OMPT-G mixed paper

Answer: $y = 2 x + 2$

The slope is $m = \dfrac{-2 - 0}{-2 - (-1)} = 2\mathclose{.}$ The point $(-1, 0)$ lies on the line, so $0 = 2 \cdot (-1) + c\mathclose{,}$ which gives $c = 2\mathclose{.}$ The line is $y = 2 x + 2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Inverted the slope (run over rise) → $y = \frac{x}{2} + \frac{1}{2}$
• Found the slope but left out the intercept → $y = 2 x$
• Got the sign of the intercept wrong → $y = 2 x - 2$

Answer: $-\dfrac{21}{2}$

Multiply both sides by $21$ to clear the denominators: $3x = 7\left(x + 6\right)\mathclose{.}$ Expand: $3 x = 7 x + 42\mathclose{.}$ Collect: $- 4 x = 42\mathclose{,}$ so $x = -\dfrac{21}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied out $7\left(x + 6\right)$ as $7 x + 6\mathclose{,}$ multiplying only the $x$ → $- \frac{3}{2}$
• Cross-multiplied the wrong way, writing $7\left(x\right) = 3\left(x + 6\right)$ → $\frac{9}{2}$

Answer: $x \ge 2$

Collect the $x$ terms on the left and the numbers on the right: $8 x \ge 16\mathclose{.}$ Divide by $8\mathclose{,}$ a positive number, so the sign stays: $x \ge 2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to divide the number side by $8$ → $x \ge 16$
• Moved the number across without changing its sign → $x \ge \frac{15}{4}$

Answer: $5$

Multiply the first by $2\mathclose{,}$ so the $y$ terms become $2 y$ and $- 2 y\mathclose{,}$ then add: $15 x = 75\mathclose{,}$ so $x = 5\mathclose{.}$ Put that into the first equation: $y + 25 = 31\mathclose{,}$ so $y = 6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the value of $y$ instead of $x$ → $6$
• Multiplied only the left-hand sides when scaling the equations → $\frac{44}{15}$

Answer: $y = \frac{x}{3}$

The second equation is $3$ times the first, so only one condition remains: $x - 3 y = 0\mathclose{,}$ which gives $y = \frac{x}{3}$ — one solution for every $x\mathclose{.}$

Common mistakes, and the answer each one gives:

• Lost the sign of the $x$ term moving it across → $y = - \frac{x}{3}$

Answer: $5$

The turning point of $ax^2+bx+c$ sits at $x = -\dfrac{b}{2a}\mathclose{.}$ Here $a = -1$ and $b = 4\mathclose{,}$ so $x = -\dfrac{4}{2 \cdot (-1)} = 2\mathclose{.}$ The maximum value is $f(2) = 5\mathclose{.}$ (Completing the square gives $f(x) = -\left(x - 2\right)^{2} + 5\mathclose{,}$ which shows the same thing.)

Common mistakes, and the answer each one gives:

• Gave the $x$-coordinate of the turning point instead of the value → $2$
• Substituted $x = \dfrac{b}{2a}\mathclose{,}$ without the minus sign → $-11$

Answer: $-3$

Factorise: $x^{2} + 7 x + 12 = \left(x + 4\right)\left(x + 3\right) = 0\mathclose{,}$ so $x = -4$ or $x = -3\mathclose{.}$ The larger is $-3\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the smaller solution → $-4$
• Read the solutions off the brackets with the signs unchanged → $4$

Answer: $\dfrac{-2 + \sqrt{7}}{3}$

Here $a = 3\mathclose{,}$ $b = 4$ and $c = -1\mathclose{,}$ so the discriminant is $b^2 - 4ac = 16 - (-12) = 28$ — positive but not a perfect square, so the formula is needed: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \dfrac{-4 \pm \sqrt{28}}{6}\mathclose{.}$ Since $\sqrt{28} = 2 \sqrt{7}\mathclose{,}$ that is $\dfrac{-4 \pm 2 \sqrt{7}}{6} = \dfrac{-2 \pm \sqrt{7}}{3}\mathclose{.}$ The larger solution is $\dfrac{-2 + \sqrt{7}}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $+b$ in the numerator instead of $-b$ → $\dfrac{2 + \sqrt{7}}{3}$
• Divided by $a$ instead of $2a$ → $\dfrac{-4 + 2\sqrt{7}}{3}$
• Gave the smaller solution → $\dfrac{-2 - \sqrt{7}}{3}$

Answer: $x = -2$

The vertical asymptote sits where the denominator is zero: $x + 2 = 0\mathclose{,}$ so $x = -2\mathclose{.}$ (The horizontal one is $y = 1\mathclose{,}$ the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the $x$-intercept) → $x = -6$
• Gave the horizontal asymptote instead → $y = 1$
• Lost the minus sign when solving the denominator → $x = 2$

Answer: $21$

Multiply both sides by $x + 7\mathclose{:}$ $4 x = 3\left(x + 7\right)\mathclose{,}$ which is $4 x = 3 x + 21\mathclose{.}$ Collect: $x = 21\mathclose{.}$ That is not $x = -7\mathclose{,}$ the value the equation forbids, so it stands.

