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. If $a, b, c$ are in A.P. and $x, y, z$ are in G.P., then the value of $x^{b-c} y^{c-a} z^{a-b}$ is
2. The sum of integers from 1 to 100 that are divisible by 2 or 5 is
3. If the AM and GM of roots of a quadratic equation are 8 and 5 respectively, then the quadratic equation is
4. The minimum value of $4^x + 4^{1-x}$, $x \in R$ is
5. If $a, b, c$ are in G.P., then $\log a, \log b, \log c$ are in
6. Find the sum of the sequence $0.6, 0.66, 0.666, \dots$ to $n$ terms.
7. If $\frac{a+bx}{a-bx} = \frac{b+cx}{b-cx} = \frac{c+dx}{c-dx}$, show that $a, b, c, d$ are in G.P.
8. The sum of an infinite G.P. is 57 and the sum of their cubes is 9747. Find the G.P.
9. The sum of two numbers is 6 times their geometric mean, show that the numbers are in the ratio $(3+2\sqrt{2}) : (3-2\sqrt{2})$.
10. The sum of $n$ terms of two arithmetic progressions are in the ratio $(3n+8):(7n+15)$. Find the ratio of their 12th terms.
11. The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.
12. Case Study: A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year.
    (i) Find the production in the 1st year.
    (ii) Find the production in the 10th year.
    (iii) Find the total production in first 7 years.