Practice bank

Every question, cut by section

All 127 questions in this paper in one place, filterable down to a single spec section when you know what you’re bad at — and mixed by default, because the real paper never tells you which section you’re in.

Section

Mode

Every pick reveals its explanation immediately here — right or wrong, and why. Switch to Drill when you want to rehearse the real paper’s pacing instead: a timed countdown, with feedback withheld until the whole set is done.

Pool

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 1

1 mark

The curve $y = 3x^2 - 12x + 12$ meets the x-axis at exactly one point. What is the value of $b^2 - 4ac$ for this quadratic?

From Quadratic inequalities, interpreted and represented graphically

Question 2

3 marks

Solve $-x^2 + 5x - 4 \geq 0$ .

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 3

2 marks

Which expression, multiplied against the denominator of $\frac{7}{5+\sqrt{3}}$ , gives a denominator with no surd left in it?

From Quadratic inequalities, interpreted and represented graphically

Question 4

3 marks

Solve $x^2 + 2x - 15 < 0$ .

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 5

2 marks

Simplify $\frac{10}{\sqrt{5}}$ , giving your answer in simplest surd form.

From Quadratic inequalities, interpreted and represented graphically

Question 6

3 marks

The curve $y = (x-4)(x-16)$ crosses the x-axis at $x=4$ and $x=16$ , and lies below the axis between these values. A question asks you to complete the statement: 'the curve is above the x-axis for $x < 4$ or ___.' What is the correct completion?

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 7

2 marks

A WMA11 question asks you to 'find the exact value of $x$ ', where the correct working leads to $x = 7 - 2\sqrt{5}$ . You write your final answer as $x = 2.53$ (3 s.f.). What happens to the marks for this part?

From Quadratic inequalities, interpreted and represented graphically

Question 8

2 marks

The curve $y = g(x)$ crosses the x-axis at $x = -1$ and $x = 6$ , and the coefficient of $x^2$ is negative. What is the solution set of $g(x) > 0$ ?

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 9

2 marks

Simplify $2\sqrt{45} + \sqrt{20}$ , giving your answer in the form $k\sqrt{5}$ .

From Quadratic inequalities, interpreted and represented graphically

Question 10

2 marks

The curve $y = f(x)$ crosses the x-axis at $x = -3$ and $x = 5$ , and the coefficient of $x^2$ is positive. What is the solution set of $f(x) < 0$ ?

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 11

3 marks

Rationalise the denominator of $\frac{3}{4-\sqrt{2}}$ .

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 12

3 marks

For $0 \leqslant x < 2\pi$ , at how many values of $x$ is $\tan x$ undefined, and what are they?

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 13

2 marks

Simplify $\sqrt{150}$ , leaving your answer as a surd in simplest form.

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 14

2 marks

What is the range of $y = 3\sin x$ ?

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 15

1 mark

Rationalise the denominator of $\frac{6}{\sqrt{2}}$ .

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 16

3 marks

What is the period of $y = \sin 2x$ ?

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 17

2 marks

Simplify $\sqrt{20} + \sqrt{45}$ , giving your answer in the form $k\sqrt{5}$ .

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 18

2 marks

The graph of $y = \sin\left(x + \frac{\pi}{6}\right)$ is obtained from $y = \sin x$ by...

From Simplifying Surds, Rationalising Denominators, and the Exact-Value Rule

Question 19

1 mark

A question states: "Given that $x = 4 + \sqrt{7}$ , find the exact value of $x$ ." Which of these should you write as your final answer?

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 20

2 marks

Which best describes the symmetry of $y = \cos x$ ?

From Solving simultaneous equations by substitution

Question 21

2 marks

Two different WMA11 topics both use the word 'substitution.' Which statement correctly distinguishes them?

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 22

2 marks

What is the period of $y = \tan x$ ?

From Solving simultaneous equations by substitution

Question 23

3 marks

Three vertices of a rectangle are given, and you need to find the fourth. Which statement is correct?

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 24

2 marks

Which of the following statements about $y = \tan x$ is TRUE?

From Solving simultaneous equations by substitution

Question 25

3 marks

Solving $y=x-1$ and $y=x^2-3x-1$ simultaneously, you correctly reach $x^2-4x=0$ and solve it to get $x=0$ or $x=4$ . What is the correct final answer?

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 26

2 marks

Which single transformation turns $y = \sin x$ into $y = \sin 2x$ ?

