ESAT Mathematics Mock Module 5 Worked Solutions

Correct Answer: C. $6$

Fastest Approach (🚀):
$x = 2$: $3A = 9 \implies A = 3$.
$x = -1$: $-3B = -6 \implies B = 2$.
$AB = 6$.

Matches Option C.

Step-by-Step Breakdown:

1. Use the licence the identity gives you

The statement is true for every $x$, so it is true for whichever value is most convenient. Pick the value that makes a bracket vanish.

2. Kill the B term

Set $x = 2$, so that $x-2 = 0$:
$A(3)+B(0) = 5(2)-1 = 9 \implies A = 3$

3. Kill the A term

Set $x = -1$, so that $x+1 = 0$:
$A(0)+B(-3) = 5(-1)-1 = -6 \implies B = 2$

4. Answer and check

$AB = 3\times 2 = 6$.
Check by comparing coefficients instead: $A+B = 5$ and $A-2B = -1$; $3+2 = 5$ and $3-4 = -1$. Both hold.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $1$ — Misread Question
  Giving $A-B$.
• B. $5$ — Misread Question
  Giving $A+B$ rather than $AB$.
• D. $-6$ — Sign Error
  Sign slip on $x = -1$, giving $B = -2$.
• E. $10$ — Algebraic Error
  Using $A = 5$, $B = 2$ from a mis-solved system.

Common Mistake (⚠️):
Solving the simultaneous system $A+B = 5$, $A-2B = -1$ by elimination. It gets there, but takes three times as long as two substitutions.

Takeaway (📌):
Strategic substitution is the identity shortcut: choose $x$ to zero a bracket, read the unknown straight off.

Correct Answer: E. Two pairs of equal sides and a pair of equal angles not between them

Fastest Approach (🚀):
Valid: $\text{SSS}$, $\text{SAS}$, $\text{ASA}$ (and $\text{AAS}$), $\text{RHS}$.
Option E is $\text{SSA}$, the ambiguous case.

Matches Option E.

Step-by-Step Breakdown:

1. List the criteria that work

$\text{SSS}$ — three sides.
$\text{SAS}$ — two sides and the included angle.
$\text{ASA}$ — two angles and a corresponding side; with two angles the third follows, so $\text{AAS}$ works too.
$\text{RHS}$ — right angle, hypotenuse and one other side.

2. Why SSA fails

Suppose two sides of lengths $a$ and $b$ are known along with an angle not between them. Placing the angle and the side $b$ fixes one arm; the side $a$ then swings like a compass and can cut the other arm at two distinct points. Both results have the same $\text{SSA}$ data but different third sides and different angles.

3. Why RHS is not a counter-example

$\text{RHS}$ looks like $\text{SSA}$ — the right angle is not between the hypotenuse and the other given side. It survives because Pythagoras fixes the third side uniquely, so no ambiguity remains.

4. And why AAA fails too

Equal angles give similar triangles, not congruent ones: shape is fixed but size is not.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. Three pairs of equal sides — Valid Criterion
  $\text{SSS}$ — valid.
• B. Two pairs of equal sides and the pair of equal angles between them — Valid Criterion
  $\text{SAS}$ with the included angle — valid.
• C. Two pairs of equal angles and a pair of equal corresponding sides — Valid Criterion
  $\text{ASA}$ or $\text{AAS}$ — valid, since the third angle follows.
• D. A right angle, equal hypotenuses and one other pair of equal sides — Valid Criterion
  $\text{RHS}$ — valid, because Pythagoras fixes the remaining side.

Common Mistake (⚠️):
Accepting $\text{SSA}$ because it lists three pieces of information. Congruence depends on which three, not how many.

Takeaway (📌):
$\text{SSS}$, $\text{SAS}$, $\text{ASA}$, $\text{RHS}$ prove congruence. $\text{SSA}$ is ambiguous and $\text{AAA}$ only gives similarity.

Correct Answer: D. $30$

Fastest Approach (🚀):
$6.1 \to 6$, $19.8 \to 20$, $3.9 \to 4$.
$\dfrac{6\times 20}{4} = 30$.

Matches Option D.

Step-by-Step Breakdown:

1. Round to one figure each

$6.1 \approx 6$, $19.8 \approx 20$, $3.9 \approx 4$.

2. Order the arithmetic to cancel

$\dfrac{6\times 20}{4}$. Cancel the $20$ against the $4$ first:
$6\times 5 = 30$

3. Check against the exact value

$\dfrac{6.1\times 19.8}{3.9} = \dfrac{120.78}{3.9} \approx 30.97$, so the estimate is within about $3\%$.

4. Which direction the estimate leans

Rounding the numerator down and the denominator up both push the estimate low, which is why the true value sits slightly above $30$. Tracking that direction turns an estimate into a bound.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $3$ — Place Value
  Estimating $\dfrac{6\times 2}{4}$ with a place-value slip on $19.8$.
• B. $12$ — Arithmetic Error
  Computing $\dfrac{6\times 20}{10}$ or similar.
• C. $300$ — Place Value
  Losing a factor of ten in the division.
• E. $24$ — Operation Error
  Estimating $6\times 4$ and ignoring the structure of the fraction.

Common Mistake (⚠️):
Rounding $3.9$ to $4$ but $19.8$ to $19$, leaving an awkward division. Round for convenience, not accuracy — the point is a one-line answer.

Takeaway (📌):
Estimate by rounding to one figure, then adjust one number so the division is exact. Note which way each rounding pushed the result.

Correct Answer: B. $320\ \text{m}$

Fastest Approach (🚀):
Trapezium with parallel sides $25$ and $7$, height $20$:
$\tfrac12(25+7)(20) = 16\times 20 = 320\ \text{m}$.

Matches Option B.

Step-by-Step Breakdown:

1. Recognise what the area means

On a speed-time graph the vertical axis is $\text{m/s}$ and the horizontal is $\text{s}$, so an area carries units of $\text{m/s}\times\text{s} = \text{m}$. The area is the distance travelled.

