Why did I lose the mark?

Why it loses marksSquaring a bracket is not the same as squaring each term. $(x+3{)}^2=x^2+6x+9$, and that missing $6x$ is the whole reason completing the square works.

Why it loses marksThe bracket contributes 9, not 0. Half of 6 is 3, and $3^2=9$ has to be taken back off the constant.

Why it loses marksThat is the solution of the opposite inequality. The parabola is above the axis outside its roots, so $>0$ selects the outside.

Why it loses marksThe discriminant is $b^2-4ac$, and both the 4 and the $c$ matter. With $a=1$ and $c=9$ the discriminant is $k^2-36$, not $k^2-9$.

Why it loses marksThe substitution solves for $\sqrt{x}$, not for $x$. Each value has to be squared, and both are positive so neither is rejected.

Why it loses marksThe four numbers are right but they are not paired, and the pairs are what the question asks for. The point $(3,-3)$ is on neither the line nor the circle.

Why it loses marksOn the whole real line $f$ is not one-one, so it has no inverse at all. Restricting the domain to $x\ge 0$ is what makes the inverse exist, and that restriction must be stated.

Why it loses marksThe function nearest the $x$ acts first. $gf(x)$ applies $f$ first and then $g$, which is the opposite of what the letters look like in order.

Why it loses marksThe rule decreases as $x$ increases, so the smallest input gives the largest output. The range runs downwards from 7, not upwards.

Why it loses marksA change inside the bracket acts on the input and does the opposite of what the sign suggests. The value $f(4)$ is now reached at $x=1$, so the graph has moved to the left.

Why it loses marksIt is the other way round: $f(2x)$ reaches a given height twice as quickly, so each x-coordinate is halved. $y=2f(x)$ is the one that doubles, and it doubles y-coordinates.

Why it loses marksThe coordinates have been subtracted the wrong way round. Gradient is rise over run, so the difference in $y$ goes on the top and the difference in $x$ on the bottom.

Why it loses marksBoth numbers have been read wrongly. The sign inside each bracket is the opposite of the coordinate it gives, and the number on the right is $r^2$, not $r$.

Why it loses marksChanging the sign is not enough. Perpendicular gradients multiply to $-1$, so each is the negative reciprocal of the other.

Why it loses marksSetting the discriminant to zero gives $k^2=16$, which has two solutions. The line can touch the circle on either side, so both values are needed.

Why it loses marksAdding the run and the rise does not give the distance. The distance is the hypotenuse of the right-angled triangle whose legs are the run and the rise.

Why it loses marksThe formula $s=r\theta$ needs the angle in radians. ${60}^{\circ }$ is not 60 radians, and the answer is larger than the whole circle, which is the giveaway.

Why it loses marksThat is the arc on its own. The boundary of a sector is the arc together with the two radii that close it.

Why it loses marksThe radius is squared in the area formula. Using $r$ instead of $r^2$ gives an answer four times too small here.

Why it loses marksThat is the area of the sector. The segment stops at the chord, so the triangle formed by the two radii and the chord has to come off.

Why it loses marks$\pi$ radians is ${180}^{\circ }$. The conversion factor has been confused with the unit itself, which makes every answer about 3 times too large.

Why it loses marksThe $-1$ in ${\sin }^{-1}x$ marks an inverse function, exactly as it does in $f^{-1}(x)$. The reciprocal of the sine is a different function altogether and is not in this syllabus.

Why it loses marksThe index sits on the function value, not on the angle. ${\sin }^2\theta$ is the conventional shorthand for $(\sin \theta {)}^2$, and the square applies to the sine, never to the angle.

Why it loses marksThe interval for $x$ has been used for $2x$. Since $x$ runs from 0 to $2\pi$, the angle $2x$ runs from 0 to $4\pi$, which contains twice as many solutions.

Why it loses marksThe size is right but the sign is wrong. ${150}^{\circ }$ is in the second quadrant, where the cosine is negative, and the minus sign has been dropped between the related angle and the answer.

Why it loses marksThe sine of an angle never leaves the interval from $-1$ to 1, so no angle has a sine of 1.5. A calculator's error message is telling you something mathematical, not something mechanical.

Why it loses marksCancelling divides by $\cos \theta$, which is zero at $\theta ={90}^{\circ }$ and $\theta ={270}^{\circ }$. At those angles the original expression is undefined, so the simplified version is not equivalent to it everywhere.

Why it loses marksThe index is one less than the position, because the first term is $a$ itself and no multiplication by $r$ has happened yet. Using $a{r}^n$ gives the term after the one you want.

