JEE Advanced (Fill in the Blanks): Mathematical Induction & Binomial Theorem

Q.1. The larger of 9950 + 10050 and 10150 is ................ (1982 - 2 Marks)

Ans.  (101)50

Sol.  Consider (101)50 – {(99)50 + (100)50} = (100 + 1)50  –  (100 – 1)50 – (100)50
= (100)50 [(1+ 0.0 1)50 – (1– 0.01)50 – 1]
= (100)50 [2 (50C1(0.01) + 50C3(0.01)3  + ....) – 1]
= (100)50 [2 (50C3(0.01)3  + ....)] >  0
∴ (101)50 > (99)50 + (100)50
∴  (101)50   is greater.

Q.2. The sum of the coefficients of the plynomial (1 + x – 3x2)2163 is ................ (1982 - 2 Marks)

Ans. -1

Sol.  If we put  x = 1 in the expansion of (1+ x – 3x2)2163 = A0 + A1x + A2x2 + ...
we will get the sum of coefficients of given polynomial, which clearly comes to be – 1.

Q.3. If (1 + ax )n = 1 + 8x + 24x2 + ....  then a = ...... and n = ............... (1983 - 2 Marks)

Ans. a = 2, n = 4

Sol.  (1 + ax)n = 1 + 8x + 24x+ ...
⇒ (1 + ax)n = 1 + nxa +

= 1 + 8 x + 24x 2+ ...
Comparing like powers of  x we get

nax = 8x ⇒ na = 8 ....(1)

....(2)

Solving (1) and (2),  n = 4,  a = 2

Q.4. Let n be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of (1 + x)n are in A.P., then the value of n is .............. (1994 -  2 Marks)

Ans. 7

Sol.  We know that for a +ve integer n (1 + x)n = nC0 + nCx + nCx2 + ......+ nCxn
ATQ coefficients of 2nd, 3nd, and 4th terms are in A.P. i.e.nC1, nC2, nC3 are in A.P.
⇒ 2.nC2 =  nC1 +  nC3

⇒ n2 – 9n + 14 = 0

⇒ (n – 7) (n – 2) = 0 ⇒  n = 7  or  2
But for the existance of 4th term, n = 7.

Q.5. The sum of the rational terms in th e expansion of     is ................ (1997 - 2 Marks)

Ans. 41

Sol.  Let Tr +1 be the general term in the expansion of

Let Tr +1will be rational if 25–r/2 and 3r/5 are rational numbers.

⇒  are integers.

⇒ r = 0 and r = 10   ⇒  T1 and T11 are rational terms.
⇒ Sum of T1  and T11 = 10C025 – 0.30 + 10C1025–5.32
= 1.32.1 + 1.1.9 = 32 + 9 = 41

The document JEE Advanced (Fill in the Blanks): Mathematical Induction & Binomial Theorem | Chapter-wise Tests for JEE Main & Advanced is a part of the JEE Course Chapter-wise Tests for JEE Main & Advanced.
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