2. Split the shape

The graph is a trapezium: a rising triangle, a flat rectangle, then a falling triangle.
acceleration: $\tfrac12\times 10\times 20 = 100\ \text{m}$
constant speed: $7\times 20 = 140\ \text{m}$
deceleration: $\tfrac12\times 8\times 20 = 80\ \text{m}$

3. Total

$100+140+80 = 320\ \text{m}$

4. The one-line route

The whole figure is a trapezium of height $20$ with parallel sides equal to the total time $25\ \text{s}$ and the constant-speed time $7\ \text{s}$:
$\tfrac12(25+7)(20) = 320\ \text{m}$

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $240\ \text{m}$ — Omitted Region
  Omitting the deceleration triangle.
• C. $500\ \text{m}$ — Shape Misread
  Multiplying $20$ by the total time of $25\ \text{s}$.
• D. $250\ \text{m}$ — Shape Misread
  Treating the whole graph as one triangle, $\tfrac12(25)(20)$.
• E. $180\ \text{m}$ — Omitted Region
  Omitting the constant-speed rectangle.

Common Mistake (⚠️):
Multiplying the maximum speed by the total time to get $500\ \text{m}$. That would be the area of a rectangle, and the car is only at $20\ \text{m/s}$ for $7$ of the $25$ seconds.

Takeaway (📌):
Gradient of a speed-time graph is acceleration; area under it is distance. On a distance-time graph the gradient is speed and the area means nothing.

Correct Answer: A. $50^{\circ}$

Fastest Approach (🚀):
Same arc $AB$; centre angle is double the circumference angle.
$\tfrac12(100^{\circ}) = 50^{\circ}$.

Matches Option A.

Step-by-Step Breakdown:

1. Identify the arc both angles stand on

Angle $AOB$ is at the centre and angle $ACB$ is at the circumference. Both are subtended by the same arc $AB$ — the minor arc, since $C$ is on the major arc.

2. Apply the theorem

The angle subtended at the centre is twice the angle subtended at the circumference on the same arc:
$\angle AOB = 2\angle ACB$
$100^{\circ} = 2\angle ACB \implies \angle ACB = 50^{\circ}$

3. Where the other theorems come from

Take $AB$ as a diameter: the centre angle is $180^{\circ}$, so the circumference angle is $90^{\circ}$ — the angle in a semicircle. Take a second point $D$ on the major arc: it also gives $50^{\circ}$, which is why angles in the same segment are equal. Both are corollaries of this one result.

4. Check the direction

The circumference angle is the smaller one. Doubling instead of halving gives $200^{\circ}$, which is impossible inside a triangle.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $100^{\circ}$ — Theorem Misapplied
  Assuming the two angles are equal.
• C. $200^{\circ}$ — Direction Reversed
  Doubling rather than halving.
• D. $80^{\circ}$ — Wrong Theorem
  Using $180^{\circ}-100^{\circ}$ and mis-halving.
• E. $130^{\circ}$ — Wrong Arc
  Halving the reflex angle $260^{\circ}$, which stands on the major arc.

Common Mistake (⚠️):
Doubling instead of halving, or using $360^{\circ}-100^{\circ}$ and then halving to get $130^{\circ}$ — that reflex angle is the one standing on the major arc, not the arc $C$ sits opposite.

Takeaway (📌):
Centre angle $=2\times$ circumference angle on the same arc. Every other angle theorem in the circle follows from it.

Correct Answer: B. $-0.5,\ -\tfrac13,\ 0.3,\ \tfrac38$

Fastest Approach (🚀):
$-\tfrac13 = -0.33\ldots$ and $\tfrac38 = 0.375$.
$-0.5 \lt -0.33 \lt 0.3 \lt 0.375$.

Matches Option B.

Step-by-Step Breakdown:

1. Convert everything to one form

$-\tfrac13 = -0.333\ldots$ and $\tfrac38 = 0.375$. The list to order is now $-0.5,\ -0.333\ldots,\ 0.3,\ 0.375$.

2. Handle the negatives

On the number line $-0.5$ lies further from zero than $-\tfrac13$, so $-0.5$ is the smaller number:
$-0.5 \lt -\tfrac13$
The larger digit gives the smaller value once the sign is negative — the single most common slip in ordering questions.

3. Handle the close pair

$\tfrac38 = 0.375$ and $0.375 \gt 0.3$, so $\tfrac38$ comes last.

4. Assemble

$-0.5 \lt -\tfrac13 \lt 0.3 \lt \tfrac38$

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $-\tfrac13,\ -0.5,\ 0.3,\ \tfrac38$ — Sign Order
  Placing $-\tfrac13$ before $-0.5$ — the negatives are the wrong way round.
• C. $-0.5,\ -\tfrac13,\ \tfrac38,\ 0.3$ — Conversion Error
  Placing $\tfrac38$ before $0.3$, treating $\tfrac38$ as less than a third.
• D. $-\tfrac13,\ -0.5,\ \tfrac38,\ 0.3$ — Sign Order
  Both errors together.
• E. $0.3,\ \tfrac38,\ -\tfrac13,\ -0.5$ — Direction Reversed
  Descending order, with the negatives also reversed.

Common Mistake (⚠️):
Ordering $-\tfrac13$ before $-0.5$ because $3$ is smaller than $5$. Comparing the digits works only for positive numbers; for negatives the order reverses.

Takeaway (📌):
Convert to a single form, then order. Negatives reverse, and a fraction close to a decimal is worth converting rather than eyeballing.

Correct Answer: B. $(2,3)$

Fastest Approach (🚀):
Check the cheapest constraints first: $x \ge 1$ rules out $(0,2)$.
$y \gt x$ rules out $(4,1)$; $x+y \le 6$ rules out $(3,4)$; $y \le 2x+1$ rules out $(1,4)$.

Matches Option B.

Step-by-Step Breakdown:

1. Order the constraints by cost

$x \ge 1$ needs one glance. $y \gt x$ needs a comparison. The two involving arithmetic come last.

2. Eliminate

$(0,2)$: $x = 0$, so $x \ge 1$ fails.
$(4,1)$: $y \gt x$ becomes $1 \gt 4$ — fails.
$(3,4)$: $x+y = 7$, so $x+y \le 6$ fails.
$(1,4)$: $2x+1 = 3$ and $y = 4$, so $y \le 2x+1$ fails.

