1. The conjugate of complex number z = 4 - 5i is ___. 2. The value of log_2 81 is ___. 3. The value of 30! / 29! is ___. 4. The number of terms in the expansion of (x - y)^3 is ___... 1. The conjugate of complex number z = 4 - 5i is ___. 2. The value of log_2 81 is ___. 3. The value of 30! / 29! is ___. 4. The number of terms in the expansion of (x - y)^3 is ___. 5. Evaluate i^30 + i^40 + i^60. 6. Find n, if nC4 = nC6. 7. Find x, if log_x 343 = 3. 8. Expand (2a + 3b)^4 using binomial theorem. 9. Prove that log_5 (14/3) - log_5 (2/15) = log_5 7. 10. If 2nP3 = 100x nP2, find n. Read more

Steps to Solve

1. Finding the conjugate of a complex number

The conjugate of a complex number $z = a + bi$ is given by $z^* = a - bi$.

For $z = 4 - 5i$, the conjugate is: $$ z^* = 4 + 5i $$

2. Calculating the logarithm value

To find $\log_{10} 81$, we can express 81 as a power of 10.

Since $81 = 3^4$, we have: $$ \log_{10} 81 = \log_{10} (3^4) = 4 \cdot \log_{10} 3 $$

Using a calculator, $\log_{10} 3 \approx 0.4771$, thus: $$ \log_{10} 81 \approx 4 \cdot 0.4771 \approx 1.9084 $$

3. Calculating the logarithm value (base 3)

To find $30! / 29!$, we can simplify: $$ \frac{30!}{29!} = 30 $$

Now, we evaluate $\log_{10} 30$.

Using a calculator: $$ \log_{10} 30 \approx 1.4771 $$

4. Finding the number of terms in the binomial expansion

In the expansion of $(x - y)^3$, the number of terms is given by $n + 1$, where $n$ is the power.

Thus, for $(x - y)^3$, we have: $$ 3 + 1 = 4 $$

5. Evaluating the complex expression

For $i^{30} + i^{40} + i^{60}$, recall that $i^n$ cycles every 4 terms:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

Thus, we calculate:

• $30 \mod 4 = 2 \implies i^{30} = -1$
• $40 \mod 4 = 0 \implies i^{40} = 1$
• $60 \mod 4 = 0 \implies i^{60} = 1$

Combining these gives: $$ i^{30} + i^{40} + i^{60} = -1 + 1 + 1 = 1 $$

6. Finding n in combinations

Given $nC_u = nC_6$, we can apply the property that $nC_k = nC_{n-k}$. Hence: $$ u = n - 6 $$

This leads us to find $n = 6 + u$.

7. Finding x in logarithmic form

Given $\log_{3} 343 = 3$, we rewrite 343 as a power of 7: $$ 7^3 = 343 \implies \log_{3} (7^3) = 3 \log_{3} 7 = 3 $$

Thus: $$ \log_{3} 7 = 1 $$

8. Expanding using binomial theorem

Using the binomial theorem to expand $(2a + 3b)^4$: $$ (2a + 3b)^4 = \sum_{k=0}^{4} \binom{4}{k} (2a)^{4-k} (3b)^{k} $$

Calculating the individual terms yields: $$ = \binom{4}{0}(2a)^4(3b)^0 + \binom{4}{1}(2a)^3(3b)^1 + \binom{4}{2}(2a)^2(3b)^2 + \binom{4}{3}(2a)^1(3b)^3 + \binom{4}{4}(2a)^0(3b)^4 $$