If C 0 , C 1 , C 2 , … C n are the binomial coefficients in the expansion of 1 + x n , then prove that: 3 · 2 - 1 2 C 1 + 3 2 · 2 2 - 1 2 2 C 2 + 3 3 · 2 3 - 1 2 3 C 3 + … … . + 3 n · 2 n - 1 2 n C n = 2 3 n - 3 n 2 n

We know, 1 + x n = ∑ r = 0 n x r C r Given, 3 · 2 - 1 2 C 1 + 3 2 · 2 2 - 1 2 2 C 2 + 3 3 · 2 3 - 1 2 3 C 3 + … … . + 3 n · 2 n - 1 2 n C n = ∑ r = 0 n 3 r C r - ∑ r = 0 n 1 2 r C r = 1 + 3 n - 1 + 1 2 n = 2 3 n - 3 n 2 n

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 12

EASY

JEE Main/Advance

IMPORTANT

Earn 100

Find the middle term(s) in the expansion of ${(1 - 2 x + x^{2})}^{n}$ .

Important Questions on Binomial Theorem

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 13

Prove that the coefficient of the middle term in the expansion of

(1 + x)^{2 n}

equal to the sum of the coefficients of the two middle terms in the expansion of

= \frac{2 n!}{{(n!)}^{2}} \times {(- 1)}^{n} \times x^{n}

.

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 14

Find the last digit, last two digits and last three digits of the number

{(27)}^{27}

.

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 15

Using binomial theorem prove that

6^{n} - 5 n

always leaves the remainder

1

when divided by

25

.

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 16

Find the last digit, last two digits and last three digits of the number

{(27)}^{27}

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 18

Find numerically greatest term(s) in the expansion of

{(3 - 5 x)}^{15}

when

x = \frac{1}{5}

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 19

Which term is numerically the greatest term in the expansion of

{(2 x + 5 y)}^{34},

when

x = 3

and

y = 2 ?

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 20

Find the term in the expansion of

{(2 x - 5)}^{6}

which have
(i) Greatest binomial coefficient
(ii) Greatest numerical coefficient
(iii) Algebraically greatest coefficient
(iv) Algebraically least coefficient

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 21

If $C_{0}, C_{1}, C_{2}, \dots C_{n}$ are the binomial coefficients in the expansion of ${(1 + x)}^{n}$ , then prove that:

$\frac{(3 \cdot 2 - 1)}{2} C_{1} + \frac{3^{2} \cdot 2^{2} - 1}{2^{2}} C_{2} + \frac{3^{3} \cdot 2^{3} - 1}{2^{3}} C_{3} + \dots \dots . + \frac{3^{n} \cdot 2^{n} - 1}{2^{n}} C_{n} = \frac{2^{3 n} - 3^{n}}{2^{n}}$