3. Confirm the survivor

$(2,3)$:
$y \le 2x+1$: $3 \le 5$ ✓
$x+y \le 6$: $5 \le 6$ ✓
$x \ge 1$: $2 \ge 1$ ✓
$y \gt x$: $3 \gt 2$ ✓

All four hold, so $(2,3)$ lies in the region.

4. If you had to sketch it

Each inequality is a half-plane. Solid boundaries for $\le$ and $\ge$, a dashed boundary for the strict $\gt$, and the region is the overlap of all four. Substitution avoids drawing four lines under time pressure.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $(1,4)$ — Constraint Missed
  Fails $y \le 2x+1$, since $4 \gt 3$.
• C. $(4,1)$ — Constraint Missed
  Fails the strict inequality $y \gt x$.
• D. $(3,4)$ — Constraint Missed
  Fails $x+y \le 6$, since $3+4 = 7$.
• E. $(0,2)$ — Constraint Missed
  Fails $x \ge 1$.

Common Mistake (⚠️):
Checking one or two inequalities and stopping. Every distractor here satisfies at least three of the four.

Takeaway (📌):
For a region defined by several inequalities, substitute rather than sketch, and check the cheapest constraint first.

Correct Answer: C. $\begin{pmatrix} 3 \\ 10 \end{pmatrix}$

Fastest Approach (🚀):
Top: $3(1)-2(0) = 3$.
Bottom: $3(2)-2(-2) = 6+4 = 10$.

Matches Option C.

Step-by-Step Breakdown:

1. Scale each vector

$3\mathbf{a} = 3\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \end{pmatrix}$
$2\mathbf{b} = 2\begin{pmatrix} 0 \\ -2 \end{pmatrix} = \begin{pmatrix} 0 \\ -4 \end{pmatrix}$

2. Subtract componentwise

$\begin{pmatrix} 3 \\ 6 \end{pmatrix}-\begin{pmatrix} 0 \\ -4 \end{pmatrix} = \begin{pmatrix} 3-0 \\ 6-(-4) \end{pmatrix} = \begin{pmatrix} 3 \\ 10 \end{pmatrix}$

3. The step that costs marks

Subtracting a negative component adds. The bottom row is $6-(-4) = 10$, not $6-4 = 2$.

4. A sanity check

$\mathbf{b}$ points straight down, so $-2\mathbf{b}$ points straight up and can only increase the vertical component. Any answer with a bottom row below $6$ must be wrong.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ — Sign Error
  Computing $6-4$ instead of $6-(-4)$.
• B. $\begin{pmatrix} 3 \\ 8 \end{pmatrix}$ — Scalar Omitted
  Using $\mathbf{b}$ rather than $2\mathbf{b}$ in the second row.
• D. $\begin{pmatrix} 1 \\ 10 \end{pmatrix}$ — Scalar Omitted
  Using $\mathbf{a}$ rather than $3\mathbf{a}$ in the first row.
• E. $\begin{pmatrix} -3 \\ -10 \end{pmatrix}$ — Order Reversed
  Computing $2\mathbf{b}-3\mathbf{a}$.

Common Mistake (⚠️):
Computing $6-4 = 2$ and answering $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ — the same double-negative slip that appears throughout vector work.

Takeaway (📌):
Scale first, then combine row by row, and read the signs of the components before subtracting.

Correct Answer: C. $30\%$

Fastest Approach (🚀):
Acid: $40+50 = 90\ \text{ml}$. Total: $300\ \text{ml}$.
$\dfrac{90}{300} = 30\%$.

Matches Option C.

Step-by-Step Breakdown:

1. Find the acid in each solution

$20\%$ of $200 = 40\ \text{ml}$
$50\%$ of $100 = 50\ \text{ml}$

2. Total the acid and the volume

acid: $40+50 = 90\ \text{ml}$
mixture: $200+100 = 300\ \text{ml}$

3. Express as a percentage

$\dfrac{90}{300} = \dfrac{3}{10} = 30\%$

4. Why the answer is not the midpoint

The plain average of $20\%$ and $50\%$ is $35\%$, which would be right only if equal volumes were mixed. Twice as much of the weak solution pulls the result towards $20\%$, and the answer lands one third of the way from $20$ to $50$ — matching the $2:1$ split of the volumes.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $35\%$ — Unweighted Average
  Averaging the two percentages, which assumes equal volumes.
• B. $25\%$ — Weights Swapped
  Weighting the wrong way round, towards the $100\ \text{ml}$ solution.
• D. $70\%$ — Operation Error
  Adding the percentages.
• E. $33\tfrac13\%$ — Wrong Base
  Dividing $100$ by $3$, or averaging over the wrong total.

Common Mistake (⚠️):
Averaging the two percentages to get $35\%$. A percentage is a rate, and rates can only be averaged when the amounts they apply to are equal.

Takeaway (📌):
Mixing questions are weighted means: multiply each rate by its own volume, add, then divide by the combined volume.

Correct Answer: D. $y = \dfrac{3}{x}$

Fastest Approach (🚀):
Two perpendicular asymptotes $\Rightarrow$ reciprocal.
Positive constant $\Rightarrow$ first and third quadrants.

Matches Option D.

Step-by-Step Breakdown:

1. Use the asymptotes to fix the family

A vertical asymptote means the function is undefined at one $x$-value; a horizontal asymptote means it approaches a fixed height without reaching it. Only a reciprocal curve $y = \dfrac{k}{x}$ has both, at $x = 0$ and $y = 0$.

2. Use the quadrants to fix the sign

Rearranged, $y = \dfrac{3}{x}$ says $xy = 3$. A positive product means $x$ and $y$ carry the same sign — both positive (first quadrant) or both negative (third). With $k = -3$ the product is negative and the curve sits in the second and fourth instead.

