Questions — Edexcel C1 (574 questions)

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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]
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]