Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Students get through Maths 2A Important Questions Inter 2nd Year Maths 2A Binomial Theorem Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2A Binomial Theorem Important Questions

Question 1.
Find the number of terms in the expression of (2x + 3y + z)7 [Mar. 14, 13, 07]
Solution:
Number of terms in (a + b + c)n are \(\frac{(n+1)(n+2)}{2}\), where n is a positive integer. Hence number of terms in (2x + 3y + z)7 are \(\frac{(7+1)(7+2)}{2}=\frac{8 \times 9}{2}\) = 36

Question 2.
Prove that C0 + 2 . C1 + 4 . C2 + 8 . C3 + …………… + 2n . Cn = 3n [A.P. Mar. 15; May 07]
Solution:
L.H.S. = C0 + 2 . C1 + 4 . C2 + 8 . C3 + …………… + 2n . Cn
= 0 + C1(2) + C2 (2)2 + C3 (23) + …………… + Cn . 2n
= (1 + 2)n = 3n
Note: (1 + x)n = C0 + C1 . x + C2 . x2 + …………… + Cn . xn

Question 3.
If 22Cr is the largest binomial coefficient in the expansion of (1 + x)22, find the value of 13Cr. [A.P. Mar. 15; May 07]
Solution:
Here n = 22 is an even integer. There is only one largest binomial coefficient and it is
nC(n/2) = 22C11 = 22Cr ⇒ r = 11
13Cr = 13C11 = 13C2 = \(\frac{13 \times 12}{1 \times 2}\) = 78

Question 4.
Write down and simplify 6th term in (\(\frac{2x}{3}\) + \(\frac{3y}{2}\))9 [May 13]
Solution:
6th term in (\(\frac{2x}{3}\) + \(\frac{3y}{2}\))9
The general term in (\(\frac{2x}{3}\) + \(\frac{3y}{2}\))9 is
Tr+1 = 9Cr (\(\frac{2x}{3}\))9-r (\(\frac{3y}{2}\))r
Put r = 5
T6 = 9C5 (\(\frac{2x}{3}\))4 (\(\frac{3y}{2}\))5
= 9C5 (\(\frac{2}{3}\))4 (\(\frac{3}{2}\))5 x4 y5
= \(\frac{9 \times 8 \times 7 \times 6}{1 \times 2 \times 3 \times 4} \frac{\left(2^{4}\right)}{3^{4}} \cdot \frac{3^{5}}{2^{5}} \cdot x^{4} y^{5}\)
= 189 x4y5

Question 5.
If the coefficients of (2r + 4)th term and (3r + 4)th term in the expansion of (1 + x)21 are equal, find r. [T.S. Mar.15]
Solution:
T2r+4 in (1 + x)21 is
= 21C2r+3 (x)2r+3 ………………… (1)
T3r+4 in (1 + x)21 is
= 21C3r+3 . (x)3r+3 ……………….. (2)
⇒ Coefficients are equal
21C2r+3 = 21C3r+3
⇒ 21 = (2r + 3) + (3r + 3)
(or)
2r + 3 = 3r + 3
⇒ 5r = 15
⇒ r = 3 (or) r = 0 .
Hence r = 0, 3.

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 6.
Fin the sum of the infinite series 1 + \(\frac{1}{3}\) + \(\frac{1.3}{3.6}\) + \(\frac{1.3.5}{3.6.9}\) + …………… [T.S. Mar. 15]
Solution:
The series can be written as
S = 1 + \(\frac{1}{1}\). \(\frac{1}{3}\) + \(\frac{1.3}{3.6}\) (\(\frac{1}{3}\))2 + \(\frac{1.3.5}{1.2.3}\) (\(\frac{1}{3}\))3 + ……………..
The series of the right is of the form
1 + \(\frac{p}{1}\) (\(\frac{x}{q}\)) + \(\frac{p(p+q)}{1.2}\) (\(\frac{x}{q}\))2 + \(\frac{p(p+q)(p+2 q)}{1.2 .3}\) (\(\frac{x}{q}\))3 + ……………
Here p = 1, q = 2, \(\frac{x}{q}\) = \(\frac{1}{3}\) ⇒ x = \(\frac{2}{3}\)
The sum of the given series
S = (1 – x)-p/q
= (1 – \(\frac{2}{3}\))-1/2 = (\(\frac{1}{3}\))-1/2 = \(\sqrt{3}\)

