Edexcel C2 (Core Mathematics 2)

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3 + 2x)^5\), giving each term in its simplest form. [4]

Find the coordinates of the stationary point on the curve with equation \(y = 2x^2 - 12x\). [4]

\(f(x) = 2x^3 + x^2 - 5x + c\), where \(c\) is a constant. Given that \(f(1) = 0\),

1. find the value of \(c\), [2]
2. factorise \(f(x)\) completely, [4]
3. find the remainder when \(f(x)\) is divided by \((2x - 3)\). [2]

Evaluate \(\int_0^1 \frac{1}{\sqrt{x}} \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [4]

1. Find the remainder when \(x^3 - 2x^2 - 4x + 8\) is divided by
   1. \(x - 3\),
   2. \(x + 2\). [3]
2. Hence, or otherwise, find all the solutions to the equation \(x^3 - 2x^2 - 4x + 8 = 0\). [4]

The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.

1. Find the coordinates of the mid-point of \(AB\). [2]

Given that \(AB\) is a diameter of the circle \(C\),

1. find an equation for \(C\). [4]

Solve

1. \(5^x = 8\), giving your answer to 3 significant figures, [3]
2. \(\log_2(x + 1) - \log_2 x = \log_2 7\). [3]

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^9\), where \(p\) is a constant. [2]

The first 3 terms are 1, 36x and \(qx^2\), where \(q\) is a constant.

1. Find the value of \(p\) and the value of \(q\). [4]

\(f(x) = 3x^3 - 5x^2 - 16x + 12\).

1. Find the remainder when \(f(x)\) is divided by \((x - 2)\). [2]

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

1. factorise \(f(x)\) completely. [4]

The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find

1. the common ratio, [2]
2. the first term, [2]
3. the sum of the first 20 terms, giving your answer to the nearest whole number. [2]

Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(3^x = 5\), [3]
2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]

1. Use the factor theorem to show that \((x + 4)\) is a factor of \(2x^3 + x^2 - 25x + 12\). [2]
2. Factorise \(2x^3 + x^2 - 25x + 12\) completely. [4]

\includegraphics{figure_1} In Figure 1, \(A(4, 0)\) and \(B(3, 5)\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(AB\), [2]
2. the coordinates of the midpoint \(P\) of \(AB\), [2]
3. an equation for the circle \(C\). [3]

1. Find the first four terms, in ascending powers of \(x\), in the bionomial expansion of \((1 + kx)^8\), where \(k\) is a non-zero constant. [3]

Given that, in this expansion, the coefficients of \(x\) and \(x^2\) are equal, find

1. the value of \(k\), [2]
2. the coefficient of \(x^3\). [1]

1. Find the first 4 terms of the expansion of \(\left(1 + \frac{x}{3}\right)^{18}\) in ascending powers of \(x\), giving each term in its simplest form. [4]
2. Use your expansion to estimate the value of \((1.005)^{18}\), giving your answer to 5 decimal places. [3]

1. Show that the equation $$5 \cos^2 x = 3(1 + \sin x)$$ can be written as $$5 \sin^2 x + 3 \sin x - 2 = 0.$$ [2]
2. Hence solve, for \(0 \leq x < 360°\), the equation $$5 \cos^2 x = 3(1 + \sin x),$$ giving your answers to 1 decimal place where appropriate. [5]

1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^{12}\), where \(p\) is a non-zero constant. [2]

Given that, in the expansion of \((1 + px)^{12}\), the coefficient of \(x\) is \((-q)\) and the coefficient of \(x^2\) is \(11q\),

1. find the value of \(p\) and the value of \(q\). [4]

The first term of a geometric series is 120. The sum to infinity of the series is 480.

1. Show that the common ratio, \(r\), is \(\frac{3}{4}\). [3]
2. Find, to 2 decimal places, the difference between the 5th and 6th terms. [2]
3. Calculate the sum of the first 7 terms. [2]

The sum of the first \(n\) terms of the series is greater than 300.

1. Calculate the smallest possible value of \(n\). [4]

\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 6\) cm, \(BC = 4\) cm and \(CA = 5\) cm.

