Edexcel C2 (Core Mathematics 2)

Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which

1. \(\cos (\theta + 75)° = 0\). [3]
2. \(\sin 2\theta ° = 0.7\), giving your answers to one decimal place. [5]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\).

1. Find the \(x\)-coordinates of \(A\) and \(B\). [3]

The shaded region \(R\) is bounded by the curve and the line.

1. Find the area of \(R\). [6]

The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

1. the common ratio of the series, [2]
2. the first term of the series, [2]
3. the sum to infinity of the series. [2]
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]

\includegraphics{figure_2} Triangle \(ABC\) has \(AB = 9\) cm, \(BC = 10\) cm and \(CA = 5\) cm. A circle, centre \(A\) and radius 3 cm, intersects \(AB\) and \(AC\) at \(P\) and \(Q\) respectively, as shown in Fig. 3.

1. Show that, to 3 decimal places, \(\angle BAC = 1.504\) radians. [3]

Calculate,

1. the area, in cm\(^2\), of the sector \(APQ\), [2]
2. the area, in cm\(^2\), of the shaded region \(BPQC\), [3]
3. the perimeter, in cm, of the shaded region \(BPQC\). [4]

\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

The manufacturer needs to minimise the surface area of a carton.

1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
2. Calculate the minimum value of \(A\). [2]
3. Prove that this value of \(A\) is a minimum. [2]

Given that \(2 \sin 2\theta = \cos 2\theta\),

1. show that \(\tan 2\theta = 0.5\). [1]
2. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2\theta ° = \cos 2\theta °\). [5]

\includegraphics{figure_4} Figure 1 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,

1. prove that \(\theta = 1 + \frac{1}{4}\pi\). [5]

Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),

1. the perimeter of shape \(X\), [2]
2. the perimeter of shape \(Y\). [3]
3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]

\includegraphics{figure_5} Figure 2 shows part of the curve with equation $$y = x^3 - 6x^2 + 9x.$$ The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).

1. Show that the equation of the curve may be written as $$y = x(x - 3)^2,$$ and hence write down the coordinates of \(A\). [2]
2. Find the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the area of \(R\). [5]

A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by $$P = \frac{2000a^t}{4 + a^t},$$ where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,

1. calculate, to 4 decimal places, the value of \(a\), [4]
2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800. [4]
3. With reference to this model, give a reason why the population of deer cannot exceed 2000. [1]

1. Given that $$(2 + x)^5 + (2 - x)^5 = A + Bx^2 + Cx^4,$$ find the values of the constants \(A\), \(B\) and \(C\). [6]
2. Using the substitution \(y = x^2\) and your answers to part (a), solve, $$(2 + x)^5 + (2 - x)^5 = 349.$$ [5]

A circle \(C\) has equation $$x^2 + y^2 - 10x + 6y - 15 = 0.$$

1. Find the coordinates of the centre of \(C\). [2]
2. Find the radius of \(C\). [2]

$$f(x) = ax^3 + bx^2 - 7x + 14, \text{ where } a \text{ and } b \text{ are constants.}$$ Given that when \(f(x)\) is divided by \((x - 1)\) the remainder is 9,

1. write down an equation connecting \(a\) and \(b\). [2]

Given also that \((x + 2)\) is a factor of \(f(x)\),

1. Find the values of \(a\) and \(b\). [4]

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),

1. Find the value of \(c\), [2]
2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]

Find the values of \(\theta\), to 1 decimal place, in the interval \(-180 \leq \theta < 180\) for which $$2 \sin^2 \theta ° - 2 \sin \theta ° = \cos^2 \theta °.$$ [8]

\includegraphics{figure_6} Figure 1 shows a gardener's design for the shape of a flower bed with perimeter \(ABCD\). \(AD\) is an arc of a circle with centre \(O\) and radius 5 m. \(BC\) is an arc of a circle with centre \(O\) and radius 7 m. \(OAB\) and \(ODC\) are straight lines and the size of \(\angle AOD\) is \(\theta\) radians.

1. Find, in terms of \(\theta\), an expression for the area of the flower bed. [3]

Given that the area of the flower bed is 15 m\(^2\),

1. show that \(\theta = 1.25\), [2]
2. calculate, in m, the perimeter of the flower bed. [3]

The gardener now decides to replace arc \(AD\) with the straight line \(AD\).

1. Find, to the nearest cm, the reduction in the perimeter of the flower bed. [2]

A geometric series is \(a + ar + ar^2 + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}.$$ [4]

The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find

1. the value of \(r\) and the value of \(a\), [5]
2. the sum to infinity of the series. [3]

\includegraphics{figure_7} Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6x - x^2 - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the line and the curve.

1. Calculate the area of \(R\). [7]

Given that \(p = \log_4 16\), express in terms of \(p\),

1. \(\log_4 2\). [2]
2. \(\log_4 (8q)\). [4]

$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ Given that the coefficient of \(x^3\) is twice the coefficient of \(x^2\) in the binomial expansion of \(f(x)\),

1. prove that \(n = 6k + 2\). [3]

Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero,

1. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). [4]

Using these values of \(k\) and \(n\),

1. expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\). Give each coefficient as an exact fraction in its lowest terms [4]

$$f(x) = 4x^3 + 3x^2 - 2x - 6.$$ Find the remainder when \(f(x)\) is divided by \((2x + 1)\). [3]

A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).

1. Write down an equation of the circle \(C\). [2]
2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
3. Find the distance between these two points. [2]

\includegraphics{figure_8} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.

