OMPT-G mixed paper

Answer: $-6$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-10 - (-4)}{-3 - (-4)} = -6$.

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $- \frac{1}{6}$
• Subtracted the coordinates in opposite orders → $6$

Answer: $2$

Expand the bracket: $2 x + 16 = 7 x + 6$. Collect the $x$ terms on one side and the numbers on the other, then divide: $x = 2$.

Common mistakes, and the answer each one gives:

• Multiplied only the $x$ by the number outside the bracket → $\frac{2}{5}$
• Moved a term across without changing its sign → $\frac{28}{5}$

Answer: $\frac{17}{3}$

Cross-multiply to clear both denominators: $7\left(x - 3\right) = 4\left(x - 1\right)$. Expand, collect the $x$ terms and divide: $x = \frac{17}{3}$.

Common mistakes, and the answer each one gives:

• Multiplied only the numerators by the other denominator → $\frac{20}{3}$
• Added the fractions instead of cross-multiplying → $\frac{25}{11}$

Answer: $-1$

Eliminate $y$ by scaling and adding, or substitute. The solution is $x = -1$, $y = 7$.

Common mistakes, and the answer each one gives:

• Solved for y instead of x → $7$

Answer: $-5$

The first equation already gives $y$, so put it straight into the second: $3x + 2\left(- x\right) = -5$. That leaves one unknown, and $x = -5$.

Common mistakes, and the answer each one gives:

• Gave the value of $y$ rather than of $x$ → $5$
• Substituted into the equation it came from, which says nothing → $-3$

Answer: $-2$

$\left(x + 2\right) \left(x + 7\right) = 0$, so $x = -2$ or $x = -7$. The larger is $-2$.

Common mistakes, and the answer each one gives:

• Gave the smaller root → $-7$
• Read the roots off with the signs unchanged → $7$

Answer: $- \frac{7}{2} + \frac{\sqrt{41}}{2}$

The discriminant is $b^2 - 4ac = 41$, which is positive but not a perfect square, so the roots are irrational and the formula is needed: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$. The larger root is $- \frac{7}{2} + \frac{\sqrt{41}}{2}$.

Common mistakes, and the answer each one gives:

• Used $+b$ in the numerator instead of $-b$ → $\frac{\sqrt{41}}{2} + \frac{7}{2}$
• Divided by $a$ instead of $2a$ → $-7 + \sqrt{41}$
• Gave the smaller root → $- \frac{7}{2} - \frac{\sqrt{41}}{2}$

Answer: $5$

The turning point of $ax^2+bx+c$ sits at $x = -\dfrac{b}{2a}$. Here $a = -3$ and $b = 30$, so $x = -\dfrac{30}{2(-3)} = 5$. (Completing the square gives $-3\left(x - 5\right)^2 + 6$, which shows the same thing.)

Common mistakes, and the answer each one gives:

• Used $\dfrac{b}{2a}$ without the minus sign → $-5$
• Gave the MAXIMUM VALUE instead of where it happens → $6$

Answer: $6$

Where the graphs meet, the two expressions are equal. Bring everything to one side: $x^{2} - 7 x + 6 = 0$, which factorises as $\left(x - 6\right) \left(x - 1\right)$. The roots are $1$ and $6$.

Common mistakes, and the answer each one gives:

• Gave the smaller root → $1$
• Read the roots straight off the brackets with the signs unchanged → $-1$

Answer: $-5$

The vertical asymptote sits where the denominator is zero: $x + 5 = 0$, so $x = -5$. (The horizontal one is $y = 1$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $-8$
• Gave the horizontal asymptote instead → $1$
• Forgot the sign when solving → $5$

Answer: $\frac{x - 4}{x - 8}$

Factorise both: $\dfrac{\left(x - 4\right) \left(x + 4\right)}{\left(x - 8\right) \left(x + 4\right)}$. The factor $x + 4$ is common to both and cancels, leaving $\frac{x - 4}{x - 8}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{4}{x + 8}$
• Cancelled the wrong bracket → $\frac{x + 4}{x - 8}$

Answer: $18 x^{2} - 36 x + 13$

Square the bracket FIRST, then multiply by $2$, then add the constant — the order matters. $\left(3 x - 3\right)^2 = 9 x^{2} - 18 x + 9$, and the whole thing comes to $18 x^{2} - 36 x + 13$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $36 x^{2} - 72 x + 31$
• Squared each term inside the bracket separately → $18 x^{2} + 13$

