OMPT-A mixed paper

Answer: $0.0028$

Multiply as whole numbers, $4 \times 7 = 28\mathclose{,}$ then count the decimal places of BOTH factors — $2 + 2 = 4$ — so the answer is $0.0028\mathclose{.}$

Common mistakes, and the answer each one gives:

• Counted the decimal places of only one factor → $0.28$

Answer: $-\dfrac{1}{3}$

The bars mean the inside is $5$ away from zero in EITHER direction, so there are two equations: $3 x + 6 = 5$ gives $x = -\dfrac{1}{3}\mathclose{,}$ and $3 x + 6 = -5$ gives $x = -\dfrac{11}{3}\mathclose{.}$ The larger is $-\dfrac{1}{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{11}{3}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $\frac{1}{3}$

Answer: $\dfrac{177}{88}$

Multiplication comes before addition, so work out $\dfrac{5}{8} \times \dfrac{9}{11} = \dfrac{45}{88}$ first, then add $\dfrac{3}{2}\mathclose{:}$ $\dfrac{177}{88}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{153}{88}$

Answer: $\dfrac{17}{99}$

Multiplication comes before addition, so work out $\dfrac{8}{9} \times \dfrac{1}{11} = \dfrac{8}{99}$ first, then add $\dfrac{1}{11}\mathclose{:}$ $\dfrac{17}{99}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{97}{1089}$

Answer: $18 x^{2} + 39 x - 45$

Multiply every term in the first bracket by every term in the second: $6x \cdot 3x = 18 x^{2}\mathclose{,}$ the two cross terms give $54 x - 15 x = 39 x\mathclose{,}$ and $-5 \cdot 9 = -45\mathclose{.}$ Together: $18 x^{2} + 39 x - 45\mathclose{.}$

Common mistakes, and the answer each one gives:

• Multiplied only the first terms and the last terms → $18 x^{2} - 45$

Answer: $\left(x - 7\right)\left(x + 1\right)$

Look for two numbers whose PRODUCT is $-7$ and whose SUM is $-6\mathclose{:}$ they are $-7$ and $1\mathclose{.}$ So $x^{2} - 6 x - 7 = \left(x - 7\right)\left(x + 1\right)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Found the right two numbers but put the opposite signs in the brackets → $\left(x - 1\right)\left(x + 7\right)$

Answer: $w = \dfrac{P - 2l}{2}$

Undo what is done to $w$ in reverse order, the same to both sides: take $2l$ off both sides, $P - 2l = 2w\mathclose{,}$ then divide by 2. So $w = \dfrac{P - 2l}{2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Divided only $2l$ by 2 → $w = P - l$
• Took off $l$ instead of $2l$ → $w = \dfrac{P - l}{2}$

Answer: $4$

On the $y$-axis $x = 0\mathclose{,}$ so $3 y = 12$ and $y = 4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the other intercept → $3$
• Divided the coefficient by the constant → $\dfrac{1}{4}$

Answer: $x > 6$

Collect the $x$ terms on the left and the numbers on the right: $- 2 x < -12\mathclose{.}$ Divide by $-2$ — a NEGATIVE number, so the inequality sign turns round: $x > 6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not reverse the inequality when dividing by a negative number → $x < 6$
• Moved the number across without changing its sign → $x > 0$

Answer: $-2$

The first equation already gives $y\mathclose{,}$ so put it straight into the second: $4 x + 3\left(- 3 x - 12\right) = -26\mathclose{.}$ Expand: $- 5 x - 36 = -26\mathclose{,}$ so $- 5 x = 10$ and $x = -2\mathclose{.}$ (Then the first equation gives $y = -6\mathclose{.})$

Common mistakes, and the answer each one gives:

• Gave the value of $y$ instead of $x$ → $-6$
• Multiplied only the first term of the bracket by $3$ → $\frac{14}{5}$

Answer: $4$

The $y$ terms are already $5 y$ and $- 5 y\mathclose{,}$ so add the two equations: $9 x = -9\mathclose{,}$ so $x = -1\mathclose{.}$ Put that into the first equation: $5 y - 5 = 15\mathclose{,}$ so $5 y = 20$ and $y = 4\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the value of $x$ instead of $y$ → $-1$

Answer: $\text{infinitely many solutions}$

The second equation's left side is $-1$ times the first's. So is its right side $(-1 \times 3 = -3)\mathclose{:}$ it is the SAME line, and every point on it is a solution.