1. Find, in surd form, the length \(AB\). [2]
2. Find, in terms of \(\pi\), the area of the badge. [2]
3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [3]

Given that \(f(x) = 15 - 7x - 2x^2\),

1. find the coordinates of all points at which the graph of \(y = f(x)\) crosses the coordinate axes. [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Calculate the coordinates of the stationary point of \(f(x)\). [3]

$$f(x) = 5 \sin 3x°, \quad 0 \leq x \leq 180.$$

1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis. [3]
2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]

Given that \(f'(x) = (2x^3 - 3x^{-2})^2 + 5\), \(x > 0\),

1. Find, to 3 significant figures, the value of \(x\) for which \(f'(x) = 5\). [3]
2. Show that \(f'(x)\) may be written in the form \(Ax^6 + \frac{B}{x^4} + C\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
3. Hence evaluate \(\int_1^2 f'(x) \, dx\). [5]

\includegraphics{figure_9} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500).$$ [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

1. Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \((1 + 3x)^n\), where \(n > 2\). [2]

Given that the coefficient of \(x^3\) in this expansion is ten times the coefficient of \(x^2\),

1. find the value of \(n\), [2]
2. find the coefficient of \(x^4\) in the expansion. [2]

1. Given that \(3 + 2 \log_2 x = \log_2 y\), show that \(y = 8x^2\). [3]
2. Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log_2 x = \log_2 (14x - 3).$$ [3]
3. Show that \(\log_2 \alpha = -2\). [1]
4. Calculate \(\log_2 \beta\), giving your answer to 3 significant figures. [3]

$$f(x) = x^3 + ax^2 + bx - 10, \text{ where } a \text{ and } b \text{ are constants.}$$ When \(f(x)\) is divided by \((x - 3)\), the remainder is 14. When \(f(x)\) is divided by \((x + 1)\), the remainder is \(-18\).

1. Find the value of \(a\) and the value of \(b\). [5]
2. Show that \((x - 2)\) is a factor of \(f(x)\). [2]

1. Using the factor theorem, show that \((x + 3)\) is a factor of $$x^3 - 3x^2 - 10x + 24.$$ [2]
2. Factorise \(x^3 - 3x^2 - 10x + 24\) completely. [4]

1. Differentiate with respect to \(x\) $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5]
2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5]

1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]

A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £\(d\). This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive.

1. Find the value of \(d\). [4]

Using your value of \(d\),

1. find the predicted profit for the year 2011. [2]

An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54 000,

1. find the predicted profit for the year 2011. [3]

1. Solve, for \(0° < x < 180°\), the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
   1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
   2. Find the exact value of \(\cos A\). [2]

1. Using the substitution \(u = 2^x\), show that the equation \(4^x - 2^{(x + 1)} - 15 = 0\) can be written in the form \(u^2 - 2u - 15 = 0\). [2]
2. Hence solve the equation \(4^x - 2^{(x + 1)} - 15 = 0\), giving your answers to 2 decimals places. [4]

The sequence \(u_1, u_2, u_3, \ldots, u_n\) is defined by the recurrence relation $$u_{n+1} = pu_n + 5, \quad u_1 = 2, \text{ where } p \text{ is a constant.}$$ Given that \(u_3 = 8\),

1. show that one possible value of \(p\) is \(\frac{1}{2}\) and find the other value of \(p\). [5]

Using \(p = \frac{1}{2}\),

1. write down the value of \(\log_2 p\). [1]

Given also that \(\log_2 q = t\),

1. express \(\log_2 \left(\frac{p^3}{\sqrt{q}}\right)\) in terms of \(t\). [3]

1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
2. Write down the coordinates of the points at which the graph meets the axes. [3]
3. Solve, for \(0 \leq x < 360°\), the equation $$\sin (x + 30°) = -\frac{1}{2}.$$ [3]

A geometric series has first term 1200. Its sum to infinity is 960.

1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
3. Write down an expression for the sum of the first \(n\) terms of the series. [2]

Given that \(n\) is odd,

1. prove that the sum of the first \(n\) terms of the series is $$960(1 + 0.25^n).$$ [2]

On a journey, the average speed of a car is \(v\) m s\(^{-1}\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by $$C = \frac{160}{v} + \frac{v^2}{100}.$$ Using this model,

1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\). [5]
2. Justify that this value of \(v\) gives a minimum value of \(C\). [2]
3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey. [3]

\includegraphics{figure_10} Figure 1 shows the cross-section \(ABCD\) of a chocolate bar, where \(AB\), \(CD\) and \(AD\) are straight lines and \(M\) is the mid-point of \(AD\). The length \(AD\) is 28 mm, and \(BC\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B\), \(C\) and \(D\) have coordinates \((7, 24)\), \((21, 24)\) and \((28, 0)\) respectively.

1. Show that the length of \(BM\) is 25 mm. [1]
2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
3. Hence calculate, in mm\(^2\), the area of the cross-section of the chocolate bar. [5]

Given that this chocolate bar has length 85 mm,

1. calculate, to the nearest cm\(^3\), the volume of the bar. [2]

\includegraphics{figure_11} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, \quad x \geq 0.$$ The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.

1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
2. Use integration to find the area of \(R\). [5]

The first three terms in the expansion, in ascending powers of \(x\), of \((1 + px)^n\), are \(1 - 18x + 36p^2x^2\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\). [7]

\includegraphics{figure_12} The circle \(C\), with centre \((a, b)\) and radius 5, touches the \(x\)-axis at \((4, 0)\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\). [1]
2. Find a cartesian equation of \(C\). [2]

A tangent to the circle, drawn from the point \(P(8, 17)\), touches the circle at \(T\).

1. Find, to 3 significant figures, the length of \(PT\). [3]