Question 7.
Find the set E of the value of x for which the binomial expansions for the (a + bx)r are valid. [Mar. 08]
Solution:
(4 + 9x)-2/3 = 4-2/3 [1 + \(\frac{9x}{4}\)]-2/3
The binomial expansion of (4 + 9x)-2/3 is valid
When |\(\frac{9x}{4}\)| < 1
⇒ |x| < \(\frac{4}{9}\)
⇒ x ∈ (\(\frac{-4}{9}\), \(\frac{4}{9}\))
i.e., E = (\(\frac{-4}{9}\), \(\frac{4}{9}\))

Question 8.
If the 2nd, 3rd and 4th terms in the expansion of (a + x)n are respectively 240, 720,
1080, find a, x, n. [T.S. Mar. 16]
Solution:
T2 = 240 ⇒ nC1 an-1 x = 240 …………….. (1)
T3 = 720 ⇒ nC2 an-2 x2 = 720 …………… (2)
T4 = 1080 ⇒ nC3 an-3 x3 = 1080 …………… (3)
\(\frac{(2)}{(1)} \Rightarrow \frac{{ }^{n} C_{2} a^{n-2} x^{2}}{{ }^{n} C_{1} a^{n-1} x}=\begin{aligned}
&720 \\
&240
\end{aligned}\)
⇒ \(\frac{n-1}{2} \frac{x}{a}\) = 3 ⇒ (n – 1)x = 6a …………………. (4)
\(\frac{(3)}{(2)} \Rightarrow \frac{{ }^{n} C_{3} a^{n-3} x^{3}}{{ }^{n} C_{2} a^{n-2} x^{2}}=\frac{1080}{720}\)
⇒ \(\frac{n-2}{3} \frac{x}{a}=\frac{3}{2}\)
⇒ 2(n – 2)x = 9a …………………… (5)
\(\frac{(4)}{(5)} \Rightarrow \frac{(n-1) x}{2(n-2) x}=\frac{6 a}{9 a} \Rightarrow \frac{n-1}{2 n-4}=\frac{2}{3}\)
⇒ 3n – 3 = 4n – 8
⇒ n = 5
From (4), (5 – 1) x = 6a ⇒ 4x = 6a
⇒ x = \(\frac{3}{2}\) a
Substitute x = \(\frac{3}{2}\) a, n = 5 in (1)
5C1 . a4 . \(\frac{3}{2}\) a = 240
5 × \(\frac{3}{2}\) a5 = 240
a5 = \(\frac{480}{15}\) = 32 = 25
∴ a = 2, x = \(\frac{3}{2}\) a = \(\frac{3}{2}\) (2) = 3
∴ a = 2, x = 3, n = 5.

Question 9.
If the coefficients of rth, (r + 1)th, and (r + 2)nd, terms in the expansion of (1 + x)n, are in A.P. then show that n2 – (4r + 1)n + 4r2 – 2 = 0. [T.S. Mar. 15, 08]
Solution:
Coefficient of Tr = nCr-1
Coefficient of Tr+1 = nCr
Coefficient of Tr+2 = nCr+1
Given nCr-1, nCr, nCr+1 are in A.P.
⇒ 2 . nCr = nCr-1 + nCr+1
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 1.1
⇒ (2n – 3r + 2) (r + 1) = (n – r) (n – r + 1)
⇒ 2nr + 2n – 3r2 – 3r + 2r + 2 = n2 – 2nr + r2 + n – r
⇒ n2 – 4nr + 4r2 – n – 2 = 0
∴ n2 – (4r + 1)n + 4r2 – 2 = 0

