(ii). The fith term in the expansion of $$ \left(1-\frac{1}{x} ight)^{10} $$ is : (a) $$ \frac{210}{x^{4}} $$ (b) $$ \frac{x^{4}}{210} $$ (c) $$ \frac{1}{x^{4}} $$ (d) None $$ \square imes \square $$ (iv). The middle term/s in the expansion of $$ \left(x+\frac{1}{x} ight)^{10} $$ is : (a) 504 (b) 252 (c) $$ \frac{1}{x^{4}} $$ (d) None (v). The value of $$ i^{6} $$, (where $$ i=\sqrt{-1} $$ ): (a) 1 (b) $$ i $$ (c) -1 (d) None (vi). The value of $$ i^{6}+\omega^{6} $$ : (a) 1 (b) 2 (c), 0 (d) None

Question

(ii). The fith term in the expansion of $$ \left(1-\frac{1}{x}ight)^{10} $$ is : (a) $$ \frac{210}{x^{4}} $$ (b) $$ \frac{x^{4}}{210} $$ (c) $$ \frac{1}{x^{4}} $$ (d) None $$ \square 	imes \square $$ (iv). The middle term/s in the expansion of $$ \left(x+\frac{1}{x}ight)^{10} $$ is : (a) 504 (b) 252 (c) $$ \frac{1}{x^{4}} $$ (d) None (v). The value of $$ i^{6} $$, (where $$ i=\sqrt{-1} $$ ): (a) 1 (b) $$ i $$ (c) -1 (d) None (vi). The value of $$ i^{6}+\omega^{6} $$ : (a) 1 (b) 2 (c), 0 (d) None

UpStudy AI · Accepted Answer

Let's address each part of the question step-by-step.

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### (ii). The fifth term in the expansion of $$ \left(1-\frac{1}{x}\right)^{10} $$

**Solution:**

The general term in the binomial expansion of $$ (a + b)^n $$ is given by:
$$
T_{k+1} = C(n, k) \cdot a^{n-k} \cdot b^k
$$
where $$ C(n, k) $$ is the binomial coefficient.

Here, $$ a = 1 $$, $$ b = -\frac{1}{x} $$, and $$ n = 10 $$.

For the **fifth term**, $$ k = 4 $$:
$$
T_5 = C(10, 4) \cdot 1^{10-4} \cdot \left(-\frac{1}{x}\right)^4 = C(10, 4) \cdot \frac{1}{x^4}
$$
$$
C(10, 4) = 210
$$
$$
T_5 = \frac{210}{x^4}
$$

**Answer:**  
**(a) $$ \frac{210}{x^{4}} $$**

---

### (iv). The middle term/s in the expansion of $$ \left(x+\frac{1}{x}\right)^{10} $$

**Solution:**

For an expansion of $$ (a + b)^{n} $$, if $$ n $$ is even, there are two middle terms; if odd, there's one middle term. Here, $$ n = 10 $$ (even), so there are two middle terms.

However, considering the symmetry in $$ \left(x + \frac{1}{x}\right)^{10} $$, both middle terms will be the same.

The general term is:
$$
T_{k+1} = C(10, k) \cdot x^{10-k} \cdot \left(\frac{1}{x}\right)^k = C(10, k) \cdot x^{10-2k}
$$
For the middle term:
$$
k = 5
$$
$$
T_6 = C(10, 5) \cdot x^{0} = C(10, 5)
$$
$$
C(10, 5) = 252
$$

**Answer:**  
**(b) 252**

---

### (v). The value of $$ i^{6} $$, where $$ i = \sqrt{-1} $$

**Solution:**

Powers of $$ i $$ cycle every 4:
$$
i^1 = i \
i^2 = -1 \
i^3 = -i \
i^4 = 1 \
i^5 = i \
i^6 = -1
$$

**Answer:**  
**(c) -1**

---

### (vi). The value of $$ i^{6} + \omega^{6} $$

**Assumption:**  
Assuming $$ \omega $$ is a primitive cube root of unity, i.e., $$ \omega^3 = 1 $$.

**Solution:**

From part (v):
$$
i^6 = -1
$$

For $$ \omega^6 $$:
$$
\omega^6 = (\omega^3)^2 = 1^2 = 1
$$

Therefore:
$$
i^6 + \omega^6 = -1 + 1 = 0
$$

**Answer:**  
**(c) 0**

---

### Summary of Answers

- **(ii)** (a)
- **(iv)** (b)
- **(v)** (c)
- **(vi)** (c)
**(ii)** (a) **(iv)** (b) **(v)** (c) **(vi)** (c)

(ii). The fith term in the expansion of is :
(a)
(b)
©
(d) None

(iv). The middle term/s in the expansion of is :
(a) 504
(b) 252
©
(d) None
(v). The value of , (where ):
(a) 1
(b)
© -1
(d) None
(vi). The value of :
(a) 1
(b) 2
©, 0
(d) None

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(ii). The fith term in the expansion of is : (a) (b) © (d) None (iv). The middle term/s in the expansion of is : (a) 504 (b) 252 © (d) None (v). The value of , (where ): (a) 1 (b) © -1 (d) None (vi). The value of : (a) 1 (b) 2 ©, 0 (d) None

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preguntas relacionadas

Latest Algebra Questions

(ii). The fith term in the expansion of is :
(a)
(b)
©
(d) None

(iv). The middle term/s in the expansion of is :
(a) 504
(b) 252
©
(d) None
(v). The value of , (where ):
(a) 1
(b)
© -1
(d) None
(vi). The value of :
(a) 1
(b) 2
©, 0
(d) None