**Chapter 4 Methods of Induction and Binomial Theorem Miscellaneous Exercise 4**

## Chapter 4 Methods of Induction and Binomial Theorem Miscellaneous Exercise 4

**(I) Select the correct answers from the given alternatives.**

**Question 1.The total number of terms in the expression of (x + y) ^{100} + (x – y)^{100} after simplification is:(A) 50(B) 51(C) 100(D) 202Answer:**

(B) 51

Hint:

**Question 2.**

**Question 3.**

**Question 4.**
Hint:

**Question 5.The number of terms in expansion of (4y + x) ^{8} – (4y – x)^{8} is(A) 4(B) 5(C) 8(D) 9Answer:**
(A) 4

Hint:

**Question 6.Hint:**

**Question 7.**

Hint:

**Question 8.In the expansion of (3x + 2) ^{4}, the coefficient of the middle term is(A) 36(B) 54(C) 81(D) 216Answer:**
(D) 216

Hint:

(3x + 2)

^{4}has 5 terms.

∴ (3x + 2)

^{4}has 3rd term as the middle term.

The coefficient of the middle term

= 6 × 9 × 4

= 216

**Question 9.**

**Question 10.If the coefficients of x ^{2} and x^{3} in the expansion of (3 + ax)^{9} are the same, then the value of a isHint:**

**(II) Answer the following.**

**Question 1.**

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., 8 + 17 + 26 + …… + [9(k + 1) – 1]

∴ P(n) is true for n = k + 1.

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

∴ P(n) is true for n = k + 1.

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

∴ P(n) is true for n = k + 1.

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

Solution:

**Question 2.**

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

**Question 3.Prove by method of inductionSolution:**

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

Using binomial theorem,

**Question 5.Solution:**

**Question 6.Solution:**

**Question 7.Solution:**

**Question 8.**

∴ The Middle term is -20.

Solution:

**Question 9.Solution:**

Solution:

**Question 10.Find the constant term in the expansion ofSolution:**

Solution:

**Question 11.Prove by method of inductionSolution:**

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

Solution:

**Question 12.If the coefficient of x ^{16} in the expansion of (x^{2} + ax)^{10} is 3360, find a.Solution:**

**Question 13.Solution:**

**Question 14.If the coefficients of x ^{2} and x^{3} in theexpansion of (3 + kx)^{9} are equal, find k.Solution:**

**Question 15.Solution:**

**Question 16.Solution:**

**Question 17.Solution:**

**Question 18.Solution:**

**Question 19.Solution:**

**Question 20.Solution:**

**Question 21.Solution:**

**Question 22.Solution:**

**Question 23.Solution:**

**Question 24.(a + bx) (1 – x) ^{6} = 3 – 20x + cx^{2} + …, then find a, b, c.Solution:**

**Question 25.The 3rd term of (1 + x) ^{n} is 36x^{2}. Find 5th term.Solution:**

**Question 26.Suppose (1 + kx) ^{n} = 1 – 12x + 60x^{2} – …… find k and n.Solution:**