3. Rule out the others

$y = 3x$ is a straight line through the origin, with no asymptotes.
$y = 3^{x}$ has the horizontal asymptote $y = 0$ but no vertical one, and lies above the $x$-axis for every $x$.
$y = x^{2}+3$ is a parabola with a minimum at $(0,3)$ and no asymptotes at all.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $y = 3x$ — Family Misread
  A straight line through the origin — no asymptotes.
• B. $y = -\dfrac{3}{x}$ — Sign Error
  The right family, but the negative constant puts it in the second and fourth quadrants.
• C. $y = 3^{x}$ — Family Misread
  One asymptote only, and always above the x-axis.
• E. $y = x^{2}+3$ — Family Misread
  A parabola, with a minimum rather than an asymptote.

Common Mistake (⚠️):
Choosing $y = 3^{x}$ because it also approaches the $x$-axis. An exponential curve has one asymptote, not two, and never enters the third quadrant.

Takeaway (📌):
Count the asymptotes to name the family, then use one sign or one point to pin down the constant.

Correct Answer: C. Rhombus

Fastest Approach (🚀):
Bisect + perpendicular $\Rightarrow$ square or rhombus.
Not equal $\Rightarrow$ rhombus.

Matches Option C.

Step-by-Step Breakdown:

1. Take the three conditions separately

Bisect each other — true for any parallelogram, so square, rectangle, rhombus and parallelogram all qualify. A kite does not: one diagonal bisects the other, but not the reverse.

At right angles — true for the square, the rhombus and the kite; false for the rectangle and the general parallelogram.

Not equal in length — the square and the rectangle have equal diagonals, so both are excluded.

2. Intersect the three

Bisecting and perpendicular leaves the square and the rhombus. Unequal diagonals rules out the square.

3. Read it back

A rhombus is a parallelogram with four equal sides. Its diagonals bisect each other because it is a parallelogram, and meet at right angles because it is equilateral — but they are equal only in the special case where all four angles are right angles, which is the square.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. Square — Over-specified
  Satisfies the first two conditions but has equal diagonals.
• B. Rectangle — Property Confused
  Diagonals bisect and are equal, but do not meet at right angles.
• D. Kite — Property Confused
  Diagonals are perpendicular, but only one bisects the other.
• E. Parallelogram — Property Confused
  Diagonals bisect each other but are not generally perpendicular.

Common Mistake (⚠️):
Choosing the kite. Its diagonals do meet at right angles, but only one of them is bisected — the axis of symmetry cuts the other in half, not the other way round.

Takeaway (📌):
Diagonals: parallelograms bisect, rhombi and kites are perpendicular, rectangles and squares are equal. The square is the only shape with all three.

Correct Answer: C. $16$

Fastest Approach (🚀):
$3.9\pi \approx 4\times 3 = 12$.
$\sqrt{17}\approx 4$.
$12+4 = 16$.

Matches Option C.

Step-by-Step Breakdown:

1. Estimate the π term

$\pi \approx 3$ and $3.9 \approx 4$, so $3.9\pi \approx 12$.

2. Estimate the surd

$17$ sits just above the perfect square $16$, so $\sqrt{17}\approx 4$.

3. Add

$12+4 = 16$

4. Check the direction of each rounding

$\pi$ was rounded down from $3.14$ and $3.9$ up to $4$; those roughly cancel, since $4\times 3 = 12$ against a true $12.25$. The surd was rounded down from $4.12$. Both effects are small and downward, so the true value should sit a little above $16$ — and $3.9\pi+\sqrt{17} = 16.37$ to two decimal places.

Matches Option C.

Why the Other Options Are Wrong (❌):

• A. $20$ — Rounding Error
  Using $\pi \approx 4$, or $\sqrt{17}\approx 8$.
• B. $12$ — Omitted Term
  Estimating the $\pi$ term only and dropping the surd.
• D. $8$ — Rounding Error
  Using $\pi \approx 1$, giving $4+4$.
• E. $24$ — Rounding Error
  Using $\sqrt{17}\approx 12$ or double-counting the first term.

Common Mistake (⚠️):
Rounding $\sqrt{17}$ to $17$, or treating $\pi$ as $\approx 1$. Every surd should go to the nearest perfect square, not to the number under the root.

Takeaway (📌):
For estimates: $\pi \approx 3$, and a surd goes to the nearest perfect square. Then note which way each rounding pushed the answer.

Correct Answer: E. $12$

Fastest Approach (🚀):
Out $4\ \text{km}$, back $4\ \text{km}$: $8\ \text{km}$ in $40$ min.
$8\times\dfrac{60}{40} = 12\ \text{km/h}$.

Matches Option E.

Step-by-Step Breakdown:

1. Read what each section means

Rising: moving away from home, at $\dfrac{4}{10}$ km per minute $= 24\ \text{km/h}$.
Level: the distance from home is not changing — the cyclist has stopped.
Falling: returning home, at $\dfrac{4}{15}$ km per minute $= 16\ \text{km/h}$.

2. Total the distance actually travelled

$4\ \text{km}$ out and $4\ \text{km}$ back is $8\ \text{km}$, even though the cyclist ends where they started.

3. Divide by the total time

$40$ minutes $= \tfrac23$ of an hour:
$\text{average speed} = \dfrac{8}{\tfrac23} = 12\ \text{km/h}$

4. Speed against velocity

The displacement is zero, so the average velocity is $0\ \text{km/h}$. Average speed uses distance travelled and is $12\ \text{km/h}$. The question asks for speed, and the stop is included in the time — which is why the answer sits below both leg speeds.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. $6$ — Displacement Used
  Using the $4\ \text{km}$ net displacement instead of the $8\ \text{km}$ travelled.
• B. $0$ — Wrong Quantity
  Giving the average velocity, which is zero over a round trip.
• C. $16$ — Single Leg
  Giving the speed of the return leg only.
• D. $8$ — Wrong Quantity
  Giving the distance in km rather than a speed.

Common Mistake (⚠️):
Averaging the two leg speeds, $\tfrac{24+16}{2} = 20\ \text{km/h}$, which ignores the stop and weights the legs equally though they take different times.

Takeaway (📌):
Distance-time: gradient is speed, flat is stopped, falling is returning. Average speed is always total distance over total time, stops included.