Answer: $- \frac{7}{2}$

Multiply both sides by $x - 3$: $4 x + 1 = 2\left(x - 3\right)$. Expand, collect and divide: $x = - \frac{7}{2}$. (Check it is not $x = 3$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $\frac{1}{4}$
• Set the denominator to zero, which is the EXCLUDED value → $3$

Answer: $-3$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-27} = -3$. So $x = -3$ and $x = -3$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $-9$
• Took the cube root but lost the sign → $3$

Answer: $3 x^{3}$

The power rule turns $3\log_{2}(x)$ into $\log_{2}(x^{3})$, and the product rule combines the two into $\log_{2}(3 x^{3})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{3} + 3$
• Multiplied by the power instead of raising to it → $9 x$

Answer: $7$

Equal logarithms with the same base have equal arguments, so $x + 9 = 16$ and $x = 7$. Check the domain: $x + 9 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $25$
• Treated the log as a multiplier and divided → $\frac{16}{9}$

Answer: $1$

Write the right-hand side as a power of $2$: $32 = 2^{5}$. With the same base on both sides the exponents must be equal, so $4 x + 1 = 5$ and $x = 1$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 32 instead of to 5 → $\frac{31}{4}$
• Forgot to move the constant in the exponent across → $\frac{5}{4}$

Answer: $3$

Write both numbers as powers of $5$: $25 = 5^{2}$ and $15625 = 5^{6}$. Then $\log_{25}15625 = \dfrac{6}{2} = 3$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $625$
• Turned the fraction of exponents upside down → $\frac{1}{3}$

Answer: $1$

$\frac{\pi}{2}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\sin\left(\frac{\pi}{2}\right) = 1$.

Common mistakes, and the answer each one gives:

• Gave the cosine instead → $0$
• Worked in degrees, reading the number as an angle in degrees → $\sin{\left(\frac{\pi}{360} \right)}$

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

Divide by $6$ first: $\cos(x) = \frac{1}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{3}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{5 \pi}{3}$
• Forgot to divide by the coefficient before taking the inverse → $0$

Answer: $3 \sin{\left(x \right)}$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $3 \sin{\left(x \right)}$.

Common mistakes, and the answer each one gives:

• Cancelled to the cosine rather than the sine → $3 \cos{\left(x \right)}$
• Left the tangent untouched → $3 \tan{\left(x \right)}$

Answer: $8$

Rearrange Pythagoras: the missing leg squared is $10^2 - 6^2 = 64$, so the leg is $\sqrt{64} = 8$.

Common mistakes, and the answer each one gives:

• Subtracted the lengths instead of their squares → $4$
• Added the squares, which is the rule for the HYPOTENUSE → $2 \sqrt{34}$

Answer: $46$

The three angles of any triangle add to $180^\circ$, so the third is $180 - 92 - 42 = 46$.

Common mistakes, and the answer each one gives:

• Used $360^\circ$, which is the angles round a POINT → $226$
• Subtracted only one of the two given angles → $88$

Answer: $3960$

A polygon with $24$ sides splits into $22$ triangles from one corner, and each triangle contributes $180^\circ$: $(24 - 2) \times 180 = 3960$.

Common mistakes, and the answer each one gives:

• Multiplied the number of sides by 180 without subtracting 2 → $4320$
• Gave the EXTERIOR angle sum, which is 360 for every polygon → $360$

Answer: $16 \pi$

Area $= \pi r^2 = \pi \times 4^2 = 16\pi$.

Common mistakes, and the answer each one gives:

• Used the circumference formula $2\pi r$ → $8 \pi$
• Doubled the radius instead of squaring it → $8 \pi$

Answer: $10$

The horizontal gap is $6$ and the vertical gap is $8$, so Pythagoras gives $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$.

Common mistakes, and the answer each one gives:

• Added the two gaps instead of using Pythagoras → $14$
• Forgot the square root → $100$

Answer: $127$

The $n$th term is $a + (n-1)d$ — the step is taken $15$ times, not $16$: $7 + 15 \times 8 = 127$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $135$
• Multiplied the first term by the step → $56$

Answer: $256$

The $n$th term is $ar^{n-1}$, so the ratio is applied $7$ times: $2 \times \left(2\right)^{7} = 256$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $512$
• Added the ratio each time instead of multiplying → $16$

Answer: $119$

The last term is $-1 + 16 \times 1 = 15$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $17 \times \dfrac{-1 + 15}{2} = 119$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $255$
• Multiplied the number of terms by the FIRST term → $-17$

Answer: $8$

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-10 - (-2)}{6 - (7)} = 8$.

Common mistakes, and the answer each one gives:

• Inverted the fraction (run over rise) → $\frac{1}{8}$
• Subtracted the coordinates in opposite orders → $-8$