Question 10.
If n is a postive integer, prove that \(\sum_{r=1}^{n} r^{3}\left(\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}\right)^{2}=\frac{(n)(n+1)^{2}(n+2)}{12}\) [Mar. 13]
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 1.2
= (n + 1)2 Σ r – 2(n + 1) Σ r2 + Σ r3
= (n + 1)2 \(\frac{(n)(n+1)}{2}\)
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 1.3

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 11.
Find the set of values of x for which the binomial expansions of the following are valid.
(i) (2 + 3x)-2/3
(ii) (5 + x)3/2
(iii) (7 + 3x)-5
(iv) (4 – \(\frac{x}{3}\))-1/2 [A.P. Mar. 17; Mar. 16; Mar. 11]
Solution:
(i) (2 + 3x)-2/3 = [2(1 + \(\frac{3}{2}\)x)]-2/3
= 2-2/3 (1 + \(\frac{3}{2}\)x)-2/3
∴ The binomial expansion of (2 + 3x)-2/3 is valid when |\(\frac{3}{2}\)x| < 1
(i.e.,) |x| < \(\frac{2}{3}\)
(i.e.,) x ∈ (-\(\frac{2}{3}\), \(\frac{2}{3}\))

ii) (5 + x)3/2 = [5 (1 + \(\frac{x}{5}\))]3/2 [T.S. Mar. 17]
= 53/2 (1 + \(\frac{x}{5}\))]3/2
∴ The binomial expansion of (5 + x)3/2 is valid when \(\frac{x}{5}\) < 1
(i.e.,) |x| < 5
(i.e.,) x ∈ (-5, 5)

iii) (7 + 3x)-5 = [7 (1 + \(\frac{3}{7}\) x)]-5
= 7-5 (1 + \(\frac{3}{7}\) x)]-5
(7 + 3x)-5 is valid when \(\frac{3x}{7}\) < 1
⇒ |x| < \(\frac{7}{3}\) ⇒ x ∈ (\(\frac{-7}{3}\), \(\frac{7}{3}\))

iv) (4 – \(\frac{x}{3}\))-1/2 = [4(1 – \(\frac{x}{3}\))]-1/2
(4 – \(\frac{x}{3}\))-1/2 is valid when \(\frac{-x}{12}\) < 1
⇒ |x| < 12
⇒ x ∈ (-12, 12)

Question 12.
Find the sum of the infinite series
\(\frac{3}{4}\) + \(\frac{3.5}{4.8}\) + \(\frac{3.5 .7}{4.8 .12}\) + …… (Mar. 11)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 120

Question 13.
If x = \(\frac{1.3}{3.6}\) + \(\frac{1.3 .5}{3.6 .9}\) + \(\frac{1.3 .5 .7}{3.6 .9 .12}\) + …… then prove that 9x2 + 24x = 11 (TS Mar. ’16, AP Mar. ’17, ’15)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 121
⇒ 3x + 4 = 3\(\sqrt{3}\)
Squaring on both sides
(3x + 4)2 = (3\(\sqrt{3}\))2
⇒ 9x2 + 24x + 16 = 27
⇒ 9x2 + 24x = 11

Question 14.
If x = \(\frac{5}{(2 !) \cdot 3}\) + \(\frac{5.7}{(3 !) \cdot 3^{2}}\) + \(\frac{5.7 .9}{(4 !) \cdot 3^{3}}\) + …… then find the value of x2 + 4x. (mar. 13)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 122
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 123

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 15.
Find the sum of the infinite series
\(\frac{7}{5}\) (1 + \(\frac{1}{10^{2}}\) + \(\frac{1.3}{1.2}\).\(\frac{1}{10^{4}}\) + \(\frac{1.3 .5}{1.2 .3}\).\(\frac{1}{10^{6}}\) + …….) (AP Mar. ‘16, May 13; Mar. ’05)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 124
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 125

