Questions C2 (1550 questions)

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Edexcel C2 2008 January Q4
9 marks Moderate -0.8
  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 \(0° \leq \theta < 360°\), the equation $$3 \sin^2 \theta - 2 \cos^2 \theta = 1,$$ giving your answers to 1 decimal place. [7]
Edexcel C2 2008 January Q5
6 marks Moderate -0.3
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]
Edexcel C2 2008 January Q6
7 marks Moderate -0.8
\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]
Edexcel C2 2008 January Q7
10 marks Moderate -0.8
\includegraphics{figure_2} In Figure 2 the curve \(C\) has equation \(y = 6x - x^2\) and the line \(L\) has equation \(y = 2x\).
  1. Show that the curve \(C\) intersects the \(x\)-axis at \(x = 0\) and \(x = 6\). [1]
  2. Show that the line \(L\) intersects the curve \(C\) at the points \((0, 0)\) and \((4, 8)\). [3]
The region \(R\), bounded by the curve \(C\) and the line \(L\), is shown shaded in Figure 2.
  1. Use calculus to find the area of \(R\). [6]
Edexcel C2 2008 January Q8
11 marks Standard +0.3
A circle \(C\) has centre \(M\) \((6, 4)\) and radius 3.
  1. Write down the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = r^2.$$ [2]
\includegraphics{figure_3} Figure 3 shows the circle \(C\). The point \(T\) lies on the circle and the tangent at \(T\) passes through the point \(P\) \((12, 6)\). The line \(MP\) cuts the circle at \(Q\).
  1. Show that the angle \(TMQ\) is 1.0766 radians to 4 decimal places. [4]
The shaded region \(TPQ\) is bounded by the straight lines \(TP\), \(QP\) and the arc \(TQ\), as shown in Figure 3.
  1. Find the area of the shaded region \(TPQ\). Give your answer to 3 decimal places. [5]
Edexcel C2 2008 January Q9
12 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is 100 m³.
  1. Show that the area \(A\) m² of the sheet metal used to make the tank is given by $$A = \frac{300}{x} + 2x^2.$$ [4]
  2. Use calculus to find the value of \(x\) for which \(A\) is stationary. [4]
  3. Prove that this value of \(x\) gives a minimum value of \(A\). [2]
  4. Calculate the minimum area of sheet metal needed to make the tank. [2]
Edexcel C2 Q1
4 marks Moderate -0.8
\(f(x) = 2x^3 - x^2 + px + 6\), where \(p\) is a constant. Given that \((x - 1)\) is a factor of \(f(x)\), find
  1. the value of \(p\), [2]
  2. the remainder when \(f(x)\) is divided by \((2x + 1)\). [2]
Edexcel C2 Q2
5 marks Easy -1.2
  1. Find \(\int \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [3]
  2. Hence evaluate \(\int_1^2 \left( 3 + 4x^3 - \frac{2}{x^2} \right) dx\). [2]
Edexcel C2 Q3
5 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a logo \(ABD\). The logo is formed from triangle \(ABC\). The mid-point of \(AC\) is \(D\) and \(BC = AD = DC = 6\) cm. \(\angle BCA = 0.4\) radians. The curve \(BD\) is an arc of a circle with centre \(C\) and radius 6 cm.
  1. Write down the length of the arc \(BD\). [1]
  2. Find the length of \(AB\). [3]
  3. Write down the perimeter of the logo \(ABD\), giving your answer to 3 significant figures. [1]
Edexcel C2 Q4
6 marks Standard +0.3
Solve $$2 \log_3 x - \log_3 (x - 2) = 2, \quad x > 2.$$ [6]
Edexcel C2 Q5
7 marks Moderate -0.8
The second and fifth terms of a geometric series are 9 and 1.125 respectively. For this series find
  1. the value of the common ratio, [3]
  2. the first term, [2]
  3. the sum to infinity. [2]
Edexcel C2 Q6
7 marks Standard +0.3
The circle \(C\), with centre \(A\), has equation $$x^2 + y^2 - 6x + 4y - 12 = 0.$$
  1. Find the coordinates of \(A\). [2]
  2. Show that the radius of \(C\) is 5. [2]
The points \(P\), \(Q\) and \(R\) lie on \(C\). The length of \(PQ\) is 10 and the length of \(PR\) is 3.
  1. Find the length of \(QR\), giving your answer to 1 decimal place. [3]
Edexcel C2 Q7
8 marks Standard +0.3
The first four terms, in ascending powers of \(x\), of the binomial expansion of \((1 + kx)^n\) are $$1 + Ax + Bx^2 + Bx^3 + \ldots,$$ where \(k\) is a positive constant and \(A\), \(B\) and \(n\) are positive integers.
  1. By considering the coefficients of \(x^2\) and \(x^3\), show that \(3 = (n - 2) k\). [4]
Given that \(A = 4\),
  1. find the value of \(n\) and the value of \(k\). [4]
Edexcel C2 Q8
10 marks Moderate -0.3
  1. Solve, for \(0 \leq x < 360°\), the equation \(\cos (x - 20°) = -0.437\), giving your answers to the nearest degree. [4]
  2. Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$3 \tan \theta = 2 \cos \theta.$$ [6]
Edexcel C2 Q9
11 marks Standard +0.3
A pencil holder is in the shape of an open circular cylinder of radius \(r\) cm and height \(h\) cm. The surface area of the cylinder (including the base) is 250 cm\(^2\).
  1. Show that the volume, V cm\(^3\), of the cylinder is given by \(V = 125r - \frac{\pi r^3}{2}\). [4]
  2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value. [3]
  3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\). [2]
  4. Calculate, to the nearest cm\(^3\), the maximum volume of the pencil holder. [2]
Edexcel C2 Q10
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2x - \frac{2}{\sqrt{x}}, \quad x > 0.$$ The point \(A(1, 5)\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B(b, 0)\), where \(b\) is a constant and \(b > 0\).
  1. Verify that \(b = 4\). [1]
The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2.
  1. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\). [4]
  2. Find the coordinates of the point \(D\). [1]
The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(AD\) and the \(x\)-axis.
  1. Use integration to find the area of \(R\). [6]
Edexcel C2 Q1
8 marks Moderate -0.8
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]
Edexcel C2 Q2
9 marks Moderate -0.3
\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]
Edexcel C2 Q3
10 marks Moderate -0.8
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]
Edexcel C2 Q4
12 marks Moderate -0.3
\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]
Edexcel C2 Q5
14 marks Standard +0.3
\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]
Edexcel C2 Q6
6 marks Moderate -0.3
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]
Edexcel C2 Q7
12 marks Moderate -0.3
\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]
Edexcel C2 Q8
12 marks Moderate -0.3
\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]
Edexcel C2 Q9
9 marks Standard +0.3
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]