Why it loses marksTo reach the nth term you add $d$ one time fewer than $n$. Writing $a+nd$ gives the $(n+1)$th term.

Why it loses marksThe formula comes from letting $r^n\to 0$, which only happens when $|r|<1$. For $r=2$ or $r=-1$ the terms do not shrink and the sum has no limit, so the formula produces a meaningless number.

Why it loses marksThe $(n-r)!$ in the denominator has been dropped. $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ counts selections, not arrangements, and dividing only by $r!$ counts arrangements instead.

Why it loses marksThe binomial coefficient has been left out. The term in $x^2$ is $\left(\genfrac{}{}{0ex}{}{6}{2}\right)(3x{)}^2$, so the 15 and the 9 both belong in the answer.

Why it loses marksThree steps of the progression separate the two terms, so the ratio between them is $r^3$, not $r$. The cube root is the common ratio, not the quotient.

Why it loses marksA derivative is a gradient. The graph of $y=5$ is a horizontal line, and a horizontal line has gradient 0. Constants do not change when $x$ changes, so they contribute nothing to the rate of change.

Why it loses marksThe power has not been reduced. The rule says multiply by the old power and then take one away from it, so the new power is $\frac{1}{2}-1=-\frac{1}{2}$.

Why it loses marksThe normal is perpendicular to the tangent, so its gradient is the negative reciprocal. Using the tangent gradient again produces a line that crosses the curve instead of meeting it at right angles.

Why it loses marksAt a stationary point $\frac{dy}{dx}$ is 0 by definition, so that substitution tells you nothing. The test uses the rate of change of the gradient, which is the second derivative.

Why it loses marksThe chain rule was half applied: the outside was differentiated but the inside was not. The bracket $2x+5$ differentiates to 2, and that factor must be multiplied in.

Why it loses marksThat is the rate of change of the area with respect to the radius, not with respect to time. A question about a circle growing asks for $\frac{dA}{dt}$, which needs the chain rule with the given $\frac{dr}{dt}$.

Why it loses marksThe division is by the new power, not the old one. Dividing by 3 instead of 4 gives an answer whose derivative is $\frac{4}{3}{x}^3$, not $x^3$.

Why it loses marksThe division by $a=2$ is missing. Differentiating $\frac{(2x+3{)}^5}{5}$ gives $2(2x+3{)}^4$ by the chain rule, which is twice the integrand.

Why it loses marksAn indefinite integral is a family of curves, not one curve. Every function of the form $x^2+x+c$ has derivative $2x+1$, and only a point on the curve can say which one is wanted.

Why it loses marksThe lower function was subtracted from the upper one the wrong way round. The line is above the curve between $x=-1$ and $x=2$, so the line must come first.

Why it loses marksA thin strip of the region becomes a disc of radius $y$, and the area of that disc is $\pi y^2$, not $\pi y$. Without the square the answer is an area, not a volume.

Why it loses marksOn that interval the curve lies below the $x$-axis, so the integral counts every strip as negative. The integral is the signed area, not the area.

Why it loses marksThe modulus is a distance, so it can never be negative. The definition has two cases and the second one is the one that removes the sign: for $x<0$, $|x|=-x$.

Why it loses marksSquaring preserves the direction of an inequality only when both sides are certainly non-negative. The left side here is negative for $x<-2.5$, so the two inequalities are not equivalent: at $x=-5$ the original reads $-5<4$, which is true, while the squared version reads $25<16$, which is false.

Why it loses marksThe remainder theorem uses the value that makes the divisor zero. Setting $2x-1=0$ gives $x=\frac{1}{2}$, not $x=1$.

Why it loses marksA repeated factor needs one fraction for each power. Two constants cannot reproduce a numerator of degree 2 against a denominator of degree 3, and the form mark is lost before any algebra is done.

Why it loses marksThe first two terms are not 'the expansion' unless the question asks for them, and the validity condition belongs to the substituted variable $u=3x$, not to $x$ itself.

Why it loses marksThe product law is ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$, which is about a product inside the logarithm. There is no law at all for the logarithm of a sum, and ${\log }_{2}6$ is not $1+2$.

Why it loses marksThe 3 is the value of the logarithm, so the definition gives $x(x+2)=2^3$. Losing the base is the commonest single error in this topic.

Why it loses marks$\ln 0.5$ is negative, so dividing by it reverses the direction of the inequality. The arithmetic is right and the inequality sign is wrong.

Why it loses marksThe linear form is $\ln y=\ln k+x\ln a$, so the intercept on the $\ln y$ axis is $\ln k$. The value of $k$ is the exponential of that intercept.