Correct Answer: E. $34$

Fastest Approach (🚀):
Co-interior $\Rightarrow$ sum to $180$.
$5x+10 = 180 \implies x = 34$.

Matches Option E.

Step-by-Step Breakdown:

1. Choose the right fact

On parallel lines cut by a transversal:
alternate angles (a Z shape) are equal
corresponding angles (an F shape) are equal
co-interior or allied angles (a C or U shape) sum to $180^{\circ}$

The two angles here are described as co-interior, so they add to $180^{\circ}$.

2. Form and solve the equation

$(3x+10)+(2x) = 180$
$5x+10 = 180$
$5x = 170$
$x = 34$

3. Check

$3(34)+10 = 112^{\circ}$ and $2(34) = 68^{\circ}$, and $112+68 = 180$. ✓

Had the question said alternate, the equation would be $3x+10 = 2x$, giving $x = -10$ — an impossible angle, and a useful sign that the wrong fact has been used.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. $38$ — Equation Error
  Solving $3x+10 = 2x+38$ or another mis-set equation.
• B. $44$ — Wrong Fact
  Treating the angles as summing to $220$, or as angles at a point.
• C. $36$ — Arithmetic Error
  Using a sum of $190$ or dropping the $+10$.
• D. $30$ — Arithmetic Error
  Using $5x = 150$ from a sum of $160$.

Common Mistake (⚠️):
Setting the two expressions equal because most parallel-line facts are equalities. Co-interior is the one that sums instead.

Takeaway (📌):
Z equal, F equal, C sums to $180^{\circ}$. Name the shape before writing the equation, and sanity-check that both angles come out positive and under $180^{\circ}$.

Correct Answer: A. $7$

Fastest Approach (🚀):
$96\div 4 = 24$, then $24-17 = 7$.

Matches Option A.

Step-by-Step Breakdown:

1. Work backwards through the chain

Forwards, the number was increased by 17, then multiplied by 4.

Backwards, undo them in the opposite order: divide by $4$ first, then subtract $17$.

2. Apply

$96\div 4 = 24$
$24-17 = 7$

3. Check forwards

$7+17 = 24$, and $24\times 4 = 96$ ✓

4. Why the order reverses

Each operation is undone by its inverse, but the last thing done must be the first thing undone. Subtracting $17$ first would give $96-17 = 79$, which is undoing an operation that had not yet been applied at that stage — and that is the error behind two of the options.

The same reversal governs changing the subject of a formula: strip the outermost operation first and work inwards.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $41$ — Inverse Wrong
  Computing $96\div 4+17$ — the wrong inverse for the addition.
• C. $24$ — Incomplete
  Stopping after undoing the multiplication.
• D. $379$ — Inverses Reversed
  Computing $96\times 4-17$ — both inverses wrong.
• E. $11$ — Order Wrong
  Undoing in the wrong order: $(96-17)\div 4$ rounded.

Common Mistake (⚠️):
Undoing in the same order as the operations were applied, giving $(96-17)\div 4$. The inverses must be applied in reverse order.

Takeaway (📌):
Reverse the operations and reverse their order. Always check by running the chain forwards.

Correct Answer: A. $17$

Fastest Approach (🚀):
$u_1 = 3$, $u_2 = 5$, $u_3 = 9$, $u_4 = 17$.

Matches Option A.

Step-by-Step Breakdown:

1. Apply the rule once at a time

$u_2 = 2(3)-1 = 5$
$u_3 = 2(5)-1 = 9$
$u_4 = 2(9)-1 = 17$

2. Count the steps, not the terms

Reaching $u_4$ from $u_1$ takes three applications of the rule, not four. Writing the index beside each value is what stops the off-by-one.

3. Term-to-term against position-to-term

This rule tells you how to get from one term to the next, so every earlier term has to be computed. A position-to-term rule such as $u_n = 2^{n}+1$ would give $u_4$ directly — and in fact describes this same sequence, since $3,5,9,17 = 2^{1}+1,\ 2^{2}+1,\ 2^{3}+1,\ 2^{4}+1$.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $9$ — Off by One
  Stopping at $u_3$.
• C. $15$ — Rule Misapplied
  Computing $2u_3-3$ or another mis-applied rule.
• D. $33$ — Off by One
  Going one step too far, to $u_5$.
• E. $11$ — Rule Misread
  Using $u_{n+1} = u_n+2n$ or another misread rule.

Common Mistake (⚠️):
Applying the rule four times and giving $u_5 = 33$, or stopping at $u_3 = 9$.

Takeaway (📌):
Follow a term-to-term rule step by step and label each index as you go. The error in these questions is almost always the count, not the arithmetic.

Correct Answer: A. $(-8,-2)$

Fastest Approach (🚀):
$-2\times 4 = -8$ and $-2\times 1 = -2$.

Matches Option A.

Step-by-Step Breakdown:

1. Enlargement about the origin

Enlarging by scale factor $k$ about the origin maps $(x,y)$ to $(kx,ky)$. Every distance from the centre is multiplied by $|k|$.

2. Apply $k = -2$

$(4,1)\mapsto(-2\times 4,\ -2\times 1) = (-8,-2)$

3. What the negative sign does geometrically

The image lands on the opposite side of the centre, twice as far out. The shape is inverted — equivalent to enlarging by $+2$ and then rotating $180^{\circ}$ about the centre. Lengths double; the shape stays similar, and orientation reverses.

4. A check that catches sign errors

The centre, the object and the image must be collinear. $(0,0)$, $(4,1)$ and $(-8,-2)$ all lie on $y = \tfrac14 x$, and the image is on the far side. ✓

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $(8,2)$ — Sign Ignored
  Ignoring the negative sign.
• C. $(-2,-0.5)$ — Reciprocal Error
  Dividing by $2$ rather than multiplying.
• D. $(-8,2)$ — Partial Sign
  Applying the sign to one coordinate only — a reflection, not an enlargement.
• E. $(2,-8)$ — Wrong Transformation
  Swapping the coordinates, as a rotation would.

Common Mistake (⚠️):
Applying the factor $2$ and ignoring the sign, giving $(8,2)$ — the correct distance but the wrong side of the centre.