Common mistakes, and the answer each one gives:

• Saw two different equations and concluded the lines never meet → $\text{no solution}$
• Assumed two lines always cross exactly once → $\text{exactly one solution}$

Answer: $(3, 3)$

The vertex starts at $(3, -2)\mathclose{.}$ Reflecting in the $x$-axis changes the sign of its $y$-coordinate, giving $(3, 2)\mathclose{;}$ shifting up adds $1\mathclose{:}$ $(3, 3)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Reflected in the $y$-axis instead → $(-3, -1)$
• Shifted up before reflecting → $(3, 1)$

Answer: $-6$

Half the coefficient of $x$ is $7\mathclose{;}$ add $7^{2} = 49$ to BOTH sides: $\left(x + 7\right)^{2} = 1\mathclose{.}$ So $x + 7 = \pm 1\mathclose{,}$ giving $x = -6$ or $x = -8\mathclose{;}$ the larger is $-6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to subtract $\frac{b}{2}$ at the end → $1$

Answer: $0$

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

Common mistakes, and the answer each one gives:

• Gave the smaller solution → $-1$
• Read the solutions off the brackets with the signs unchanged → $1$

Answer: $\dfrac{x + 8}{x + 7}$

Factorise both: $\dfrac{\left(x - 5\right) \left(x + 8\right)}{\left(x - 5\right) \left(x + 7\right)}\mathclose{.}$ The factor $x - 5$ is common to both and cancels, leaving $\dfrac{x + 8}{x + 7}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Cancelled the wrong bracket, keeping the common one → $\frac{x - 5}{x + 7}$
• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{3 x - 40}{2 x - 35}$

Answer: $-1$

Multiply both sides by $x - 2\mathclose{:}$ $3 x - 3 = 2\left(x - 2\right)\mathclose{,}$ which is $3 x - 3 = 2 x - 4\mathclose{.}$ Collect: $x = -1\mathclose{.}$ That is not $x = 2\mathclose{,}$ the value the equation forbids, so it stands.

Common mistakes, and the answer each one gives:

• Ignored the denominator and solved $3 x - 3 = 2$ → $\frac{5}{3}$
• Multiplied out $2\left(x - 2\right)$ as $2 x - 2\mathclose{,}$ multiplying only the $x$ → $1$

Answer: $(-2, -7)$

Factorise the numerator: $x^{2} - 3 x - 10 = \left(x + 2\right)\left(x - 5\right)\mathclose{.}$ The factor $\left(x + 2\right)$ cancels, so $f(x) = x - 5$ everywhere EXCEPT at $x = -2\mathclose{,}$ where the original is $\frac{0}{0}$ and undefined. The hole is at $x = -2\mathclose{,}$ at the height the simplified formula gives there: $y = -2 - 5 = -7\mathclose{.}$ So it is $(-2, -7)\mathclose{.}$

Common mistakes, and the answer each one gives:

• Read the hole off the factor $\left(x + 2\right)$ with the wrong sign → $(2, -3)$
• Gave the point where the graph meets the $x$-axis → $(5, 0)$

Answer: $x = 4$

Square both sides: $x + 60 = \left(x + 4\right)^{2}\mathclose{,}$ so $x^{2} + 7 x - 44 = 0$ and $x = 4$ or $x = -11\mathclose{.}$ SQUARING CAN ADD SOLUTIONS, so check both in the original: at $x = 4$ both sides are $8\mathclose{;}$ at $x = -11$ the right-hand side is $-7\mathclose{,}$ negative, while a square root never is. Only $x = 4$ survives.

Common mistakes, and the answer each one gives:

• Kept both solutions of the squared equation without checking them → $x = -11 \text{ or } x = 4$
• Gave the solution that fails the check → $x = -11$

Answer: $\dfrac{9 x - 7}{3 x - 2}$

Write $y = \dfrac{2 x - 7}{3 x - 9}$ and solve for $x\mathclose{:}$ $y\left(3 x - 9\right) = 2 x - 7\mathclose{,}$ so $3 x y - 9 y = 2 x - 7\mathclose{.}$ Gather the $x$ terms: $x\left(3 y - 2\right) = 9 y - 7\mathclose{,}$ so $x = \dfrac{9 y - 7}{3 y - 2}\mathclose{.}$ Swap the letters: $f^{-1}(x) = \dfrac{9 x - 7}{3 x - 2}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Gave the reciprocal $\frac{1}{f(x)}$ instead of the inverse → $\frac{3 x - 9}{2 x - 7}$
• Swapped $a$ and $d$ but did not change the signs of $b$ and $c$ → $\frac{- 9 x - 7}{3 x + 2}$

Answer: $-16$

The leading term of a product is the product of the leading terms: $\left(2 x\right)^{3} \cdot (-2x) = 8x^{3} \cdot (-2x) = -16x^{4}\mathclose{.}$ So the leading coefficient is $-16\mathclose{.}$

Common mistakes, and the answer each one gives:

• Did not raise the coefficient to its power → $-4$
• Took the leading term of $\left(7 - 2x\right)$ as $+2x$ → $16$

Answer: $2$

At $x = 0\mathclose{,}$ $f(0) = a \cdot b^{0} = a\mathclose{,}$ so $a = 5\mathclose{.}$ Then $5 \cdot b^{3} = 40\mathclose{,}$ so $b^{3} = 8$ and, as $b > 0\mathclose{,}$ $b = 2\mathclose{:}$ the graph is multiplied by $2$ at every step of $1$ to the right.

