Questions — Edexcel (10514 questions)

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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 P2 2022 June Q1
4 marks Easy -1.2
Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left(2 + \frac{3}{8}x\right)^{10}$$ Give each coefficient as an integer. [4]
Edexcel P2 2022 June Q2
8 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows the graph of $$y = 1 - \log_{10}(\sin x) \quad 0 < x < \pi$$ where \(x\) is in radians. The table below shows some values of \(x\) and \(y\) for this graph, with values of \(y\) given to 3 decimal places.
\(x\)0.511.522.53
\(y\)1.3191.0011.2231.850
  1. Complete the table above, giving values of \(y\) to 3 decimal places. [2]
  2. Use the trapezium rule with all the \(y\) values in the completed table to find, to 2 decimal places, an estimate for $$\int_{0.5}^{3} \left(1 - \log_{10}(\sin x)\right) dx$$ [3]
  3. Use your answer to part (b) to find an estimate for $$\int_{0.5}^{3} \left(3 + \log_{10}(\sin x)\right) dx$$ [3]
Edexcel P2 2022 June Q3
7 marks Moderate -0.8
  1. Show that the following statement is false: "\((n + 1)^3 - n^3\) is prime for all \(n \in \mathbb{N}\)" [2]
  2. Given that the points \(A(1, 0)\), \(B(3, -10)\) and \(C(7, -6)\) lie on a circle, prove that \(AB\) is a diameter of this circle. [5]
Edexcel P2 2022 June Q4
6 marks Standard +0.3
In this question you must show all stages of your working. Give your answers in fully simplified surd form. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations \begin{align} a - b &= 8
\log_5 a + \log_5 b &= 3 \end{align} [6]
Edexcel P2 2022 June Q5
6 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve, for \(-180° < \theta \leq 180°\), the equation $$3\tan(\theta + 43°) = 2\cos(\theta + 43°)$$ [6]
Edexcel P2 2022 June Q6
8 marks Moderate -0.3
In a geometric sequence \(u_1, u_2, u_3, \ldots\)
  • the common ratio is \(r\)
  • \(u_2 + u_3 = 6\)
  • \(u_4 = 8\)
  1. Show that \(r\) satisfies $$3r^2 - 4r - 4 = 0$$ [3]
Given that the geometric sequence has a sum to infinity,
  1. find \(u_1\) [3]
  2. find \(S_∞\) [2]
Edexcel P2 2022 June Q7
7 marks Standard +0.3
$$f(x) = Ax^3 + 6x^2 - 4x + B$$ where \(A\) and \(B\) are constants. Given that
  • \((x + 2)\) is a factor of \(f(x)\)
  • \(\int_{-3}^{5} f(x)dx = 176\)
Find the value of \(A\) and the value of \(B\). [7]
Edexcel P2 2022 June Q8
8 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curve has equation $$y = 256x^4 - 304x - 35 + \frac{27}{x^2} \quad x \neq 0$$
  1. Find \(\frac{dy}{dx}\) [3]
  2. Hence find the coordinates of the stationary points of the curve. [5]
Edexcel P2 2022 June Q9
9 marks Moderate -0.8
A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k\lambda^t$$ where
  • \(N\) grams is the amount of carbon-14 currently present in the item
  • \(k\) grams was the initial amount of carbon-14 present in the item
  • \(t\) is the number of years since the item was made
  • \(\lambda\) is a constant, with \(0 < \lambda < 1\)
  1. Sketch the graph of \(N\) against \(t\) for \(k = 1\) [2]
Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,
  1. show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. [2]
Given that Item A
  • is known to have had 15 grams of carbon-14 present initially
  • is thought to be 3250 years old
  1. calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item A. [2]
Item B is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.
  1. Use algebra to calculate the age of Item B to the nearest 100 years. [3]
Edexcel P2 2022 June Q10
12 marks Standard +0.3
The circle \(C\) has centre \(X(3, 5)\) and radius \(r\) The line \(l\) has equation \(y = 2x + k\), where \(k\) is a constant.
  1. Show that \(l\) and \(C\) intersect when $$5x^2 + (4k - 26)x + k^2 - 10k + 34 - r^2 = 0$$ [3]
Given that \(l\) is a tangent to \(C\),
  1. show that \(5r^2 = (k + p)^2\), where \(p\) is a constant to be found. [3]
\includegraphics{figure_2} The line \(l\)
  • cuts the \(y\)-axis at the point \(A\)
  • touches the circle \(C\) at the point \(B\)
as shown in Figure 2. Given that \(AB = 2r\)
  1. find the value of \(k\) [6]
Edexcel C2 Q1
4 marks Easy -1.2
Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3 + 2x)^5\), giving each term in its simplest form. [4]
Edexcel C2 Q2
6 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.
  1. Find the coordinates of the mid-point of \(AB\). [2]
Given that \(AB\) is a diameter of the circle \(C\),
  1. find an equation for \(C\). [4]
Edexcel C2 Q3
7 marks Moderate -0.3
Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(3^x = 5\), [3]
  2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]
Edexcel C2 Q4
7 marks Moderate -0.3
  1. Show that the equation $$5 \cos^2 x = 3(1 + \sin x)$$ can be written as $$5 \sin^2 x + 3 \sin x - 2 = 0.$$ [2]
  2. Hence solve, for \(0 \leq x < 360°\), the equation $$5 \cos^2 x = 3(1 + \sin x),$$ giving your answers to 1 decimal place where appropriate. [5]
Edexcel C2 Q5
8 marks Moderate -0.8
\(f(x) = x^3 - 2x^2 + ax + b\), where \(a\) and \(b\) are constants. When \(f(x)\) is divided by \((x - 2)\), the remainder is 1. When \(f(x)\) is divided by \((x + 1)\), the remainder is 28.
  1. Find the value of \(a\) and the value of \(b\). [6]
  2. Show that \((x - 3)\) is a factor of \(f(x)\). [2]
Edexcel C2 Q6
8 marks Moderate -0.3
The second and fourth terms of a geometric series are 7.2 and 5.832 respectively. The common ratio of the series is positive. For this series, find
  1. the common ratio, [2]
  2. the first term, [2]
  3. the sum of the first 50 terms, giving your answer to 3 decimal places, [2]
  4. the difference between the sum to infinity and the sum of the first 50 terms, giving your answer to 3 decimal places. [2]
Edexcel C2 Q7
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 8\) cm, \(AC = 11\) cm and \(\angle BAC = 0.7\) radians. The arc \(BD\), where \(D\) lies on \(AC\), is an arc of a circle with centre \(A\) and radius 8 cm. The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(BC\) and \(CD\) and the arc \(BD\). Find
  1. the length of the arc \(BD\), [2]
  2. the perimeter of \(R\), giving your answer to 3 significant figures, [4]
  3. the area of \(R\), giving your answer to 3 significant figures. [5]
Edexcel C2 Q8
12 marks Moderate -0.3
\includegraphics{figure_2} The line with equation \(y = 3x + 20\) cuts the curve with equation \(y = x^2 + 6x + 10\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). [5]
The shaded region \(S\) is bounded by the line and the curve, as shown in Figure 2.
  1. Use calculus to find the exact area of \(S\). [7]