Takeaway (📌):
Enlargement about the origin: multiply both coordinates by $k$. Negative $k$ means the image is inverted through the centre, not reflected in an axis.

Correct Answer: D. $0.38$

Fastest Approach (🚀):
$\tfrac38 = 0.375$; $37.5\% = 0.375$; $\tfrac{15}{40} = \tfrac38$.
$0.38$ is the odd one.

Matches Option D.

Step-by-Step Breakdown:

1. Convert each to a decimal

$\tfrac38 = 3\div 8 = 0.375$
$0.375$ — already decimal
$37.5\% = 37.5\div 100 = 0.375$
$0.38$ — already decimal
$\tfrac{15}{40}$: divide top and bottom by $5$ to get $\tfrac38 = 0.375$

2. Compare

Four of them are exactly $0.375$. Only $0.38$ differs, by $0.005$.

3. Why it is worth converting rather than eyeballing

$0.38$ and $0.375$ look almost identical, and $\tfrac{15}{40}$ looks unrelated to $\tfrac38$ until it is simplified. Neither similarity nor difference in appearance is a reliable guide, which is exactly what the question is testing.

4. The conversions worth knowing instantly

$\tfrac18 = 0.125$, $\tfrac14 = 0.25$, $\tfrac38 = 0.375$, $\tfrac12 = 0.5$, $\tfrac58 = 0.625$, $\tfrac34 = 0.75$, $\tfrac13 = 0.333\ldots$, $\tfrac15 = 0.2$.

The eighths recur constantly and are the ones most often converted wrongly under time pressure.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $\tfrac38$ — Equivalent
  Equals $0.375$.
• B. $0.375$ — Equivalent
  Equals $0.375$.
• C. $37.5\%$ — Equivalent
  Equals $0.375$.
• E. $\tfrac{15}{40}$ — Equivalent
  Simplifies to $\tfrac38$, so equals $0.375$.

Common Mistake (⚠️):
Assuming $\tfrac{15}{40}$ is different because it looks different. Simplify every fraction before comparing anything.

Takeaway (📌):
Convert all forms to decimals to compare. A percentage is a fraction over $100$; a fraction is a division.

Correct Answer: E. the $9$th

Fastest Approach (🚀):
$n^{2}-3n-40 = 0 \implies (n-8)(n+5) = 0 \implies n = 8$.
$n = 8$ gives exactly $40$, so take $n = 9$.

Matches Option E.

Step-by-Step Breakdown:

1. Solve the boundary equation

$n^{2}-3n = 40$
$n^{2}-3n-40 = 0$
$(n-8)(n+5) = 0$
$n = 8$ or $n = -5$, and only $n = 8$ is a valid position.

2. Read the boundary carefully

At $n = 8$: $64-24 = 40$. The question asks for the first term to exceed $40$, and $40$ does not exceed itself.

3. Test the next term

At $n = 9$: $81-27 = 54 \gt 40$. ✓

So the $9$th term is the first above $40$.

4. Why a position-to-term rule matters here

The rule gives any term directly, so nothing before the $8$th ever has to be computed. Had this been a term-to-term rule, all eight earlier terms would have been needed first.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. the $8$th — Boundary Error
  Giving the solution of the equation, which equals $40$ rather than exceeding it.
• B. the $6$th — Misread Question
  Solving $n^{2}-3n = 18$ or misreading the threshold.
• C. the $7$th — Under-shoot
  Testing $n = 7$, which gives $28$.
• D. the $10$th — Off by One
  Overshooting by one after spotting the boundary.

Common Mistake (⚠️):
Answering the $8$th because the equation gave $n = 8$. That is the term equal to $40$, not the first one past it — the whole question turns on strict inequality.

Takeaway (📌):
Solve the equality, then test the integers on each side. When the boundary lands exactly on an integer, 'exceeds' and 'reaches' give different answers.

Correct Answer: E. $18$

Fastest Approach (🚀):
$n$-gonal prism: $3n$ edges, $2n$ vertices, $n+2$ faces.
$3\times 6 = 18$.

Matches Option E.

Step-by-Step Breakdown:

1. Count from the cross-section

A hexagonal prism is two hexagons joined by vertical edges.
edges of the two hexagons: $6+6 = 12$
vertical edges, one per corner: $6$
total: $18$

2. The general rule

For a prism whose cross-section is an $n$-gon:
faces $= n+2$ (the $n$ rectangles plus two ends)
vertices $= 2n$
edges $= 3n$

With $n = 6$: $8$ faces, $12$ vertices, $18$ edges.

3. Check with Euler's formula

For any convex polyhedron, $F-E+V = 2$:
$8-18+12 = 2$ ✓

That check is worth running whenever a count is uncertain — it fails immediately if a face or an edge has been double-counted.

Matches Option E.

Why the Other Options Are Wrong (❌):

• A. $12$ — Omitted Edges
  The number of vertices, or the two hexagons with the joining edges omitted.
• B. $20$ — Counting Error
  Adding faces and vertices, or an uncounted sketch.
• C. $24$ — Over-count
  Using $4n$, as though each corner carried two joining edges.
• D. $8$ — Wrong Quantity
  The number of faces.

Common Mistake (⚠️):
Counting $12$ — the two hexagons only — and forgetting the six edges that join them. Or answering $12$ because that is the vertex count.

Takeaway (📌):
$n$-gonal prism: $3n$ edges, $2n$ vertices, $n+2$ faces. Confirm any count with $F-E+V = 2$.

Correct Answer: A. $2\ \text{km}$

Fastest Approach (🚀):
$8\times 25\,000 = 200\,000\ \text{cm}$.
$\div 100\,000 = 2\ \text{km}$.

Matches Option A.

Step-by-Step Breakdown:

1. Apply the scale

$1:25\,000$ means one unit on the map is $25\,000$ of the same unit in reality. The map distance is in centimetres, so the real distance is in centimetres too:
$8\times 25\,000 = 200\,000\ \text{cm}$

2. Convert to kilometres

$1\ \text{km} = 1000\ \text{m} = 100\,000\ \text{cm}$

$\dfrac{200\,000}{100\,000} = 2\ \text{km}$

3. Why the scale carries no units

A ratio compares like with like, so $1:25\,000$ holds whether the measurement is in centimetres, inches or anything else. That is why the conversion belongs at the end, after the multiplication, rather than in the middle.