Common mistakes, and the answer each one gives:

• Gave $b^3\mathclose{,}$ the growth over all $3$ steps → $8$
• Shared the growth out by dividing by $3$ instead of taking the root → $\dfrac{8}{3}$

Answer: $\dfrac{7}{4}$

Write the right-hand side as a power of $2\mathclose{:}$ $8 = 2^{3}\mathclose{.}$ With the same base on both sides the exponents must be equal, so $4 x - 4 = 3$ and $x = \dfrac{7}{4}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Set the exponent equal to $8$ instead of to $3$ → $3$
• Forgot to move the constant in the exponent across → $\frac{3}{4}$

Answer: $134$

$\log_{5}(\text{something}) = 3$ means $\text{something} = 5^{3} = 125\mathclose{.}$ So $x - 9 = 125$ and $x = 134\mathclose{.}$ (Check: $x - 9$ is then positive, as a logarithm needs.)

Common mistakes, and the answer each one gives:

• Multiplied the base by $3$ instead of raising it to the power $3$ → $24$
• Forgot to add $9$ at the end → $125$
• Raised the exponent to the power of the base, $3^{5}$ → $252$

Answer: $8x\left(x^{2} - 8\right)^{3}$

Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside: $4\left(x^{2} - 8\right)^{3} \cdot 2x = 8x\left(x^{2} - 8\right)^{3}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Forgot to multiply by the derivative of the inside → $4\left(x^{2} - 8\right)^{3}$
• Differentiated the inside and raised THAT to the power → $16 x^{4}$

Answer: $6 x^{2} - 20$

Put $f(x)$ in place of every $x$ in the outer function: $g(f(x)) = 3\left(2 x^{2} - 5\right) - 5\mathclose{.}$ Altogether $g(f(x)) = 6 x^{2} - 20\mathclose{.}$

Common mistakes, and the answer each one gives:

• Composed the wrong way round, working out $f(g(x))$ → $18 x^{2} - 60 x + 45$
• Multiplied $f(x)$ by $g(x)$ instead of composing them → $6 x^{3} - 10 x^{2} - 15 x + 25$

Answer: $\dfrac{- 4 x^{2} + 10 x + 4}{\left(x^{2} + 1\right)^{2}}$

The quotient rule is $\dfrac{u'v - uv'}{v^2}$ — and the ORDER of the two products matters, unlike the product rule. With $u = 4 x - 5$ and $v = x^{2} + 1\mathclose{:}$ $u' = 4$ and $v' = 2x\mathclose{.}$ The numerator is $4\left(x^{2} + 1\right) - \left(4 x - 5\right) \cdot 2x = - 4 x^{2} + 10 x + 4\mathclose{,}$ so $f'(x) = \dfrac{- 4 x^{2} + 10 x + 4}{\left(x^{2} + 1\right)^{2}}\mathclose{.}$

Common mistakes, and the answer each one gives:

• Wrote $uv' - u'v$ in the numerator, the two products swapped → $\dfrac{4 x^{2} - 10 x - 4}{\left(x^{2} + 1\right)^{2}}$
• Differentiated top and bottom separately → $\frac{2}{x}$

Answer: $y = - 7 x - 6$

The slope is the derivative there: $f'(x) = 4 x - 3\mathclose{,}$ so $m = f'(-1) = -7\mathclose{.}$ The point is $(-1, f(-1)) = (-1, 1)\mathclose{.}$ So $y = -7\left(x + 1\right) + 1\mathclose{,}$ which is $y = - 7 x - 6\mathclose{.}$

Common mistakes, and the answer each one gives:

• Used the $y$-value $f(-1)$ as the slope → $y = x + 2$
• Wrote $y = f'(a)\,x + f(a)\mathclose{,}$ leaving out the shift to $x = a$ → $y = - 7 x + 1$

Answer: $2$

$f'(x) = 3 x^{2} + 6 x - 24 = 3\left(x + 4\right)\left(x - 2\right)\mathclose{,}$ which is zero at $x = -4$ and $x = 2\mathclose{.}$ $f''(x) = 6 x + 6\mathclose{,}$ and $f''(2) = 18 > 0\mathclose{,}$ so $x = 2$ is the local minimum.

Common mistakes, and the answer each one gives:

• Gave the local MAXIMUM instead → $-4$
• Gave the $y$-value rather than the $x$-coordinate → $-28$

Answer: $\text{increasing and getting less steep}$

$f'(-5) = 9$ is positive, so $f$ is increasing. $f''(-5) = -12\mathclose{:}$ the slope itself is falling, and a positive slope that is falling gets less steep.

Common mistakes, and the answer each one gives:

• Read the direction from the sign of $f(-5)$ instead of $f'(-5)$ → $\text{decreasing and getting less steep}$