Question 16.
For n = 0, 1, 2, 3, ….n, prove that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 126
(TS Mar. ’15)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 127
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 128

Question 17.
If x = \(\frac{1}{5}\) + \(\frac{1.3}{5.10}\) + \(\frac{1.3 .5}{5.10 .15}\) + …… ∞ find 3x2 + 6x. (May. ’14, ’07, ’06; May. ’11)
Solution:
Given that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 129
⇒ 3(1 + x)2 = 5
⇒ 3x2 + 6x + 3= 5
⇒ 3x2 + 6x = 2

Question 18.
Write the expansion or (2a + 3b)6.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 131

Question 19.
Find the 5th term in the expansion of (3x – 4y)7.
Solution:
T5 = T4 + 1
= 7C4 (3x)7 – 4 (-4y)4
= 35.27x3. 256y4
= 241920 x3 y4

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 20.
Find the 4th term from the end in the expansion (2a + 5b)8.
Solution:
(2a + 5b)8 expansion contain 9 terms. The fourth term from the end is 6th term from the beginning.
8C5 (2a)8 – 5 (5b)5
= 8C5 (2a)8 – 5 (5b)5
= 8C5 . 23. 55 a3 b5

Question 21.
Find the middle term of the following expansions
(i) (3a – 5b)6
(ii) (2x + 3y)7
Solution:
i) Here n = 6 (even)
∴ \(\frac{n}{2}\) + 1 = \(\frac{6}{2}\) + 1 = 4th term is the middle term
∴ T4 = T3 + 1
= 6C3 (3a)6 – 3 (-5b)3,
= –6C3. 33. 53. a3 b3

ii) Here n = 7 (odd)
\(\frac{\mathrm{n}+1}{2}\) = \(\frac{7+1}{2}\) = 4, \(\frac{\mathrm{n}+3}{2}\) = \(\frac{7+3}{2}\) = 5
∴ 4th, 5th terms are middle terms.
∴ T4 = T3 + 1 = 7C3 (2x)7 – 3 (3y)3 = 7C3 24 33.x4.y3
T5 = T4 + 1 = 7C4(2x)7-4(3y)4 = 7C4. 23.34. x3. y4

Question 22.
n is a positive integer then prove that
Solution:
i) Co + C1 + C2 + …….. + Cn = 2n
ii) a) Co + C2 + C4 + … + Cn = 2n – 1 if n is even
(b) Co + C2 + C4 + …. + Cn – 1 = 2n – 1 if n is odd.
iii) (a) C1 + C3 + C5 + …. + Cn – 1 = 2n – 1 if n is even.
(b) C1 + C3 + C5 + …. + Cn – 1 = 2n – 1 if n is odd.
Solution:
We know (1 + x)n = nC0 + nC1 x + nC2 x2+ …… + nCn xn
= C0 + C1x + C2x2 + …… + Cnxn

Inter 2nd Year Maths 2A Binomial Theorem Important Questions 28
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 29

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 23.
Prove that C0 + 3.C1 + 5.C2 + ……… +(2n + 1). Cn = (2n + 2). 2n – 1.
Solution:
Let S = C0 + 3.C1 + 5.C2 + …… + (2n + 1). Cn —— (1)
By writing the terms in (1) in the reverse older, we get
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 30

Question 24.
Find the numerically greatest term in the binomial expansion of (1 – 5x)12 when x = \(\frac{2}{3}\).
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 31
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 32

Question 25.
Compute numerically ireatist term (s) in the expansionly of (3x – 5y)n when x = \(\frac{3}{4}\), y = \(\frac{2}{7}\) and n = 17
Solution:
Given x = \(\frac{3}{4}\), y = \(\frac{2}{7}\) and n = 17
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 33
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 34

