Questions

1. Question 1

   Using binomial theorem find the value of the following to three places of decimals:

   (i)

   $\sqrt{99}$

   (ii)

   $(1.03)^{1/3}$

   (iii)

   $\dfrac{1}{\sqrt[3]{252}}$

   (iv)

   $\dfrac{\sqrt{7}}{\sqrt{8}}$

   SolutionTheory
2. Question 2

   Find the remainder when $8^{100}$ is divided by 7.

   SolutionTheory
3. Question 3

   Find the remainder when $2^{100}$ is divided by 3.

   SolutionTheory
4. Question 4

   Using the binomial theorem, find which number is larger:

   (i)

   $19^{10} + 20^{10}$ or $21^{10}$

   (ii)

   $29^{15} + 30^{15}$ or $31^{15}$

   SolutionTheory
5. Question 5

   Using the binomial theorem, show that:

   (i)

   $5^7 + 7^5$ is divisible by 36

   (ii)

   $(17)^{15} + (13)^7$ is divisible by 6

   (iii)

   $(21)^9 + (19)^{11}$ is divisible by 20

   (iv)

   $(31)^4 + (29)^6$ is divisible by 30

   (v)

   $(101)^5 + (99)^7$ is divisible by 100

   SolutionTheory
6. Question 6

   Using the binomial theorem, find the remainder when $3^{101}$ is divided by 8.

   SolutionTheory
7. Question 7

   Using the binomial theorem, find the last digit of the number $(32)^{32}$.

   SolutionTheory
8. Question 8

   Using the binomial theorem, show that $7^n - 6n$ leaves remainder 1 when divided by 6 for all positive integers $n$.

   SolutionTheory
9. Question 9

   By using Binomial Theorem show that for each $n \in \mathbb{N}$, $5^n - 1$ is divisible by 4.

   SolutionTheory
10. Question 10

    By using Binomial Theorem show that for each $n \in \mathbb{N}$, $5^n - 2^n$ is divisible by 3.

    SolutionTheory
11. Question 11

    Show that $a^2 + (a+2)^2 + (a+4)^2 + 1$ is divisible by 12, whenever $a$ is an odd integer.

    SolutionTheory
12. Question 12

    A company expects its annual revenue to grow at a fixed rate of 6% per year. The revenue in year 1 is $R =$ Rs. 10,000,000. Estimate the company's revenue after 4 years using the binomial theorem for small growth rates.

    SolutionTheory
13. Question 13

    A bank offers a compound interest rate of 10% per year. Zafar invests Rs. 2,000,000 for 4 years. How much will his investment be worth at the end of 4 years?

    SolutionTheory
14. Question 14

    Zaid is organizing a sports competition with 8 teams. Every team plays against every other team exactly once. How many matches will be played in total? Use Pascal's triangle to solve this.

    SolutionTheory