Binomial Theorem — Mathematics textbook solutions

4.2 Binomial Theorem for a Positive Integral Index

7 q

Worked · 7

4.2 SolvedEx.1 $(x^2 + 3y)^5$
4.2 SolvedEx.2 $\left(2x - \frac{y}{2}\right)^5$
4.2 SolvedEx.3 $\left(\sqrt{5} + \sqrt{3}\right)^4$
4.2 SolvedEx.4 $\left(\sqrt{2}+1\right)^5 - \left(\sqrt{2}-1\right)^5$
4.2 SolvedEx.5 $(99)^4$
4.2 SolvedEx.6 $(2.02)^5$
4.2 SolvedEx.7 $(2x-1)^5 + 5(2x-1)^4(1-x) + 10(2x-1)^3(1-x)^2 + 10(2x-1)^2(1-x)^3 + 5(2x-1)(1-x)^4 + (1-x)^5$

17 q

Expand.
Ex 4.2 Q1(i) $\left(\sqrt{3} + \sqrt{2}\right)^4$
Ex 4.2 Q1(ii) $\left(\sqrt{5} - \sqrt{2}\right)^5$
Expand.
Ex 4.2 Q2(i) $(2x^2 + 3)^4$
Ex 4.2 Q2(ii) $\left(2x - \frac{1}{x}\right)^6$
Find the value of
Ex 4.2 Q3(i) $\left(\sqrt{3}+1\right)^4 - \left(\sqrt{3}-1\right)^4$
Ex 4.2 Q3(ii) $\left(2+\sqrt{5}\right)^5 + \left(2-\sqrt{5}\right)^5$
Prove that
Ex 4.2 Q4(i) $\left(\sqrt{3}+\sqrt{2}\right)^6 + \left(\sqrt{3}-\sqrt{2}\right)^6 = 970$
Ex 4.2 Q4(ii) $\left(\sqrt{5}+1\right)^5 - \left(\sqrt{5}-1\right)^5 = 352$
Using binomial theorem, find the value of
Ex 4.2 Q5(i) $(102)^4$
Ex 4.2 Q5(ii) $(1.1)^5$
Using binomial theorem, find the value of
Ex 4.2 Q6(i) $(9.9)^3$
Ex 4.2 Q6(ii) $(0.9)^4$
Without expanding, find the value of
Ex 4.2 Q7(i) $(x+1)^4 - 4(x+1)^3(x-1) + 6(x+1)^2(x-1)^2 - 4(x+1)(x-1)^3 + (x-1)^4$
Ex 4.2 Q7(ii) $(2x-1)^4 + 4(2x-1)^3(3-2x) + 6(2x-1)^2(3-2x)^2 + 4(2x-1)^1(3-2x)^3 + (3-2x)^4$
Ex 4.2 Q8 $(1.02)^6$
Ex 4.2 Q9 $(1.01)^5$
Ex 4.2 Q10 $(0.9)^6$

