Binomial Theorem

1 BT LEVEL−I 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 3 (A) 56   4 3 (C) 56   4

4.

4

4 (B) 56   3

5

2 (D) 56   5

6.

 x2 x3  1n x n The coefficient of x in  1  x    ....   2! 3! n! 

 nn

(B)

n! 1

2

  is  

 2 n

(D) –

n!2

n! 1

n!2

1  The sum of coefficients of even powers of x in the expansion of  x   x  11 11 (A) 11  11C5 (B)  C6 2 (C) 11 11 C 5  11 C 6 (D) 0



9.

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

(A)

7.

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 st nd (C) 31 and 32 terms (D) 28th and 29th terms 5

5.

5

11

is



1  1  The number of irrational terms in the expansion of  5 8  2 6      (A) 97 (B) 98 (C) 96 (D) 99

100

is equal to;

In the expansion of (1 + ax)n, n  N, then the coefficient of x and x2 are 8 and 24 respectively. Then (A) a = 2, n = 4 (B) a = 4, n = 2

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 (n + 5) (C) 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 th th (C) 5 and 6 (D) 6th and 7th

14.

The expression n C 0  4.n C1  4 2 C 2  ..........4 n C n , equals

n

(A) 2 2n (B) 2 3n 15.

n

(C) 5 n (D) None of these

2 60 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 (1  x  3 x 2 ) 2163 is (A) 1 (B) –1 (C) 0 (D) None of these

17.

The value of 1 

18.

1   The sum of the rational terms in the expansion of  2  3 5     

n

n n   C  C   1  n 2  ………. 1  n n  is equal to C1  C n1     n n 1 (n  1) n 1 (n  1) n1 (n  1) n (A) (B) (C) (D) (n  1)! (n  1)! n! n!



C1 n C0

10

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 nCr–1 + nCr–2 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 =

23.

Let R = 5 5  11 = ………………

24.

23n –7n –1 is divisible by …………





2n1

1 is …………… 5

and f = R –[R] where [.] denotes the greatest integer function, then Rf

3 25.

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

26.

2  x If the rth term in the expansion of   2  3 x 

10

contains x4, then r is equal to …………

27.

1.nC1 + 2.nC2 + 3.nC3 + ……..+ n.nCn is equal to nn  1 n (A) .2 (B) 2n+1 –3 4 (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 2 C 0  2 2

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 (C) a= 2

32.

C1 C C C  2 3 2  2 4 3  ...........  2n 1 n is equal to ……… 2 3 4 n 1

(B) n= 5 (D) n =4

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

1   ax  2  , then a and b are connected by the relation bx   (A) a = 1/b (B) a = 2/b (C) ab = 1 (D) ab = 2

LEVEL−II 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 1.3.5.  ( 2n  1).2 n 1.3.5.  ( 2n  1).2 n (A) (B) n! n! n ! 1.3.5.  ( 2n  1) 1.3.5.  ( 2n  1) (C) (D) n! n! n ! 8

3.

 1  If 6 term in the expansion of  8 / 3  x 2 log10 x  is 5600, then x is equal to x  (A) 5 (B) 4 (C) 8 (D) 10 th

4

4.

If coefficient of x2 y3 z4 in (x + y +z)n is A, then coefficient of x4y4z is nA (A) 2A (B) 2 A (C) (D) none of these 2

5.

The coefficient of x6 in {(1 + x)6 + (1 + x)7 + ……….+ (1 + x)15} is (A) 16C9 (B) 16C5 – 6C5 16 (C) C6 –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 9 (C) 2 (D) none of these

7.

The remainder of 7103 when divided by 25 is……………

8.

2  The term independent of x in the expansion of  1  2x   is………… x 

3

9.

10.

1  1  The number of irrational terms in the expansion of  2 2  3 10      (A) 47 (B) 56 (C) 50 (D) 48

55

is;

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

1    ax  2  , then a and b are connected by the relation bx   (A) a= 1/b (B) a =2/b (C) ab= 1 (D) ab=2 20

11.

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

a x r

r

then a2 is equal to;

r 0

(A) 210 (C) 220 12.

(B) 620 (D) none of these

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

n  1n

(B)

n!

n  1n1

(D)

n! n

13.

Value of

 r 0

(A) 2n (C) 2–n + 1

n

C r  sin 2

n  1n1 n  1! n  1n n  1!

r , is equal to; 2

(B) 2n –1 (D) 2n –1 –1

Pn 1 is equal Pn

5

n

14.

If a  b  1 , then



n

Cr ar bn  r equals

r 0

(B) n

(A) 1

15.

(C) na

 3 2n   , n  N is  8 

If { x } denotes the fractional part of x , then  (A) 3/8 (B) 7/8 (C) 1/8

16.

(D) nb

(D) None of these.

The coefficient of x m in : (1  x) m  (1  x) m1  .......(1  x) n , m  n is (A)

n 1

C m 1

(B)

n 1

(C) n C m

C m 1

(D) n C m 1

5

17.

