**Answer:** $\cos A = \frac{105^2 + 92^2 - 75^2}{2 \times 105 \times 92}$ oe; $0.717598...soi$; $A = 44.14345...°$ soi $[0.770448553...]$; $\frac{1}{2} \times 92 \times 105 \times \sin(\text{their } A)$; $3360$ or $3361$ to $3365$

**Marks:** M1 | A1 | A1 | M1 | A1 | [5]

**Guidance:** or $\cos B = \frac{75^2 + 92^2 - 105^2}{2 \times 75 \times 92}$ oe; $0.2220289...soi$; $B = 77.1717719...°$ soi $[1.346901422]$; or $\cos C = \frac{105^2 + 75^2 - 92^2}{2 \times 105 \times 75}$ oe; $0.519746...soi$; $C = 58.6847827...°$ soi $[1.024242678...]$; ignore minor errors due to premature rounding for second A1 condone $A$, $B$ or $C$ wrongly attributed or $\frac{1}{2} \times 75 \times 92 \times \sin(\text{their } B)$ or $\frac{1}{2} \times 75 \times 105 \times \sin(\text{their } C)$; or M3 for $\sqrt{136(136-75)(136-105)(136-92)}$; A2 for correct answer $3360$ or $3363-3364$

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## Question 8(i)

**Answer:** Curve of correct shape in both quadrants; through $(0, 1)$ shown on graph or in commentary

**Marks:** M1 | A1 | [2]

**Guidance:** SC1 for curve correct in 1st quadrant and touching $(0,1)$ or identified in commentary

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## Question 8(ii)

**Answer:** $5x - 1 = \frac{\log_{10} 500000}{\log_{10} 3}$ or $5x - 1 = \log_3 500\,000$; $x = \left(\frac{\log_{10} 500000}{\log_{10} 3} + 1\right) \div 5$; $[x = ]$ $2.588$ to $2.59$

**Marks:** M1 | M1 | A1 | [3]

**Guidance:** condone omission of base 10; use of logs in other bases may earn full marks; $x = (\log_3 500000 + 1) \div 5$; oe; or B3 www; if unsupported, B3 for correct answer to 3 sf or more www

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## Question 9(i)

**Answer:** $\left(\frac{\sin \theta}{\cos \theta}\right) = 1$ oe; $\frac{\cos \theta}{\cos \theta}$; $\sin \theta = \cos^2 \theta$ and completion to given result

**Marks:** M1 | A1 | [2]

**Guidance:** www

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## Question 9(ii)

**Answer:** $\sin^2 \theta + \sin \theta - 1[= 0]$; $[\sin \theta =] \frac{-1 \pm \sqrt{5}}{2}$ oe may be implied by correct answers; $[\theta =] 38.17...$ or $38.2$ and $141.83..., 141.8$ or $142$

**Marks:** M1 | A1 | A1 | [3]

**Guidance:** allow 1 on RHS if attempt to complete square; may be implied by correct answers; ignore extra values outside range or in radians; A0 if extra values in range or in radians; condone $y^2 + y - 1 = 0$; mark to benefit of candidate; ignore any work with negative root & condone omission of negative root with no comment eg M1 for $0.618...$; if unsupported, B1 for one of these, B2 for both. If both values correct with extra values in range, then B1. NB $0.6662$ and $2.4754$ if working in radian mode earns MIAIA0; NB $0.6662$ and $2.4754$ to 3sf or more

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## Question 10(i)

**Answer:** at $A$ $y = 3$; $\frac{dy}{dx} = 2x - 4$; their $\frac{dy}{dx} = 2 \times 4 - 4$; grad of normal $= -\frac{1}{\text{their 4}}$; $y - 3 = (-\frac{1}{4}) \times (x - 4)$ oe isw; substitution of $y = 0$ and completion to given result with at least 1 correct interim step www

**Marks:** B1 | B1 | M1* | M1dep* | A1 | A1 | [6]

**Guidance:** must follow from attempt at differentiation; or substitution of $x = 16$ to obtain $y = 0$; correct interim step may occur before substitution

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## Question 10(ii)

**Answer:** at B, $x = 3$; $F[x] = \frac{x^3}{3} - \frac{4x^2}{2} + 3x$; $F[4] - F[\text{their 3}]$; area of triangle $= 18$ soi; area of region $= 19\frac{1}{3}$ oe isw