6 q

Worked · 6

23 q

In the following expansions, find the indicated term.
Ex 4.3 Q1(i) $\left(2x^{2}+\frac{3}{2x}\right)^{8}$
Ex 4.3 Q1(ii) $\left(x^{2}-\frac{4}{x^{3}}\right)^{11}$
Ex 4.3 Q1(iii) $\left(\frac{4x}{5}-\frac{5}{2x}\right)^{9}$
Ex 4.3 Q1(iv) $\left(\frac{1}{3}+a^{2}\right)^{12}$
Ex 4.3 Q1(v) $\left(3a+\frac{4}{a}\right)^{13}$
In the following expansions, find the indicated coefficients.
Ex 4.3 Q2(i) $x^{3}$
Ex 4.3 Q2(ii) $x^{8}$
Ex 4.3 Q2(iii) $x^{9}$
Ex 4.3 Q2(iv) $x^{-3}$
Ex 4.3 Q2(v) $x^{-20}$
Find the constant term (term independent of $x$ ) in the expansion of
Ex 4.3 Q3(i) $\left(2x+\frac{1}{3x^{2}}\right)^{9}$
Ex 4.3 Q3(ii) $\left(x-\frac{2}{x^{2}}\right)^{15}$
Ex 4.3 Q3(iii) $\left(\sqrt{x}-\frac{3}{x^{2}}\right)^{10}$
Ex 4.3 Q3(iv) $\left(x^{2}-\frac{1}{x}\right)^{8}$
Ex 4.3 Q3(v) $\left(2x^{2}-\frac{5}{x}\right)^{9}$
Find the middle terms in the expansion of
Ex 4.3 Q4(i) $\left(\frac{x}{y}+\frac{y}{x}\right)^{12}$
Ex 4.3 Q4(ii) $\left(x^{2}+\frac{1}{x}\right)^{7}$
Ex 4.3 Q4(iii) $\left(x^{2}-\frac{2}{x}\right)^{8}$
Ex 4.3 Q4(iv) $\left(\frac{x}{a}-\frac{a}{x}\right)^{10}$
Ex 4.3 Q4(v) $\left(x^{4}-\frac{1}{x^{3}}\right)^{11}$
Ex 4.3 Q5 $(k+x)^{8}$
Ex 4.3 Q6 $x^{6}$
Ex 4.3 Q7 $x^{2}$

4 q

Worked · 4

20 q

State, by writing first four terms, the expansion of the following, where $|x| < 1$
Ex 4.4 Q1(i) $(1+x)^{-4}$
Ex 4.4 Q1(ii) $(1-x)^{1/3}$
Ex 4.4 Q1(iii) $(1-x^{2})^{-3}$
Ex 4.4 Q1(iv) $(1+x)^{-1/5}$
Ex 4.4 Q1(v) $(1+x^{2})^{-1}$
State, by writing first four terms, the expansion of the following, where $|b| < |a|$
Ex 4.4 Q2(i) $(a-b)^{-3}$
Ex 4.4 Q2(ii) $(a+b)^{-4}$
Ex 4.4 Q2(iii) $(a+b)^{1/4}$
Ex 4.4 Q2(iv) $(a-b)^{-1/4}$
Ex 4.4 Q2(v) $(a+b)^{-1/3}$
Simplify first three terms in the expansion of the following
Ex 4.4 Q3(i) $(1+2x)^{-4}$
Ex 4.4 Q3(ii) $(1+3x)^{-1/2}$
Ex 4.4 Q3(iii) $(2-3x)^{1/3}$
Ex 4.4 Q3(iv) $(5+4x)^{-1/2}$
Ex 4.4 Q3(v) $(5-3x)^{-1/3}$
Use binomial theorem to evaluate the following upto four places of decimals.
Ex 4.4 Q4(i) $\sqrt{99}$
Ex 4.4 Q4(ii) $\sqrt[3]{126}$
Ex 4.4 Q4(iii) $\sqrt[4]{16.08}$
Ex 4.4 Q4(iv) $(1.02)^{-5}$
Ex 4.4 Q4(v) $(0.98)^{-3}$

4 q

Worked · 4

4.5 SolvedEx.1 $C_0 + C_1 + C_2 + \dots + C_{10} = 1024$
4.5 SolvedEx.2 $C_0 + C_2 + C_4 + \dots + C_{12} = C_1 + C_3 + C_5 + \dots + C_{11} = 2048$
4.5 SolvedEx.3 $C_1 + 2C_2 + 3C_3 + 4C_4 + \dots + nC_n = n \cdot 2^{n-1}$
4.5 SolvedEx.4 $C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \dots + \frac{C_n}{n+1} = \frac{2^{n+1}-1}{n+1}$

