List of practice Questions

If P and Q are positive integers such that \(P^2 = Q^2 + 13\), find the value of the product \(PQ\).
If $ x = 2 $, what is the value of $ 3x^2 - 5x + 7 $?
If \[ \frac{1}{a(b + c)} + \frac{1}{b(c + a)} + \frac{1}{c(a + b)} = k, \text{ then the value of } k \text{ is:} \]
Solve the cubic equation: \[ x^3 - \frac{15}{2} x^2 + 18x + 20 = 0 \]

Pick the CORRECT eigenvalue(s) of the matrix [A] from the following choices.

\[ [A] = \begin{bmatrix} 6 & 8 \\ 4 & 2 \end{bmatrix} \]

Given the equation: $$ 3^{x^2} = 27 \times 9^x $$ Find the value of: $$ \frac{2^{x^2}}{(2^x)^2} $$
If $ x + \frac{1}{x} = 3 $, what is the value of $ x^2 + \frac{1}{x^2} $?
A runs $\dfrac{3}{2}$ times as fast as B. If A gives B a start of 40 m, how far must the winning post be from the starting point, so that A and B reach at the same time?
A boat goes 3.5 km upstream and then returns. Total time taken is 1 hour and 12 minutes. If the speed of the current is 1 km/h, then find the speed of the boat in still water.
If $a>b$ and $c<0$, then which of the following is true?

Let \(a = 30!\), \(b = 50!\), and \(c = 100!\). Consider the following numbers: \(\log_a c\), \(\log_c a\), \(\log_b a\), \(\log_a b\)
Which one of the following inequalities is CORRECT?

If x satisfies the equation 4^{8x} = 256, then x is equal to .