Why did I lose the mark?

Why it loses marksA prime has exactly two factors. 1 has only one factor, itself, so it fails the definition. This is a standard one-mark trap.

Why it loses marksGuessing from memory rather than listing or factorising. The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12, so the highest is 12 - twice the answer given.

Why it loses marks36 is not a multiple of 24: 24 x 1.5 = 36, and a multiple needs a whole number. The LCM is 72.

Why it loses marksGoing on for ever is not what makes a number irrational - never repeating is. A recurring decimal is a fraction in disguise.

Why it loses marksThis counts the elements in the intersection twice. When the sets overlap, the sum is too big by n(A intersection B).

Why it loses marksThe minus sign in the index means reciprocal, not negative. It moves the power to the bottom of a fraction; it does not change the sign of the answer.

Why it loses marksThe root sign on its own gives the positive root only. The $\pm$ belongs to solving an equation such as $x^2=36$, not to evaluating a root.

Why it loses marksMultiplying the powers does not multiply the base. The base stays the same and the indices are added.

Why it loses marksA bracket means a power of a power, so the indices are multiplied: 3 x 2 = 6. Adding them is the rule for multiplying two powers of the same base, which is a different question.

Why it loses marksThe denominator of a fractional index is a root, not a divisor. $\frac{2}{3}$ means cube root then square.

Why it loses marksThe square factor taken outside the root is $\sqrt{25}=5$, not 25. The number in front of a surd is its square root.

Why it loses marksRoots of different numbers cannot be added that way. The rule $\sqrt{a}\times \sqrt{b}=\sqrt{ab}$ is for multiplication only.

Why it loses marksCambridge states that candidates are expected to write fractions in their simplest form, so a common factor left in the answer loses the final mark even though the value is correct.

Why it loses marksA 15% increase makes the amount larger, so the change is added. The multiplier route removes the decision altogether, which is why it is safer under pressure.

Why it loses marksMultiplying the rate by the number of years is simple interest. Compound interest multiplies by the same factor once for each year, so it needs a power.

Why it loses marksThe 20% was calculated on the original price, not on 96. Taking a percentage off the new figure always gives an answer that is too small.

Why it loses marksWriting a recurring decimal over 100 treats it as if it terminated. The repeating block needs a denominator of 99, because subtracting gives $100x-x=99x$.

Why it loses marksThe addition was done first because it is written first. Division has a higher priority, so it must be done before the addition.

Why it loses marksThe multiplication was done before the division. Division and multiplication have equal priority, so the calculation is worked from left to right.

Why it loses marksThe numerators and the denominators were both added. The denominator is the size of the part, not a quantity, so it must not be added.

Why it loses marksThe minus sign was treated as part of the number being squared. Without brackets the index applies to the 3 only.

Why it loses marksThe comparison was made on the magnitudes and the signs were ignored. On the negative side of the number line, the larger magnitude is the smaller value.

Why it loses marks$A$ must be at least 1 and less than 10, and 47 is too big. The number is equal to 47000 but it is not written in standard form.

Why it loses marksTwo faults: a number smaller than 1 needs a negative index, and the index is the number of places the point moves, which is 4 rather than the 3 zeros.

Why it loses marksThe indices were multiplied. When two powers of ten are multiplied the indices are added.

Why it loses marksThe $A$ values were added without making the powers of ten match. $7\times {10}^4$ is only a tenth of ${10}^5$, so it contributes 0.7, not 7.

Why it loses marksA negative index moves the point to the left, making the number smaller. The point was moved in the direction a positive index would take it.

Why it loses marksThe band reaches half a unit, not a whole one. A length of $7.4$ m still rounds down to $7$ m, so $8$ m is too far out.

Why it loses marksThe two leading zeros are place holders, not significant figures, so the first significant figure is the 4. Counting from the decimal point gives the wrong two digits.

Why it loses marksThat is the exact value, not an estimate. The syllabus method is to write each number correct to 1 significant figure first, and the mark is for that step.

Why it loses marksHalving $1$ gives $0.5$, but the rounding unit here is not $1$. The second significant figure is in the tens column, so the unit is $10$.

Why it loses marksThe rounding unit for 1 decimal place is $0.1$, so half a unit is $0.05$ and not $0.1$. The error is the half-unit step again.