Common mistakes, and the answer each one gives:

• Ignored the denominator and solved $4 x = 3$ → $\frac{3}{4}$
• Multiplied out $3\left(x + 7\right)$ as $3 x + 7\mathclose{,}$ multiplying only the $x$ → $7$

Answer: $2$

$x^{3} \div x = x^{2}\mathclose{;}$ subtract $x^{3} - 2 x^{2}\mathclose{,}$ leaving $- x^{2} - x + 8\mathclose{.}$ $- x^{2} \div x = - x\mathclose{;}$ subtract $- x^{2} + 2 x\mathclose{,}$ leaving $8 - 3 x\mathclose{.}$ $- 3 x \div x = -3\mathclose{;}$ subtract $6 - 3 x\mathclose{,}$ leaving $2\mathclose{:}$ the quotient is $x^{2} - x - 3$ and the remainder $2\mathclose{.}$ Check it with the remainder theorem: the remainder on dividing by $x - 2$ is the value at $x = 2\mathclose{,}$ and $f(2) = 2\mathclose{.}$

Common mistakes, and the answer each one gives:

• Put in $x = -2$ instead of $x = 2$ → $-10$
• Gave the constant term of the quotient → $-3$

Answer: $y = \left(x - 1\right)^{3} + 5$

Shifting 1 unit to the right replaces $x$ by $x - 1\mathclose{;}$ shifting 5 units up adds $5\mathclose{:}$ $y = \left(x - 1\right)^{3} + 5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Shifted the wrong way horizontally → $y = \left(x + 1\right)^{3} + 5$
• Shifted the wrong way vertically → $y = \left(x - 1\right)^{3} - 5$

Answer: $\left(x - 1\right)^{2} + 5 \text{ for } x \ge 1$

Write $y = \sqrt{x - 5} + 1$ and undo the steps in reverse order: take $1$ off, $y - 1 = \sqrt{x - 5}\mathclose{;}$ square, $\left(y - 1\right)^{2} = x - 5\mathclose{;}$ then $x = \left(y - 1\right)^{2} + 5\mathclose{.}$ Swap the letters: $f^{-1}(x) = \left(x - 1\right)^{2} + 5\mathclose{.}$ A square root is never negative, so $f$ only takes values $y \ge 1$ — and those are the $x$ the inverse is defined for: $x \ge 1\mathclose{.}$

Common mistakes, and the answer each one gives:

• Squared both sides before taking the $1$ off → $x^{2} + 4$
• Undid the steps in the original order instead of the reverse order → $\left(x + 5\right)^{2} - 1$

Answer: $10$

The highest power in the product comes from the highest powers of the factors: $\left(x^{2}\right)^{2} \cdot \left(2x^{3}\right)^{2}$ has degree $4 + 6 = 10\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added only the exponents outside the brackets → $4$
• Multiplied the degrees instead of adding them → $24$

Answer: $\frac{x^{3}}{4}$

The power rule turns $3\log_{10}(x)$ into $\log_{10}(x^{3})\mathclose{,}$ and the quotient rule combines the two into $\log_{10}\left(\frac{x^{3}}{4}\right)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Subtracted the arguments instead of dividing them → $x^{3} - 4$
• Divided the wrong way round → $\frac{4}{x^{3}}$
• Multiplied by the power instead of raising to it → $\frac{3 x}{4}$

Answer: $3$

Write both numbers as powers of $3\mathclose{:}$ $9 = 3^{2}$ and $729 = 3^{6}\mathclose{.}$ Then $\log_{9}(729) = \dfrac{6}{2} = 3\mathclose{,}$ because $9$ raised to that power is $3^{6} = 729\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided the two numbers → $81$
• Turned the fraction of exponents upside down → $\frac{1}{3}$

Answer: $57$

The three angles of any triangle add to $180^\circ\mathclose{,}$ so the third is $180 - 96 - 27 = 57\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $360^\circ\mathclose{,}$ the angles round a POINT → $237$
• Subtracted only one of the two given angles → $84$

Answer: $- \frac{1}{2}$

$\frac{2\pi}{3}$ is a special angle, so the value is exact. Its reference value is $\frac{1}{2}\mathclose{,}$ and the quadrant fixes the sign: $\cos\left(\frac{2\pi}{3}\right) = - \frac{1}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the right size but the wrong sign for that quadrant → $\frac{1}{2}$
• Gave the sine instead → $\frac{\sqrt{3}}{2}$

Answer: $13$

The cosine rule: $c^2 = a^2 + b^2 - 2ab\cos C\mathclose{,}$ and $\cos 120^\circ = -\dfrac{1}{2}\mathclose{.}$ So $c^2 = 49 + 64 + 56 = 169\mathclose{,}$ and $c = \sqrt{169} = 13\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $+2ab\cos C$ instead of $-2ab\cos C$ → $\sqrt{57}$
• Forgot to take the square root → $169$

Answer: $5$

The amplitude is the size of the number in front, $|-5| = 5\mathclose{:}$ the graph swings $5$ above and below its middle line.