7 q

Practice · 7

37 q

Practice · 10

Misc I Q1 $(x+y)^{100} + (x-y)^{100}$
1. A. $50$
2. B. $51$
3. C. $100$
4. D. $202$
Misc I Q2 $(1+x)^{2n}$
1. A. $(n-1)^{\text{th}}$
2. B. $n^{\text{th}}$
3. C. $(n+1)^{\text{th}}$
4. D. $(n+2)^{\text{th}}$
Misc I Q3 $(x^2 - 2x)^{10}$
1. A. $-1680$
2. B. $1680$
3. C. $3360$
4. D. $6720$
Misc I Q4 $x$
1. A. ${}^{11}C_{5}$
2. B. ${}^{10}C_{5}$
3. C. ${}^{10}C_{4}$
4. D. ${}^{10}C_{7}$
Misc I Q5 $(4y+x)^8 - (4y-x)^8$
1. A. $4$
2. B. $5$
3. C. $8$
4. D. $9$
Misc I Q6 ${}^{14}C_{1} + {}^{14}C_{3} + {}^{14}C_{5} + \dots + {}^{14}C_{11}$
1. A. $2^{14} - 1$
2. B. $2^{14} - 14$
3. C. $2^{12}$
4. D. $2^{13} - 14$
Misc I Q7 ${}^{11}C_{2} + {}^{11}C_{4} + {}^{11}C_{6} + {}^{11}C_{8}$
1. A. $2^{10} - 1$
2. B. $2^{10} - 11$
3. C. $2^{10} + 12$
4. D. $2^{10} - 12$
Misc I Q8 $(3x+2)^4$
1. A. $36$
2. B. $54$
3. C. $81$
4. D. $216$
Misc I Q9 $8^{\text{th}}$
1. A. $7$
2. B. $120$
3. C. ${}^{10}C_{8}$
4. D. $210$
Misc I Q10 $x^2$
1. A. $-\frac{7}{9}$
2. B. $-\frac{9}{7}$
3. C. $\frac{7}{9}$
4. D. $\frac{9}{7}$

Practice · 27

Misc II Q4 $(3x^2 + 2y)^5$
Misc II Q5 $\left(\frac{2x}{3} - \frac{3}{2x}\right)^4$
Misc II Q6 $\left(9x^2 - \frac{y^3}{6}\right)^4$
Misc II Q7 $\left(2x^2 + \frac{1}{x}\right)^{12}$
Find the middle term (s) in the expansion of
Misc II Q8(i) $\left(\frac{2a}{3} - \frac{3}{2a}\right)^6$
Misc II Q8(ii) $\left(x - \frac{1}{2y}\right)^{10}$
Misc II Q8(iii) $(x^2 + 2y^2)^7$
Misc II Q8(iv) $\left(\frac{3x^2}{2} - \frac{1}{3x}\right)^9$
Find the coefficients of
Misc II Q9(i) $x^6$
Misc II Q9(ii) $x^{60}$
Find the constant term in the expansion of
Misc II Q10(i) $\left(\frac{4x^2}{3} + \frac{3}{2x}\right)^9$
Misc II Q10(ii) $\left(2x^2 - \frac{1}{x}\right)^{12}$
Misc II Q12 $x^{16}$
Misc II Q13 $\left(x + \frac{b}{x}\right)^6$
Misc II Q14 $x^2$
Misc II Q15 $\left(x^3 + \frac{k}{x^8}\right)^{11}$
Misc II Q16 $x^6$
Misc II Q17 $\left(2x - \frac{x^2}{4}\right)^9$
Misc II Q18 $\left(1 - \frac{2x}{3}\right)^{-\frac{1}{2}}$
Misc II Q19 $(1-x)^{-\frac{1}{4}}$
Misc II Q20 $(5 + 4x)^{-\frac{1}{2}}$
Misc II Q21 $\sqrt[3]{995}$
Misc II Q22 $\frac{1}{4.08}$
Misc II Q23 $x$
Misc II Q24 $(a + bx)(1 - x)^6 = 3 - 20x + cx^2 + \dots$
Misc II Q25 $3^{\text{rd}}$
Misc II Q26 $(1+kx)^n = 1 - 12x + 60x^2 - \dots$