Question 26.
Find the largest binomial coefficients (s) in the expansion of
(i) (1 + x)19
(ii) (1 + x)24
Solution:
(i) Here n = 19 is an odd integer. Hence the largest binomial coefficients are
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 35
(ii) Here n = 24 is an even integer. Hence the largest binomial coefficient is
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 36

Question 27.
If 22Cr is the largest binomial coefficient in the expansion of (1 + x)22, find the value of 13Cr. (A.P. Mar’16, May ‘11)
Solution:
Here n = 22 is an even integer. There is only one largest binomial coefficient and it is
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 37

Question 28.
Find the 7th term in the expansion of \(\left(\frac{4}{x^{3}}+\frac{x^{2}}{2}\right)^{14}\)
Solution:
The general term in the expansion of
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 38

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 29.
Find the 3rd term from the end in the expansion of \(\left(x^{-2 / 3}-\frac{3}{x^{2}}\right)^{8}\)
Solution:
Comparing with (X + a)n, we get
X = x-2/3, a = \(\frac{-3}{x^{2}}\), n = 8
In the given expansion \(\left(x^{-2 / 3}-\frac{3}{x^{2}}\right)^{8}\), we have n + 1 = 8 + 1 = 9 terms
Hence the 3rd term from the end is 7th term from the beginning.
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 39

Question 30.
Find the coefficient of x9 and x10 in the expansion of \(\left(2 x^{2}-\frac{1}{x}\right)^{2 c}\)
Solution:
If we write X = 2x2 and a = –\(\frac{1}{x}\), then the general term in the expansion of
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 40
Since r = \(\frac{31}{3}\) which is impossible since r must be a positive integer. Thus ‘there is no term containing x9 in the expansion of the given expression. In otherwords the coefficient of x9 is ‘0’.
Now, to find the coefficient of x10.
put 40 – 3r = 10
⇒ r = 10
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 41

Question 31.
Find the term independent of x (that is the constant term) in the expansion of
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 42
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 43

Question 32.
If the coefficients of x10 in the expansion of \(\left(a x^{2}+\frac{1}{b x}\right)^{11}\) is equal to the coefficient of x-10 in the expansion of \(\left(a x-\frac{1}{b x^{2}}\right)^{11}\) ; find the relation between a and b where a and b are real numbers.
Solution:
The general term in the expansion of \(\left(a x^{2}+\frac{1}{b x}\right)^{11}\) is
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 44
To find the coefficient of x10, put
22 – 3r = 10 ⇒ 3r = 12 ⇒ r = 4
Hence the coefficient of x10 in
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 45
Given that the coefficients are equal.
Hence from (1) and (2), we get
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 46

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 33.
If the kth term is the middle term in the expansion of \(\left(x^{2}-\frac{1}{2 x}\right)^{20}\), find Tk and Tk + 3.
Solution:
The general term in the expansion of
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 47
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 48

Question 34.
If the coefficients of (2r + 4)th and (r – 2)nd terms in the expansion of (1 + x)18 are equal, find r.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 50

Question 35.
Prove that 2.C0 + 7.C1 + 12.C2 + …… + (5n + 2)Cn = (5n + 4)2n – 1.
Solution:
First method:
The coefficients of C0, C1, C2, …., Cn are in A.P. with first term a = 2, C.d (d) = 5
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 51
Second method:
General term in LH.S.
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 52
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 53

Question 36.
Prove that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 54
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 55

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 37.
For n = 0, 1, 2, 3 , n, prove that C0. Cr + C1. Cr + 1 + C2. Cr + 2 + ……. + Cn – r. Cn = 2nCn + 1. (T.S. Mar. ’15)
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 56
Solution:
We know that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 57
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 58
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 59

Question 38.
Prove that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 60
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 61

Question 39.
Find the numerically greatest term (s) in the expansion of

i) (2 + 3x)10 when x = \(\frac{11}{8}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 62
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 63

ii) (3x – 4y)14 when x = 8, y = 3.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 64
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 65

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 40.
Prove that 62n – 35n – 1 is divisible by 1225 for all natural numbers of n.
Solution:
62n – 35n – 1 = (36)n – 35n – 1
= (35 + 1)n – 35n – 1
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 66
Hence 62n – 35n – 1 is divisible by 1225 for all integral values of n.