From Solving simultaneous equations by substitution

Question 27

2 marks

After substituting a line into a quadratic curve, you correctly reach $x^2+3x+10=0$ . What does this tell you about the line and the curve?

From Sine, Cosine and Tangent Graphs — Period vs Domain Confusion

Question 28

1 mark

The graph of $y = \sin x$ repeats itself exactly every $2\pi$ . What is the name for that repeat-length of $2\pi$ ?

From Solving simultaneous equations by substitution

Question 29

2 marks

To solve $2x + y = 7$ and $y = x^2 - x - 1$ simultaneously by substitution, what should you do first?

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 30

4 marks

$f''(x) = 4 - 6x$ . Given that $f'(1) = 3$ and that the curve $y = f(x)$ passes through $(1, 2)$ , find $f(x)$ .

From Solving simultaneous equations by substitution

Question 31

2 marks

Solving $y = x - 2$ and $y = x^2 - 4x + 2$ by substitution, you correctly reach $x^2 - 5x + 4 = 0$ and solve it to get $x = 1$ or $x = 4$ . What is the correct final answer?

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 32

1 mark

Why does the mark scheme lose marks for integrating $f''(x)$ all the way to $f(x)$ before finding either constant, even when the final answer eventually comes out right?

From Solving simultaneous equations by substitution

Question 33

1 mark

Substituting a line into a quadratic curve, you correctly reach $x^2 - 2x + 5 = 0$ . What does this tell you about the line and the curve?

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 34

1 mark

The curve $y = f(x)$ passes through the point $(1, -8)$ . Which equation should this fact be written as, to find a constant in $f(x)$ ?

From Solving simultaneous equations by substitution

Question 35

1 mark

Solving $y = 3x - 1$ and $y = x^2 + 2x - 4$ simultaneously by substitution, which equation do you get after eliminating $y$ ?

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 36

3 marks

$f''(x) = 6x$ and $f'(2) = 1$ . What is $f'(x)$ ?

From The Sine Rule, the Cosine Rule, and the Ambiguous Case

Question 37

3 marks

Triangle $PQR$ : $PQ = 12$ cm, $QR = 8$ cm, angle $QPR = 27°$ . Given that angle $PRQ$ is obtuse, find angle $PRQ$ (1 d.p.).

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 38

3 marks

Given $f''(x)$ , a gradient condition $f'(a) = m$ , and a point $(p, q)$ on $y = f(x)$ , which sequence of steps is correct?

From The Sine Rule, the Cosine Rule, and the Ambiguous Case

Question 39

2 marks

Triangle $PQR$ has $PQ = 9$ cm, $PR = 6$ cm, and angle $PQR = 40°$ (opposite side $PR$ ). What can be said about the number of triangles this data describes, before doing any further arithmetic?

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 40

2 marks

You are given $f''(x)$ , a gradient condition $f'(a) = m$ , and a point $(p, q)$ on the curve $y = f(x)$ . How many constants of integration do you need, and when does each one first exist?

From The Sine Rule, the Cosine Rule, and the Ambiguous Case

Question 41

3 marks

Triangle $ABC$ has $a = 9$ cm, $b = 10$ cm, $c = 15$ cm. A student finds $\cos C = \frac{a^2+b^2-c^2}{2ab} \approx -0.244$ and stops, saying they must have made a sign error since a cosine can't be negative here. Are they right?

From Integrate Twice, Find Two Constants — Sequencing a Two-Stage Problem

Question 42

2 marks

$f'(x) = 4x - 5$ , and the curve $y = f(x)$ passes through the point $(2, 1)$ . What is the constant of integration?

From The Sine Rule, the Cosine Rule, and the Ambiguous Case

Question 43

1 mark

A triangle question gives one angle, the side opposite it, and one other side — then asks for the angle opposite that OTHER side, using the sine rule. What should you expect before you even start calculating?

From Graph Transformations Described in Words

Question 44

2 marks

Which single transformation maps $y = \sin x$ to $y = 3\sin x$ — the specification's own worked example?

From The Sine Rule, the Cosine Rule, and the Ambiguous Case

Question 45

2 marks

A triangle question gives all three side lengths and asks for one of the angles. Which rule applies, and why can the sine rule not be used at all here?

From Graph Transformations Described in Words

Question 46

3 marks

The curve $y=f(x)$ has a single minimum at $(2,-5)$ , no other turning points, and $y \to \infty$ as $x \to \pm\infty$ . For how many values of the constant $k$ does the curve $y = f(x) + k$ touch the x-axis at exactly one point (rather than crossing it twice, or missing it entirely)?

