Questions — Edexcel C2 (579 questions)

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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]
Edexcel C2 Q10
11 marks Standard +0.3
  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]
Edexcel C2 Q11
4 marks Easy -1.2
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]
Edexcel C2 Q12
6 marks Moderate -0.3
$$f(x) = ax^3 + bx^2 - 7x + 14, \text{ where } a \text{ and } b \text{ are constants.}$$ 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]
Edexcel C2 Q13
7 marks Moderate -0.8
$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ 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]
Edexcel C2 Q14
8 marks Standard +0.3
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]
Edexcel C2 Q15
10 marks Standard +0.3
\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]
Edexcel C2 Q16
12 marks Moderate -0.3
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]
Edexcel C2 Q17
12 marks Moderate -0.3
\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]
Edexcel C2 Q18
6 marks Moderate -0.8
Given that \(p = \log_4 16\), express in terms of \(p\),
  1. \(\log_4 2\). [2]
  2. \(\log_4 (8q)\). [4]
Edexcel C2 Q19
11 marks Standard +0.8
$$f(x) = \left(1 + \frac{x}{k}\right)^n, \quad k, n \in \mathbb{N}, \quad n > 2.$$ 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]
Edexcel C2 Q20
3 marks Moderate -0.8
$$f(x) = 4x^3 + 3x^2 - 2x - 6.$$ Find the remainder when \(f(x)\) is divided by \((2x + 1)\). [3]
Edexcel C2 Q21
9 marks Moderate -0.3
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]
Edexcel C2 Q22
7 marks Standard +0.3
\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]
Edexcel C2 Q23
8 marks Moderate -0.8
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]
Edexcel C2 Q24
10 marks Moderate -0.8
$$f(x) = 5 \sin 3x°, \quad 0 \leq x \leq 180.$$
  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]
Edexcel C2 Q25
11 marks Standard +0.3
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]