4. The useful shortcut for this scale

At $1:25\,000$, one centimetre is $250\ \text{m}$, so $4\ \text{cm}$ is $1\ \text{km}$. Eight centimetres is therefore $2\ \text{km}$ — the same answer with no large numbers at all.

5. Going the other way

To find a map distance from a real one, divide by the scale instead. Confusing the two directions gives answers thousands of times too large or too small, which is what Option D is built from.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $20\ \text{km}$ — Conversion Error
  Dividing by $10\,000$ instead of $100\,000$.
• C. $0.2\ \text{km}$ — Conversion Error
  Dividing by $10^{6}$.
• D. $200\ \text{km}$ — Conversion Error
  Converting before scaling, or dividing by 1000.
• E. $3.125\ \text{km}$ — Direction Reversed
  Dividing by the scale rather than multiplying.

Common Mistake (⚠️):
Converting to kilometres before applying the scale, or dividing by $100$ instead of $100\,000$. There are $100\,000$ centimetres in a kilometre, not $1000$.

Takeaway (📌):
Multiply the map distance by the scale to get the real distance in the same unit, then convert once at the end.

Correct Answer: B. $2n-3$

Fastest Approach (🚀):
Twice $n$ is $2n$; three less than it is $2n-3$.

Matches Option B.

Step-by-Step Breakdown:

1. Build it in pieces

'Twice a number $n$' is $2n$.
'Three less than' that quantity means subtract $3$ from it: $2n-3$.

2. Why the order in the words is misleading

The phrase says 'three' before 'twice a number', but the three is being taken away from the doubled number. So the expression is $2n-3$, not $3-2n$.

The test that settles it: substitute a value. With $n = 10$, twice the number is $20$, and three less than $20$ is $17$. Only $2n-3$ gives $17$; $3-2n$ gives $-17$.

3. The other phrasings worth distinguishing

'three less than twice $n$' — $2n-3$
'twice, three less than $n$' or 'twice the result of subtracting three from $n$' — $2(n-3)$
'three more than twice $n$' — $2n+3$

The bracket in the second changes the answer, because the subtraction happens before the doubling.

4. Always check with a number

Substituting a convenient value into each candidate is faster than parsing the English, and it is decisive.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $3-2n$ — Order Reversed
  Subtracting the doubled number from three — the reverse.
• C. $2(n-3)$ — Bracket Added
  Subtracts first and then doubles.
• D. $3n-2$ — Numbers Swapped
  Multiplies by three and subtracts two.
• E. $2n+3$ — Sign Wrong
  Three more, not three less.

Common Mistake (⚠️):
Writing $3-2n$ because 'three' comes first in the sentence. 'Less than' means subtract from the quantity that follows it.

Takeaway (📌):
Build the expression in pieces, then test it with a number. 'Less than' reverses the reading order; a bracket changes which operation happens first.

Correct Answer: D. A cylinder standing on one of its circular ends

Fastest Approach (🚀):
Circular plan $\Rightarrow$ circular cross-section.
Rectangular elevations $\Rightarrow$ constant width with height.

Matches Option D.

Step-by-Step Breakdown:

1. What the plan tells you

Looking straight down, the outline is a circle. So the widest horizontal cross-section is circular — which rules out the cuboid and the square-based pyramid immediately.

2. What the elevations tell you

Looking from the front, the outline is a rectangle: the width does not change as the solid rises. A cone would narrow to a point and show a triangle; a sphere would show a circle from every direction.

3. The only survivor

A cylinder standing on a circular end: circle from above, and the same rectangle from the front and from the side, since it looks identical from every horizontal direction.

4. The case worth noticing

Lay the same cylinder on its side and the views swap over: the plan becomes a rectangle and one elevation becomes a circle. The orientation is part of the answer, not an afterthought.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. A cone — Elevation Misread
  Circular plan, but the elevations are triangles.
• B. A sphere — View Misread
  Every view of a sphere is a circle.
• C. A cuboid — Plan Misread
  Rectangular elevations, but a rectangular plan too.
• E. A square-based pyramid — Plan Misread
  A square plan and triangular elevations.

Common Mistake (⚠️):
Choosing the cone, on the grounds that it too has a circular plan. Its elevations are triangles, because it narrows with height.

Takeaway (📌):
Plan from above, front and side elevations from the two horizontal directions. Work through them in turn and eliminate; a solid that survives all three is the answer.

Correct Answer: B. $\tfrac14$

Fastest Approach (🚀):
$3\ \text{h} = 180\ \text{min}$.
$\dfrac{45}{180} = \dfrac14$.

Matches Option B.

Step-by-Step Breakdown:

1. Convert to a common unit

$3$ hours $= 3\times 60 = 180$ minutes.

2. Write the fraction

The quantity being expressed goes on top; what it is being expressed as a fraction of goes on the bottom:
$\dfrac{45}{180}$

3. Simplify

$\dfrac{45}{180} = \dfrac{9}{36} = \dfrac{1}{4}$

Or in one step, since $180 = 4\times 45$.

4. Sanity-check

Forty-five minutes is three quarters of an hour, and there are three hours, so it should be a quarter of the whole. ✓

5. Why the units must match first

Writing $\dfrac{45}{3}$ compares minutes with hours and means nothing — it is Option A, and it is the error the question exists to catch. A fraction of one quantity by another is only meaningful when both are measured the same way.

Matches Option B.

Why the Other Options Are Wrong (❌):

• A. $\tfrac{45}{3}$ — Units Not Converted
  Comparing minutes with hours directly.
• C. $\tfrac{1}{15}$ — Inverted
  Using $\dfrac{3}{45}$ and simplifying.
• D. $\tfrac{3}{4}$ — Wrong Denominator
  Forty-five minutes as a fraction of one hour, not three.
• E. $\tfrac{4}{1}$ — Inverted
  Inverting the comparison: writing $3$ hours as a fraction of $45$ minutes, $\tfrac{180}{45} = 4$.