Common mistakes, and the answer each one gives:

• Gave the vertical shift instead → $2$
• Gave the coefficient of $x$ instead → $3$
• Kept the minus sign; an amplitude is a distance → $-5$

Answer: $-\frac{\pi}{3}$

$\sin\left(-\frac{2\pi}{3}\right) = - \frac{\sqrt{3}}{2}\mathclose{,}$ and $\arcsin$ gives back the ONE angle in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ with that value: $-\frac{\pi}{3}\mathclose{.}$ That is not $-\frac{2\pi}{3}\mathclose{,}$ which lies outside the range.

Common mistakes, and the answer each one gives:

• Cancelled $\arcsin$ against $\sin$ → $-\frac{2\pi}{3}$
• Gave the value of $\sin\mathclose{,}$ not an angle → $- \frac{\sqrt{3}}{2}$

Answer: $x = \frac{7\pi}{6} \text{ or } x = \frac{11\pi}{6}$

Divide by $6\mathclose{:}$ $\sin(x) = - \frac{1}{2}\mathclose{.}$ One angle is $\alpha = \arcsin\left(- \frac{1}{2}\right) = -\frac{\pi}{6}\mathclose{;}$ sine is also equal at $\pi - \alpha\mathclose{,}$ and both are brought into $[0, 2\pi)\mathclose{.}$ So $x = \frac{7\pi}{6} \text{ or } x = \frac{11\pi}{6}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the second solution as $\pi + \alpha$ instead of $\pi - \alpha$ → $x = \frac{5\pi}{6} \text{ or } x = \frac{11\pi}{6}$
• Gave only the first solution → $x = \frac{7\pi}{6}$

Answer: $x = \frac{5\pi}{4} \text{ or } x = \frac{7\pi}{4}$

Divide by $4\mathclose{:}$ $\sin(x) = - \frac{\sqrt{2}}{2}\mathclose{.}$ One angle is $\alpha = \arcsin\left(- \frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}\mathclose{;}$ sine is also equal at $\pi - \alpha\mathclose{,}$ and both are brought into $[0, 2\pi)\mathclose{.}$ So $x = \frac{5\pi}{4} \text{ or } x = \frac{7\pi}{4}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Took the second solution as $\pi + \alpha$ instead of $\pi - \alpha$ → $x = \frac{3\pi}{4} \text{ or } x = \frac{7\pi}{4}$
• Gave only the first solution → $x = \frac{5\pi}{4}$

Answer: $40$

The other two angles are equal and share what is left of $180^\circ\mathclose{:}$ $\dfrac{180 - 100}{2} = 40\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not halve what was left → $80$
• Halved the top angle instead → $50$

Answer: $25$

Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs. $7^2 + 24^2 = 625\mathclose{,}$ and $\sqrt{625} = 25\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the legs instead of their squares → $31$
• Subtracted the squares, which finds a LEG, not the hypotenuse → $\sqrt{527}$

Answer: $5$

The horizontal gap is $|11 - 7| = 4$ and the vertical gap is $|5 - 2| = 3\mathclose{,}$ so Pythagoras gives $\sqrt{4^2 + 3^2} = \sqrt{25} = 5\mathclose{.}$

Common mistakes, and the answer each one gives:

• Added the two gaps instead of using Pythagoras → $7$
• Forgot the square root → $25$

Answer: $135$

The slope is the tangent of that angle: $\tan\theta = -1\mathclose{,}$ so $\theta = 135^\circ\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the angle in radians → $\frac{3\pi}{4}$
• Gave the acute angle with the axis → $45$

Answer: $y = \frac{4 x}{3} + \frac{47}{3}$

The radius to $P$ runs from the centre $(-3, -5)$ to $(-11, 1)\mathclose{,}$ so its slope is $\dfrac{6}{-8} = -\dfrac{3}{4}\mathclose{.}$ The tangent is perpendicular to it, with slope $\dfrac{4}{3}\mathclose{.}$ Through $P\mathclose{:}$ $y = \dfrac{4}{3}\left(x + 11\right) + 1\mathclose{,}$ which is $y = \frac{4 x}{3} + \frac{47}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the slope of the radius itself → $y = - \frac{3 x}{4} - \frac{29}{4}$
• Turned the radius's slope over but kept its sign → $y = - \frac{4 x}{3} - \frac{41}{3}$

Answer: $18$

The common difference is $d = -4\mathclose{,}$ and term number $n$ is $5 + (n - 1) \times (-4)\mathclose{.}$ Setting that equal to $-63\mathclose{:}$ $(n - 1) \times (-4) = -68\mathclose{,}$ so $n - 1 = 17$ and $n = 18\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to add 1 at the end → $17$

Answer: $-48$

The common ratio is $r = -12 \div 6 = -2\mathclose{.}$ The $n$th term is $ar^{n-1}\mathclose{,}$ so the ratio is applied $3$ times: $6 \times \left(-2\right)^{3} = -48\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used $ar^n\mathclose{,}$ applying the ratio one time too many → $96$
• Added the ratio each time instead of multiplying → $0$