From The Sine Rule, the Cosine Rule, and the Ambiguous Case

Question 47

2 marks

$\sin\theta = 0.6$ , and $0° < \theta < 180°$ . How many values of $\theta$ satisfy this, and what are they (1 d.p.)?

From Graph Transformations Described in Words

Question 48

2 marks

The curve $y = f(x)$ , where $f(x) = x^3 - 4x$ , passes through $(1, -3)$ . What point does the curve $y = f(-x)$ pass through?

From Sketching cubic graphs from factored form

Question 49

2 marks

A real WMA11 question sketches $f(x) = (3x+20)(x+6)(2x-3)$ — three simple roots at $x=-\tfrac{20}{3}$ , $x=-6$ and $x=\tfrac{3}{2}$ , positive leading coefficient. For which values of $x$ is $f(x) > 0$ ?

From Graph Transformations Described in Words

Question 50

1 mark

The curve $y=f(x)$ passes through $(1,-3)$ . Which point must lie on the curve $y=-f(x)$ ?

From Sketching cubic graphs from factored form

Question 51

2 marks

$(x+4)(x-1)^2 = 0$ . How many DISTINCT real values of $x$ satisfy this equation?

From Graph Transformations Described in Words

Question 52

2 marks

Which single transformation maps $y = \cos x$ to $y = \cos 2x$ ?

From Sketching cubic graphs from factored form

Question 53

3 marks

$g(x) = -(x+3)(x-2)^2$ . What is the y-intercept, and which way does the curve open?

From Graph Transformations Described in Words

Question 54

3 marks

A question asks you to fully describe the single transformation that maps $y = \frac{6}{x}$ to $y = \frac{6}{x-2}$ , and to give the equations of the new asymptotes. Which response would score full marks?

From Sketching cubic graphs from factored form

Question 55

3 marks

Which is a correct, fully-labelled description of the graph of $y = (x+1)(x-2)^2$ ?

From Graph Transformations Described in Words

Question 56

2 marks

The curve $y = f(x)$ has a single turning point at $(3, -2)$ . What are the coordinates of the turning point of $y = 2f(x)$ ?

From Sketching cubic graphs from factored form

Question 57

2 marks

What is the y-intercept of the curve $y = (x+2)(x-1)(x-4)$ ?

From Graph Transformations Described in Words

Question 58

2 marks

The curve $y = f(x)$ has a single turning point at $(3, -2)$ . What are the coordinates of the turning point of $y = f(x + 5)$ ?

From Sketching cubic graphs from factored form

Question 59

2 marks

$f(x) = (x-3)^2(x+1)$ . At $x = 3$ , does the curve $y = f(x)$ cross the x-axis or touch it — and why?

From Graph Transformations Described in Words

Question 60

2 marks

The curve $y = f(x)$ has a single turning point at $(3, -2)$ . What are the coordinates of the turning point of $y = f(x) - 5$ ?

From Sketching cubic graphs from factored form

Question 61

1 mark

The curve $y = (x+2)(x-1)(x-4)$ meets the x-axis at three points. What are their x-coordinates?

From The hidden quadratic — substitution from an indices/exponential equation

Question 62

3 marks

Solving a hidden-quadratic equation using the substitution $u=2^x$ , you correctly reach $4u^2+4u-3=0$ and then $u=\dfrac{1}{2}$ or $u=-\dfrac{3}{2}$ . What is the complete, correctly-justified solution for $x$ ?

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 63

2 marks

$\frac{dy}{dx} = \frac{4x^3-9}{x^2}$ and the curve $y=f(x)$ passes through $(1,2)$ . A student splits and integrates correctly to reach $y = 2x^2 + 9x^{-1} + c$ , then substitutes the point to find $c=-9$ , giving the final answer $y = 2x^2 + 9x^{-1} - 9$ . What did this student do right that the research bank's own four-series evidence says many real candidates get wrong?

From The hidden quadratic — substitution from an indices/exponential equation

Question 64

4 marks

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (Verbatim WMA11 instruction, verified attached to real substitution-style questions; the equation below is VERIDIAN-original, not a reproduction.) Solve $3 \times 4^x - 11 \times 2^x - 4 = 0$ .

Which response is certain to earn full marks?

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 65

2 marks

Which of these, once split into separate powers of x, produces a term that needs a technique beyond what P1 covers?