Common Mistake (⚠️):
Writing $\dfrac{45}{3}$ without converting. Mixed units make the fraction meaningless, however it is then simplified.

Takeaway (📌):
Convert to a common unit, then put the stated quantity over the quantity it is a fraction of, and simplify.

Correct Answer: A. $3x^{3}y^{2}$

Fastest Approach (🚀):
Numerator: $27x^{6}y^{3}$.
$\dfrac{27}{9} = 3$; $x^{6-3} = x^{3}$; $y^{3-1} = y^{2}$.

Matches Option A.

Step-by-Step Breakdown:

1. Expand the bracket

The outer index applies to every factor inside, the number included:
$\left(3x^{2}y\right)^{3} = 3^{3}\times\left(x^{2}\right)^{3}\times y^{3} = 27x^{6}y^{3}$

Note $\left(x^{2}\right)^{3} = x^{6}$ — indices multiply when a power is raised to a power.

2. Divide

Coefficients divide; indices subtract:
$\dfrac{27}{9} = 3$
$x^{6-3} = x^{3}$
$y^{3-1} = y^{2}$

$= 3x^{3}y^{2}$

3. The two places this goes wrong

Forgetting to cube the 3. Writing $3x^{6}y^{3}$ for the numerator leaves the coefficient untouched and gives Option D after dividing.

Forgetting the invisible index on $y$. In the denominator $y$ means $y^{1}$, so the subtraction is $3-1 = 2$, not $3-0 = 3$.

4. The laws in play

$a^{m}\times a^{n} = a^{m+n}$
$\dfrac{a^{m}}{a^{n}} = a^{m-n}$
$\left(a^{m}\right)^{n} = a^{mn}$
$(ab)^{n} = a^{n}b^{n}$

The last of those is the one being tested by the coefficient.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $3x^{6}y^{3}$ — Incomplete
  Expanding the numerator but not dividing.
• C. $9x^{3}y^{2}$ — Arithmetic Error
  Dividing $27$ by $3$ instead of $9$.
• D. $\dfrac{x^{3}y^{2}}{3}$ — Coefficient Ignored
  Not raising the coefficient to the power.
• E. $27x^{3}y^{2}$ — Coefficient Ignored
  Dividing the letters but not the coefficient.

Common Mistake (⚠️):
Leaving the coefficient uncubed, or treating a bare $y$ as having index zero. Every factor inside the bracket takes the outer power, and a bare letter carries an index of one.

Takeaway (📌):
Raise every factor including the number, multiply indices for a power of a power, then subtract indices to divide.

Correct Answer: D. $75^{\circ}$

Fastest Approach (🚀):
Opposite angles sum to $180^{\circ}$.
$180^{\circ}-105^{\circ} = 75^{\circ}$.

Matches Option D.

Step-by-Step Breakdown:

1. Identify the opposite pair

In quadrilateral $ABCD$ the vertices run round the circle in order, so $B$ and $D$ are opposite, as are $A$ and $C$.

2. Apply the theorem

Opposite angles of a cyclic quadrilateral sum to $180^{\circ}$:
$\angle ABC+\angle ADC = 180^{\circ}$
$\angle ADC = 180^{\circ}-105^{\circ} = 75^{\circ}$

3. Where the theorem comes from

The two angles stand on the two arcs $AC$, which together make the full circle. By the angle-at-the-centre theorem they are half of the two centre angles, and those centre angles sum to $360^{\circ}$ — so the two circumference angles sum to $180^{\circ}$.

4. The immediate corollary

An exterior angle of a cyclic quadrilateral equals the interior angle at the opposite vertex, since both are $180^{\circ}$ minus the same angle. That is the same fact wearing a different hat.

Matches Option D.

Why the Other Options Are Wrong (❌):

• A. $105^{\circ}$ — Wrong Theorem
  Treating opposite angles as equal, as in a parallelogram.
• B. $52.5^{\circ}$ — Wrong Theorem
  Halving, as though this were the angle at the centre.
• C. $95^{\circ}$ — Arithmetic Error
  Using $200^{\circ}$ as the sum.
• E. $255^{\circ}$ — Wrong Sum
  Subtracting from $360^{\circ}$.

Common Mistake (⚠️):
Assuming opposite angles are equal, as they are in a parallelogram. On a circle they are supplementary instead.

Takeaway (📌):
Cyclic quadrilateral: opposite angles sum to $180^{\circ}$, and each exterior angle equals the opposite interior angle.

Correct Answer: A. $-9$

Fastest Approach (🚀):
$(-2)(7) = -14$, then $5-14 = -9$.

Matches Option A.

Step-by-Step Breakdown:

1. Substitute with brackets

$v = u+at = 5+(-2)(7)$

Writing the negative value in brackets is what prevents the sign being lost or misapplied.

2. Evaluate

$(-2)\times 7 = -14$
$v = 5+(-14) = 5-14 = -9$

3. What the answer means

The acceleration is negative, so the object is slowing. Starting at $5$ and losing $2$ per second, it reaches zero after $2.5$ seconds and is travelling at $-9$ in the opposite direction by $7$ seconds. A negative answer is expected here, not a warning sign.

4. Order of operations

Multiplication before addition, so $at$ is evaluated first. Computing $u+a$ and then multiplying by $t$ gives $(5-2)\times 7 = 21$, which is a different formula entirely.

5. Why $19$ is offered

Treating $a$ as $+2$ gives $5+14 = 19$. Dropping a minus sign is the single most common substitution error, and bracketing is the habit that stops it.

Matches Option A.

Why the Other Options Are Wrong (❌):

• B. $19$ — Sign Lost
  Treating the acceleration as positive.
• C. $9$ — Sign Error
  Sign error at the final subtraction.
• D. $-19$ — Sign Error
  Computing $-(5+14)$.
• E. $70$ — Order of Operations
  Computing $(u+a)t$ — addition before multiplication.

Common Mistake (⚠️):
Losing the minus sign on $a$ and computing $5+14$. Substitute negatives inside brackets, every time.

Takeaway (📌):
Bracket negative values on substitution, and respect the order of operations — multiplication before addition.