Questions — Edexcel AS Paper 1 (150 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. \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 \(R\).
Edexcel AS Paper 1 Specimen Q13
  1. The growth of pond weed on the surface of a pond is being investigated.
The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.
  1. State the surface area of the pond covered by the weed at the start of the investigation.
  2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
  3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
  4. Evaluate the model in light of this information, giving a reason for your answer.
Edexcel AS Paper 1 Specimen Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-30_673_819_246_623} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = ( x - 2 ) ^ { 2 } ( x + 3 )\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel AS Paper 1 Q1
  1. Find
$$\int \left( \frac { 1 } { 2 } x ^ { 2 } - 9 \sqrt { x } + 4 \right) d x$$ giving your answer in its simplest form.
Edexcel AS Paper 1 Q2
2. Use a counter example to show that the following statement is false. $$\text { " } n ^ { 2 } - n + 5 \text { is a prime number, for } 2 \leq n \leq 6 \text { " }$$
Edexcel AS Paper 1 Q3
  1. 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 \(\overrightarrow { A B } = - 3 \mathbf { i } + 4 \mathbf { j }\), find
    a. the values of \(x\) and \(y\)
    b. a unit vector in the direction of \(\overrightarrow { A B }\).
  2. The line \(l _ { 1 }\) has equation \(2 x - 3 y = 9\)
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
5. A student is asked to solve the equation $$\log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } = 1$$ The student's attempt is shown $$\begin{aligned} \log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } & = 1
x - \sqrt { x - 2 } & = 3 ^ { 1 }
x - 3 & = \sqrt { x - 2 }
( x - 3 ) ^ { 2 } & = x - 2
x ^ { 2 } - 7 x + 11 & = 0
x = \frac { 7 + \sqrt { 5 } } { 2 } \quad \text { or } \quad x & = \frac { 7 - \sqrt { 5 } } { 2 } \end{aligned}$$ a. Identify the error made by this student, giving a brief explanation.
b. Write out the correct solution.
Edexcel AS Paper 1 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-07_629_835_306_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} 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. $$\mathrm { h } ( t ) = 7 + 21 t - 4.9 t ^ { 2 } , t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1.
Using the model,
a. give a physical interpretation of the meaning of the constant term 7 in the model.
b. find the time taken after the stone is thrown for it to reach ground level.
c. Rearrange \(\mathrm { h } ( t )\) into the form \(A - B ( t - C ) ^ { 2 }\), where \(A , B\) and \(C\) are constants to be found.
d. Using your answer to part cor otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached.
Edexcel AS Paper 1 Q7
7. In a triangle \(P Q R , P Q = 20 \mathrm {~cm} , P R = 10 \mathrm {~cm}\) and angle \(Q P R = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(P Q R\) is \(80 \mathrm {~cm} ^ { 2 }\).
a. Show that the two possible values of \(\cos \theta = \pm \frac { 3 } { 5 }\) Given that \(Q R\) is the longest side of the triangle,
b. find the exact perimeter of the triangle \(P Q R\), giving your answer as a simplified surd.
Edexcel AS Paper 1 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-11_691_1098_365_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a solid cuboid \(A B C D E F G H\).
\(A B = x \mathrm {~cm} , B C = 2 x \mathrm {~cm} , A E = h \mathrm {~cm}\)
The total surface area of the cuboid is \(180 \mathrm {~cm} ^ { 2 }\).
The volume of the cuboid is \(V \mathrm {~cm} ^ { 3 }\).
a. Show that \(V = 60 x - \frac { 4 x ^ { 3 } } { 3 }\) Given that \(x\) can vary,
b. 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 \(V\).
c. Find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
Edexcel AS Paper 1 Q9
9. $$f ( x ) = - 2 x ^ { 3 } - x ^ { 2 } + 4 x + 3$$ a. Use the factor theorem to show that ( \(3 - 2 x\) ) is a factor of \(\mathrm { f } ( x )\).
b. Hence show that \(\mathrm { f } ( x )\) can be written in the form \(\mathrm { f } ( x ) = ( 3 - 2 x ) ( x + a ) ^ { 2 }\) where \(a\) is an integer to be found. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-15_657_1024_278_450} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
c. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
i. \(\mathrm { f } ( x ) \leq 0\)
ii. \(\mathrm { f } \left( \frac { x } { 2 } \right) = 0\)
Edexcel AS Paper 1 Q10
10. Prove, from the first principles, that the derivative of \(5 x ^ { 2 }\) is \(10 x\).
Edexcel AS Paper 1 Q11
11. The first 3 terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + k x ) ^ { 10 }\) are given by $$1 + 15 x + p x ^ { 2 }$$ where \(k\) and \(p\) are constants.
a. Find the value of \(k\)
b. Find the value of \(p\)
c. Given that, in the expansion of \(( 1 + k x ) ^ { 10 }\), the coefficient of \(x ^ { 4 }\) is \(q\), find the value of \(q\).
Edexcel AS Paper 1 Q12
12. a. Explain mathematically why there are no values of \(\theta\) that satisfy the equation $$( 3 \cos \theta - 4 ) ( 2 \cos \theta + 5 ) = 0$$ b. 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.)
Edexcel AS Paper 1 Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-19_694_1246_344_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The value of a sculpture, \(\pounds V\), is modelled by the equation \(V = A p ^ { t }\), where \(A\) and \(p\) are constants and \(t\) is the number of years since the value of the painting was first recorded on \(1 ^ { \text {st } }\) January 1960. The line \(l\) shown in Figure 4 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) for \(t \geq 0\). The line \(l\) passes through the point \(\left( 0 , \log _ { 10 } 20 \right)\) and \(\left( 50 , \log _ { 10 } 2000 \right)\).
a. Write down the equation of the line \(l\).
b. Using your answer to part a or otherwise, find the values of \(A\) and \(p\).
c. With reference to the model, interpret the values of the constant \(A\) and \(p\).
d. Use your model, to predict the value of the sculpture, on \(1 { } ^ { \text {st } }\) January 2020, giving your answer to the nearest pounds.
Edexcel AS Paper 1 Q14
14. A curve with centre \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0$$ a. i. State the coordinates of \(C\).
ii. Find the radius of the circle, giving your answer as \(r = n \sqrt { 2 }\).
b. The line \(l\) is a tangent to the circle and has gradient - 7 . Find two possible equations for \(l\), giving your answers in the form \(y = m x + c\).
Edexcel AS Paper 1 Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-24_712_972_296_598} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve \(y = 2 x + \frac { 8 } { x ^ { 2 } } - 5 , x > 0\).
The point \(A \left( 4 , \frac { 7 } { 2 } \right)\) lies on C . The line \(l\) is the tangent to \(C\) at the point A .
The region \(\boldsymbol { R }\), shown shaded in figure 5 is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(\boldsymbol { R }\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(Total for Question 15 is 9 marks)