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 Q19
14 marks Easy -1.2
\(f(x) = 9 - (x - 2)^2\)
  1. Write down the maximum value of \(f(x)\). [1]
  2. Sketch the graph of \(y = f(x)\), showing the coordinates of the points at which the graph meets the coordinate axes. [5]
The points \(A\) and \(B\) on the graph of \(y = f(x)\) have coordinates \((-2, -7)\) and \((3, 8)\) respectively.
  1. Find, in the form \(y = mx + c\), an equation of the straight line through \(A\) and \(B\). [4]
  2. Find the coordinates of the point at which the line \(AB\) crosses the \(x\)-axis. [2]
The mid-point of \(AB\) lies on the line with equation \(y = kx\), where \(k\) is a constant.
  1. Find the value of \(k\). [2]
Edexcel C1 Q20
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 Q21
5 marks Easy -1.3
\(y = 7 + 10x^{\frac{3}{2}}\).
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find \(\int y \, dx\). [3]
Edexcel C1 Q22
8 marks Moderate -0.8
  1. Given that \(3^x = 9^{y-1}\), show that \(x = 2y - 2\). [2]
  2. Solve the simultaneous equations \begin{align} x &= 2y - 2,
    x^2 &= y^2 + 7. \end{align} [6]
Edexcel C1 Q23
11 marks Moderate -0.8
The straight line \(l_1\) with equation \(y = \frac{3}{2}x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \((5, -3)\). The straight line \(l_2\) is perpendicular to \(l_1\) and passes through \(Q\).
  1. Calculate the coordinates of the mid-point of \(PQ\). [3]
  2. Find an equation for \(l_2\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integer constants. [4]
The lines \(l_1\) and \(l_2\) intersect at the point \(R\).
  1. Calculate the exact coordinates of \(R\). [4]
Edexcel C1 Q24
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 Q25
7 marks Moderate -0.8
Find the set of values for \(x\) for which
  1. \(6x - 7 < 2x + 3\), [2]
  2. \(2x^2 - 11x + 5 < 0\), [4]
  3. both \(6x - 7 < 2x + 3\) and \(2x^2 - 11x + 5 < 0\). [1]
Edexcel C1 Q26
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 17 000 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 Q27
10 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((4, 6)\) and \((12, 2)\) respectively. The straight line \(l_1\) passes through \(A\) and \(B\).
  1. Find an equation for \(l_1\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(l_2\) passes through the origin and has gradient \(-4\).
  1. Write down an equation for \(l_2\). [1]
The lines \(l_1\) and \(l_2\) intercept at the point \(C\).
  1. Find the exact coordinates of the mid-point of \(AC\). [5]
Edexcel C1 Q28
7 marks Moderate -0.8
For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),
  1. find \(\frac{dy}{dx}\). [2]
The point \(A\), on the curve \(C\), has \(x\)-coordinate 1.
  1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
Edexcel C1 Q29
6 marks Easy -1.2
The sum of an arithmetic series is $$\sum_{r=1}^{n} (80 - 3r).$$
  1. Write down the first two terms of the series. [2]
  2. Find the common difference of the series. [1]
Given that \(n = 50\),
  1. find the sum of the series. [3]
Edexcel C1 Q30
7 marks Moderate -0.8
  1. Solve the equation \(4x^2 + 12x = 0\). [3]
\(f(x) = 4x^2 + 12x + c\), where \(c\) is a constant.
  1. Given that \(f(x) = 0\) has equal roots, find the value of \(c\) and hence solve \(f(x) = 0\). [4]
Edexcel C1 Q31
7 marks Standard +0.3
Solve the simultaneous equations \begin{align} x - 3y + 1 &= 0,
x^2 - 3xy + y^2 &= 11. \end{align} [7]
Edexcel C1 Q32
4 marks Standard +0.3
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 Q33
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 Q34
8 marks Standard +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]
  3. Prove that the sum of the first \(n\) terms of the series is a perfect square. [3]
Edexcel C1 Q35
7 marks Moderate -0.3
\(f(x) = x^2 - kx + 9\), 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 Q36
9 marks Moderate -0.8
The curve \(C\) with equation \(y = f(x)\) is such that $$\frac{dy}{dx} = 3\sqrt{x} + \frac{12}{\sqrt{x}}, \quad x > 0.$$
  1. Show that, when \(x = 8\), the exact value of \(\frac{dy}{dx}\) is \(9\sqrt{2}\). [3]
The curve \(C\) passes through the point \((4, 30)\).
  1. Using integration, find \(f(x)\). [6]
Edexcel C1 Q37
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 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 Q38
7 marks Standard +0.3
A sequence is defined by the recurrence relation $$u_{n+1} = \sqrt{\frac{u_n}{2} + \frac{a}{u_n}}, \quad n = 1, 2, 3, \ldots,$$ where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u_1 = 3\), find the values of \(u_2\), \(u_3\) and \(u_4\), giving your answers to 2 decimal places. [3]
  2. Given instead that \(u_1 = u_2 = 3\),
    1. calculate the value of \(a\), [3]
    2. write down the value of \(u_5\). [1]
Edexcel C1 Q39
6 marks Easy -1.3
The points \(A\) and \(B\) have coordinates \((1, 2)\) and \((5, 8)\) respectively.
  1. Find the coordinates of the mid-point of \(AB\). [2]
  2. Find, in the form \(y = mx + c\), an equation for the straight line through \(A\) and \(B\). [4]
Edexcel C1 Q40
6 marks Moderate -0.8
Giving your answers in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational numbers, find
  1. \((3 - \sqrt{8})^2\), [3]
  2. \(\frac{1}{4 - \sqrt{8}}\). [3]
Edexcel C1 Q41
8 marks Moderate -0.8
The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m,
  1. form a linear inequality in \(x\). [2]
Given that the area of the pitch must be greater than 4800 m²,
  1. form a quadratic inequality in \(x\). [2]
  2. by solving your inequalities, find the set of possible values of \(x\). [4]
Edexcel C1 Q42
9 marks Moderate -0.8
The curve \(C\) has equation \(y = x^2 - 4\) and the straight line \(l\) has equation \(y + 3x = 0\).
  1. In the space below, sketch \(C\) and \(l\) on the same axes. [3]
  2. Write down the coordinates of the points at which \(C\) meets the coordinate axes. [2]
  3. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\). [4]
Edexcel C1 Q43
5 marks Moderate -0.3
\(f(x) = \frac{(x^2 - 3)^2}{x^3}, x \neq 0\).
  1. Show that \(f(x) \equiv x - 6x^{-1} + 9x^{-3}\). [2]
  2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]