Why it loses marks$\ln (-5)$ is not defined: the argument of a logarithm must be positive. The inverse relationship only holds for $x>0$.

Why it loses marksThe reciprocal and the inverse are different functions. $\sec \theta$ is a value, ${\cos }^{-1}\theta$ is an angle, and $\sec 0.5$ would be 2 if it existed at all - which it does not, since $\sec \theta$ never lies between $-1$ and $1$.

Why it loses marksThe identity has no case-by-case version. It comes from dividing ${\sin }^2\theta +{\cos }^2\theta =1$ by ${\cos }^2\theta$, and the plus sign is fixed by that division. The quadrant changes the sign of $\sec \theta$ itself, not the sign in the identity.

Why it loses marksThe sign is the wrong way round. The cosine expansion reverses the sign inside the bracket, so a plus in the bracket produces a minus in the expansion. Testing with $A=B={60}^{\circ }$ shows it: the left side is $-\frac{1}{2}$, and the wrong right side gives $\frac{1}{4}+\frac{3}{4}=1$.

Why it loses marksThis is illegal when $\cos \theta =0$, and it throws away those solutions. It also divides the $\sin 2\theta$ term by $\cos \theta$, which is not $\tan 2\theta$ - the angle is different. In $0^{\circ }\le \theta \le {360}^{\circ }$ the division loses $\theta ={90}^{\circ }$ and ${270}^{\circ }$.

Why it loses marksThe sign inside the bracket is wrong. Matching $R\sin (\theta +\alpha )$ against the expression gives $R\cos \alpha =3$ and $R\sin \alpha =-4$, so the sine of $\alpha$ is negative and the bracket must be $\theta -\alpha$. Check at $\theta =0$: the original is $-4$, while $5\sin {53.13}^{\circ }=+4$.

Why it loses marksThe chain rule was applied to the outside function and then forgotten. $\ln 2x=\ln 2+\ln x$, and $\ln 2$ is a constant, so the derivative is $\frac{1}{x}$. The chain rule gives the same thing: $\frac{1}{2x}\times 2=\frac{1}{x}$.

Why it loses marksThe exponential differentiates to itself only when the exponent is exactly $x$. Here the inside function is $3x$, whose derivative is 3, and the chain rule requires that factor.

Why it loses marksThe derivatives of the two factors have been multiplied. The product rule adds two terms, each of which keeps one original factor: $u\frac{dv}{dx}+v\frac{du}{dx}$. The wrong answer also happens to be constant, which is impossible for a curve that is clearly not a straight line.

Why it loses marksThe $y^2$ term was differentiated as though $y$ were the variable of differentiation. Because $y$ is a function of $x$, the chain rule contributes a factor of $\frac{dy}{dx}$, and without it the equation cannot give a gradient at all.

Why it loses marksThe normal is perpendicular to the tangent, so its gradient is the negative reciprocal, $-\frac{1}{3}$. Writing the tangent's gradient again gives a line that lies along the curve's direction instead of across it.

Why it loses marksDifferentiating $e^{2x}$ brings a factor of 2 down from the exponent. The integral needs the reciprocal factor to cancel it, exactly as the $\frac{1}{a}$ in the standard result says.

Why it loses marksThe derivative of the bottom is 2, not 1, so the logarithm result brings a factor $\frac{1}{2}$. This is the same missing factor as the exponential case.

Why it loses marksThe standard result carries $\frac{1}{a}$ in front, and here $a=3$. Without it the derivative of the answer is three times too big.

Why it loses marksThat is the power rule applied to $\sin x$ as though it were a variable. The square has to be removed with the double-angle identity before anything can be integrated.

Why it loses marksDifferentiating $\sin 2x$ gives $2\cos 2x$, which is no simpler, and the integral you are left with is worse than the one you started with.

Why it loses marksA sign change locates the root inside an interval; it does not find it. 2.5 is the midpoint of the interval and is not a root of the equation.

Why it loses marksThat is $x^3$ on its own, not $x$. An iterative formula needs a single $x$ on the left, so the rearrangement has to finish by taking the cube root.

Why it loses marksOne approximation says nothing about accuracy. The sequence has to settle: two successive values that agree at the accuracy asked for, and only then is the answer known to that many figures.

Why it loses marksA sign change proves a root exists, but the absence of one proves nothing. The curve can dip below the axis and come back up, crossing it twice inside the interval.

Why it loses marksIn degrees the equation being solved is not $x=\cos x$ but $x=\cos \left(x^{\circ }\right)$, which has a different root. The iteration still converges, which is what makes the error hard to spot.