Question 41.
Suppose that n is a natural number and I, F are respectively the integral part and fractional part of (7 + \(\sqrt{3}\))n. Then show that
(i) I is an odd integer
(ii) (I + F) (I – F) = 1
Solution:
Given that (7 + 4\(\sqrt{3}\))n = I + F where I is an integer and 0 < F < 1
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 67
= 2k, where k is a positive integer —— (1)
Thus I + F + f is n even integer.
Since I is an integer, we get that F + f is an integer. Also since 0 < F < 1 and 0 < f < 1
⇒ 0 < F + f < 2
∵ F + 1 is an integer
We get F + f = 1
(i.e.,) I – F = f ——— (2)

(i) From (1) I + F + f = 2k
⇒ f = 2k – 1, an odd integer.
(ii) (I + F) (I – F) = (I + F) f
= (7 + 4\(\sqrt{3}\))n (7 – 4\(\sqrt{3}\))n
= (49 – 48)n = 1.

Question 42.
Find the coefficient of x6 in (3 + 2x + x2)6.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 68
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 69

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 43.
If n is a positive integer, then prove that
Co + \(\frac{C_{1}}{2}\) + \(\frac{C_{2}}{3}\) + ….. + \(\frac{C_{n}}{n+1}\) = \(\frac{2^{n+1}-1}{n+1}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 70

Question 44.
If n is a positive integer and x is any nonzero real number, then prove that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 71
(May. ’14, May 13, ’05)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 72
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 73

Question 45.
Prove that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 74
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 75
Now we can find the term independent of in the L.H.S. of (1).
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 76
Suppose n is an even integer, say n = 2k. Then from (2),
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 77
When n is odd:
Observe that the expansion in the numerator of (2) contains only even powers of x.
∴ If n is odd, then there is no constant term in (2) (i.e.,) the term indep. of x in
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 78

Question 46.
Find the set E of the value of x for which the binomial expansions for the following are valid
(i) (3 – 4x)3/4
(ii) (2 + 5)x-1/2
(iii) (7 – 4x)-5
(iv) (4 + 9x)-2/3
(iv) (a + bx)r (Mar. ’08)
Solution:
i) (3 – 4x)3/4 = 33/4\(\left(1-\frac{4 x}{3}\right)^{3 / 4}\)
The binomial expansion of (3 – 4x)3/4 is valid, when \(\frac{4 x}{3}\) < 1
i.e., |x| < \(\frac{3}{4}\)
i.e., E = \(\left(\frac{-3}{4}, \frac{3}{4}\right)\)

ii) (2 + 5x)-1/2 = 2-1/2\(\left(1+\frac{5 x}{2}\right)^{-1 / 2}\)
The binomial expansion of (2 + 5x)-1/2 is valid when |\(\frac{5 x}{3}\)| < 1 ⇒ |x| < \(\frac{2}{5}\)
i.e., E = (-\(\frac{2}{5}\), \(\frac{2}{5}\))

iii) (7 – 4x)-5 = 7-5\(\left(1-\frac{4 x}{7}\right)^{-5}\)
The binomial expansion of (7 – 4x)-5 is valid when \(\frac{4 x}{7}\) < 1 ⇒ |x| < \(\frac{7}{4}\)
i.e., E = \(\left(\frac{-7}{4}, \frac{7}{4}\right)\)

iv) (4 + 9x)-2/3 = 4-2/3 \(\left(1+\frac{9 x}{4}\right)^{-2 / 3}\)
The binomial expansipn of (4 + 9x)-2/3 is valid
When \(\frac{9 x}{4}\) < 1
⇒ |x| < \(\frac{4}{9}\)
⇒ x ∈ \(\left(\frac{-4}{9}, \frac{4}{9}\right)\)
i.e., E = \(\left(\frac{-4}{9}, \frac{4}{9}\right)\)

v) For any non zero reals a and b, the set of x for which the binomial expansion of (a + bx)r is valid when r ∉ Z+ ∪ {0}, is \(\left(-\frac{|a|}{|b|}, \frac{|a|}{|b|}\right)\)

