Questions AS Paper 1 (378 questions)

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AQA AS Paper 1 2022 June Q1
1 marks Easy -1.8
Express as a single logarithm $$\log_{10} 2 - \log_{10} x$$ Circle your answer. [1 mark] \(\log_{10} (2 + x)\) \quad \(\log_{10} (2 - x)\) \quad \(\log_{10} (2x)\) \quad \(\log_{10} \left(\frac{2}{x}\right)\)
AQA AS Paper 1 2022 June Q2
1 marks Easy -1.8
The graph of the function \(y = \cos \frac{1}{2}x\) for \(0° \leq x \leq 360°\) is one of the graphs shown below. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_2}
AQA AS Paper 1 2022 June Q3
3 marks Easy -1.2
Find the coefficient of the \(x^3\) term in the expansion of \(\left(3x + \frac{1}{2}\right)^4\) [3 marks]
AQA AS Paper 1 2022 June Q4
5 marks Standard +0.3
Find all the solutions of the equation $$\cos^2 \theta = 10 \sin \theta + 4$$ for \(0° < \theta < 360°\), giving your answers to the nearest degree. Fully justify your answer. [5 marks]
AQA AS Paper 1 2022 June Q5
3 marks Moderate -0.8
Express \(3x^3 + 5x^2 - 27x + 10\) in the form \((x - 2)(ax^2 + bx + c)\), where \(a\), \(b\) and \(c\) are integers. [3 marks]
AQA AS Paper 1 2022 June Q6
9 marks Moderate -0.3
\(AB\) is a diameter of a circle where \(A\) is \((1, 4)\) and \(B\) is \((7, -2)\)
  1. Find the coordinates of the midpoint of \(AB\). [1 mark]
  2. Show that the equation of the circle may be written as $$x^2 + y^2 - 8x - 2y = 1$$ [4 marks]
  3. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(DEC\). [4 marks]
AQA AS Paper 1 2022 June Q7
6 marks Standard +0.3
A curve has equation \(y = a^2 - x^2\), where \(a > 0\) The area enclosed between the curve and the \(x\)-axis is 36 units. Find the value of \(a\). Fully justify your answer. [6 marks]
AQA AS Paper 1 2022 June Q8
11 marks Standard +0.3
A curve has equation $$y = x^3 - 6x + \frac{9}{x}$$
  1. Show that the \(x\) coordinates of the stationary points of the curve satisfy the equation $$x^4 - 2x^2 - 3 = 0$$ [3 marks]
  2. Deduce that the curve has exactly two stationary points. [3 marks]
  3. Find the coordinates and nature of the two stationary points. Fully justify your answer. [4 marks]
  4. Write down the equation of a line which is a tangent to the curve in two places. [1 mark]
AQA AS Paper 1 2022 June Q9
5 marks Standard +0.3
Integers \(m\) and \(n\) are both odd. Prove that \(m^2 + n^2\) is a multiple of 2 but not a multiple of 4 [5 marks]
AQA AS Paper 1 2022 June Q10
9 marks Standard +0.8
Curve \(C\) has equation \(y = \frac{\sqrt{2}}{x^2}\)
  1. Find an equation of the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) [4 marks]
  2. Show that the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) is also a normal to the curve at a different point. \includegraphics{figure_10} [5 marks]
AQA AS Paper 1 2022 June Q11
1 marks Easy -2.0
A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, \(v\) m s\(^{-1}\), changes over time, \(t\) seconds. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_11}
AQA AS Paper 1 2022 June Q12
1 marks Easy -1.8
A horizontal force of 30 N causes a crate to travel with an acceleration of 2 m s\(^{-2}\), in a straight line, on a smooth horizontal surface. Find the weight of the crate. Circle your answer. [1 mark] 15 kg \quad 15g N \quad 15 N \quad 15g kg
AQA AS Paper 1 2022 June Q13
3 marks Moderate -0.8
Two points \(A\) and \(B\) lie in a horizontal plane and have coordinates \((-2, 7)\) and \((3, 19)\) respectively. A particle moves in a straight line from \(A\) to \(B\) under the action of a constant resultant force of magnitude 6.5 N Express the resultant force in vector form. [3 marks]
AQA AS Paper 1 2022 June Q14
3 marks Moderate -0.8
A ball is released from rest from a height \(h\) metres above horizontal ground and falls freely downwards. When the ball reaches the ground, its speed is \(v\) m s\(^{-1}\), where \(v \leq 10\) Show that $$h \leq \frac{50}{g}$$ [3 marks]
AQA AS Paper 1 2022 June Q15
5 marks Moderate -0.3
Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane. A force of \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) N is applied to \(P\). A force of \(\begin{bmatrix} 8 \\ 15 \end{bmatrix}\) N is applied to \(Q\).
