Questions AS Paper 1 (378 questions)

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Edexcel AS Paper 1 Specimen Q5
5 marks Moderate -0.3
Given that $$f(x) = 2x + 3 + \frac{12}{x^2}, \quad x > 0$$ show that \(\int_1^{2\sqrt{2}} f(x)\,dx = 16 + 3\sqrt{2}\) [5]
Edexcel AS Paper 1 Specimen Q6
4 marks Moderate -0.5
Prove, from first principles, that the derivative of \(3x^2\) is \(6x\). [4]
Edexcel AS Paper 1 Specimen Q7
5 marks Moderate -0.8
  1. Find the first \(3\) terms, in ascending powers of \(x\), of the binomial expansion of $$\left(2 - \frac{x}{2}\right)^7$$ giving each term in its simplest form. [4]
  2. Explain how you would use your expansion to give an estimate for the value of \(1.995^7\) [1]
Edexcel AS Paper 1 Specimen Q8
5 marks Moderate -0.3
\includegraphics{figure_1} A triangular lawn is modelled by the triangle \(ABC\), shown in Figure 1. The length \(AB\) is to be \(30\text{m}\) long. Given that angle \(BAC = 70°\) and angle \(ABC = 60°\),
  1. calculate the area of the lawn to \(3\) significant figures. [4]
  2. Why is your answer unlikely to be accurate to the nearest square metre? [1]
Edexcel AS Paper 1 Specimen Q9
5 marks Standard +0.3
Solve, for \(360° \leqslant x < 540°\), $$12\sin^2 x + 7\cos x - 13 = 0$$ Give your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]
Edexcel AS Paper 1 Specimen Q10
4 marks Standard +0.3
The equation \(kx^2 + 4kx + 3 = 0\), where \(k\) is a constant, has no real roots. Prove that $$0 \leqslant k < \frac{3}{4}$$ [4]
Edexcel AS Paper 1 Specimen Q11
3 marks Standard +0.3
  1. Prove that for all positive values of \(x\) and \(y\) $$\sqrt{xy} \leqslant \frac{x + y}{2}$$ [2]
  2. Prove by counter example that this is not true when \(x\) and \(y\) are both negative. [1]
Edexcel AS Paper 1 Specimen Q12
4 marks Moderate -0.3
A student was asked to give the exact solution to the equation $$2^{2x+4} - 9(2^x) = 0$$ The student's attempt is shown below: $$2^{2x+4} - 9(2^x) = 0$$ $$2^{2x} + 2^4 - 9(2^x) = 0$$ Let \(2^x = y\) $$y^2 - 9y + 8 = 0$$ $$(y - 8)(y - 1) = 0$$ $$y = 8 \text{ or } y = 1$$ $$\text{So } x = 3 \text{ or } x = 0$$
  1. Identify the two errors made by the student. [2]
  2. Find the exact solution to the equation. [2]
Edexcel AS Paper 1 Specimen Q13
7 marks Standard +0.3
  1. Factorise completely \(x^3 + 10x^2 + 25x\) [2]
  2. Sketch the curve with equation $$y = x^3 + 10x^2 + 25x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. [2]
The point with coordinates \((-3, 0)\) lies on the curve with equation $$y = (x + a)^3 + 10(x + a)^2 + 25(x + a)$$ where \(a\) is a constant.
  1. Find the two possible values of \(a\). [3]
Edexcel AS Paper 1 Specimen Q14
13 marks Moderate -0.3
\includegraphics{figure_2} A town's population, \(P\), is modelled by the equation \(P = ab^t\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log_{10} P\) for the population over a period of 100 years. The line \(l\) meets the vertical axis at \((0, 5)\) as shown. The gradient of \(l\) is \(\frac{1}{200}\).
  1. Write down an equation for \(l\). [2]
  2. Find the value of \(a\) and the value of \(b\). [4]
  3. With reference to the model interpret
    1. the value of the constant \(a\),
    2. the value of the constant \(b\).
    [2]
  4. Find
    1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
    2. the number of years it takes the population to reach \(200\,000\), according to the model.
    [3]
  5. State two reasons why this may not be a realistic population model. [2]
Edexcel AS Paper 1 Specimen Q15
8 marks Challenging +1.2
\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]
Edexcel AS Paper 1 Specimen Q16
10 marks Standard +0.3
\includegraphics{figure_4} Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool \(ABCDEA\) consists of a rectangular section \(ABDE\) joined to a semicircular section \(BCD\) as shown in Figure 4. Given that \(AE = 2x\) metres, \(ED = y\) metres and the area of the pool is \(250\text{m}^2\),
  1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2x + \frac{250}{x} + \frac{\pi x}{2}$$ [4]
  2. Explain why \(0 < x < \sqrt{\frac{500}{\pi}}\) [2]
  3. Find the minimum perimeter of the pool, giving your answer to \(3\) significant figures. [4]
Edexcel AS Paper 1 Specimen Q17
10 marks Standard +0.3
A circle \(C\) with centre at \((-2, 6)\) passes through the point \((10, 11)\).
  1. Show that the circle \(C\) also passes through the point \((10, 1)\). [3]
The tangent to the circle \(C\) at the point \((10, 11)\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \((10, 1)\) meets the \(y\) axis at the point \(Q\).
