

Q.39

is-

(A) 252

(B) – 252

(C) 210

(D) – 210

If the middle term in the expansion of n

n!

(B)

2n !

(C)

2n ! 2 !n !

(D)

2n !

  2 1   x    is 924 x6, then n = x  

n !n !

(A) 10

(B) 12

(C) 14

(D) None of these

The coefficient of the term independent of y in 3n

Q.40

  1     is the expansion of  y  2     y  

The greatest coefficient in the expansion of (1+ x)10 is(A)

(A) 3nCn–1 (–1)n-1

(B) 3n Cn

(C) 3n Cn(–1)n

(D) None of these (C)

Q.34

The number of integral terms in the expansion of (51/2 + 7 1/6)642 is -

Question based on

Q.35

(A) 106

(B) 108

(C) 103

(D) 109

Q.41

The

10! 5! 6!

(B)

10!

10! (5! )

2

(D) None of these

5! 7! middle

term

in

the

expansion

of

(1 – 3x + 3x  – x )  is 2

3 6

(A) 18C10x10 (B) 18C9(–x)9

Middle Term

(C) 18C9 x9 (D) – 18C10 x10

Middle term in the expansion of (x 2 – 2x)10 will  be (A) 10C4x17 24

(B) – 10C5 25 x15

(C) – 10C4 24 x17

(D) 10C5 24 x15

Question based on

Q.42 Q.36

The

middle

term

in

the

expansion

9

  3 x 3     is  x2  6      (A)

189 8

x2,

21 16

x7

189 2 21 7 (C) –  x , –  x 8 16

(B)

189

21 x2, –  x7 8 16

of

Term from end

The 5th  term from the end in the expansion of 9

 x 3 2      is  2 x3      (A) 63x3 (C)

(D) None of these

672 x18

(B) –

252 x3

(D) None of these

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3

n

Q.43

1     If in the expansion of  21 / 3  1/ 3  , the ratio 3   

Q.51

C0C1 + C 1C2 + C2C3 + ....+ Cn–1Cn is equal to-

of 6 th terms from beginning and from the end is

(A)

1/6, then the value of n is (A) 5

Q.44

(C)

(D) None of these

Binomial Coefficient

Q.52

If (1+ x) n = C 0 + C 1x + C 2x2 + C 3x3 + ....+ C nxn, (A) 2n+1

(B) 2n–1

(C) 2 + 1

(D) 2  – 1

n

If (1+ x) n = C 0 + C 1x + C 2x2 + ...+ C nxn, then the

2n !

(D)

(n  1)! n !

n 1 1 Cn C0 –  nC1 + n C2 – ......+ (–1) n = 2 3 n 1

(B) 1/n 1

n 1

(D)

1 n 1

  1  In the expansion of (1 + x) 1   , the term   x  n

independent of x is-

(A) 2n (n + 1)

(B) 2 n-1 (n + 1)

(A) C 02 + 2 C12 + .....+ (n +1) C 2n

(C) 2n-1 (n + 2)

(D) 2 n (n + 2)

(B) (C0 + C1 + …..+ C n)2 (C) C 02 + C12 + .....+ C 2n

If C0, C 1, C 2, ....., C 15 are coefficients of different

(D) None of these

(B) 2 14

(C) 2 7

Q.54

(D) 28

If (1+ x)n = C0 + C1x + C2x2  +...+ Cnxn, then C0Cr  + C1Cr +1 + C2Cr+2 + .....+ C n–r Cn is equal to-

If (1+ x)n  = 1 + C 1x + C2x2  + ....+ C nxn, then

(A)

C1 + C3 + C5 + ..... is equal to(A) 2n

(B) 2 n – 1 (C) 2n + 1

  1

n! 

 n !

1 2 ! ( n  2) !

1 4!(n  4) !

(D) 2 n–1

 ...... 

(C)

(B) 2 n–1

1! (n  1) ! (A)

2

+

1 3!( n  3)!

(C) 2n+1

+

1 5!( n  5)!

n

n!

(C) 0

( n  r )!(n  r )! 2n! n !(n  r )!

(B)

(D)

2n! n !( n  r )! 2n! (n  1) !(n  1)!

  is

n ! 

Q.55

If (1+ x)n = C0 + C1x + C2x2 + ...Cn. xn then the value of C0 + 3C 1 + 5C2 + .....+ (2n + 1) C n is-

(A) 2n 1

2n!