Why it loses marksThe displacement from A to B is the position vector of B minus the position vector of A. Subtracting the other way round gives the negative vector, which reverses the direction and changes every later answer.

Why it loses marksThe angle between two lines depends only on their directions, and position vectors depend on where the origin happens to be.

Why it loses marksAdding the lengths of two journeys is only the length of the direct route when the two vectors point the same way. In general the direct route is shorter, which is the triangle inequality.

Why it loses marks"Similar" is not "a scalar multiple". Parallelism is an exact test: one direction vector must equal a constant times the other, with the same constant in every component.

Why it loses marksSkew means not parallel and not meeting. Two parallel lines also never meet, but they are parallel, not skew. The decision is a two-part test and both parts must be shown.

Why it loses marksA unit vector must have magnitude exactly 1. Dividing by 3 when the magnitude is 7 gives a vector of magnitude $\frac{7}{3}$, and leaving $\overrightarrow{a}$ unchanged leaves its original magnitude.

Why it loses marksThe constant is added inside the logarithm, so exponentiating turns it into a multiplier, not an additive term. $e^{x^2+c}=e^c{e}^{x^2}$, and $e^c$ is just a new constant.

Why it loses marksThe initial condition is a statement about the solution, not about the differential equation. Substituting it into the equation tells you nothing and destroys the information you need.

Why it loses marksTwo constants that differ by a fixed amount can be combined into one, so the second carries no extra information. Carrying both invites an algebra slip and hides which constant you are finding.

Why it loses marksDifferentiating $\ln |3y+1|$ gives $\frac{3}{3y+1}$, which is three times too big. The linear coefficient has to come out as a factor of $\frac{1}{3}$.

Why it loses marksThe constant has to carry the sign, and the wording says the volume decreases. Taking k positive with a plus sign builds growth into the model, and no later substitution can repair it.

Why it loses marksAn exponential decay never reaches zero in finite time: at every finite t the value is positive, however small. "Gone" is not something the model predicts.

Why it loses marks$\frac{\pi }{4}$ is only the reference angle with the real axis. The point $(-2,-2)$ is in the third quadrant, so the angle from the positive real axis is $\frac{\pi }{4}-\pi =-\frac{3\pi }{4}$.

Why it loses marksModuli do not add when the complex numbers do. Take $z_1=1$ and $z_2=i$: $|z_1+z_2|=|1+i|=\sqrt{2}$, but $|z_1|+|z_2|=2$.

Why it loses marksA non-zero complex number has two square roots. $(-3-2i{)}^2=5+12i$ as well, so giving one root is an incomplete answer.

Why it loses marks$i^2=-1$, so $3i\times (-3i)=-9i^2=+9$. The product of a complex number and its conjugate is never negative.

Why it loses marksNon-real roots occur in conjugate pairs, so the count of non-real roots is even. A real cubic has three real roots or one real root and a conjugate pair.

Why it loses marksMultiplying only the numerator changes the value of the fraction. The conjugate must multiply the numerator and the denominator.

Why it loses marksA force with a vertical component changes the reaction. Pulling a block upwards at an angle reduces $R$ below $mg$; pushing down at an angle increases it.

Why it loses marksForces are vectors, so they combine as a triangle rather than as numbers. Adding the magnitudes is only correct when the forces act in the same direction.

Why it loses marks$F=\mu R$ is the maximum friction, reached only at limiting equilibrium. A block at rest on rough ground often has far less friction than the maximum available.

Why it loses marksThey act on the same body, and a third-law pair acts on two different bodies. The forces in a pair are also of the same type.

Why it loses marksTension in a string can only pull the particle towards the other end. A string that would have to push simply goes slack.

Why it loses marksResolving horizontally and vertically is not wrong in principle, but it puts both $R$ and $F$ into both equations and invites sign errors. Along and perpendicular to the plane is the method the mark scheme expects.

Why it loses marksBeing at rest is a statement about the velocity, not about the position. A particle can be instantaneously at rest at any point of its path, and a question that asks where it is at that instant expects you to substitute the time back into the displacement.

Why it loses marks24 m/s is the final velocity v, not the acceleration. The acceleration is how fast the velocity changes, which here is 24 divided by 8.

Why it loses marksArea below the time axis is displacement in the negative direction, so it subtracts. Adding it inflates the displacement and gives the distance travelled instead.

Why it loses marksThe constant of integration has been dropped. The acceleration alone does not determine the velocity; you also need to know the velocity at one instant.