1. Show that \(\cos A = \frac{3}{4}\). [3]
2. Hence, or otherwise, find the exact value of \(\sin A\). [2]

1. Show that the equation \(3 \sin^2 \theta - 2 \cos^2 \theta = 1\) can be written as \(5 \sin^2 \theta = 3\). [2]
2. Hence solve, for \(0° \leq \theta \leq 360°\), the equation \(3 \sin^2 \theta - 2 \cos^2 \theta = 1\), giving your answer to 1 decimal place. [7]

\(f(x) = x^3 - 2x^2 + ax + b\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \((x - 2)\), the remainder is 1. When \(f(x)\) is divided by \((x + 1)\), the remainder is 28.

1. Find the value of \(a\) and the value of \(b\). [6]
2. Show that \((x - 3)\) is a factor of \(f(x)\). [2]

Solve, for \(0 \leq x \leq 180°\), the equation $$\sin(x + 10°) = \frac{\sqrt{3}}{2}.$$ [4]

1. \(\cos 2x = -0.9\), giving your answers to 1 decimal place. [4]

\includegraphics{figure_2} In Figure 2 \(OAB\) is a sector of a circle, radius 5 m. The chord \(AB\) is 6 m long.

1. Show that \(\cos A\hat{O}B = \frac{7}{25}\). [2]
2. Hence find the angle \(A\hat{O}B\) in radians, giving your answer to 3 decimal places. [1]
3. Calculate the area of the sector \(OAB\). [2]
4. Hence calculate the shaded area. [3]

The curve \(C\) has equation \(y = x\sqrt{x^2 + 1}, \quad 0 \leq x \leq 2\).

1. Copy and complete the table below, giving the values of \(y\) to 3 decimal places at \(x = 1\) and \(x = 1.5\).

   \(x\)	0	0.5	1	1.5	2
   \(y\)	0	0.530			6

   [2]
2. Use the trapezium rule, with all the \(y\) values from your table, to find an approximation for the value of \(\int_0^2 x\sqrt{x^2 + 1} \, dx\), giving your answer to 3 significant figures. [4]

\includegraphics{figure_2} Figure 2 shows the curve \(C\) with equation \(y = x\sqrt{x^2 + 1}\), \(0 \leq x \leq 2\), and the straight line segment \(l\), which joins the origin and the point \((2, 6)\). The finite region \(R\) is bounded by \(C\) and \(l\).

1. Use your answer to part (b) to find an approximation for the area of \(R\), giving your answer to 3 significant figures. [3]

Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \(a = 3b\), \(\log_3 a + \log_3 b = 2\). Give your answers as exact numbers. [6]

The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive. For this series, find

1. the common ratio, [2]
2. the first term, [2]
3. the sum of the first 50 terms, giving your answer to 3 decimal places, [2]
4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. [2]

A river, running between parallel banks, is 20 m wide. The depth, \(y\) metres, of the river measured at a point \(x\) metres from one bank is given by the formula $$y = \frac{1}{10}x(20 - x), \quad 0 \leq x \leq 20.$$

1. Complete the table below, giving values of \(y\) to 3 decimal places.

   \(x\)	0	4	8	12	16	20
   \(y\)	0		2.771			0

   [2]
2. Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. [4]

Given that the cross-sectional area is constant and that the river is flowing uniformly at 2 m s⁻¹,

1. estimate, in m³, the volume of water flowing per minute, giving your answer to 3 significant figures. [2]

The speed, \(v\) m s⁻¹, of a train at time \(t\) seconds is given by \(v = \sqrt{(1.2^t - 1)}, \quad 0 \leq t \leq 30.\) The following table shows the speed of the train at 5 second intervals.

1. Complete the table, giving the values of \(v\) to 2 decimal places. [3]

The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int_0^{30} \sqrt{(1.2^t - 1)} \, dt.$$

1. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\). [3]

1. Find, to 3 significant figures, the value of \(x\) for which \(8^x = 0.8\). [2]
2. Solve the equation \(2 \log_3 x - \log_3 7x = 1\). [4]

\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]

The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.

1. Calculate the value of \(\theta\). [4]

\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 8\) cm, \(AC = 11\) cm and \(\angle BAC = 0.7\) radians. The arc \(BD\), where \(D\) lies on \(AC\), is an arc of a circle with centre \(A\) and radius 8 cm. The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(BC\) and \(CD\) and the arc \(BD\). Find

1. the length of the arc \(BD\), [2]
2. the perimeter of \(R\), giving your answer to 3 significant figures, [4]
3. the area of \(R\), giving your answer to 3 significant figures. [5]

In the triangle \(ABC\), \(AB = 8\) cm, \(AC = 7\) cm, \(\angle ABC = 0.5\) radians and \(\angle ACB = x\) radians.