  1. Calculate, to the nearest degree, the acute angle between the two forces. [2 marks]
  2. The particles begin to move under the action of the respective forces. \(P\) and \(Q\) have the same mass. \(P\) has an acceleration of magnitude 5 m s\(^{-2}\) Find the magnitude of the acceleration of \(Q\). [3 marks]
AQA AS Paper 1 2022 June Q16
6 marks Moderate -0.3
Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of \(u\) m s\(^{-1}\) Jermaine is walking 0.2 m s\(^{-1}\) more slowly than Meena. When Jermaine is \(d\) metres behind Meena he starts to run with a constant acceleration of 2 m s\(^{-2}\), for a time of \(t\) seconds, until he reaches her.
  1. Show that $$d = t^2 - 0.2t$$ [4 marks]
  2. When Jermaine's speed is 7.8 m s\(^{-1}\), he reaches Meena. Given that \(u = 1.4\) find the value of \(d\). [2 marks]
AQA AS Paper 1 2022 June Q17
8 marks Moderate -0.8
\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$
  1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
  2. The car has a mass of 1500 kg The caravan has a mass of 850 kg When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N
    1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
    2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
  3. State one assumption you have made about the tow bar. [1 mark]
AQA AS Paper 1 2023 June Q1
1 marks Easy -1.8
At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer. [1 mark] \(-10\) \quad \(-0.1\) \quad \(0.1\) \quad \(10\)
AQA AS Paper 1 2023 June Q2
1 marks Easy -1.8
Identify the expression below which is equivalent to \(\left(\frac{2x}{5}\right)^{-3}\) Circle your answer. [1 mark] \(\frac{8x^3}{125}\) \quad \(\frac{125x^3}{8}\) \quad \(\frac{125}{8x^3}\) \quad \(\frac{8}{125x^3}\)
AQA AS Paper 1 2023 June Q3
3 marks Moderate -0.3
The coefficient of \(x^2\) in the binomial expansion of \((1 + ax)^6\) is \(\frac{20}{3}\) Find the two possible values of \(a\) [3 marks]
AQA AS Paper 1 2023 June Q4
5 marks Moderate -0.3
It is given that \(5\cos^2 \theta - 4\sin^2 \theta = 0\)
  1. Find the possible values of \(\tan \theta\), giving your answers in exact form. [3 marks]
  2. Hence, or otherwise, solve the equation $$5\cos^2 \theta - 4\sin^2 \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1°\) in the interval \(0° \leq \theta \leq 360°\) [2 marks]
AQA AS Paper 1 2023 June Q5
7 marks Moderate -0.3
  1. Given that \(y = x\sqrt{x}\), find \(\frac{dy}{dx}\) [2 marks]
  2. The line, \(L\), has equation \(6x - 2y + 5 = 0\) \(L\) is a tangent to the curve with equation \(y = x\sqrt{x} + k\) Find the value of \(k\) [5 marks]
AQA AS Paper 1 2023 June Q6
6 marks Moderate -0.8
  1. The curve \(C_1\) has equation \(y = 2x^2 - 20x + 42\) Express the equation of \(C_1\) in the form $$y = a(x - h)^2 + c$$ where \(a\), \(b\) and \(c\) are integers. [3 marks]
  2. Write down the coordinates of the minimum point of \(C_1\) [1 mark]
  3. The curve \(C_1\) is mapped onto the curve \(C_2\) by a stretch in the \(y\)-direction. The minimum point of \(C_2\) is at \((5, -4)\) Find the equation of \(C_2\) [2 marks]
AQA AS Paper 1 2023 June Q7
5 marks Moderate -0.8
Points \(P\) and \(Q\) lie on the curve with equation \(y = x^4\) The \(x\)-coordinate of \(P\) is \(x\) The \(x\)-coordinate of \(Q\) is \(x + h\)
  1. Expand \((x + h)^4\) [2 marks]
  2. Hence, find an expression, in terms of \(x\) and \(h\), for the gradient of the line \(PQ\) [1 mark]
  3. Explain how to use the answer from part (b) to obtain the gradient function of \(y = x^4\) [2 marks]
AQA AS Paper 1 2023 June Q8
7 marks Standard +0.3
  1. Show that $$\int_1^a \left(6 - \frac{12}{\sqrt{x}}\right) dx = 6a - 24\sqrt{a} + 18$$ [3 marks]
  2. The curve \(y = 6 - \frac{12}{\sqrt{x}}\), the line \(x = 1\) and the line \(x = a\) are shown in the diagram below. The shaded region \(R_1\) is bounded by the curve, the line \(x = 1\) and the \(x\)-axis. The shaded region \(R_2\) is bounded by the curve, the line \(x = a\) and the \(x\)-axis. \includegraphics{figure_8} It is given that the areas of \(R_1\) and \(R_2\) are equal. Find the value of \(a\) Fully justify your answer. [4 marks]