More about Equations: practice questions

6 questions · 12 marks · free, no account · Read the notes first

1. Question 1Multiple choice · 1 mark

   Solve the simultaneous equations $y = x + 1$ and $y = x^2 - 1$.

   A

   $(-2, -1)$ and $(1, 2)$

   B

   $(2, 3)$ only

   C

   $(2, 3)$ and $(-1, 0)$

   D

   $(1, 2)$ and $(-1, 0)$

2. Question 2Multiple choice · 1 mark

   The line $y = 2x + k$ is a tangent to the curve $y = x^2$. Find $k$.

   A

   $1$

   B

   $-1$

   C

   $0$

   D

   $-4$

3. Question 3Multiple choice · 1 mark

   Solve $4^x - 6(2^x) + 8 = 0$.

   A

   $x = 1$ or $x = 2$

   B

   $x = 2$ or $x = 4$

   C

   $x = 1$ only

   D

   $x = 2$ or $x = 3$

4. Question 4Multiple choice · 1 mark

   Solve $x - 2 = \dfrac{3}{x}$.

   A

   $x = -3$ or $x = 1$

   B

   $x = 3$ only

   C

   $x = 1$ or $x = 3$

   D

   $x = 3$ or $x = -1$

5. Question 54 marks

   Solve $\log(x + 3) + \log x = 1$.

   Show worked solution

   $$\log[x(x + 3)] = 1 \qquad \text{1M}$$ $$x^2 + 3x - 10 = 0 \qquad \text{1M}$$

   $x = 2$ or $x = -5$ 1A

   Since $\log x$ is undefined when $x = -5$, it is rejected. So $x = 2$. 1A

6. Question 64 marks

   Solve the simultaneous equations $x + y = 5$ and $x^2 + y^2 = 13$.

   Show worked solution

   From the linear equation, $y = 5 - x$. Substitute: $x^2 + (5 - x)^2 = 13$ 1M

   $$x^2 - 5x + 6 = 0 \qquad \text{1A}$$

   $x = 2$ or $x = 3$ 1A

   The solutions are $(2, 3)$ and $(3, 2)$. 1A