From The hidden quadratic — substitution from an indices/exponential equation

Question 66

4 marks

Solve $5^{2x} - 6 \times 5^x + 5 = 0$ for $x$ .

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 67

1 mark

$\frac{dy}{dx} = \frac{x^3-2}{x^2}$ . Once split, why can this be integrated term-by-term using spec 5.2's rule, with no technique beyond P1 needed?

From The hidden quadratic — substitution from an indices/exponential equation

Question 68

2 marks

An equation contains the terms $2^{x+2}$ and $2^{x-1}$ . Using the substitution $u = 2^x$ , what do these two terms become?

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 69

3 marks

Write $\frac{5x^4 + 2}{x^3}$ as the sum of two separate terms, each a power of x.

From The hidden quadratic — substitution from an indices/exponential equation

Question 70

2 marks

Substituting $u = 3^x$ into an equation, you correctly reach the quadratic $9u^2+26u-3=0$ and solve it to get $u=\frac{1}{9}$ or $u=-3$ . What is the correct next step?

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 71

2 marks

Two candidates correctly split and integrate $\frac{dy}{dx} = \frac{6x^2 - 4}{x^2}$ . Candidate 1 writes $y = 6x + 4x^{-1}$ . Candidate 2 writes $y = 6x + 4x^{-1} + c$ . No point on the curve is given anywhere in the question. Which is correct?

From The hidden quadratic — substitution from an indices/exponential equation

Question 72

1 mark

If $u = 2^x$ , what is $2^{x+3}$ in terms of $u$ ?

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 73

2 marks

Write $\frac{8x^5 - 6}{2x^3}$ as the sum of two separate terms, each a power of x.

From The hidden quadratic — substitution from an indices/exponential equation

Question 74

1 mark

If $u = 5^x$ , what is $5^{2x}$ in terms of $u$ ?

From Splitting an Algebraic Fraction into Separate Terms Before Integrating

Question 75

1 mark

To find $\int \frac{5x^3 - 2}{x^2}\,dx$ , what is the correct first step?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 76

2 marks

In which of these does using the discriminant require first checking that the stated constant cannot take a specific 'forbidden' value, or the equation stops being a quadratic?

From The "hidden" higher-degree equation, reached via differentiation

Question 77

2 marks

Which of these equations is a "hidden quadratic in x²", solvable by substituting $u = x^2$ ?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 78

3 marks

For which values of the constant $m$ does the equation $x - \dfrac{6}{x} = m$ have no real solutions for $x$ (with $x \neq 0$ )?

From The "hidden" higher-degree equation, reached via differentiation

Question 79

3 marks

A curve has a stationary point at $x = 2$ , where $y = 7$ . What is the equation of the NORMAL to the curve at this point?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 80

4 marks

The equation $2mx^2 - 4mx + 3 = 0$ , where $m$ is a non-zero constant, has at least one real root. Which is the complete set of possible values of $m$ ?

From The "hidden" higher-degree equation, reached via differentiation

Question 81

2 marks

A student, solving a differentiation question, reaches $6x^4 - 5x^2 - 6 = 0$ and writes: " $x^2 = \frac{3}{2}$ or $x^2 = -\frac23$ , so $x = \frac32$ ." What is the single biggest error in this response?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 82

3 marks

The equation $5rx^2 + 20rx + 2 = 0$ , where $r$ is a non-zero constant, has no real roots. A student's working reads: ' $(20r)^2 - 4(5r)(2) = 20r^2 - 40r < 0$ , so $0 < r < 2$ .' What is the actual correct range?

From The "hidden" higher-degree equation, reached via differentiation

Question 83

4 marks

The curve $y = x^3 + \dfrac{2}{x}$ ( $x \neq 0$ ) has $\dfrac{dy}{dx} = 5$ at certain points. Find all values of $x$ for which this is true.

From The Discriminant with a Parameter — "No Real Roots" Style

Question 84

4 marks

The equation $qx^2 - 10qx + 9 = 0$ , where $q$ is a non-zero constant, has two distinct real roots. Which is the complete set of possible values of $q$ ?

From The "hidden" higher-degree equation, reached via differentiation

Question 85

1 mark

A student correctly finds that $f'(\frac13) = 0$ for a curve $y = f(x)$ . What can be said about the tangent to the curve at $x = \frac13$ ?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 86

2 marks

A different question states: "the curve does not cross the x-axis." Does this include or exclude the case where the curve touches the x-axis at exactly one point?

