1st PUC Maths Question Bank Chapter 8 Binomial Theorem

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Karnataka 1st PUC Maths Question Bank Chapter 8 Binomial Theorem

Question 1.
State and prove Binomial theorem.
Answer:

Some observations in a binomial theorem:
(1) The expansion of {a + b)ⁿ has (n + 1) terms
(2) The coefficients ⁿC_r occurring in the binomial theorem are known as binomial coefficients.
(3) The indices of V go on decreasing and that of ‘a’ go on increasing by 1 at each stage.
i.e., for each term: index of a + index of b-n.
(4) Since ⁿC_r=ⁿC_n__r we have
ⁿC₀ =ⁿC_n, ⁿC_x =ⁿC_n__r and so on.
Thus the coefficients of the terms equidistant from the beginning and the end in a binomial theorem are equal.
(5) General term in (a + b)ⁿ: T_r+1 – ⁿC_r a^n–rb^r
(6) Middle terms in (a + b)ⁿ

• When ‘n’ is even, the middle term
  \(=\left(\frac{n}{2}+1\right)^{t h} \text { term }\)
• When ‘n’ is odd ,the middle term are
  \(\frac{1}{2}(n+1)^{n} \text { term an } \frac{1}{2}(n+3)^{n} \text { term }\)

(7) Taking a = x and b = -y in the expansion, we get (x-y)ⁿ =[x + (-y)]ⁿ

Question 2.
Expand each of the expression
(i)\(\left(x^{2}+\frac{3}{x}\right)^{4}\)
(ii)\((1-2 x)^{5}\)
(iii)\(\left(\frac{2}{x}-\frac{x}{2}\right)^{5}\)
(iv)\((2 x-3)^{6}\)
(v)\(\left(\frac{x}{3}+\frac{1}{x}\right)^{5}\)
(vi)\(\left(x+\frac{1}{x}\right)^{6}\)
Answer:

Question 3.
Using binomial theorem, evaluate each of the following:
(i) (98)⁵
(ii) (96)³
(iii) (102)⁵
(iv) (101)⁴
(v) (99)⁵
Answer:

Question 4.
Which is longer \((1.01)^{1000000} \text { or } 10,000 ?\)
Answer:

Question 5.
Using Binomial theorem, indicate which number is larger \( (1.1)^{10000} \text { or } 1000 ?\)
Answer:
Spilitting 1.01 and using Binomial theorem write the first few terms, we have.

Question 6.
Find (a + b)⁴ – (a -b)⁴. Hence evaluate
\((\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}\)
Answer:

Question 7.
Find (x +1)⁶ +(x -1)⁶. Hence or otherwise evaluate
\((\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}\)
Answer:

Question 8.
Evaluate:
\((\sqrt{3}+\sqrt{2})^{6}+(\sqrt{3}-\sqrt{2})^{6}\)
Answer:

Question 9.
Find the value of
\(\left(a^{2}+\sqrt{a^{2}-1}\right)^{4}+\left(a^{2}-\sqrt{a^{2}-1}\right)^{4}\)
Answer:

Question 10.
Show that 9ⁿ⁺¹ – 8n – 9 is divisible by 64, whenever ‘n’ is a positive integer.
Answer:

Question 11.
Using binomial theorem, prove that 6ⁿ -5n always leaves remainder 1 when
divided by 25.
Answer:

Question 12.
Prove that
\(\sum_{n=0}^{n} y \cdot c_{n}=4\)
Answer:

Question 13.
Find the 4^th term in the expansion of (x-2 y)¹².
Answer:

Question 14.
Find the 13^th term in the expansion of
\(\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x \neq 0 \)
Answer:

Question 15.
Write the general term in the expression of
(i) (x² -y)⁶
(ii) (x²-yx)¹²,x≠0
Answer:

Question 16.
Find the coefficient of
(i) x⁵ in (x + 3)⁸
(ii) a⁵b⁷ in (a – 2b)¹²
(iii) x⁶y³ in (x + 2y)⁹
Answer:

