NBT NBT Mathematics: Exponents, Surds and Equations 4 — Questions and Answers
Question 1: Solve for $x$ and $y$: $2x + 3y = 12$ and $x = y + 1$
- $x = 3, y = 2$ (Correct answer)
- $x = 2, y = 1$
- $x = 4, y = 3$
- $x = 5, y = 4$
Correct answer: $x = 3, y = 2$
Substituting $x = y+1$: $2(y+1)+3y=12 \Rightarrow 5y=10 \Rightarrow y=2$, so $x=3$.
Question 2: Which of the following is equivalent to $a^{1/2} \cdot a^{3/2}$?
- $a^2$ (Correct answer)
- $a^{3/4}$
- $a^{5/4}$
- $a^{1/3}$
Correct answer: $a^2$
$a^{1/2+3/2} = a^{4/2} = a^2$.
Question 3: Rationalise and simplify: $\dfrac{\sqrt{3}}{\sqrt{3}+1}$
- $\dfrac{3-\sqrt{3}}{2}$ (Correct answer)
- $\dfrac{\sqrt{3}-1}{2}$
- $\dfrac{3+\sqrt{3}}{2}$
- $3-\sqrt{3}$
Correct answer: $\dfrac{3-\sqrt{3}}{2}$
Multiply by $\dfrac{\sqrt{3}-1}{\sqrt{3}-1}$: $\dfrac{\sqrt{3}(\sqrt{3}-1)}{3-1} = \dfrac{3-\sqrt{3}}{2}$.
Question 4: Solve: $x^2 - 4 = 0$
- $x = 2$ or $x = -2$ (Correct answer)
- $x = 4$ or $x = -4$
- $x = 2$ only
- $x = \sqrt{4}$
Correct answer: $x = 2$ or $x = -2$
$x^2 = 4 \Rightarrow x = \pm 2$.
Question 5: Simplify: $3\sqrt{8} - \sqrt{2}$
- $5\sqrt{2}$ (Correct answer)
- $2\sqrt{2}$
- $6\sqrt{2}$
- $3\sqrt{6}$
Correct answer: $5\sqrt{2}$
$3\sqrt{8} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}$, so $6\sqrt{2} - \sqrt{2} = 5\sqrt{2}$.
Question 6: If $9^x = 3^{x+4}$, what is $x$?
- $x = 4$ (Correct answer)
- $x = 2$
- $x = 8$
- $x = 3$
Correct answer: $x = 4$
$9^x = 3^{2x}$; equating: $2x = x + 4 \Rightarrow x = 4$.
Question 7: Which is the correct factorised form of $x^2 - 9$?
- $(x-3)(x+3)$ (Correct answer)
- $(x-9)(x+1)$
- $(x-3)^2$
- $(x+9)(x-1)$
Correct answer: $(x-3)(x+3)$
$x^2 - 9$ is a difference of two squares: $(x-3)(x+3)$.
NBT NBT Mathematics: Exponents, Surds and Equations 4
Solve for $x$ and $y$: $2x + 3y = 12$ and $x = y + 1$