**Marks:** B1 | M1* | M1*dep | B1 | A1 | [5]

**Guidance:** may be embedded; condone one error, must be three terms. ignore $+ c$; dependent on integration attempted; may be embedded in final answer; 19.3 or better

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## Question 11(i)(A)

**Answer:** $2A + D = 25$ oe; $4A + 6D = 250$ oe; $D = 50,$; $A = -12.5$ oe

**Marks:** B1 | B1 | B1 | B1 | [4]

**Guidance:** condone lower-case $a$ and $d$

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## Question 11(i)(B)

**Answer:** $\frac{50}{2}(2 \times \text{their } A + 49 \times \text{their } D) = [60\,625]$ or $\frac{20}{2}(2 \times \text{their } A + 19 \times \text{their } D) = [9250]$; their "$S_{50} - S_{20}$"; $51\,375$ cao

**Marks:** M1 | M1 | A1 | [3]

**Guidance:** or $a = \text{their } A + 20D$; $S_{50} = \frac{30}{2}(a + l)$ oe with $l = \text{their } A + 49D$; $S_{50} = \frac{30}{2}(2 \times \text{their } 987.5 + 29 \times \text{their } 50)$

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## Question 11(ii)

**Answer:** $a\left(\frac{r^2 - 1}{r - 1}\right) = 25$ or $\frac{a(r^4 - 1)}{r - 1} = 250$; $\frac{a(r^2 - 1)}{r - 1} = \frac{250}{a\frac{(r^2 - 1)}{(r-1)}}$ and completion to given result www; use of $r^4 - 1 = (r^2 - 1)(r^2 + 1)$ to obtain $r^2 + 1 = 10$ www; $r = \pm 3$; $a = 6.25$ or $-12.5$ oe

**Marks:** B1 | M1 | M1 | A1 | A1 | [5]

**Guidance:** allow $a(1 + r)$ as the denominator in the quadruple-decker fraction; at least one correct interim step required; or multiplication and rearrangement of quadratic to obtain $r^4 - 10r^2 + 9 = 0$ oe with all three terms on one side; $r^2 = x$ oe may be used; or M1 for valid alternative algebraic approaches eg using $a(1 + r) = 25$ and $ar^2(1 + r) = 225$; or B2 for all four values correct, B1 for both $r$ values or both $a$ values or one pair of correct values if second M mark not earned; or A1 for one correct pair of values of $r$ and $a$

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## Question 12(i)

**Answer:** $\log_{10} p = \log_{10} d + \log_{10} 10^{kt}$; $\log_{10} p = \log_{10} d + kt$ www

**Marks:** M1 | A1 | [2]

**Guidance:** condone omission of base; if unsupported, B2 for correct equation

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## Question 12(ii)

**Answer:** $2.02, 2.13, 2.23$; plots correct; ruled line of best fit

**Marks:** B1 | B1ft. | B1 | [3]

**Guidance:** allow given to more sig figs; to nearest half square; y-intercept between 1.65 and 1.7 and at least one point on or above the line and at least one point on or below the line; 2.022304623..., 2.129657673, 2.229707433; fit their plots; must cover range from $x = 9$ to 49

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## Question 12(iii)

**Answer:** $0.0105$ to $0.0125$ for $k$; $1.66$ to $1.69$ for $\log_{10} d$ or $45.7$ to $49.0$ for $a$; $\log_{10} p = \text{their } kt + \text{their } \log_{10} d$; $p = \text{their } "47.9 \times 10^{0.0115x}"$ or $10^{0.6785 + 0.0115x}$ or

**Marks:** B1 | B1 | B1 | B1 | [4]

**Guidance:** must be connected to $k$; must be connected to $a$; must be a correct form for equation of line and with their y-intercept and their gradient (may be found from graph or from table, must be correct method); as above, "$47.9$" and "$0.0115$" must follow from correct method

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## Question 12(iv)

**Answer:** $45.7$ to $49.0$ million

**Marks:** 1 | [1]

**Guidance:** 'million' needed, not just the value of $p$

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## Question 12(v)

**Answer:** reading from graph at $2.301...$; their $54$; $2014$ cao

**Marks:** M1* | M1dep* | A1 | [3]

**Guidance:** or $\log_{10} 200 = "\log_{10} d + kt"$; eg for their $t = \frac{\log 200 - 1.68}{0.0115}$; or M1 for their $t = \frac{\log \frac{200}{47.9}}{0.0115}$; if unsupported, allow B3 only if consistent with graph