  1. Show that the distance \(PQ\) is \(58\) explaining your method clearly. [7]
Edexcel AS Paper 1 Q1
Easy -1.2
Find $$\int(\frac{1}{2}x^2 - 9\sqrt{x} + 4) dx$$ giving your answer in its simplest form.
Edexcel AS Paper 1 Q2
Easy -1.8
Use a counter example to show that the following statement is false. "\(n^2 - n + 5\) is a prime number, for \(2 \leq n \leq 6\)"
Edexcel AS Paper 1 Q3
5 marks Moderate -0.8
Given that the point \(A\) has position vector \(x\mathbf{i} - \mathbf{j}\), the point \(B\) has position vector \(-2\mathbf{i} + y\mathbf{j}\) and \(\overrightarrow{AB} = -3\mathbf{i} + 4\mathbf{j}\), find
  1. the values of \(x\) and \(y\) [3]
  2. a unit vector in the direction of \(\overrightarrow{AB}\). [2]
Edexcel AS Paper 1 Q4
Moderate -0.8
The line \(l_1\) has equation \(2x - 3y = 9\) The line \(l_2\) passes through the points \((3, -1)\) and \((-1, 5)\) Determine, giving full reasons for your answer, whether lines \(l_1\) and \(l_2\) are parallel, perpendicular or neither.
Edexcel AS Paper 1 Q5
4 marks Moderate -0.3
A student is asked to solve the equation $$\log_3 x - \log_3 \sqrt{x - 2} = 1$$ The student's attempt is shown $$\log_3 x - \log_3 \sqrt{x - 2} = 1$$ $$x - \sqrt{x - 2} = 3^1$$ $$x - 3 = \sqrt{x - 2}$$ $$(x - 3)^2 = x - 2$$ $$x^2 - 7x + 11 = 0$$ $$x = \frac{7 + \sqrt{5}}{2} \text{ or } x = \frac{7 - \sqrt{5}}{2}$$
  1. Identify the error made by this student, giving a brief explanation. [1]
  2. Write out the correct solution. [3]
Edexcel AS Paper 1 Q6
9 marks Moderate -0.8
\includegraphics{figure_1} A stone is thrown over level ground from the top of a tower, \(X\). The height, \(h\), in meters, of the stone above the ground level after \(t\) seconds is modelled by the function. $$h(t) = 7 + 21t - 4.9t^2, \quad t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1. Using the model,
  1. give a physical interpretation of the meaning of the constant term 7 in the model. [1]
  2. find the time taken after the stone is thrown for it to reach ground level. [3]
  3. Rearrange \(h(t)\) into the form \(A - B(t - C)^2\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
  4. Using your answer to part c or otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached. [2]
Edexcel AS Paper 1 Q7
7 marks Standard +0.3
In a triangle \(PQR\), \(PQ = 20\) cm, \(PR = 10\) cm and angle \(QPR = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(PQR\) is 80 cm\(^2\).
  1. Show that the two possible values of \(\cos \theta = \pm \frac{3}{5}\) [4]
Given that \(QR\) is the longest side of the triangle,
  1. find the exact perimeter of the triangle \(PQR\), giving your answer as a simplified surd. [3]
Edexcel AS Paper 1 Q8
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a solid cuboid \(ABCDEFGH\). \(AB = x\) cm, \(BC = 2x\) cm, \(AE = h\) cm The total surface area of the cuboid is 180 cm\(^2\). The volume of the cuboid is \(V\) cm\(^3\).
  1. Show that \(V = 60x - \frac{4x^3}{3}\) [4]
Given that \(x\) can vary,
  1. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\). [5]
  2. Find the maximum value of \(V\), giving your answer to the nearest cm\(^3\). [2]
Edexcel AS Paper 1 Q9
9 marks Standard +0.3
\(f(x) = -2x^3 - x^2 + 4x + 3\)
  1. Use the factor theorem to show that \((3 - 2x)\) is a factor of \(f(x)\). [2]
  2. Hence show that \(f(x)\) can be written in the form \(f(x) = (3 - 2x)(x + a)^2\) where \(a\) is an integer to be found. [4]
\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve with equation \(y = f(x)\).
  1. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
    1. \(f(x) \leq 0\)
    2. \(f'(\frac{x}{2}) = 0\)
    [3]
Edexcel AS Paper 1 Q10
Easy -1.2
Prove, from the first principles, that the derivative of \(5x^2\) is \(10x\).
Edexcel AS Paper 1 Q11
6 marks Moderate -0.3
The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \((1 + kx)^{10}\) are given by $$1 + 15x + px^2$$ where \(k\) and \(p\) are constants.
  1. Find the value of \(k\) [2]
  2. Find the value of \(p\) [2]
  3. Given that, in the expansion of \((1 + kx)^{10}\), the coefficient of \(x^4\) is \(q\), find the value of \(q\). [2]
Edexcel AS Paper 1 Q12
5 marks Standard +0.3
  1. Explain mathematically why there are no values of \(\theta\) that satisfy the equation $$(3\cos\theta - 4)(2\cos\theta + 5) = 0$$ [2]
  2. Giving your solutions to one decimal place, where appropriate, solve the equation $$3\sin y + 2\tan y = 0 \quad \text{for } 0 \leq y \leq \pi$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) [3]