Questions — Edexcel C2 (579 questions)

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Edexcel C2 Q9
12 marks Moderate -0.3
\includegraphics{figure_2} The curve \(C\), shown in Fig. 2, represents the graph of $$y = \frac{x^2}{25}, x \geq 0.$$ 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]
Edexcel C2 Q1
5 marks Moderate -0.8
f(x) = px³ + 6x² + 12x + q. Given that the remainder when f(x) is divided by (x - 1) is equal to the remainder when f(x) is divided by (2x + 1),
  1. find the value of p. [4]
Given also that q = 3, and p has the value found in part (a),
  1. find the value of the remainder. [1]
Edexcel C2 Q2
6 marks Moderate -0.3
Figure 1 \includegraphics{figure_1} 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]
Edexcel C2 Q3
7 marks Moderate -0.8
  1. Expand (2√x + 3)². [2]
  2. Hence evaluate $$\int_1^{2^2} (2\sqrt{x} + 3)^2 \, dx$$, giving your answer in the form a + b√2, where a and b are integers. [5]
Edexcel C2 Q4
7 marks Standard +0.8
The first three terms in the expansion, in ascending powers of x, of (1 + px)ⁿ, are 1 - 18x + 36p²x². Given that n is a positive integer, find the value of n and the value of p. [7]
Edexcel C2 Q5
8 marks Moderate -0.8
Find all values of θ in the interval 0 ≤ θ < 360 for which
  1. cos (θ + 75)° = 0. [3]
  2. sin 2θ° = 0.7, giving your answers to one decimal place. [5]
Edexcel C2 Q6
9 marks Moderate -0.8
Given that log₂ x = a, find, in terms of a, the simplest form of
  1. log₂ (16x), [2]
  2. log₂ \(\left(\frac{x⁴}{2}\right)\). [3]
  1. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]
Edexcel C2 Q7
9 marks Moderate -0.8
The curve C has equation \(y = \cos \left(x + \frac{\pi}{4}\right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch C. [2]
  2. Write down the exact coordinates of the points at which C meets the coordinate axes. [3]
  1. Solve, for x in the interval \(0 \leq x \leq 2\pi\), $$\cos \left(x + \frac{\pi}{4}\right) = 0.5,$$ giving your answers in terms of π. [4]
Edexcel C2 Q8
12 marks Moderate -0.3
Figure 2 \includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = x³ - 6x² + 9x.$$ 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
12 marks Standard +0.3
\includegraphics{figure_3} Fig. 3 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, ∠BAC = 1.504 radians. [3]
Calculate,
  1. the area, in cm², of the sector APQ, [2]
  2. the area, in cm², of the shaded region BPQC, [3]
  3. the perimeter, in cm, of the shaded region BPQC. [4]
END
Edexcel C2 Q1
7 marks Moderate -0.3
f(x) = x³ + ax² + bx - 10, where a and b 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]
Edexcel C2 Q2
8 marks Standard +0.3
  1. Write down the first four terms of the binomial expansion, in ascending powers of x, of \((1 + ax)^n\), where \(n > 2\). [2]
Given that, in this expansion, the coefficient of x is 8 and the coefficient of x² is 30,
  1. find the value of n and the value of a, [4]
  2. find the coefficient of x³. [2]
Edexcel C2 Q3
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 Q4
11 marks Standard +0.3
Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),
  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 \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
  3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]
Edexcel C2 Q5
11 marks Standard +0.3
\includegraphics{figure_1} 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 mm², 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 cm³, the volume of the bar. [2]
Edexcel C2 Q6
9 marks Standard +0.3
\includegraphics{figure_2} 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 Q7
14 marks Standard +0.3
Find all the values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which
  1. \(\cos(\theta - 10°) = \cos 15°\), [3]
  2. \(\tan 2\theta = 0.4\), [5]
  3. \(2 \sin \theta \tan \theta = 3\). [6]
Edexcel C2 Q1
6 marks Moderate -0.8
  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]
Edexcel C2 Q2
7 marks Standard +0.3
\(f(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.
  1. Given that f(n) has a remainder of 3 when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
  2. Show that f(n) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
  3. Hence show that f(n) is divisible by 3 for all positive integer values of \(n\). [2]
Edexcel C2 Q3
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 Q4
7 marks Moderate -0.8
Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after \(x\) years, the total value of an initial £1 investment is £\(y\), where $$y = 1.04^x.$$
  1. Sketch the graph of \(y = 1.04^x\), \(x \geq 0\). [2]
  2. Calculate, to the nearest £, the total value of an initial £800 investment after 10 years. [2]
  3. Use logarithms to find the number of years it takes to double the total value of any initial investment. [3]
Edexcel C2 Q5
10 marks Standard +0.3
The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P\) (ln 2, \(p + 2q\)) on \(C\), the gradient is 5.
  1. Find the value of \(p\) and the value of \(q\). [5]
The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
  1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately 53.8 [5]
Edexcel C2 Q6
11 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).
  1. Show that \(p = 6\). [1]
  2. Find an equation of the tangent to \(C\) at \(A\). [4]
The curve \(C\) has a maximum at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\). [2]
The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.
  1. Find the area of \(R\). [4]
Edexcel C2 Q7
12 marks Moderate -0.3
\includegraphics{figure_1} 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 \(\theta = 1 + \frac{1}{2}\pi\). [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
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 \(n = 6k + 2\). Given also that the coefficients of \(x^4\) and \(x^5\) are equal and non-zero, [3]
  2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\), [4]
  3. 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]