From The "hidden" higher-degree equation, reached via differentiation

Question 87

2 marks

Solving $2x^4 - 11x^2 + 5 = 0$ , a student substitutes $u = x^2$ to get $2u^2 - 11u + 5 = 0$ , which factorises to $(2u - 1)(u - 5) = 0$ , giving $u = \frac{1}{2}$ or $u = 5$ . What are ALL the values of $x$ ?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 88

1 mark

A question states: "the equation ... has no real roots." Which inequality does this translate to?

From The "hidden" higher-degree equation, reached via differentiation

Question 89

1 mark

$f(x) = 5x^3 - \dfrac{2}{x^2}$ . What is $f'(x)$ ?

From The Discriminant with a Parameter — "No Real Roots" Style

Question 90

1 mark

The equation $kx^2 - 6x + 3 = 0$ is described as a quadratic equation. What condition on $k$ does that description already require, before $b^2 - 4ac$ is even written down?

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 91

4 marks

The curve $y = 2x^3 - \dfrac{5}{x^2}$ has derivative $\dfrac{dy}{dx} = 6x^2 + \dfrac{10}{x^3}$ . What is the gradient of the curve at the point where $x = 1$ ?

From Sector Area, Arc Length, and Composite Perimeter

Question 92

2 marks

A sector has area 40 m² and angle 2.4 radians at the centre. What is the radius, to 2 decimal places?

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 93

3 marks

A question asks: 'Find the exact value of the gradient of the curve at the point where $x = 3$ .' A student correctly reaches $\left.\dfrac{dy}{dx}\right|_{x=3} = \dfrac{11}{7}$ , then writes the final answer as $1.57$ . What happens to the marks?

From Sector Area, Arc Length, and Composite Perimeter

Question 94

1 mark

Part (a) asked for the length of the major arc AB, and you correctly used 2π − θ before multiplying by r. Part (b) says: 'Hence find the perimeter of the region bounded by the major arc AB and the two radii OA, OB.' You have not yet checked whether part (a)'s arithmetic was right. What is the best approach for part (b)?

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 95

2 marks

A student correctly works out that $\dfrac{d}{dx}(5x^4 - 9) = 20x^3$ , then changes their final answer to $\dfrac{dy}{dx} = 20x^3 + c$ before submitting it. What is wrong with this?

From Sector Area, Arc Length, and Composite Perimeter

Question 96

2 marks

A question states: 'Give your answer as an exact multiple of π.' A sector has radius 8 cm and angle π/3 radians. Which of these is the exact area?

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 97

3 marks

Differentiate $y = \dfrac{2}{x^2}$ .

From Sector Area, Arc Length, and Composite Perimeter

Question 98

2 marks

A composite badge design has a sector, centre O, joined to two triangles, one on each side. P and Q are the two far corners of those triangles, and POQ is a straight line of length 13 cm passing through O. A student writes: 'POQ passes through the centre, so it must be the circle's diameter — radius = 6.5 cm.' What is wrong with this claim?

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 99

2 marks

$y = 3x^5 + 11$ . A student correctly finds $\dfrac{dy}{dx} = 15x^4$ , then writes the final line as $\dfrac{dy}{dx} = 15x^4 + c$ . Is the '+c' correct?

From Sector Area, Arc Length, and Composite Perimeter

Question 100

3 marks

The REFLEX angle AOB at the centre of a circle of radius 5 cm is 4.2 radians. What is the area of the MINOR sector OAB?

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 101

3 marks

What is $\dfrac{d}{dx}\left(\dfrac{2}{x^2}\right)$ ?

From Sector Area, Arc Length, and Composite Perimeter

Question 102

2 marks

A sector OAB has radius 6 cm and angle 1.3 radians. Find the perimeter of the sector OAB (3 s.f. if not exact).

From Differentiating Negative and Fractional Indices, Then Finding an Exact Gradient

Question 103

1 mark

Before applying the power rule to $y = 4\sqrt{x}$ , which rewritten form is correct?

From Sector Area, Arc Length, and Composite Perimeter

Question 104

2 marks

A sector has radius 7 cm and angle 1.5 radians. What is its area?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 105

2 marks

$C$ lies on the perpendicular bisector of $AB$ , and you're asked to find $C$ given a fixed distance $AC$ . Which point must be used as the right-angle vertex when setting up Pythagoras?