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 47.
Find the
(i) 9th term of \(\left(2+\frac{x}{3}\right)^{-5}\)
(ii) 10th term of \(\left(1-\frac{3 x}{4}\right)^{4 / 5}\)
(iii) 8th term of \(\left(1-\frac{5 x}{2}\right)^{-3 / 5}\)
(iv) 6th term of \(\left(3+\frac{2 x}{3}\right)^{3 / 2}\)

(i) 9th term of \(\left(2+\frac{x}{3}\right)^{-5}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 79
we get X = \(\frac{x}{6}\), n = 5
The general term in the binomial expansion of (1 + x)-n is
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 80

ii) 10th term of \(\left(1-\frac{3 x}{4}\right)^{4 / 5}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 81

iii) 8th term of \(\left(1-\frac{5 x}{2}\right)^{-3 / 5}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 82

iv) 6th term of \(\left(3+\frac{2 x}{3}\right)^{3 / 2}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 83
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 84

Question 48.
Write the first 3 terms in the expansion of
(i) \(\left(1+\frac{x}{2}\right)^{-5}\)
(ii) (3 + 4x)-2/3
(iii) (4 – 5x)-1/2

i) \(\left(1+\frac{x}{2}\right)^{-5}\)
Solution:
We have
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 85

ii) (3 + 4x)-2/3
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 86

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

iii) (4 – 5x)-1/2
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 87
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 88

iv) (2 – 3x)-1/3
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 89

Question 49.
Find the coefficient of x12 in \(\frac{1+3 x}{(1-4 x)^{4}}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 90
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 91

Question 50.
Find coeff. of x6 in the expansion of (1 – 3x)-2/5
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 92

Question 51.
Find the sum of the infinite series
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 93
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 94

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 52.
Find the sum of the series
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 95
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 96
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 97

Question 53.
If x = \(\frac{1}{5}\) + \(\frac{1.3}{5.10}\) + \(\frac{1.3 .5}{5.10 .15}\) + ……. ∞ then find 3x2 + 6x. (Mar. ’14, ’07, ’06; May. ’11)
Solution:
Given that
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 98
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 99

Question 54.
Find an approximate value of
i) \(\frac{1}{\sqrt[3]{999}}\)
ii) (627)1/4
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 100
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 101

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 55.
If |x| is so small that x3 and higher powers or x can be neglected, find approximate value of \(\frac{(4-7 x)^{1 / 2}}{(3+5 x)^{3}}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 102

Question 56.
Find an approximate value of \(\sqrt[6]{63}\) correct to 4 decimal places.
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 103

Question 57.
If |x| is so small thát x2 and higher powers of x may be neglected, then find an approximate value of
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 104
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 105
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 106

Question 58.
If |x| is so small that x4 and higher powers of x may be neglected, then find the approximate value of
\(\sqrt[4]{x^{2}+81}\) – \(\sqrt[4]{x^{2}+16}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 107
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 108

Question 59.
Suppose that x and y are positive and x is very small when compared to y. Then find an approximate value of
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 109
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 110
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 111

Inter 2nd Year Maths 2A Binomial Theorem Important Questions

Question 60.
Expand \(5 \sqrt{5}\) in increasing powers of \(\frac{4}{5}\)
Solution:
Inter 2nd Year Maths 2A Binomial Theorem Important Questions 112