Why it loses marksA negative acceleration means the velocity is decreasing, not that the speed is. A particle moving in the negative direction with a negative acceleration has a velocity that is becoming more negative, so its speed is increasing.

Why it loses marksMomentum is mass times velocity, and the velocity is negative when the body moves in the negative direction. Using the speed instead of the velocity gives a momentum with the wrong sign, and every total built on it is wrong.

Why it loses marksThe mass has been used in place of the momentum. A stationary body has velocity zero, so its momentum is mass times zero.

Why it loses marksThe mass of the second body has been left out of the right-hand side. Coalescing means one body of the combined mass moving with one velocity, so both masses must appear.

Why it loses marksA negative velocity after an impact is a genuine result: it means the body is now moving in the direction opposite to the positive direction you chose, so it has rebounded. Changing the sign to make it look tidier loses the mark.

Why it loses marksKinetic energy is only conserved in a perfectly elastic impact, which is not part of this syllabus. When bodies coalesce the kinetic energy always decreases, so an energy equation gives the wrong answer.

Why it loses marksMass and weight are different quantities with different units. A mass of 6 kg has a weight of $6g$ newtons, which is 60 N with $g=10$.

Why it loses marksNewton's first law says constant velocity means no resultant force, not no forces. The driving force and the resistances are both there and they are equal in size.

Why it loses marksFriction opposes the motion. Moving up the plane, friction acts down the plane, so it adds to the component of the weight instead of subtracting from it.

Why it loses marksThat would leave no resultant force, so the particle would not accelerate. While the system is moving, the tension differs from the weight on both sides.

Why it loses marksThe reaction is perpendicular to the surface, so it balances only the component of the weight perpendicular to the surface, which is $mg\cos \theta$.

Why it loses marksEach resistance belongs to one body and appears once. The whole-system equation already accounts for every resistance on every part of the system.

Why it loses marks$h$ in $mgh$ is the vertical height, not the distance along the slope. The block rises $5\sin {30}^{\circ }=2.5m$.

Why it loses marksOnly the component of the force along the motion does work, and that component is $50\cos {60}^{\circ }=25N$. The other component is perpendicular to the displacement and does no work.

Why it loses marksThe normal reaction is perpendicular to the surface, and the block moves along the surface, so the angle between the force and the displacement is 90°. $\cos {90}^{\circ }=0$.

Why it loses marksFriction has done work against the motion, so some of the potential energy has gone into heating the surfaces. The speed at the bottom is less than $\sqrt{2gh}$.

Why it loses marks$P=Fv$ rearranges to $F=\frac{P}{v}$, not $Pv$. The power must also be in watts, so 30 kW is 30000 W.

Why it loses marksA kilowatt is a thousand watts. Leaving the power in kW changes every force and acceleration that follows from it.

Why it loses marksThe vertical axis of a histogram is frequency density, not frequency. The 30 observations are the area of the bar, so the height is the area divided by the width.

Why it loses marksThat expression is the variance, ${\sigma }^2$. The standard deviation is its square root, and leaving the answer as a variance loses the final mark.

Why it loses marksSubtracting a constant moves every value by the same amount, so it shifts the mean but leaves the spread exactly as it was.

Why it loses marksThe midpoint of a class is a representative value for that class, not the median. The median is the value with half the observations below it, and it usually falls somewhere inside the middle class.

Why it loses marksThat is two numbers, not a comparison. There is no comparison word, no unit, and nothing about spread.

Why it loses marksThat is the number of ordered selections. A committee does not have an order, and every committee has been counted 3! = 6 times, once for each order of its three members.

Why it loses marks240 is the number of arrangements in which they ARE next to each other. The question asks for the arrangements in which they are apart, which is the complement of that.

Why it loses marksSwapping the two Ls, or the two Os, produces the same word, so 5040 counts every word four times over.

Why it loses marksThat counts only the committees with exactly one woman. The committees with 2 women and with 3 women also satisfy 'at least one', and they have been left out.

Why it loses marksOnce the pair is glued together there are only 4 objects to arrange, not 5, and the pair itself has 2 orders. Both corrections are needed.

Why it loses marksThe totals are not equally likely. A total of 7 happens in six ways and a total of 2 in only one, so the equally likely outcomes are the 36 pairs of faces.

Why it loses marksWithout replacement the second probability is conditional. One red counter has gone, so only 4 of the remaining 7 counters are red.

Why it loses marksMutually exclusive means the events cannot both happen, so $P(A\cap B)=0$. Independence needs $P(A\cap B)=P(A)\times P(B)$, which is not 0 when both probabilities are greater than 0.