Why it loses marksThe two parts are in different units, so they are not being compared like with like. 250 is a number of grams and 1.5 is a number of kilograms.

Why it loses marksThe 60 is shared between 5 parts, not 2. Dividing by the ratio numbers gives shares that are in the wrong ratio and do not add to 60.

Why it loses marksThe two speeds were held for different lengths of time, so they cannot be averaged directly. Average speed is total distance divided by total time.

Why it loses marks15 minutes is a quarter of an hour, 0.25, not 0.15. A time in hours must be written as a decimal fraction of an hour.

Why it loses marksDividing turns pounds into fewer dollars, which cannot be right when one pound is worth more than one dollar. The direction of the conversion is wrong.

Why it loses marksThe 0.25 is a quarter of an hour. The part of a time after the decimal point is a fraction of an hour, not a number of minutes.

Why it loses marksMoney is written to two decimal places because there are 100 cents in a dollar. The calculator dropped the trailing zero, not the cent.

Why it loses marksSubtracting the minutes, 15 - 40, goes below zero, so an hour has to be borrowed. Borrowing from the hours turns 10 15 into 09 75.

Why it loses marksMinutes do not stop at 100. There are 60 minutes in an hour, so 75 minutes is 1 hour 15 minutes.

Why it loses marksDividing turns pounds into fewer dollars, which cannot be right when one pound buys more than one dollar. The direction of the conversion is wrong.

Why it loses marksThat is linear decay: it takes 15% of the original price every year. Exponential decay takes 15% of the value at the start of each year, and that value is shrinking, so the total fall is less than 9000 dollars.

Why it loses marksA rise of 12% has multiplier 1.12. A fall needs $1-0.12=0.88$, and using 1.12 grows a value that should be shrinking - the answer comes out bigger than the starting value.

Why it loses marksRounding in the middle of a calculation carries the error forward, and with a power the error grows with every period. The final value will not match the rounded answer.

Why it loses marksOne division undoes one year. Two years means the multiplier was applied twice, so it has to be undone twice.

Why it loses marksThe square was applied to the 3 without its sign. Written without brackets, $-3^2$ means $-(3^2)=-9$, but the value being squared is the whole number $-3$, so $(-3{)}^2=9$.

Why it loses marksA number written next to a letter means multiply, not add. This convention is what makes the notation work, and misreading it makes every substitution question wrong.

Why it loses marksThe sentence takes 5 away from $n$, so the letter comes first. $5-n$ describes a different quantity - 'n less than 5' - and gives a negative answer as soon as $n$ is bigger than 5.

Why it loses marksOnly part of the right-hand side was divided by 2, and the order of operations makes $2l÷2$ come before the subtraction. Whatever you do to one side must be done to the whole of the other side.

Why it loses marksLike terms must have the same letter part including the index. $a$ and $a^2$ are different quantities, so the two terms cannot be collected into one.

Why it loses marksAdding like terms adds the coefficients, not the powers. $3x$ and $2x$ are like terms, so they collect to $5x$.

Why it loses marksThe 3 outside the bracket multiplies every term inside it, not just the first.

Why it loses marksA squared bracket is the bracket multiplied by itself, so the two middle terms $3x+3x=6x$ cannot be left out.

Why it loses marksThe $x$ divides both terms as well as the 3, so taking out only the 3 leaves a common factor behind and the expression is not fully factorised.

Why it loses marksThe x's here are terms of a sum, not factors of a product, so they cannot be cancelled. Cancelling removes a common factor from the whole of the top and the whole of the bottom.

Why it loses marksThe division by 3 has to apply to the whole of $y-7$, not just to the 7. Undo the operations in reverse order: subtract 7 first, then divide by 3.

Why it loses marksMultiplying powers of the same base adds the indices. Multiplying them is the rule for a power of a power, $(x^3{)}^4$, which is a different question.

Why it loses marksThe outside power applies to everything inside the bracket, so the coefficient 5 is squared as well. Squaring only the letter gives an answer five times too small.

Why it loses marksA negative index means a reciprocal, not a negative value. Nothing in the answer is negative: the power moves to the bottom of a fraction.

Why it loses marksThe indices are subtracted, and subtracting a negative adds: $5-(-2)=7$, not 3. Dividing by $a^{-2}$ is the same as multiplying by $a^2$, so the index grows.

Why it loses marksThe minus sign was lost while collecting the $x$ terms. $5=5x+21$ leads to $5x=-16$, so the answer is negative. Dropping a sign at this step is the single commonest error in the topic.

Why it loses marksEvery term has to be multiplied by the common denominator, including the 7 on the right-hand side. Multiplying one term only leaves an equation that is not equivalent to the original.

Why it loses marksDividing by $x$ assumes that $x$ is not zero, and it destroys the root $x=0$. A quadratic equation has two roots and both are needed.

Why it loses marksDividing by a negative number reverses the inequality sign. The student divided correctly but kept the original direction, so every value in the answer is wrong.

Why it loses marksThe 3 was subtracted instead of being undone. To isolate $5x$ you must add 3 to both sides, because the 3 was taken away from $5x$ in the original formula.

Why it loses marksThe common difference gives the 3, but $3\times 1=3$, and the first term is 5. The constant has to be found by substituting $n=1$, not ignored.

Why it loses marksThe sequence decreases, so the coefficient of $n$ is negative. The common difference is -3, and using +3 makes the sequence increase instead.

Why it loses marksThe second difference 2 does give $n^2$, but the constant is not the first term: $n^2$ already contributes 1 when $n=1$. Subtract $n^2$ from each term and read off what is left.

Why it loses marksA number is a term only if the equation has a whole-number solution. $3n+2=100$ gives $n=32.67$, and being close to a term is not the same as being one.

Why it loses marks$\propto$ is not an equals sign. Dropping k turns the relationship into one particular case and throws away the constant that the question is about.

Why it loses marksThe constant was found correctly, $5\times 4=20$, but it was put in the wrong place. Inverse proportion divides by x, so the 20 belongs on top of a fraction.

Why it loses marksThe pair was substituted into $y=kx$ instead of $y=k{x}^2$. The x in the formula is squared, so the 3 has to be squared as well.

Why it loses marksA straight line only shows direct proportion when it passes through the origin. This one crosses the y-axis at 1, so doubling x does not double y.

Why it loses marksThat is direct proportion. Inverse proportion works the other way: doubling one quantity halves the other, because the product $xy$ stays constant.

Why it loses marksThe gradient is the rate at which the distance changes, which is the speed. The distance is read from the vertical axis at a particular time; it is not the gradient.

Why it loses marksArea and gradient are two different measurements of the same graph. On a speed-time graph the gradient is the acceleration and the area is the distance travelled.

Why it loses marksA horizontal line means the speed is constant, so the acceleration is zero. The object is still moving, and the area under that section is still distance travelled.

Why it loses marksThe gradient is the change in the vertical quantity divided by the change in the horizontal one. Inverting it gives the reciprocal, which is a different rate: minutes per kilometre instead of kilometres per minute.

Why it loses marksThe factors are $(x+1)(x-3)$, and a root is the value that makes a bracket zero. Reading the numbers straight out of the brackets without changing the sign gives the wrong pair.

Why it loses marksThe turning point lies on the line of symmetry, which is halfway between the roots: $x=\frac{-1+3}{2}=1$. The value $x=3$ is where the curve crosses the $x$-axis, not where it turns.

Why it loses marksIt never crosses. As $x$ becomes large the curve gets closer and closer to the axis without reaching it, which is what an asymptote means. The same is true of $y=\frac{a}{x}+b$ and the line $y=b$.

Why it loses marksA sketch shows the shape and the labelled features, not a scaled plot. Time spent plotting to scale is time not spent labelling the intercepts, the turning point and the asymptotes, and those labels are where the marks are.

Why it loses marksThe constant 5 was left unchanged. A constant is a horizontal line and a horizontal line has gradient zero, so the 5 differentiates to 0, not to itself.

Why it loses marksThe derivative returns gradients, not points. At a stationary point $\frac{dy}{dx}=0$, so this substitution gives 0 for every curve in the paper.

Why it loses marksThe sign has been read the wrong way round. A negative second derivative means the gradient is falling, so the curve is about to turn downwards: that is a maximum.

Why it loses marksThe y-coordinate was taken as zero instead of being worked out from the curve. The point where $x=3$ is $(3,4)$, not $(3,0)$, so the tangent line is shifted down by 4.

Why it loses marksThe bracket was dropped. The whole input is multiplied by 3, so $3(x+1)$ is $3x+3$, not $3x+1$.

Why it loses marksThe $-1$ in $f^{-1}(x)$ is notation for the inverse function, not a power of the function. $\frac{1}{3x-5}$ is $\frac{1}{f(x)}$, the reciprocal, which does not undo f.

Why it loses marksThe order was reversed. $fg(x)$ means $f(g(x))$, so g is applied first. The expression given is $gf(x)$, which is a different function.

Why it loses marksThat is the domain. A square can never be negative, so the outputs stop at 0 and the two words have been swapped.

Why it loses marksThe pair is ordered: the first number is the $x$-coordinate and the second is the $y$-coordinate. $(2,5)$ is 2 across and 5 up, while $(5,2)$ is 5 across and 2 up - a different point in the same quadrant.

Why it loses marksThe student has worked out $2\times 2-3$, ignoring the sign of $x$. Multiplying a negative number by a positive one gives a negative answer, so $2\times (-2)=-4$.

Why it loses marksThe equation fixes the $x$-coordinate at 3 and says nothing about $y$, so the line runs vertically through $(3,0)$. Horizontal lines have the form $y=c$.

Why it loses marksThe 2 is the gradient, not the intercept. In $y=mx+c$ the intercept is $c$, the number standing on its own, which here is $-3$.

Why it loses marksThat is the reciprocal: the horizontal change over the vertical change. On a line that rises 4 while travelling 2 across it gives $\frac{1}{2}$ instead of 2.

Why it loses marksThe line falls from left to right, so the gradient must be negative. The student has worked out $\frac{4-2}{5-1}$, subtracting the $y$ values in the opposite order to the $x$ values.

Why it loses marksOn $x=3$ the run between any two points is 0, and dividing by 0 is undefined. It is a horizontal line that has gradient 0.

Why it loses marksThe 3 is the $y$-intercept, the value of $y$ when $x=0$. The gradient is the number multiplying $x$.

Why it loses marksThe two distances are the legs of a right-angled triangle, not parts of the segment. Adding them measures a journey along the legs instead of along the hypotenuse, so the answer is always too big.

Why it loses marksThe square applies to the whole bracket. The bracket works out to $-2$, and squaring a negative number gives a positive answer.

Why it loses marksHalving one endpoint gives the point a quarter of the way along the segment. A midpoint is the average of the two ends, so the coordinates have to be added before they are halved.

Why it loses marksThat is the sum of the two coordinates. The answer has been left twice as far from the origin as it should be, because the division by 2 was missed.

Why it loses marks3.6055... rounds up at the third significant figure, because the digit after it is 5. Writing 3.6 gives only two significant figures, so the accuracy asked for has not been reached.

Why it loses marksThe $x$-term is $-2x$, so the sign travels with the number. Reading only the size of the coefficient turns a falling line into a rising one.

Why it loses marksThe $y$-values were subtracted in one order and the $x$-values in the other. Mixing the orders flips the sign of the gradient.

Why it loses marksThe $y$-coordinate of a point is the intercept only when the point lies on the $y$-axis. Here $x=1$, so the point has to be substituted.

Why it loses marksOnly the $y$-term was divided by 4. Every term on both sides has to be divided, including the constant 8.

Why it loses marksThe gradient and the intercept have both been left unsimplified. Cambridge expects a fully simplified equation.

Why it loses marksThe gradient has had its sign changed, which is what you do for a perpendicular line. Parallel lines keep exactly the same gradient, sign included.

Why it loses marksOnly the sign was changed. The gradient also has to be turned upside down, because the two gradients must multiply to $-1$.

Why it loses marksThe coefficient of $x$ is 3, but $y$ is not on its own, so 3 is not the gradient of the given line. The line has to be rearranged first.

Why it loses marksTwo lines with the same equation are the same line, and $(4,1)$ is not on $y=2x+3$ because $2\times 4+3=11$, not 1.

Why it loses marksThe perpendicular bisector cuts the segment at right angles at its midpoint, so it passes through $(3,3)$ and not through either end.