1  

equal to -

Q.50

(n  1)! (n  1) !

n !( n  1)!

value of C0 + 2C 1 + 3C2 + ....+(n +1)C n is -

(A) 215

Q.49

2n !

2n !

n

C0+ C2+ C4 + ...+ C 14 is equal to-

Q.48

(B)

n

terms in the expansion of (1 + x) 15, then

Q.47

n

(C)

Q.53

Q.46

n! n!

(A) n

then the value of C 1+ C2+ C3 + ...+ Cn is-

Q.45

2n !

(B) 7

(C) 9 Question based on

If (1+ x)n = C0 + C1x + C2x2+ ...+ Cnxn, then

(B)

2

(D) 2 –n+1

+ .......=

n 1

n!

(D) None of these

The value of 8C0 + 8C2 + 8C4+ 8C6+ 8C8 is(A) 32

(B) 64

(C) 128

(D) 256

Q.56

(A) n.2n

(B) (n –1). 2 n

(C) (n + 2).2 n–1

(D) (n + 1).2 n

If C0, C1, C2.......Cn are binomial coefficients in the expansion of (1 + x)n, n   N, then C1 2 (A) (C)

+

C3 4

+

2 n 1  1 n 1 2n  1 n 1

C5 6

+…..+

Cn n 1

is equal to-

(B) (n + 1) . 2 n+1 (D) None of these

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4

Q.57

If (1 + x) n = C0 + C1x + C2x2 + ....+ C nxn, then for n odd, C12 + C32 + C52 + .....+ C n2 is equal to (A) 22n-2

Q.58

(B) 2 n

(C)

( 2n)! 2( n!)

2

(D)

(2n )! ( n!)

2

If (1+ x + x 2)2n = a0+ a1x + a2x2  +....then the value of a0 – a 1 + a2 – a3 + .... is(A) 2n

Q.59

(B) 3n

(C) 1

(D) 0

The sum of the coefficients in the expansion of (a + 2b + c) 10 is (A) 410

Q.60

(B) 310

(C) 2 10

(D) 104

The sum of coefficients of even powers of x in the expansion of (1+ x + x 2 + x3)5 is (A) 512

(B) – 512

(C) 215

(D) None of these

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5

LEVEL- 2 Q.1

If (1 + x – 2x 2)6 = 1 + C 1x + C2x2 + C3x3 + ....+ C12

x 12,

Q.8

then the value of C 2+ C 4+ C 6 + ...+ C 12

is -

Q.2

Q.3

(A)

(A) 30

(B) 32

(C) 31

(D) None of these

(C) Q.9

If C0, C1, C2, ....C n are binomial coefficients of

n 1

x

(B)

n  r  1 r

x

r  n  r

(D)

r  1

x

x

If (1+ x) n = C0 + C1x + C2x2 + ...+ C nxn, then the (A) n(n + 1) 2 n–2

(B) n(n + 1) 2 n–1

equals-

(C) n(n + 1) 2 n

(D) None of these

(A) –n.2n–1

(B) 0

(C) 2n–1. (2 – n)

(D) None of these

If C r

Q.10

Q.11

a/2 n

The sum of coefficients of odd powers of x in the expansion of (1+ x) n is -

stands for nCr , then the sum of first (n+1)

(A) 2n + 1

(B) 2n – 1

(C) 2n

(D) 2n–1

The sum of C02 – C12 + C22  – ....+ (–1) n Cn2 where n is an even integer, is-

(B) na

(C) 0

(A) 2nCn

(D) None of these

(B) (–1)n

5

(A) 252

(B) 252

(C) 452

(D) 532

Q.12

(A) 0, 6

(B) 0, 2 6

(C) 1, 6

(D) 0

b   3

C2,.....Cn  denote the binomial

 (r  1) C r  is -

Q.13

(A) n 2n

(B) (n+1) 2 n–1

(C) (n +2) 2 n–1

(D) (n + 2) 2 n–2

If 1  r  n–1 then

n–1C

+

n–2C

r

+ .....+ r Cr

equalsterm in the expansion of contains

same

powers

of Q.14

(A) 9

(B) 10

(C) 8

(D) 6

(A) nCr

(B) nCr+1

(C) n+1Cr

(D) None of these

The coefficient of 1/x in the expansion of

  

(1+ x)n 1 

(A)