Question 17.
Find a, if the 17^th and 18^th terms of the expansion (2 +a)⁵⁰ are equal.
Answer:

Question 18.
In the expansion of (1+ a)^m+n, prove that coefficients of a^m and aⁿ are equal.
Answer:

Question 19.
Prove that the coefficient of xⁿ in the expansion of (1 + x)²ⁿ is twice the coefficient of xⁿ in the expansion of (1+x)^2n-1
Answer:
In (1+x)²ⁿ we have

From (1) and (2) we get, the coefficient of xⁿ in (1 + x)²ⁿ is twice the coefficient of xⁿ in (1 + x)^2n-1.

Question 20.
Find a positive value of m for which the coefficient of x² in the expansion (1 + x)^m is 6.
Answer:

Question 21.
Find the middle terms in the expansions of
(i) \(\left(3-\frac{x^{3}}{6}\right)^{7}\)
(ii)\(\left(\frac{x}{3}+9 y\right)^{10}\)
Answer:

Question 22.
Show that the middle term in the expansion of \((1+x)^{2 n} \text { is } \frac{1 \cdot 3 \cdot 5 \ldots(2 n-1)}{\lfloor n} 2^{n} x^{n} \)where ‘n’ is a positive integer.
Answer:

Question 23.
The second, third and fourth terms in the binomial expansion (x + a)ⁿ are 240, 720 and 1080, respectively. Find x, a and n
Answer:

Question 24.
The coefficients of three consecutive terms in the expansion of (1 + a)ⁿ are in the ratio 1:7: 42. Find n
Answer:

Question 25.
The coefficients (r-1)^th, r^th,and (r + 1)^th, terms in the expansion of (x + 1)^th, are in the ratio 1:3:5. Find n and r.
Answer:

Question 26.
Find the term independent of x in the expansion of \(\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}\)
Answer:

Question 27.
Find the term independent of x in the expansion of \(\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x>0\)
Answer:

Question 28.
Find a, 6 and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729,7290 and 30375, respectively.
Answer:
Given: T_x = 729, T₂ = 7290 and T₃ = 30375
∴ aⁿ=729……………….(1)

Question 29.
Find a if the coefficients of x² and x³ in the expansion of (3 +ax)⁹ are equal.
Answer:

Question 30.
If the coefficients of (r – 5)^th and (2r -1)^th terms in the expansion of (1 + x)³⁴ are equal, find r.
Answer:

Question 31.
If the coefficients of a^r-1, a^r and a^r+1 in the expansion of (1 + a)ⁿ are in arithmetic progression, prove that n² – n(4r +1) + 4r² -2 = 0.
Answer:

Question 32.
Show that the coefficient of the middle term in the expansion of (1 + x)²ⁿ is equal to the sum of the coefficients of two middle terms in the expansion of (1 + x)^2n-1.
Answer:

Question 33.
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of
\(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n} \text { is } \sqrt{6}: 1\)
Answer:

Question 34.
Find the r^th term from the end in the expansion of (x + a)ⁿ.
Answer:
r^th term from the end in (x + a)ⁿ

Question 35.
If a and b are distinct integers, prove that a-b is a factor of aⁿ -bⁿ, whenever n is a positive integer.
Answer:

Question 36.
The sum of the coefficients of the first three terms in the expansion of
\(\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m \)being a natural numbers, is 559. Find the term of the expansion containing x³.
Answer:

Question 37.
Find the coefficient of x⁵ in the product (1 + 2x)⁶(1 – x)⁷ using binomial theorem.
Answer:

Question 38.
Find the coefficient of a⁴ in the product (1 + 2a)⁴(2-a)^s using binomial theorem.
Answer:

Question 39.
Expand using Binomial theorem
\(\left(1+\frac{x}{2}-\frac{2}{x}\right)^{4}, x \neq 0\)
Answer:

Question 40.
Find the expansion of (3x² – 1ax + 3a²)³ using binomial theorem.
Answer: