Questions — Edexcel (10514 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 P1 2018 Specimen Q10
12 marks Moderate -0.3
\includegraphics{figure_4} The triangle \(XYZ\) in Figure 4 has \(XY = 6\) cm, \(YZ = 9\) cm, \(ZX = 4\) cm and angle \(ZXY = a\). The point \(W\) lies on the line \(XY\). The circular arc \(ZW\), in Figure 4, is a major arc of the circle with centre \(X\) and radius 4 cm.
  1. Show that, to 3 significant figures, \(a = 2.22\) radians. [2]
  2. Find the area, in cm\(^2\), of the major sector \(XZWX\). [3]
The region, shown shaded in Figure 4, is to be used as a design for a logo. Calculate
  1. the area of the logo [3]
  2. the perimeter of the logo. [4]
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(16^{-1}\). [1]
  2. Find the value of \(16^{-\frac{1}{2}}\). [2]
Edexcel C1 Q2
8 marks Easy -1.8
  1. Given that \(y = 5x^3 + 7x + 3\), find
    1. \(\frac{dy}{dx}\), [3]
    2. \(\frac{d^2y}{dx^2}\). [1]
  2. Find \(\int \left(1 + 3\sqrt{x} - \frac{1}{x^2}\right) dx\). [4]
Edexcel C1 Q3
4 marks Moderate -0.8
Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]
Edexcel C1 Q4
6 marks Moderate -0.3
Solve the simultaneous equations $$x + y = 2$$ $$x^2 + 2y = 12.$$ [6]
Edexcel C1 Q5
6 marks Easy -1.2
The \(r\)th term of an arithmetic series is \((2r - 5)\).
  1. Write down the first three terms of this series. [2]
  2. State the value of the common difference. [1]
  3. Show that \(\sum_{r=1}^n (2r - 5) = n(n - 4)\). [3]
Edexcel C1 Q6
6 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve crosses the \(x\)-axis at the points \((2, 0)\) and \((4, 0)\). The minimum point on the curve is \(P(3, -2)\). In separate diagrams sketch the curve with equation
  1. \(y = -f(x)\), [3]
  2. \(y = f(2x)\). [3]
On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.
Edexcel C1 Q7
10 marks Moderate -0.8
The curve \(C\) has equation \(y = 4x^2 + \frac{5-x}{x}\), \(x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate \(1\).
  1. Show that the value of \(\frac{dy}{dx}\) at \(P\) is \(3\). [5]
  2. Find an equation of the tangent to \(C\) at \(P\). [3]
This tangent meets the \(x\)-axis at the point \((k, 0)\).
  1. Find the value of \(k\). [2]
Edexcel C1 Q8
9 marks Moderate -0.8
\includegraphics{figure_2} The points \(A(1, 7)\), \(B(20, 7)\) and \(C(p, q)\) form the vertices of a triangle \(ABC\), as shown in Figure 2. The point \(D(8, 2)\) is the mid-point of \(AC\).
  1. Find the value of \(p\) and the value of \(q\). [2]
The line \(l\), which passes through \(D\) and is perpendicular to \(AC\), intersects \(AB\) at \(E\).
  1. Find an equation for \(l\), in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
  2. Find the exact \(x\)-coordinate of \(E\). [2]
Edexcel C1 Q9
11 marks Moderate -0.3
The gradient of the curve \(C\) is given by $$\frac{dy}{dx} = (3x - 1)^2.$$ The point \(P(1, 4)\) lies on \(C\).
  1. Find an equation of the normal to \(C\) at \(P\). [4]
  2. Find an equation for the curve \(C\) in the form \(y = f(x)\). [5]
  3. Using \(\frac{dy}{dx} = (3x - 1)^2\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2x\). [2]
Edexcel C1 Q10
12 marks Moderate -0.3
Given that $$f(x) = x^2 - 6x + 18, \quad x \geq 0,$$
  1. express \(f(x)\) in the form \((x - a)^2 + b\), where \(a\) and \(b\) are integers. [3]
The curve \(C\) with equation \(y = f(x)\), \(x \geq 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
  1. Sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). [4]
The line \(y = 41\) meets \(C\) at the point \(R\).
  1. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q\sqrt{2}\), where \(p\) and \(q\) are integers. [5]
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(8^{-1}\). [1]
  2. Find the value of \(8^{-\frac{2}{3}}\). [2]
Edexcel C1 Q2
5 marks Easy -1.2
Given that \(y = 6x - \frac{4}{x^2}\), \(x \neq 0\),
  1. find \(\frac{dy}{dx}\), [2]
  2. find \(\int y \, dx\). [3]
Edexcel C1 Q3
6 marks Moderate -0.8
\(x^2 - 8x - 29 = (x + a)^2 + b\), where \(a\) and \(b\) are constants.
  1. Find the value of \(a\) and the value of \(b\). [3]
  2. Hence, or otherwise, show that the roots of $$x^2 - 8x - 29 = 0$$ are \(c \pm d\sqrt{5}\), where \(c\) and \(d\) are integers to be found. [3]
Edexcel C1 Q4
5 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 origin \(O\) and through the point \((6, 0)\). The maximum point on the curve is \((3, 5)\). On separate diagrams, sketch the curve with equation
  1. \(y = 3f(x)\), [2]
  2. \(y = f(x + 2)\). [3]
On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
Edexcel C1 Q5
6 marks Moderate -0.5
Solve the simultaneous equations $$x - 2y = 1,$$ $$x^2 + y^2 = 29.$$ [6]
Edexcel C1 Q6
8 marks Moderate -0.8
Find the set of values of \(x\) for which
  1. \(3(2x + 1) > 5 - 2x\), [2]
  2. \(2x^2 - 7x + 3 > 0\), [4]
  3. both \(3(2x + 1) > 5 - 2x\) and \(2x^2 - 7x + 3 > 0\). [2]
Edexcel C1 Q7
8 marks Moderate -0.8
  1. Show that \(\frac{(3 - \sqrt{x})^2}{\sqrt{x}}\) can be written as \(9x^{-\frac{1}{2}} - 6 + x^{\frac{1}{2}}\). [2]
Given that \(\frac{dy}{dx} = \frac{(3 - \sqrt{x})^2}{\sqrt{x}}\), \(x > 0\), and that \(y = \frac{2}{3}\) at \(x = 1\),
  1. find \(y\) in terms of \(x\). [6]
Edexcel C1 Q8
10 marks Moderate -0.8
The line \(l_1\) passes through the point \((9, -4)\) and has gradient \(\frac{1}{3}\).
  1. Find an equation for \(l_1\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]
The line \(l_2\) passes through the origin \(O\) and has gradient \(-2\). The lines \(l_1\) and \(l_2\) intersect at the point \(P\).
  1. Calculate the coordinates of \(P\). [4]
Given that \(l_1\) crosses the \(y\)-axis at the point \(C\),
  1. calculate the exact area of \(\triangle OCP\). [3]
Edexcel C1 Q9
13 marks Moderate -0.8
An arithmetic series has first term \(a\) and common difference \(d\).
  1. Prove that the sum of the first \(n\) terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4]
Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
  1. Find the amount Sean repays in the 21st month. [2]
Over the \(n\) months, he repays a total of £5000.
  1. Form an equation in \(n\), and show that your equation may be written as $$n^2 - 150n + 5000 = 0.$$ [3]
  2. Solve the equation in part (c). [3]
  3. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem. [1]
Edexcel C1 Q10
11 marks Moderate -0.8
The curve \(C\) has equation \(y = \frac{1}{3}x^3 - 4x^2 + 8x + 3\). The point \(P\) has coordinates \((3, 0)\).
  1. Show that \(P\) lies on \(C\). [1]
  2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [5]
Another point \(Q\) also lies on \(C\). The tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\).
  1. Find the coordinates of \(Q\). [5]
Edexcel C1 Q1
3 marks Easy -1.2
Factorise completely $$x^3 - 4x^2 + 3x.$$ [3]
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]