Why it loses marksA parallelogram is defined by having two pairs of parallel sides, and a square has two pairs. The shape name does not stop a shape belonging to another family as well.

Why it loses marksA diameter passes through the centre and joins two points of the circle, so it is twice as long. The line from the centre to the circumference is the radius.

Why it loses marksAn obtuse angle stops at 180°. Anything between 180° and 360° is reflex, and 214° is past the straight line.

Why it loses marksA trapezium has exactly one pair of parallel sides. Two pairs makes the shape a parallelogram, and a rectangle is one, so a rectangle is not a trapezium.

Why it loses marksA bearing has three figures and is measured clockwise from north at the point named second, which here is B. The answer 25 is neither three figures nor measured at B.

Why it loses marksThe question says 'using a ruler and pair of compasses only', so the angles have to come out of the construction rather than be set with a protractor.

Why it loses marksMultiplying by the scale turns a drawing length into a real length. Going from real life to the drawing means dividing, and the two lengths must be in the same unit before either is done.

Why it loses marksThe scale compares like with like, so 1 cm stands for 25 000 cm. Reading the scale number as kilometres is out by a factor of 100 000.

Why it loses marksThe arcs are the only evidence that the triangle was constructed rather than guessed. Cambridge states that construction arcs must be shown.

Why it loses marksA scale factor of 2.5 multiplies every length, and an area is built from two lengths, so the area grows by ${2.5}^2=6.25$, not by 2.5.

Why it loses marksThe order of the letters carries the matching. ABC similar to EDF says A matches E and B matches D, which is not the correspondence in the question. 'The same shape' also just restates 'similar' instead of giving a reason.

Why it loses marksEqual angles alone are not enough - the sides must also be in the same ratio. A square and a 2 cm by 5 cm rectangle both have four right angles, yet they are not similar.

Why it loses marksSurface area is two-dimensional and volume is three-dimensional, so they cannot share a scale factor. Lengths in the ratio 2 : 3 give areas in the ratio 4 : 9 and volumes in the ratio 8 : 27.

Why it loses marksFolding a parallelogram along a line joining the midpoints of opposite sides does not make the halves match, and neither diagonal works either. Parallel sides give you a parallelogram, not a mirror line.

Why it loses marksOnly the vertical and the horizontal line have been counted. Both diagonals also fold the square onto itself, so there are 4 lines in total.

Why it loses marksTwo matching halves is line symmetry, which is a reflection, not a rotation. Turn the triangle through 180° and the apex points downwards, so it does not look the same.

Why it loses marksThe order counts the positions in a full turn where the shape looks the same, not the number of corners. After a quarter turn the rectangle is taller than it is wide, so it has not landed on itself.

Why it loses marksA reason has to name a geometrical property. 'They look the same' is a description of the diagram, not a reason, and the reason is where the mark is.

Why it loses marksAlternate angles are on opposite sides of the transversal. Two angles on the same side of the transversal, between the parallel lines, are co-interior and add to 180° instead of being equal.

Why it loses marks360 divided by the number of sides gives the exterior angle, not the interior angle. The interior angle is what is left of the straight line: 180 - 60.

Why it loses marks180° is the triangle sum. A quadrilateral is two triangles, because a diagonal splits it in two, so its angles add to 360°.

Why it loses marksOnly the two angles opposite the equal sides are equal. The third angle is whatever is left of 180°, which here is 180 - 70 - 70 = 40°.

Why it loses marksA right angle at the circumference has to be justified. It comes from the angle standing on a diameter, not from the way the diagram is drawn.

Why it loses marksThe angle at the centre is the larger of the two, so the angle at the circumference is half of it. Multiplying makes it bigger, and an angle of a triangle cannot be ${216}^{\circ }$.

Why it loses marksThe alternate segment theorem makes the two angles equal, not supplementary. The pair that adds to ${180}^{\circ }$ is the angle in the alternate segment and the angle in the other segment.

Why it loses marksThe perpendicular distance from the centre to a chord is shorter than the radius. The radius is the hypotenuse of a right-angled triangle whose other two sides are half the chord and the distance from the centre.

Why it loses marksThe length conversion is x100, but an area is measured in two directions, so the factor has to be squared. A square metre is 100 cm by 100 cm, not 100 cm by 1 cm.