If n is odd, then C 02 – C12 + C 22 – C 32 + ....+ (–1)n C 2n = (B) 1

C1 ,

r  0

a and b, then the value of r is -

(A) 0

C0 ,

n

21

a  

If

value of

values of a0 and a6 are -

  a 3   b  

(D) (–1) . Cn/2

coefficients in the expansion of (1+x) n, then the

If (2x – 3x 2)6 = a 0 + a 1x + a 2x2 + ...+a 12x12, then

If the (r

n

5

The value of ( 5 + 1)  –( 5 – 1)  is -

+1) th

2nC

n/2 n

(C) 2nCn–1

Q.7

n  r  1

C0 – 2.C1+ 3.C2 – 4.C 3 + .....+ (–1) n .(n + 1)C n

(A)

Q.6

is equal to

value of 12 C1 + 22 C2 + 32 C3 + ...+ n 2 Cn is

aC0 – (a + d) C1+ (a + 2d)C2 – (a + 3d)C3 + ... is-

Q.5

Tr

different terms in the expansion of (1+ x) n then

terms of the series

Q.4

Tr 1

In the expansion of (1+ x) n,

(C) 

(D)

(C) n!

1 

n

 is -

x 

n! ( n  1)!( n  1)!

( 2n ) ! (2n  1)!(2n  1)!

(B)

(2n ) ! ( n  1)!( n  1)!

(D) None of these

2

( n / 2) !

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6

Q.15

If 6th term in the expansion of

Q.21

8

consecutive terms in the expansion of (1+ x) n,

  1    8 / 3  x 2 log10 x   is 5600, then x i s equal to x  

Q.16

(A) 8

(B) 10

(C) 9

(D) None of these

(C)

Q.22

second term, then the value of n is (B) 9

(C) 10

(D) None of these

a2  a3

is equal to-

(B)

(D)

1

a2

2 a2  a3 2a 3 a2  a3

If the coefficients of four consecutive terms in the

a1 a1  a 2

,

a2 a2  a3

,

a3 a3  a 4

are in -

series

(A) A.P.

(B) G.P.

[3.nC0 – 8.nC1 + 13. nC2 – 18. nC3 + .....+ (n +1)]

(C) H.P.

(D) None of these

The

sum

of

the

terms

of

the

Q.23

(A) 3. 2 n –5n.2n-1 (C)

3.2 n +

5n.2n-1

(B) 0

If (1+ x) n = C0 + C1x + C2x2 + .....+ C nxn, then C0C2 + C1C3 + C2C4 + ......+ C n–2 Cn =

(D) None of these (A)

If sum of all the coefficients in the expansion of (x3/2 + x-1/3) n  is 128, then the coefficient of x 5

(C)

is (A) 35

(B) 45

(C) 7

(D) None of these

Q.24

(A)

(C)

is -

 (2n )!     n!  

(B)

2n ! (n  2) ! (n  2)!

(B)

(D)

2n ! n !( n  2)! 2n ! (n  1) ! (n  2)!

2n ! ( n !) 2

(B)

(2n  2) ! n ! (n  1)!

(D)

( 2n  2)! [(n  1)!]2 (2n )! n !(n  1)!

( 2n )! ( n )! ( n )!

Q.25

2

(C) 

( n  1)!( n  2)!

(1+ x) 2n+2  is -

If (1 + x) n = C0 + C1x + C 2x2 + ....+ C n.xn then

(A) 1

2n !

The greatest coefficient in the expansion of

the value of C 0Cn + C 1Cn–1 + C2Cn–2 + ....+ C nC0

(D) (2n)2

1     The middle term in the expansion of  x     2 x  1.3.5.....(2n  3) n!

Find the value (183  73  3.18.7.25)

(C)

6

3  6.243.2 15.814 20.27.8 15.9.16 6.3.32 64 (A) 1

(B) 5

(C) 25

(D)100

2n

is(A)

Q.20

a2  a3 2a 2

a3  a4

expansion of (1+ x)n  are a 1,a2,a3  and a4, then

(A) 8

is-

Q.19

a2

If the coefficient of third term in the expansion of

a3

+

a1  a 2

(A)

n

Q.18

a1

then

1     2 / 3  x  1/ 3  is 27 more than the coefficient of x   

Q.17

If a1, a2, a3, a4 are the coefficients of any four

Q.26

(B)

1.3.5.....(2n  1) n!

1.3.5.....(2n  1) n!

