Questions C1 (1562 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
4 marks Moderate -0.8
The sequence of positive numbers \(u_1, u_2, u_3, \ldots\) is given by $$u_{n+1} = (u_n - 3)^2, \quad u_1 = 1.$$
  1. Find \(u_2\), \(u_3\) and \(u_4\). [3]
  2. Write down the value of \(u_{20}\). [1]
Edexcel C1 Q3
5 marks Easy -1.3
The line \(L\) has equation \(y = 5 - 2x\).
  1. Show that the point \(P(3, -1)\) lies on \(L\). [1]
  2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
Edexcel C1 Q4
5 marks Easy -1.2
Given that \(y = 2x^2 - \frac{6}{x}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q5
6 marks Easy -1.2
  1. Write \(\sqrt{45}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
  2. Express \(\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [5]
Edexcel C1 Q6
9 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the points \((0, 3)\) and \((4, 0)\) and touches the \(x\)-axis at the point \((1, 0)\). On separate diagrams, sketch the curve with equation
  1. \(y = f(x + 1)\), [3]
  2. \(y = 2f(x)\), [3]
  3. \(y = f\left(\frac{1}{2}x\right)\). [3]
On each diagram show clearly the coordinates of all the points at which the curve meets the axes.
Edexcel C1 Q7
13 marks Easy -1.2
On Alice's 11th birthday she started to receive an annual allowance. The first annual allowance was £500 and on each following birthday the allowance was increased by £200.
  1. Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was £1200. [1]
  2. Find the amount of Alice's annual allowance on her 18th birthday. [2]
  3. Find the total of the allowances that Alice had received up to and including her 18th birthday. [3]
When the total of the allowances that Alice had received reached £32 000 the allowance stopped.
  1. Find how old Alice was when she received her last allowance. [7]
Edexcel C1 Q8
7 marks Moderate -0.3
The curve with equation \(y = f(x)\) passes through the point \((1, 6)\). Given that $$f'(x) = 3 + \frac{5x^2 + 2}{x^4}, \quad x > 0,$$ find \(f(x)\) and simplify your answer. [7]
Edexcel C1 Q9
12 marks Moderate -0.8
\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation $$y = (x - 1)(x^2 - 4).$$ The curve cuts the \(x\)-axis at the points \(P\), \((1, 0)\) and \(Q\), as shown in Figure 2.
  1. Write down the \(x\)-coordinate of \(P\) and the \(x\)-coordinate of \(Q\). [2]
  2. Show that \(\frac{dy}{dx} = 3x^2 - 2x - 4\). [3]
  3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point \((-1, 6)\). [2]
The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point \((-1, 6)\).
  1. Find the exact coordinates of \(R\). [5]
Edexcel C1 Q10
11 marks Moderate -0.8
\(x^2 + 2x + 3 \equiv (x + a)^2 + b\).
  1. Find the values of the constants \(a\) and \(b\). [2]
  2. Sketch the graph of \(y = x^2 + 2x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes. [3]
  3. Find the value of the discriminant of \(x^2 + 2x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). [2]
The equation \(x^2 + kx + 3 = 0\), where \(k\) is a constant, has no real roots.
  1. Find the set of possible values of \(k\), giving your answer in surd form. [4]
Edexcel C1 Q1
4 marks Easy -1.2
Find \(\int (6x^2 + 2x + x^{-2}) \, dx\), giving each term in its simplest form. [4]
Edexcel C1 Q2
4 marks Moderate -0.8
Find the set of values of \(x\) for which $$x^2 - 7x - 18 > 0.$$ [4]
Edexcel C1 Q3
5 marks Easy -1.2
On separate diagrams, sketch the graphs of
  1. \(y = (x + 3)^2\), [3]
  2. \(y = (x + 3)^2 + k\), where \(k\) is a positive constant. [2]
Show on each sketch the coordinates of each point at which the graph meets the axes.