Why it loses marksA litre is a cube 10 cm by 10 cm by 10 cm. The factor is cubed, not squared.

Why it loses marksThe conversion from km to m is x1000, not x10 000. The answer given is ten times too big.

Why it loses marksThe two lengths are in different units. Dividing metres by centimetres gives a meaningless answer, and 0.08 of a piece is obviously wrong.

Why it loses marksThe half is missing. A triangle is half of a rectangle with the same base and height, so the area must be halved.

Why it loses marksThe slant side is not the height. The area formula needs the perpendicular height, which is the shorter 4.8 cm.

Why it loses marksThe formula uses the parallel sides, not the other pair. The two sides that never meet are the ones to add, and the height is the perpendicular distance between them.

Why it loses marksThat counts the join twice and includes an edge that is inside the shape. Only the edges on the outside boundary are part of the perimeter.

Why it loses marksThe diameter has been used as the radius. The radius is 7 cm, so the true area is four times smaller than this.

Why it loses marksThe fraction of the area has been used where a length was asked for. An arc is a fraction of the circumference $2\pi r$, so this answer is far too big and comes out in cm2.

Why it loses marksThe fraction has been squared as well as the radius. Only the radius is squared, so this answer is about a third of the true area.

Why it loses marksThe area formula has been used for a length. Circumference is $2\pi r$, and $25\pi$ is the area, in cm2.

Why it loses marksThe one third has been left out, so this is the volume of the cylinder on the same base with the same height. A cone is exactly one third of that cylinder.

Why it loses marksThe perpendicular height has been used where the formula needs the slant height. The sloping edge is longer than the height, so the answer is too small.

Why it loses marksOnly the curved surface has been found. A closed cylinder also has two circular ends, and page 2 does not give that part of the formula, so it has to be added by hand.

Why it loses marksThe radius has been squared where the formula needs it cubed. $r^3$ grows much faster than $r^2$, so this answer is far too small.

Why it loses marksOnly the curved arc has been found. A perimeter is the distance all the way round the outside, and for a semicircle that includes the straight diameter closing the shape.

Why it loses marksThe hole has been added instead of subtracted. A hole removes material, so its area comes off the area of the whole shape.

Why it loses marksThe cone has been treated as a cylinder. A cone is one third of the cylinder on the same base with the same height, so its volume is $32\pi$ cm3 and not $96\pi$.

Why it loses marksThe removed cone has been given a radius of 6 cm, the same as the whole cone. It is similar to the whole cone, so its radius is in the same ratio as its height.

Why it loses marksThe theorem squares the sides before adding and takes the square root at the end. Adding the sides first answers a different question, and 17 happens to be part of the 8, 15, 17 triple, which makes the error look plausible.

Why it loses marksA right angle has to be shown by the converse test, not by eye. The test compares the square of the longest side with the sum of the squares of the other two.

Why it loses marksAdding the gaps gives the length of the two steps along the grid, not the straight line. The straight line is the hypotenuse of the triangle those steps make.

Why it loses marksThe base diagonal lies flat on the bottom of the cuboid. The diagonal through the cuboid is longer, because it also climbs the height.

Why it loses marks0.5556 is the value of the sine of the angle, not the angle itself. The angle needs the inverse function, and the answer 0.5556 has no meaning as an angle in this triangle.

Why it loses marksThe hypotenuse is opposite the right angle, always. The side next to the marked angle that is not the hypotenuse is the adjacent, and confusing the two swaps cosine for sine.

Why it loses marksRounding the ratio to two decimal places before multiplying moves the answer by 0.04 cm. The question allows a rounded final answer, not rounded working.

Why it loses marks1/sqrt(2) is the exact value of sin 45 and cos 45. The exact values are a short fixed list, and mixing up which surd belongs to which angle loses the mark even when the decimal is close.

Why it loses marksSine is positive in the second quadrant. Only the cosine changes sign there, so the value is the same as $\sin {30}^{\circ }$ and keeps its plus sign.

Why it loses marksThe calculator only ever returns the acute angle, so one solution is given where the range allows two. The second crossing of the line $y=\frac{1}{2}$ has been missed.