1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. [3]

Given that there are two possible values of \(x\),

1. find these values of \(x\), giving your answers to 2 decimal places. [3]

The curve \(C\) has equation \(y = 2x^3 - 5x^2 - 4x + 2\).

1. Find \(\frac{dy}{dx}\). [2]
2. Using the result from part (a), find the coordinates of the turning points of \(C\). [4]
3. Find \(\frac{d^2y}{dx^2}\). [2]
4. Hence, or otherwise, determine the nature of the turning points of \(C\). [2]

\includegraphics{figure_3} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3. The point \(A\) has coordinates \((1, -2)\) and the mid-point \(M\) of \(AB\) has coordinates \((3, 1)\). The line \(l\) passes through the points \(M\) and \(P\).

1. Find an equation for \(l\). [4]

Given that the \(x\)-coordinate of \(P\) is 6,

1. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is \(-1\). [1]
2. find an equation for the circle. [4]

\includegraphics{figure_2} The line with equation \(y = 3x + 20\) cuts the curve with equation \(y = x^2 + 6x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.

1. Use calculus to find the exact area of \(S\). [7]

The circle \(C\), with centre at the point \(A\), has equation \(x^2 + y^2 - 10x + 9 = 0\). Find

1. the coordinates of \(A\), [2]
2. the radius of \(C\), [2]
3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. [2]

Given that the line \(l\) with gradient \(\frac{7}{T}\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),

1. find an equation of the line which passes through \(A\) and \(T\). [3]

1. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0° \leq \theta \leq 360°\) for which \(5 \sin (\theta + 30°) = 3\). [4]
2. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0° \leq \theta \leq 360°\) for which \(\tan^2 \theta = 4\). [5]

A trading company made a profit of £50 000 in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio \(r, r > 1\). The model therefore predicts that in 2007 (Year 2) a profit of £50 000r will be made.

1. Write down an expression for the predicted profit in Year \(n\). [1]

The model predicts that in Year \(n\), the profit made will exceed £200 000.

1. Show that \(n > \frac{\log 4}{\log r} + 1\). [3]

Using the model with \(r = 1.09\),

1. find the year in which the profit made will first exceed £200 000, [2]
2. find the total of the profits that will be made by the company over the 10 years from 2006 to 2015 inclusive, giving your answer to the nearest £10 000. [3]

\includegraphics{figure_3} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the stage is \(2x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2x\) metres. The perimeter of the stage is 80 m.

1. Show that the area, \(A\) m², of the stage is given by $$A = 80x - \left(2 + \frac{\pi}{2}\right)x^2.$$ [4]
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value. [4]
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\). [2]
4. Calculate, to the nearest m², the maximum area of the stage. [2]

1. A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac{a(1-r^n)}{1-r}.$$ [4]

Mr King will be paid a salary of £35 000 in the year 2005. Mr King's contract promises a 4% increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.

1. Find, to the nearest £100, Mr King's salary in the year 2008. [2]

Mr King will receive a salary each year from 2005 until he retires at the end of 2024.

1. Find, to the nearest £1000, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024. [4]

\includegraphics{figure_3} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = -2x^2 + 4x\) and the line \(y = \frac{3}{2}\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find

1. the \(x\)-coordinates of the points \(A\) and \(B\), [4]
2. the exact area of \(R\). [6]

1. Sketch, for \(0 \leq x \leq 2\pi\), the graph of \(y = \sin\left(x + \frac{\pi}{6}\right)\). [2]
2. Write down the exact coordinates of the points where the graph meets the coordinate axes. [3]
3. Solve, for \(0 \leq x \leq 2\pi\), the equation \(\sin\left(x + \frac{\pi}{6}\right) = 0.65\), giving your answers in radians to 2 decimal places. [5]

\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).

1. Find the exact area of \(R\). [8]
2. Use calculus to show that \(y\) is increasing for \(x > 2\). [4]

\includegraphics{figure_4} Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm².

1. Show that the volume, \(V\) cm³, of the brick is given by \(V = 200x - \frac{4x^3}{3}\). [4]

Given that \(x\) can vary,

1. use calculus to find the maximum value of \(V\), giving your answer to the nearest cm³. [5]
2. Justify that the value of \(V\) you have found is a maximum. [2]