Questions covered in Hard

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. The remainder when $7^{103}$ is divided by 25 is
2. The value of $\sum_{r=0}^n r \cdot ^nC_r$ is
3. The number of terms in the expansion of $(1 - 2x + x^2)^{20}$ is
4. If the coefficients of $T_r, T_{r+1}, T_{r+2}$ in the expansion of $(1+x)^n$ are in A.P., then
5. The coefficient of $x^n$ in the expansion of $(1+x)^{2n}$ is
6. Find the term independent of $x$ in the expansion of $(\sqrt{x} + \frac{1}{3x^2})^{10}$.
7. Using Binomial Theorem, prove that $6^n - 5n$ always leaves remainder 1 when divided by 25.
8. Find the middle term in the expansion of $(\frac{x}{3} + 9y)^{10}$.
9. Find the coefficient of $a^4$ in the product $(1+2a)^4 (2-a)^5$.
10. Find the value of $(\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6$.
11. Find the coefficient of $x^5$ in the expansion of the product $(1+2x)^6 (1-x)^7$.
12. Case Study: Consider the binomial expansion of $(x + \frac{1}{x})^{2n}$.
    (i) Write the general term $T_{r+1}$.
    (ii) How many terms are there in the expansion?
    (iii) Find the term independent of $x$.