Edexcel C1 Q4
5 marks Easy -1.2
A sequence \(a_1, a_2, a_3, \ldots\) is defined by $$a_1 = 3,$$ $$a_{n+1} = 3a_n - 5, \quad n \geq 1.$$
  1. Find the value \(a_2\) and the value of \(a_3\). [2]
  2. Calculate the value of \(\sum_{r=1}^5 a_r\). [3]
Edexcel C1 Q5
7 marks Easy -1.2
Differentiate with respect to \(x\)
  1. \(x^4 + 6\sqrt{x}\), [3]
  2. \(\frac{(x + 4)^3}{x}\). [4]
Edexcel C1 Q6
4 marks Easy -1.3
  1. Expand and simplify \((4 + \sqrt{3})(4 - \sqrt{3})\). [2]
  2. Express \(\frac{26}{4 + \sqrt{3}}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]
Edexcel C1 Q7
7 marks Moderate -0.3
An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term \(a\) km and common difference \(d\) km. He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period. Find the value of \(a\) and the value of \(d\). [7]
Edexcel C1 Q8
6 marks Moderate -0.8
The equation \(x^2 + 2px + (3p + 4) = 0\), where \(p\) is a positive constant, has equal roots.
  1. Find the value of \(p\). [4]
  2. For this value of \(p\), solve the equation \(x^2 + 2px + (3p + 4) = 0\). [2]
Edexcel C1 Q9
8 marks Moderate -0.8
Given that \(f(x) = (x^2 - 6x)(x - 2) + 3x\),
  1. express \(f(x)\) in the form \(a(x^2 + bx + c)\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Hence factorise \(f(x)\) completely. [2]
  3. Sketch the graph of \(y = f(x)\), showing the coordinates of each point at which the graph meets the axes. [3]
Edexcel C1 Q10
10 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\), \(x \neq 0\), passes through the point \((3, 7\frac{1}{2})\). Given that \(f'(x) = 2x + \frac{3}{x^2}\),
  1. find \(f(x)\). [5]
  2. Verify that \(f(-2) = 5\). [1]
  3. Find an equation for the tangent to \(C\) at the point \((-2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
Edexcel C1 Q11
15 marks Moderate -0.3
The line \(l_1\) passes through the points \(P(-1, 2)\) and \(Q(11, 8)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [4]
The line \(l_2\) passes through the point \(R(10, 0)\) and is perpendicular to \(l_1\). The lines \(l_1\) and \(l_2\) intersect at the point \(S\).
  1. Calculate the coordinates of \(S\). [5]
  2. Show that the length of \(RS\) is \(3\sqrt{5}\). [2]
  3. Hence, or otherwise, find the exact area of triangle \(PQR\). [4]
Edexcel C1 Q1
4 marks Easy -1.2
Given that $$y = 4x^3 - 1 + 2x^{-1}, \quad x > 0,$$ find \(\frac{dy}{dx}\). [4]
Edexcel C1 Q2
4 marks Easy -1.3
  1. Express \(\sqrt{108}\) in the form \(a\sqrt{3}\), where \(a\) is an integer. [1]
  2. Express \((2 - \sqrt{3})^2\) in the form \(b + c\sqrt{3}\), where \(b\) and \(c\) are integers to be found. [3]
Edexcel C1 Q3
6 marks Moderate -0.8
Given that \(f(x) = \frac{1}{x}\), \(x \neq 0\),
  1. sketch the graph of \(y = f(x) + 3\) and state the equations of the asymptotes. [4]
  2. Find the coordinates of the point where \(y = f(x) + 3\) crosses a coordinate axis. [2]
Edexcel C1 Q4
7 marks Moderate -0.5
Solve the simultaneous equations $$y = x - 2,$$ $$y^2 + x^2 = 10.$$ [7]
Edexcel C1 Q5
4 marks Moderate -0.3
The equation \(2x^2 - 3x - (k + 1) = 0\), where \(k\) is a constant, has no real roots. Find the set of possible values of \(k\). [4]