5

1 1     The expansion  x  x 3  1 2    x  x 3  1 2  is a polynomial of degree ……    









n

18.

19.

1   In the expansion of  x 3  2  , n  N, if the sum of the coefficients of x5 and x10 is 0 then n x   is (A) 25 (B) 20 (C) 15 (D) none of these

The sum

1 10 C0 – 10C1 + 2. 10C2 –22 . 10C3 + ……+ 29. 10C10 is equal to 2

1 2 1 (C) .310 2 (A)

20.

(B) 0 (D) none of these

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

n

are 135, 30 and 10/3

LEVEL−III

100

1.

The co-efficient of x53 in the expansion



100

C m ( x  3)100 m 2 m is

m 0

(A) 100C53 (C) 65C53 2.

(B) - 100C53 (D) 100C65

If n is an even natural number and coefficient of xr in the expansion of then (A) r  n/2 (C) r 

n2 2

(B) r 

n2 2

(D) r  n

1  x n 1 x

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

6

n1 2

3.

Let n be an odd natural number and A =



1 n

n

. Then value of

Cr (B) n( A+1)

r 1

(A) n( A-1) nA (C) 2 4.

(B)

2n 1  1 for odd values of n only n!

(D) none of these 300 300 is ………

5.

The greater of two numbers 300! and

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

 r n   C r  r C m  is equal to;  r 1  m  0 

 

(A) 2n –1 (C) 3n –2n

(B) 3n -1 (D) none of these n

Value of

r 

n

Cr



2

is equal to

r 0

n  2n C n 2 2 2n n  Cn (C) 2

(A) n . 2nCn

(B)

(C) n2 . 2nCn n

If



r n

r 1

Cr

n

= , then value of

 r 0

1 n

Cr

n 2 n (C) 2 (A)

is equal to; (B)

2 n

(D) none of these n

10.

Value of



n

C r cos rx  sinn  r x is;

r 0

(A)2n –1 sin nx (C) 2n cos nx 11.

r is equal to Cr

1 1 1    .......... is equal to 1!n  1! 3!n  3 ! 5!n  5 ! 2 n1 for even values of n only n! 2 n1 (C) for all n  N n!

9.

r 1

n

(D) nA

(A)

8.



Value of

 i

0i jn n –3

(A) n.2 (C) n(n –1) . 2n –3

(B) 2n –1 cos nx (D) 2n sin nx n

C j is; (B) (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 n (C) (n +1)2 (D) n2n + 1

13.

Value of

2n

r 

2n

Cr .

r 0

2

(A)

 2n

2

 n  1  2

(B)

 2n  1 2n  2  22 n1  2n 2  2 n  1

(C)

14.

n 1

1 is equal to r2





2 n1

and f = R – [R]; where [  ] denotes G. I. F., then R  f is equal to

2n

(B) 112 n1 (D) 11

(A) 11 (C) 112n1 15.

Value of

 2n  1 2n  2 

(D) None of these

 2n  1 2n  2 

If R = 5 3  8

22 n1  2n 2  n  1  2

 

n

2

Ci  n C j  is

0 i  j  n

2n

(A) n. Cn  2 2 n

(B)  n  1 2 n Cn  22 n

(C)  n  1 2 n Cn  2 2 n

(D)  n  1 2 n Cn  2 2 n

16.

The remainder when 7103 is divided by 25 is (A) 0 (B) 18 (C) 16 (D) 9

17.

The number 101100  1 is divisible by (A) 10 (C) 103

18.





Integral part of 5 5  11

(B) 102 (D) 104

2 n1

is

(A) Even (C) Neither

(B) Odd (D) Can’t Say

19.

Let f (n)  10 n  3  4 n  2  5; 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

n

r 2n

  r  1  r 0

(A) 8 (C) 6

Cr 

28  1 , then ‘n’ is 6 (B) 4 (D) 5

8

ANSWERS LEVEL −I 1. 5. 9. 13. 17. 21. 25. 29.

C B A C B 239 − 219 3n  1 2 n 1 3 1 n1

2. 6. 10. 14. 18. 22.

B B C C 41 50 C6 344 (2x)6

3. 7. 11. 15. 19. 23.

B D B A 12 42n+1

4. 8. 12. 16. 20. 24.

A A −56 B n+2 Cr 49

26.

3

27.

C

28.

A

30.

D

31.

C

32.

C

D A B B 7

2. 6. 10. 14. 18.

A B C A C

3. 7. 11. 15. 19.

D −7 A C A

4. 8. 12. 16. 20.

C

B 300! B A A, B, C, D

2. 6. 10. 14. 18.

D B A C. A

3. 7. 11. 15. 19.

B B C D B

4. 8. 12. 16. 20.

C B C B D

LEVEL −II 1. 5. 9. 13. 17.

3

C0  2  3C1

A A D

LEVEL −III 1. 5. 9. 13. 17.