Questions — Edexcel C1 (574 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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Edexcel C1 Q2
7 marks Standard +0.3
Solve the simultaneous equations $$x - 3y + 1 = 0,$$ $$x^2 - 3xy + y^2 = 11.$$ [7]
Edexcel C1 Q3
5 marks Moderate -0.8
The first three terms of an arithmetic series are \(p\), \(5p - 8\), and \(3p + 8\) respectively.
  1. Show that \(p = 4\). [2]
  2. Find the value of the 40th term of this series. [3]
Edexcel C1 Q4
7 marks Moderate -0.8
\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]
Given that \(k = 4\),
  1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]
Edexcel C1 Q5
7 marks Moderate -0.8
$$\frac{dy}{dx} = 5 + \frac{1}{x^2}.$$
  1. Use integration to find \(y\) in terms of \(x\). [3]
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]
Edexcel C1 Q6
4 marks Moderate -0.5
A container made from thin metal is in the shape of a right circular cylinder with height \(h\) cm and base radius \(r\) cm. The container has no lid. When full of water, the container holds 500 cm³ of water. Show that the exterior surface area, \(A\) cm², of the container is given by $$A = \pi r^2 + \frac{1000}{r}.$$ [4]
Edexcel C1 Q7
13 marks Moderate -0.3
\includegraphics{figure_1} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 1.
  1. Find the coordinates of the centre of the circle. [2]
  2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
  3. Find an equation of tangent to the circle at \(B\). [3]
The line \(l\) passes through \(A\) and the origin.
  1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]
Edexcel C1 Q1
5 marks Easy -1.3
  1. Solve the inequality $$3x - 8 > x + 13.$$ [2]
  2. Solve the inequality $$x^2 - 5x - 14 > 0.$$ [3]
Edexcel C1 Q2
5 marks Easy -1.2
Given that \(2^x = \frac{1}{\sqrt{2}}\) and \(2^y = 4\sqrt{2}\),
  1. find the exact value of \(x\) and the exact value of \(y\), [3]
  2. calculate the exact value of \(2^{y-x}\). [2]
Edexcel C1 Q3
6 marks Moderate -0.8
  1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]
The equation \(x^2 + 2kx + 81 = 0\) has equal roots.
  1. Find the possible values of \(k\). [2]
Edexcel C1 Q4
8 marks Moderate -0.8
In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \((280 + x)\) phones will be sold in the second month, \((280 + 2x)\) in the third month, and so on. Using this model with \(x = 5\), calculate
    1. the number of phones sold in the 36th month, [2]
    2. the total number of phones sold over the 36 months. [2]
The shop sets a sales target of 17000 phones to be sold over the 36 months. Using the same model,
  1. find the least value of \(x\) required to achieve this target. [4]
Edexcel C1 Q5
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).
  1. Write down the coordinates of \(A\). [1]
  2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
  3. Show that \(\angle PAQ\) is a right angle. [4]
Edexcel C1 Q6
11 marks Moderate -0.8
\includegraphics{figure_2} The points \(A (3, 0)\) and \(B (0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 2.
  1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]
The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.
  1. Find the length of \(BC\) in terms of \(k\). [2]
Given that the length of \(BC\) is 10 and using your answer to part (b),
  1. find the value of \(k\), [4]
  2. find the coordinates of \(D\). [2]
Edexcel C1 Q7
14 marks Moderate -0.3
The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),
  1. find \(y\) in terms of \(x\). [4]
  2. Verify that \(C\) passes through the point \((4, 0)\). [2]
  3. Find an equation of the tangent to \(C\) at \(P\). [3]
The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
  1. Calculate the exact \(x\)-coordinate of \(Q\). [5]
Edexcel C1 Q1
5 marks Easy -1.2
  1. Given that \(8 = 2^k\), write down the value of \(k\). [1]
  2. Given that \(4^x = 8^{2-x}\), find the value of \(x\). [4]
Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q3
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{1}{3}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q4
8 marks Moderate -0.8
  1. By completing the square, find in terms of \(k\) the roots of the equation $$x^2 + 2kx - 7 = 0.$$ [4]
  2. Prove that, for all values of \(k\), the roots of \(x^2 + 2kx - 7 = 0\) are real and different. [2]
  3. Given that \(k = \sqrt{2}\), find the exact roots of the equation. [2]
Edexcel C1 Q5
9 marks Moderate -0.8
The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).
  1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]
The lines \(l_1\) and \(l_2\) intersect at the point \(A\).
  1. Calculate, as exact fractions, the coordinates of \(A\). [3]
  2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]
Edexcel C1 Q6
8 marks Moderate -0.8
Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays £500. Her payments then increase by £50 each year, so that she pays £550 in the second year, £600 in the third year, and so on.
  1. Find the amount that Anne will pay in the 40th year. [2]
  2. Find the total amount that Anne will pay in over the 40 years. [2]
Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in £890 and his payments then increase by £\(d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
  1. find the value of \(d\). [4]
Edexcel C1 Q7
13 marks Moderate -0.8
\includegraphics{figure_1} The points \(A\) and \(B\) have coordinates \((2, -3)\) and \((8, 5)\) respectively, and \(AB\) is a chord of a circle with centre \(C\), as shown in Fig. 1.
  1. Find the gradient of \(AB\). [2]
The point \(M\) is the mid-point of \(AB\).
  1. Find an equation for the line through \(C\) and \(M\). [5]
Given that the \(x\)-coordinate of \(C\) is 4,
  1. find the \(y\)-coordinate of \(C\), [2]
  2. show that the radius of the circle is \(\frac{5\sqrt{17}}{4}\). [4]
Edexcel C1 Q8
5 marks Standard +0.3
\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2\(x\) 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]
Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q3
5 marks Easy -1.2
  1. Solve the inequality \(3x - 8 > x + 13\). [2]
  2. Solve the inequality \(x^2 - 5x - 14 > 0\). [3]
Edexcel C1 Q4
6 marks Moderate -0.3
  1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{(k^2 - c)}\). [4]
The equation \(x^2 + 2kx + 81 = 0\) has equal roots.
  1. Find the possible values of \(k\). [2]
Edexcel C1 Q5
7 marks Standard +0.3
Solve the simultaneous equations \(x - 3y + 1 = 0\), \(x^2 - 3xy + y^2 = 11\). [7]