Why it loses marks$\frac{1}{2}$ is $\cos {60}^{\circ }$, and also $\sin {30}^{\circ }$. At ${30}^{\circ }$ the adjacent side is the long one, $\sqrt{3}$, so the cosine must be the larger of the two values.

Why it loses marksSign slip in the rearrangement. The 1 crosses to the other side as $-1$, so the cosine is negative and the solutions are in the second and third quadrants, not the first and fourth.

Why it loses marksThe period of tangent is ${180}^{\circ }$, not ${360}^{\circ }$, so a second branch crosses the same line ${180}^{\circ }$ later and it is inside the range.

Why it loses marksThe inverse sine only returns the acute angle. Between 0 and 180 degrees there is a second angle with the same sine, ${180}^{\circ }-{45.6}^{\circ }={134.4}^{\circ }$, and it also fits the triangle given.

Why it loses marksThe square root has been left out. 83.27 is $a^2$, the square of the side, and a squared length in cm2 is never the length of a side.

Why it loses marksThe angle in the area formula has to be the one included between the two sides you multiply. Any other angle of the triangle gives a number that is not the area.

Why it loses marksThere is no angle anywhere in this question, so there is no matching side-and-angle pair and the sine rule cannot start. Worse, the inverse sine would return the acute angle and hide the fact that the largest angle is obtuse.

Why it loses marksA base edge has been used where the base diagonal belongs. The horizontal side of the right-angled triangle is the line from A to the point directly under G, which is the diagonal of the base, 9.43 cm long.

Why it loses marksThe centre of the base is not 5 cm from a vertex. 5 cm is the distance from the centre to the midpoint of an edge. The distance to a vertex is half the base diagonal, $5\sqrt{2}=7.07$ cm, so the slant edge must be longer than 13 cm.

Why it loses marksThe wrong perpendicular has been used. The two planes meet along AB, so the lines perpendicular to AB are the one from the centre of the base to the midpoint of AB, 5 cm, and the one from the apex to that midpoint.

Why it loses marksThe two lengths have been added instead of combined at right angles. Adding the cross-section diagonal to the length of the prism gives a number longer than any line inside the solid can be.

Why it loses marksNo direction is given. A 90 degree clockwise turn sends $(2,5)$ to $(5,-2)$, and a 90 degree anticlockwise turn sends it to $(-5,2)$. One angle, two different images, so the description is incomplete.

Why it loses marksThe scale factor alone does not fix an enlargement. Any centre at all would give a different image of the same size, so a description without a centre cannot be marked correct.

Why it loses marksThe two numbers have been swapped. The top number of a column vector is the movement across and the bottom number is the movement up or down.

Why it loses marksLengths are multiplied by the scale factor but areas are multiplied by its square. Doubling the lengths of a shape makes it four times as big in area, not twice.

Why it loses marksThe top number is the movement across and the bottom number is the movement up, so 3 right and 4 up is a different journey from 4 right and 3 up.

Why it loses marksThe magnitude is the hypotenuse of a right-angled triangle, not the sum of the two components. Adding them would be walking along the two shorter sides instead of the direct route.

Why it loses marksAdding two vectors means following one movement and then the other, which is why the triangle law works. There is no vector multiplication in this course.

Why it loses marksParallel means one vector is a scalar multiple of the other, so the same number must multiply both components. Bigger numbers prove nothing.

Why it loses marksThe arrow runs from A to B, so the vector is the position of B minus the position of A. Taking them the other way round gives the opposite vector, and every later line then carries the wrong sign.

Why it loses marksThe diagram is not drawn to scale and appearance is not a reason. Parallel needs a scalar multiple, which is the only evidence the mark scheme accepts.

Why it loses marksA scalar multiple proves only that the two segments are parallel. Nothing yet links the line through A and B to the line through C and D, so the four points need not be on one line at all.

Why it loses marksThe fraction is $\frac{m}{m+n}=\frac{2}{3}$, because AP is two parts out of the three that make up AB. Using the other fraction puts P on the wrong side of the midpoint and gives the coefficients of $𝐚$ and $𝐛$ the wrong way round.

Why it loses marksOnly the first position vector has been halved. The midpoint is the average of both ends, so each of them is halved.

Why it loses marksThe denominator is the total number of outcomes, which is 6, not the number of outcomes you are not interested in. Only the numerator counts the faces that give the event.