(D) None of these

The term independent of x in the expansion of 4

3

  1    1   x    x   is  x    x  (A) –3

(B) 0

(C) 1

(D) 3

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7

Q.27

If the sum of the coefficients in the expansion of (2 x 2 – 2x +1)51 vanishes, then the value of 

n

Q.34

If (1 + x)  =

Q.28

   

then 1 

(B) –1

(C) 1

(D) –2

If the sum of coefficients in the binomial

(A)

expansion of (x + y)n  is 4096 then greatest coefficient in the expansion is (A) 922

(B) 942

(C) 787

(D) 924

(C)

Q.35 Q.29

The value of

(nC2 –2. nC3+3.nC4 –4.nC5  +

.....) is

equal to -

Q.30

(A) 1

(B) 0

(C) –1

(D) None of these

n

Cr .( x ) r

r  0

is (A) 2

n

n

n     C1     n C2    . 1   ..... 1  C n  = n  nC  C0    n C1   n – 1    n

n – 1

( n – 1) ! (n  1)

(B)

n

n!

(D)

( n  1)

n – 1

( n – 1) ! (n  1)

n 1

n!

The term independent of x in the expansion of 10

x 1 x  1       2 / 3 1/ 3  is  x  x  1 x  x1/ 2   (A) T4 = 180

(B) T 5 = – 210

(C) T4 = – 180

(D) T5 = 210

The sum of 12 terms of the series 12C

1.

1 3

+ 12C2 .

1 9

12

 4  (A)    – 1  3 

+ 12C3 .

1 27

+ .... is -

12

 3  (B)    – 1  4 

12

 3  (C)   + 1  4  Q.31

(D) None of these

If n = 10 then ( C 02 – C12 + C 22 – C 32 + ..... + (–1)n ( C 2n )) equals-

Q.32

(A) (–1)5.10C5

(B) 0

(C) 10C5

(D) (–1)6 10C6

If n = 11 then ( C 02 – C12 + C 22 – C 32 + ..... + (–1)n ( C 2n ) equals-

Q.33

(A) (–1)5. 10C5

(B) 0

(C) 10C6

(D) None of these

The sum of (n +1) terms of the series C02+ 3C12 + 5C22 + ..... is (A) 2n–1Cn-1

(B) 2n–1 Cn

(C) 2(n +1) 2n–1Cn

(D) None of these

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8

LEVEL- 3 Q.1

If the coefficients of T r , Tr+1, T r+2  terms of

Q.9

integral terms is -

(1+ x)14 are in A.P., then r (A) 6 Q.2

(B) 7

(C) 8

The value of x , for which the



log expansion of  2 2

( 9 x 1  7 )

(D) 9 6 th term

2

Q.10

in the

1



(1 / 5) log 2 ( 3

x 1

  1)   

(A) 4, 3

(B) 0, 3

(C) 0, 2

(D) 1, 2

Q.11

Q.4

(C) 31C6 – 21C6

(D) 30C5 + 20C5

Q.12

(x2 – x – 2) 5 is-

Q.5

(B) – 82

(C) – 81

(D) 0

Q.13

In the expansion of (1+3x+2x 2)6 the coefficient of x11 is (A) 144

(B) 288

(C) 216

(D) 576

Let R = (5

5 + 11)2n+1 and ƒ = R – [R], where [.]

(A) 42n+1

(B) 42n

(C) 42n–1

(D) 4 –2n

The greatest integer less than or equal to

(A) 196

(B) 197

(C) 198

(D) 199

The number of integral terms in the expansion of

Q.14

(A) 106

(B) 108

(C) 103

(D) 109

The remainder when 599 is divides by 13 is (A) 6

(B) 8

(C) 9

(D) 10

When 2301 is divided by 5, the least positive remainder is (A) 4

Q.6

Coefficients of of (x +

x r  [0

3) n–1

(B) 8

(C) 2

(D) 6

 r  (n–1)] in the expansion

+ (x + 3)n–2  (x + 2) +

Q.15

If the sum of the coefficients in the expansion of

(x + 3)n–3 (x + 2)2 + ........ + (x + 2) n–1 -

(1 – 3x + 10x2)n  is a and if the sum of the

(A) nCr  (3r  – 2n)

(B) nCr  (3n–r  –2n–r )

coefficients in the expansion of (1 + x 2)n is b,

(C) nCr  (3r  + 2n–r )

(D) None of these

then -

n

Q.7

(D) 131

(51/2 + 71/6)642 is -

The coefficient of x5 in the expansion of (A) – 83

(C) 130

( 2 + 1)6 is -

(1+ x)21 + (1+ x)22 + ...........+ (1+ x) 30 is (B) 9C5

(B) 129

R.ƒ is -

The coefficient of x5 in the expansion of (A) 51C5

(A) 128

denotes the greatest integer function. The value of

7

is 84 is equal to -

Q.3

In the expansion of (51/2 + 71/8)1024, the number of

If x + y = 1, then

 r 2 n Cr x r y n r   equals -

(A) a = 3b

(B) a = b3

(C) b = a3

(D) none of these

r  0

Q.8

(A) nxy

(B) nx (x+ yn)

(C) nx (nx+y)

(D) None of these.