From Sector Area, Arc Length, and Composite Perimeter

Question 106

3 marks

The angle AOB at the centre O of a circle is 2.4 radians. A question asks for the length of the MAJOR arc AB. What angle, in radians (3 s.f.), should be used in $s = r\theta$ ?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 107

2 marks

$P$ , $Q$ , $R$ are three vertices of a rectangle $PQRS$ , given in order round the rectangle. Which method is confirmed, on a real script, to be the more reliable way to find $S$ ?

From Sector Area, Arc Length, and Composite Perimeter

Question 108

2 marks

A sector has radius 10 cm and angle 60°. A student writes $s = r\theta = 10 \times 60 = 600$ cm. What is wrong, and what is the correct arc length (3 s.f.)?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 109

1 mark

To find $PQ^2$ where $P(7, 3)$ and $Q(11, 15)$ , which is the correct calculation?

From Sector Area, Arc Length, and Composite Perimeter

Question 110

1 mark

A sector of a circle has radius 4 cm and angle 0.9 radians at the centre. What is the length of the arc?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 111

3 marks

A mark scheme awards 3 marks for 'prove angle $ABC = 90°$ ': M1 for a correct gradient method, A1 for both gradients correct, and a second A1 — dependent on both previous marks — for the concluding statement. A student writes: 'gradient $AB = -3$ , gradient $BC = 3$ . $-3 \times 3 = -9$ .' What is the correct assessment?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 112

1 mark

Part (a) asked for the minimum value of a quadratic. You made an arithmetic slip and wrote 7, when the correct value is 3. Part (b) begins "Hence...". What is the best thing to do?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 113

3 marks

Line $l$ has equation $2x - 5y + 10 = 0$ . Which is the equation of the line through the origin, perpendicular to $l$ ?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 114

2 marks

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Solve $2x^2 - 7x - 15 = 0$ .

Which response is certain to earn the method mark on the question above?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 115

2 marks

Line $l$ passes through $A(1, 4)$ and $B(5, -4)$ . What is the gradient of a line perpendicular to $l$ ?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 116

2 marks

A quadratic $f(x) = ax^2 + bx + c$ has $a > 0$ , and its minimum point lies strictly below the x-axis. What must be true of $b^2 - 4ac$ ?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 117

2 marks

Line $l$ has equation $3x + 4y = 12$ . What is the gradient of a line perpendicular to $l$ ?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 118

3 marks

Express $3x^2 + 12x + 5$ in the form $a(x + b)^2 + c$ , where $a$ , $b$ and $c$ are constants.

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 119

3 marks

A mark scheme awards 3 marks for 'prove that $AB$ is perpendicular to $BC$ ': M1 for a correct method for at least one gradient, A1 for both gradients correct, and a second A1 — dependent on BOTH previous marks — for an explicit concluding statement. A student's complete working reads: 'gradient $AB = 3$ , gradient $BC = -\frac{1}{3}$ . $3 \times (-\frac{1}{3}) = -1$ .' Nothing else is written. How many of the 3 marks have they earned?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 120

4 marks

The equation $px^2 + 4px + 3 = 0$ , where $p$ is a non-zero constant, has two distinct real roots. Which is the complete set of possible values of $p$ ?

From Perpendicularity Proofs, Rectangle/Point-Construction Problems, and Gradient-Pythagoras Combinations

Question 121

1 mark

A line has equation $y = 2x - 5$ . What is the gradient of a line perpendicular to it?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 122

2 marks

How many times does the curve $y = -3x^2 + 12x - 12$ meet the x-axis, and why?

From Quadratic inequalities, interpreted and represented graphically

Question 123

2 marks

Based on what real WMA11 examiner reports record about this exact topic, which of the following is the most reliable habit for protecting the marks on a quadratic-inequality question?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 124

3 marks

The equation $kx^2 + 6x + 2 = 0$ , where $k$ is a constant, has two distinct real roots. Which is the complete set of possible values of $k$ ?

From Quadratic inequalities, interpreted and represented graphically

Question 125

4 marks

Which is the correct solution set for $x^2 - x - 6 \geq 0$ and $x < 0$ , taken together?

From Quadratic Functions, the Discriminant, and Solving Quadratic Equations

Question 126

2 marks

The expression $x^2 + 6x + 1$ can be written as $(x + 3)^2 - 8$ . What are the coordinates of the turning point of the curve $y = x^2 + 6x + 1$ , and how do you know?

From Quadratic inequalities, interpreted and represented graphically

Question 127

3 marks

Which is the correct solution set for $x^2 - 9 \leq 0$ and $x > 1$ , taken together?