Questions AS Paper 1 (363 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel AS Paper 1 Specimen Q4
4. $$f ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } + 2 x - 6$$
  1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Hence show that 3 is the only real root of the equation \(\mathrm { f } ( x ) = 0\)
Edexcel AS Paper 1 Specimen Q5
5. Given that
show that \(\int _ { 1 } ^ { 2 \sqrt { 2 } } \mathrm { f } ( x ) \mathrm { d } x = 16 + 3 \sqrt { 2 }\) $$\mathrm { f } ( x ) = 2 x + 3 + \frac { 12 } { x ^ { 2 } } , \quad x > 0$$
Edexcel AS Paper 1 Specimen Q6
  1. Prove, from first principles, that the derivative of \(3 x ^ { 2 }\) is \(6 x\).
  2. (a) 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.
    (b) Explain how you would use your expansion to give an estimate for the value of \(1.995 ^ { 7 }\)
Edexcel AS Paper 1 Specimen Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-10_609_675_262_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A triangular lawn is modelled by the triangle \(A B C\), shown in Figure 1. The length \(A B\) is to be 30 m long. Given that angle \(B A C = 70 ^ { \circ }\) and angle \(A B C = 60 ^ { \circ }\),
  1. calculate the area of the lawn to 3 significant figures.
  2. Why is your answer unlikely to be accurate to the nearest square metre?
Edexcel AS Paper 1 Specimen Q9
  1. Solve, for \(360 ^ { \circ } \leqslant x < 540 ^ { \circ }\),
$$12 \sin ^ { 2 } x + 7 \cos x - 13 = 0$$ 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
  1. The equation \(k x ^ { 2 } + 4 k x + 3 = 0\), where \(k\) is a constant, has no real roots.
Prove that $$0 \leqslant k < \frac { 3 } { 4 }$$
Edexcel AS Paper 1 Specimen Q11
  1. (a) Prove that for all positive values of \(x\) and \(y\)
$$\sqrt { x y } \leqslant \frac { x + y } { 2 }$$ (b) Prove by counter example that this is not true when \(x\) and \(y\) are both negative.
Edexcel AS Paper 1 Specimen Q12
12. A student was asked to give the exact solution to the equation $$2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0$$ The student's attempt is shown below: $$\begin{aligned} & 2 ^ { 2 x + 4 } - 9 \left( 2 ^ { x } \right) = 0
& 2 ^ { 2 x } + 2 ^ { 4 } - 9 \left( 2 ^ { x } \right) = 0
& \text { Let } \quad 2 ^ { x } = y
& y ^ { 2 } - 9 y + 8 = 0
& ( y - 8 ) ( y - 1 ) = 0
& y = 8 \text { or } y = 1
& \text { So } x = 3 \text { or } x = 0 \end{aligned}$$
  1. Identify the two errors made by the student.
  2. Find the exact solution to the equation.
Edexcel AS Paper 1 Specimen Q13
  1. (a) Factorise completely \(x ^ { 3 } + 10 x ^ { 2 } + 25 x\)
    (b) Sketch the curve with equation
$$y = x ^ { 3 } + 10 x ^ { 2 } + 25 x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. 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.
(c) Find the two possible values of \(a\).
Edexcel AS Paper 1 Specimen Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-20_777_1319_315_370} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A town's population, \(P\), is modelled by the equation \(P = a b ^ { t }\), 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. Find the value of \(a\) and the value of \(b\).
  3. With reference to the model interpret
    1. the value of the constant \(a\),
    2. the value of the constant \(b\).
  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 200000 , according to the model.
  5. State two reasons why this may not be a realistic population model.
Edexcel AS Paper 1 Specimen Q15
15. Diagram not drawn to scale \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-22_725_844_251_623} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C _ { 1 }\), shown in Figure 3, has equation \(y = 4 x ^ { 2 } - 6 x + 4\).
The point \(P \left( \frac { 1 } { 2 } , 2 \right)\) lies on \(C _ { 1 }\)
The curve \(C _ { 2 }\), also shown in Figure 3, has equation \(y = \frac { 1 } { 2 } x + \ln ( 2 x )\).
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.)
Edexcel AS Paper 1 Specimen Q16
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b084faa-a680-4f35-bb5c-a4edf5171b5f-24_458_604_285_751} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the plan view of the design for a swimming pool.
The shape of this pool \(A B C D E A\) consists of a rectangular section \(A B D E\) joined to a semicircular section \(B C D\) as shown in Figure 4. Given that \(A E = 2 x\) metres, \(E D = y\) metres and the area of the pool is \(250 \mathrm {~m} ^ { 2 }\),
  1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2 x + \frac { 250 } { x } + \frac { \pi x } { 2 }$$
  2. Explain why \(0 < x < \sqrt { \frac { 500 } { \pi } }\)
  3. Find the minimum perimeter of the pool, giving your answer to 3 significant figures.
Edexcel AS Paper 1 Specimen Q17
  1. 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 \(( 10,1 )\).
