Questions — Edexcel C1 (574 questions)

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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]
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]