Questions — Edexcel (10514 questions)

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Edexcel C2 Q4
9 marks Moderate -0.8
Given that \(\log_2 x = a\), find, in terms of \(a\), the simplest form of
  1. \(\log_2 (16x)\), [2]
  2. \(\log_2 \left( \frac{x^4}{2} \right)\). [3]
  3. 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 Q5
10 marks Standard +0.3
  1. Given that \(3 \sin x = 8 \cos x\), find the value of \(\tan x\). [1]
  2. Find, to 1 decimal place, all the solutions of \(3 \sin x - 8 \cos x = 0\) in the interval \(0 \leq x < 360°\). [3]
  3. Find, to 1 decimal place, all the solutions of \(3 \sin^2 y - 8 \cos y = 0\) in the interval \(0 \leq y < 360°\). [6]
Edexcel C2 Q6
10 marks Standard +0.3
$$f(x) = \frac{(x^2 - 3)^2}{x^3}, \quad x \neq 0.$$
  1. Show that \(f(x) = x - 6x^{-1} + 9x^{-3}\). [2]
  2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]
  3. Verify that the graph of \(y = f(x)\) has stationary points at \(x = \pm\sqrt{3}\). [2]
  4. Determine whether the stationary value at \(x = \sqrt{3}\) is a maximum or a minimum. [3]
Edexcel C2 Q7
11 marks Moderate -0.3
\includegraphics{figure_1} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac{1}{3}x^2 - \frac{1}{4}x^3\). 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. 1, is bounded by \(C\) and the \(x\)-axis.
  1. Find the area of \(R\). [4]
Edexcel C2 Q8
12 marks Standard +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 \(S_n = \frac{a(1 - r^n)}{1 - r}\). [4]
The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
  1. Find, to 3 significant figures, the sum of the first twenty terms of \(G\). [3]
  2. Find the sum to infinity of \(G\). [2]
Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
  1. Find the exact value of the common ratio of this series. [3]
Edexcel C2 Q1
4 marks Moderate -0.3
Find the coefficient of \(x^2\) in the expansion of $$(1 + x)(1 - x)^6.$$ [4]
Edexcel C2 Q2
5 marks Moderate -0.8
A geometric series has common ratio \(\frac{1}{3}\). Given that the sum of the first four terms of the series is 200,
  1. find the first term of the series, [3]
  2. find the sum to infinity of the series. [2]
Edexcel C2 Q3
7 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) where $$f(x) = 4 + 5x + kx^2 - 2x^3,$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A\), \(B\) and \(C\). Given that \(A\) has coordinates \((-4, 0)\),
  1. show that \(k = -7\), [2]
  2. find the coordinates of \(B\) and \(C\). [5]
Edexcel C2 Q4
8 marks Moderate -0.8
    1. Sketch the curve \(y = \sin (x - 30)°\) for \(x\) in the interval \(-180 \leq x \leq 180\).
    2. Write down the coordinates of the turning points of the curve in this interval. [4]
  2. Find all values of \(x\) in the interval \(-180 \leq x \leq 180\) for which $$\sin (x - 30)° = 0.35,$$ giving your answers to 1 decimal place. [4]
Edexcel C2 Q5
9 marks Moderate -0.3
  1. Evaluate $$\log_3 27 - \log_3 4.$$ [4]
  2. Solve the equation $$4^x - 3(2^{x+1}) = 0.$$ [5]
Edexcel C2 Q6
9 marks Moderate -0.3
$$f(x) = 2 - x + 3x^{\frac{1}{2}}, \quad x > 0.$$
  1. Find \(f'(x)\) and \(f''(x)\). [3]
  2. Find the coordinates of the turning point of the curve \(y = f(x)\). [4]
  3. Determine whether the turning point is a maximum or minimum point. [2]
Edexcel C2 Q7
10 marks Moderate -0.3
The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.
  1. Show that \(\angle PQR = 90°\). [4]
Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),
  1. find the coordinates of the centre of \(C\), [3]
  2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]
Edexcel C2 Q8
10 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(PQ\) subtends an angle of \(120°\) at the centre of the circle.
  1. Find the exact length of the major arc \(PQ\). [2]
  2. Show that the perimeter of the shaded minor segment is given by \(k(2\pi + 3\sqrt{3})\) cm, where \(k\) is an integer to be found. [4]
  3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle. [4]
Edexcel C2 Q9
13 marks Moderate -0.3
The finite region \(R\) is bounded by the curve \(y = 1 + 3\sqrt{x}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).
  1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\). [6]
  2. Use integration to find the exact area of \(R\) in the form \(a + b\sqrt{2}\). [5]
  3. Find the percentage error in the estimate made in part (a). [2]
Edexcel C2 Q1
4 marks Easy -1.2
Expand \((3 - 2x)^4\) in ascending powers of \(x\) and simplify each coefficient. [4]
Edexcel C2 Q2
4 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows triangle \(PQR\) in which \(PQ = x\), \(PR = 7 - x\), \(QR = x + 1\) and \(\angle PQR = 60°\). Using the cosine rule, find the value of \(x\). [4]
Edexcel C2 Q3
6 marks Moderate -0.3
Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [6]
Edexcel C2 Q4
8 marks Standard +0.8
Find all values of \(x\) in the interval \(0 \leq x < 360°\) for which $$2\sin^2 x - 2\cos x - \cos^2 x = 1.$$ [8]
Edexcel C2 Q5
8 marks Moderate -0.8
  1. Sketch the curve \(y = 5^{x-1}\), showing the coordinates of any points of intersection with the coordinate axes. [2]
  2. Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5^{x-1}\) intersects
    1. the straight line \(y = 10\),
    2. the curve \(y = 2^x\). [6]
Edexcel C2 Q6
9 marks Moderate -0.3
\(f(x) = 2x^3 + 3x^2 - 6x + 1\).
  1. Find the remainder when \(f(x)\) is divided by \((2x - 1)\). [2]
    1. Find the remainder when \(f(x)\) is divided by \((x + 2)\).
    2. Hence, or otherwise, solve the equation $$2x^3 + 3x^2 - 6x - 8 = 0,$$ giving your answers to 2 decimal places where appropriate. [7]
Edexcel C2 Q7
9 marks Standard +0.3
  1. Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac{a(1-r^n)}{1-r}.$$ [4]
  2. Evaluate \(\sum_{r=1}^{12} (5 \times 2^r)\). [5]
Edexcel C2 Q8
13 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the curve with equation \(y = 5 + x - x^2\) and the normal to the curve at the point \(P(1, 5)\).
  1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
  2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again. [2]
  3. Show that the area of the shaded region bounded by the curve and the straight line \(PQ\) is \(\frac{4}{3}\). [6]
Edexcel C2 Q9
14 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows the circle \(C\) with equation $$x^2 + y^2 - 8x - 10y + 16 = 0.$$
  1. Find the coordinates of the centre and the radius of \(C\). [3]
\(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).
  1. Find the coordinates of \(P\) and \(Q\). [3]
The chord \(PQ\) subtends an angle of \(\theta\) at the centre of \(C\).
  1. Using the cosine rule, show that \(\cos \theta = \frac{7}{25}\). [4]
  2. Find the area of the shaded minor segment bounded by \(C\) and the chord \(PQ\). [4]
Edexcel C2 Q1
4 marks Moderate -0.8
A circle has the equation \(x^2 + y^2 - 6y - 7 = 0\).
  1. Find the coordinates of the centre of the circle. [2]
  2. Find the radius of the circle. [2]
Edexcel C2 Q2
5 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,
  1. find the length \(OA\), [2]
  2. find the area of the shaded segment. [3]