    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 \(Q\).
  2. Show that the distance \(P Q\) is 58 explaining your method clearly.
Edexcel AS Paper 1 Specimen Q1
  1. A curve has equation
$$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.
    VIIIV SIHI NI JIIYM IONOOVIUV SIHI NI JIIAM ION OOVI4V SIHI NI JIIIM I ON OO
Edexcel AS Paper 1 Specimen Q2
  1. The quadrilateral \(O A B C\) has \(\overrightarrow { O A } = 4 \mathbf { i } + 2 \mathbf { j } , \overrightarrow { O B } = 6 \mathbf { i } - 3 \mathbf { j }\) and \(\overrightarrow { O C } = 8 \mathbf { i } - 20 \mathbf { j }\).
    1. Find \(\overrightarrow { A B }\).
    2. Show that quadrilateral \(O A B C\) is a trapezium.
Edexcel AS Paper 1 Specimen Q3
  1. A tank, which contained water, started to leak from a hole in its base.
The volume of water in the tank 24 minutes after the leak started was \(4 \mathrm {~m} ^ { 3 }\) The volume of water in the tank 60 minutes after the leak started was \(2.8 \mathrm {~m} ^ { 3 }\) The volume of water, \(V \mathrm {~m} ^ { 3 }\), in the tank \(t\) minutes after the leak started, can be described by a linear model between \(V\) and \(t\).
  1. Find an equation linking \(V\) with \(t\). Use this model to find
    1. the initial volume of water in the tank,
    2. the time taken for the tank to empty.
  3. Suggest a reason why this linear model may not be suitable.
Edexcel AS Paper 1 Specimen Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-08_755_775_248_662} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = g ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).
  1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
  2. State the largest root of the equation $$g ( x + 1 ) = 0$$
  3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
  4. state the range of possible values for \(k\).
Edexcel AS Paper 1 Specimen Q5
5. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12$$
  1. Using the factor theorem, explain why \(\mathrm { f } ( x )\) is divisible by \(( x + 3 )\).
  2. Hence fully factorise \(\mathrm { f } ( x )\).
  3. Show that \(\frac { x ^ { 3 } + 3 x ^ { 2 } - 4 x - 12 } { x ^ { 3 } + 5 x ^ { 2 } + 6 x }\) can be written in the form \(A + \frac { B } { x }\) where \(A\) and \(B\) are integers to be found.
Edexcel AS Paper 1 Specimen Q6
  1. (i) Use a counter example to show that the following statement is false.
$$" n ^ { 2 } - n - 1 \text { is a prime number, for } 3 \leqslant n \leqslant 10 \text {." }$$ (ii) Prove that the following statement is always true.
"The difference between the cube and the square of an odd number is even."
For example \(5 ^ { 3 } - 5 ^ { 2 } = 100\) is even.
\includegraphics[max width=\textwidth, alt={}, center]{fa7abe9f-f5c0-4578-afd1-73176c717536-12_2255_51_314_1978}
Edexcel AS Paper 1 Specimen Q7
  1. (a) Expand \(\left( 1 + \frac { 3 } { x } \right) ^ { 2 }\) simplifying each term.
    (b) Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of
$$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.
(c) Hence find the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 3 } { x } \right) ^ { 2 } \left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$
Edexcel AS Paper 1 Specimen Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-16_607_983_255_541} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve with equation \(y = \sqrt { x } , x \geqslant 0\)
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = a\), where \(a\) is a constant. Given that the area of \(R\) is 10
  1. find, in simplest form, the value of
    1. \(\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x\)
    2. \(\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x\)
  2. show that \(a = 2 ^ { k }\), where \(k\) is a rational constant to be found.
Edexcel AS Paper 1 Specimen Q9
  1. Find any real values of \(x\) such that
$$2 \log _ { 4 } ( 2 - x ) - \log _ { 4 } ( x + 5 ) = 1$$
Edexcel AS Paper 1 Specimen Q10
  1. A circle \(C\) has centre \(( 2,5 )\). Given that the point \(P ( - 2,3 )\) lies on \(C\).
    1. find an equation for \(C\).
    The line \(l\) is the tangent to \(C\) at the point \(P\). The point \(Q ( 2 , k )\) lies on \(l\).
  2. Find the value of \(k\).
Edexcel AS Paper 1 Specimen Q11
  1. (i) Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation,
$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
(ii) (a) A student's attempt at the question
"Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x
7 \times \frac { \sin x } { \cos x } = 8 \sin x
7 \sin x = 8 \sin x \cos x
7 = 8 \cos x
\cos x = \frac { 7 } { 8 }
x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$
Edexcel AS Paper 1 Specimen Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-24_798_792_246_639} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.
  1. Using algebra, find the \(x\) coordinate of point \(A\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-24_636_780_1585_644} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).