10

Q.16

 20C k  =

k  0

(1 + x)n  –nx –1 is divisible by (where n  N)–

(A) 219 +

(A) 2x3

(C) 20C10

2

(C) x

(B) 2x

1 2

20

C10

(B) 219 (D) none of these

(D) All of these

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9

Q.17

If n   N such that (7 + 4 3 )n = I + F,

Q.22

where I   N and 0 < F < I, Then the value of

expansion of (3 + 5x) 11 when x =

(I + F) (1 – F) is (A) 0 Q.18

(C) 7 2n

(B) 1

The value of

95

C4 +

The value of numerically greatest term in the

(D) 2 2n

(A) 55  310

(B) 110  39

(C) 55  38

(D) 55  39

 100 j C 3  is -

j 1

Q.23 100

(A) C5

(B)

(C) 99C4

(D) 100C5

C4

The value of numerically greatest term in the expansion of (3x + 2) 9 when x = 3/2 (A)

(C) 7 

0  ƒ  1, then I equals (A) (B) (C) (D) Q.20

7  313

If (5 + 2 6 ) n = I + ƒ, where I  N, n  N and

1

5

5

95

Q.19

1

(B) 7 313

2 314

(D)

7  314 2

–f

 f

Question based on Statements (Q. 24 - 26)

1

Each of the questions given below consist of Statement – I

–f 1  f

and Statement – II. Use the following Key to choose the

1

–f 1  f  1 1  f

If (15Cr  +

(A) If both Statement- I and Statement- II are true, and Statement - II is the correct explanation of

+f

Statement– I.

15

Cr–1) (15C15–r  +

15

C16 –r ) = ( 16C13)2, then

the value of r is (A) r = 3

(B) r = 2

(C) r = 4

(D) none of these

(B) If both Statement - I and Statement - II are true but Statement - II is not the correct explanation of Statement – I. (C) If Statement - I is true but Statement - II is false (D) If Statement - I is false but Statement- II is true.

Passage Based Questions (Q. 21 - 23)

The numerically greatest term in the expansion of (x + a)n is given by

n 1 x 1 a

Q.24

(1 + x + x 2 + ......+ x 10 )5 is 51.

= k (say)

Statement II

to 22n–1 –

numerically greatest term (b) If k is not an integer. Let m is its integral part

Q.21

Q.25

1 5

two together) is equal

( 2n )! 2.(n !) 2

Statement

I

: The coefficient of x 5  in the

expansion of (1 + x 2)5 (1 + x)4 is 120.

The numerically greatest term in the expansion of (3 – 5x)15 when x =

: The sum of the products of

nC , nC , nC ..... nC  (taken 1 2 n 0

(a) If k is an integer then Tk   and Tk+1  are the

then Tm+1 is the numerically greatest term.

Statement I : The number of terms in

Statement II : The sum of the coefficients in the

expansion of (1 + 2x – 3y + 5z) 3 is 125.

is –

n

(A) T4

(B) T 5 & T6

(C) T4 & T5

(D) T6

Q.26

Statement I :

 K . (nCK )2  n . 2n–1Cn–1

K 1

Statement II : If 22003 is divided by 15 the

remainder is 1.

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10

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.   A

C

C

D

C

C

A

A

A

C

C

D

A

C

A

C

D

B

B

B

Ques. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

C

A

A

C

B

A

C

C

C

D

C

D

C

B

B

B

C

B

B

B

Ques. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

B

B

B

D

C

B

D

B

B

C

C

C

C

A

D

C

C

C

A

A

Ans.

Ans.

LEVEL- 2 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

C

B

C

B   B

A

A

C

A

D

D

C

B

B

B

B

B

A

B

A

Ques.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Ans.

C

A

C

B

B

B

C

D

A

A

A

B

C

C

D

LEVEL- 3 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

D

D

C

C

D

B

C

C

B

A

B

B

B

C

B

A

B

B

C

A

21

22

23

24

25

26

C

D

A

D

D

C

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11