Questions — Edexcel C2 (579 questions)

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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]
Edexcel C2 Q3
8 marks Moderate -0.3
\includegraphics{figure_2} Figure 2 shows the curves with equations \(y = 7 - 2x - 3x^2\) and \(y = \frac{2}{x}\). The two curves intersect at the points \(P\), \(Q\) and \(R\).
  1. Show that the \(x\)-coordinates of \(P\), \(Q\) and \(R\) satisfy the equation $$3x^3 + 2x^2 - 7x + 2 = 0.$$ [2] Given that \(P\) has coordinates \((-2, -1)\),
  2. find the coordinates of \(Q\) and \(R\). [6]
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Expand \((1 + x)^4\) in ascending powers of \(x\). [2]
  2. Using your expansion, express each of the following in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers.
    1. \((1 + \sqrt{2})^4\)
    2. \((1 - \sqrt{2})^8\) [7]
Edexcel C2 Q5
9 marks Moderate -0.3
  1. Describe fully a single transformation that maps the graph of \(y = 3^x\) onto the graph of \(y = (\frac{1}{3})^x\). [1]
  2. Sketch on the same diagram the curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\), showing the coordinates of any points where each curve crosses the coordinate axes. [3]
The curves \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\) intersect at the point \(P\).
  1. Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt{2}\). [5]
Edexcel C2 Q6
9 marks Standard +0.3
A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),
  1. find the values of \(a\) and \(b\), [6]
  2. find the coordinates of the other stationary point of the curve. [3]
Edexcel C2 Q7
9 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows part of the curve \(y = \text{f}(x)\) where $$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$
  1. Solve the equation \(\text{f}(x) = 0\). [3]
  2. Find \(\int \text{f}(x) \, dx\). [3]
  3. Find the area of the shaded region bounded by the curve \(y = \text{f}(x)\), the \(x\)-axis and the line \(x = 2\). [3]
Edexcel C2 Q8
10 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos(2x - \frac{\pi}{6}) = \frac{1}{2}.$$ [7]
Edexcel C2 Q9
12 marks Standard +0.3
The second and fifth terms of a geometric series are \(-48\) and \(6\) respectively.
  1. Find the first term and the common ratio of the series. [5]
  2. Find the sum to infinity of the series. [2]
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2^{6-n}\). [5]
Edexcel C2 Q1
4 marks Easy -1.2
A geometric series has first term 75 and second term \(-15\).
  1. Find the common ratio of the series. [2]
  2. Find the sum to infinity of the series. [2]
Edexcel C2 Q2
5 marks Moderate -0.3
A circle has the equation $$x^2 + y^2 + 8x - 4y + k = 0,$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of the circle. [2]
Given that the \(x\)-axis is a tangent to the circle,
  1. find the value of \(k\). [3]
Edexcel C2 Q3
6 marks Standard +0.8
\includegraphics{figure_1} Figure 1 shows a circle of radius \(r\) and centre \(O\) in which \(AD\) is a diameter. The points \(B\) and \(C\) lie on the circle such that \(OB\) and \(OC\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(OBC\) is \(\frac{1}{6}r^2(3\sqrt{3} - \pi)\). [6]
Edexcel C2 Q4
6 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = \sin 2x\) and \(y = \tan \frac{x}{2}\) for \(x\) in the interval \(0 \leq x \leq 360°\). [4]
  2. Hence state how many solutions exist to the equation $$\sin 2x = \tan \frac{x}{2},$$ for \(x\) in the interval \(0 \leq x \leq 360°\) and give a reason for your answer. [2]
Edexcel C2 Q5
7 marks Moderate -0.3
  1. Find the value of \(a\) such that $$\log_a 27 = 3 + \log_a 8.$$ [3]
  2. Solve the equation $$2^{x+3} = 6^{x-1},$$ giving your answer to 3 significant figures. [4]