Questions C1 (1562 questions)

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All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
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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 Specimen Q1
3 marks Easy -1.2
Calculate \(\sum_{r=1}^{20} 5 + 2r\) [3]
Edexcel C1 Specimen Q2
4 marks Easy -1.2
Find \(\int 5x + 3\sqrt{x} \, dx\) [4]
Edexcel C1 Specimen Q3
4 marks Easy -1.3
  1. Express \(\sqrt{80}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
  2. Express \((4 - \sqrt{5})^2\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [3]
Edexcel C1 Specimen Q4
5 marks Moderate -0.5
The points \(A\) and \(B\) have coordinates \((3, 4)\) and \((7, -6)\) respectively. The straight line \(l\) passes through \(A\) and is perpendicular to \(AB\). Find an equation for \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C1 Specimen Q5
6 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{f}(x)\). The curve crosses the coordinate axes at the points \((0, 1)\) and \((3, 0)\). The maximum point on the curve is \((1, 2)\). On separate diagrams, sketch the curve with equation
  1. \(y = \text{f}(x + 1)\), [3]
  2. \(y = \text{f}(2x)\). [3]
On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes.
Edexcel C1 Specimen Q6
9 marks Moderate -0.8
  1. Solve the simultaneous equations $$y + 2x = 5,$$ $$2x^2 - 3x - y = 16.$$ [6]
  2. Hence, or otherwise, find the set of values of \(x\) for which $$2x^2 - 3x - 16 > 5 - 2x$$ [3]
Edexcel C1 Specimen Q7
9 marks Moderate -0.8
Ahmed plans to save £250 in the year 2001, £300 in 2002, £350 in 2003, and so on until the year 2020. His planned savings form an arithmetic sequence with common difference £50.
  1. Find the amount he plans to save in the year 2011. [2]
  2. Calculate his total planned savings over the 20 year period from 2001 to 2020. [3]
Ben also plans to save money over the same 20 year period. He saves £\(A\) in the year 2001 and his planned yearly savings form an arithmetic sequence with common difference £60. Given that Ben's total planned savings over the 20 year period are equal to Ahmed's total planned savings over the same period,
  1. calculate the value of \(A\). [4]
Edexcel C1 Specimen Q8
11 marks Easy -1.2
Given that $$x^2 + 10x + 36 = (x + a)^2 + b$$ where \(a\) and \(b\) are constants,
  1. find the value of \(a\) and the value of \(b\). [3]
  2. Hence show that the equation \(x^2 + 10x + 36 = 0\) has no real roots. [2]
The equation \(x^2 + 10x + k = 0\) has equal roots.
  1. Find the value of \(k\). [2]
  2. For this value of \(k\), sketch the graph of \(y = x^2 + 10x + k\), showing the coordinates of any points at which the graph meets the coordinate axes. [4]
Edexcel C1 Specimen Q9
11 marks Easy -1.2
The curve \(C\) has equation \(y = \text{f}(x)\) and the point \(P(3, 5)\) lies on \(C\). Given that $$\text{f}(x) = 3x^2 - 8x + 6,$$
  1. find \(\text{f}'(x)\). [4]
  2. Verify that the point \((2, 0)\) lies on \(C\). [2]
The point \(Q\) also lies on \(C\), and the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  1. Find the \(x\)-coordinate of \(Q\). [5]
Edexcel C1 Specimen Q10
13 marks Moderate -0.8
The curve \(C\) has equation \(y = x^3 - 5x + \frac{2}{x}\), \(x \neq 0\). The points \(A\) and \(B\) both lie on \(C\) and have coordinates \((1, -2)\) and \((-1, 2)\) respectively.
  1. Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\). [5]
  2. Show that an equation for the normal to \(C\) at \(A\) is \(4y = x - 9\). [4]
The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).
  1. Find the length of \(PQ\). [4]
Edexcel C1 Q1
6 marks Moderate -0.3
  1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
  2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142}(7r + 2)\). [3]
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]