Questions C1 (1442 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 C1 2013 June Q3
3. (a) Find the value of \(8 ^ { \frac { 5 } { 3 } }\)
(b) Simplify fully \(\frac { \left( 2 x ^ { \frac { 1 } { 2 } } \right) ^ { 3 } } { 4 x ^ { 2 } }\)
Edexcel C1 2013 June Q4
4. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 4
a _ { n + 1 } & = k \left( a _ { n } + 2 \right) , \quad \text { for } n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 2\),
  2. find the two possible values of \(k\).
Edexcel C1 2013 June Q5
5. Find the set of values of \(x\) for which
  1. \(2 ( 3 x + 4 ) > 1 - x\)
  2. \(3 x ^ { 2 } + 8 x - 3 < 0\)
Edexcel C1 2013 June Q6
6. The straight line \(L _ { 1 }\) passes through the points \(( - 1,3 )\) and \(( 11,12 )\).
  1. Find an equation for \(L _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) has equation \(3 y + 4 x - 30 = 0\).
  2. Find the coordinates of the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
Edexcel C1 2013 June Q7
7. A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
  1. Find the value of \(N\). The company then plans to continue to make 600 mobile phones each week.
  2. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
Edexcel C1 2013 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-09_369_956_287_504} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 3 ) ^ { 2 } ( x - 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(( 1,0 )\), touches it at \(( - 3,0 )\) and crosses the \(y\)-axis at \(( 0 , - 9 )\)
  1. In the space below, sketch the curve \(C\) with equation \(y = \mathrm { f } ( x + 2 )\) and state the coordinates of the points where the curve \(C\) meets the \(x\)-axis.
  2. Write down an equation of the curve \(C\).
  3. Use your answer to part (b) to find the coordinates of the point where the curve \(C\) meets the \(y\)-axis.
Edexcel C1 2013 June Q9
9. $$f ^ { \prime } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) ^ { 2 } } { x ^ { 2 } } , \quad x \neq 0$$
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 9 x ^ { - 2 } + A + B x ^ { 2 }\),
    where \(A\) and \(B\) are constants to be found.
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\). Given that the point \(( - 3,10 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\),
  3. find \(\mathrm { f } ( x )\).
Edexcel C1 2013 June Q10
  1. Given the simultaneous equations
$$\begin{aligned} & 2 x + y = 1
& x ^ { 2 } - 4 k y + 5 k = 0 \end{aligned}$$ where \(k\) is a non zero constant,
  1. show that $$x ^ { 2 } + 8 k x + k = 0$$ Given that \(x ^ { 2 } + 8 k x + k = 0\) has equal roots,
  2. find the value of \(k\).
  3. For this value of \(k\), find the solution of the simultaneous equations.
Edexcel C1 2013 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-15_592_1394_274_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(H\) with equation \(y = \frac { 3 } { x } + 4 , x \neq 0\).
  1. Give the coordinates of the point where \(H\) crosses the \(x\)-axis.
  2. Give the equations of the asymptotes to \(H\).
  3. Find an equation for the normal to \(H\) at the point \(P ( - 3,3 )\). This normal crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  4. Find the length of the line segment \(A B\). Give your answer as a surd.
Edexcel C1 2014 June Q1
Factorise fully \(25 x - 9 x ^ { 3 }\)
\includegraphics[max width=\textwidth, alt={}, center]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-02_37_42_2700_1909}
Edexcel C1 2014 June Q4
4. Given that \(y = 2 x ^ { 5 } + \frac { 6 } { \sqrt { } x } , x > 0\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)
Edexcel C1 2014 June Q5
5. Solve the equation $$10 + x \sqrt { 8 } = \frac { 6 x } { \sqrt { 2 } }$$ Give your answer in the form \(a \sqrt { } b\) where \(a\) and \(b\) are integers.
Edexcel C1 2014 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-08_917_1322_239_303} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres.
Given that the perimeter of the garden is greater than 40 m ,
  1. show that \(x > 1.7\) Given that the area of the garden is less than \(120 \mathrm {~m} ^ { 2 }\),
  2. form and solve a quadratic inequality in \(x\).
  3. Hence state the range of the possible values of \(x\).
Edexcel C1 2014 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-10_869_1073_267_440} \captionsetup{labelformat=empty} \caption{Diagram NOT to scale}
\end{figure} Figure 2 Figure 2 shows a right angled triangle \(L M N\). The points \(L\) and \(M\) have coordinates ( \(- 1,2\) ) and ( \(7 , - 4\) ) respectively.
  1. Find an equation for the straight line passing through the points \(L\) and \(M\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that the coordinates of point \(N\) are ( \(16 , p\) ), where \(p\) is a constant, and angle \(L M N = 90 ^ { \circ }\),
  2. find the value of \(p\). Given that there is a point \(K\) such that the points \(L , M , N\), and \(K\) form a rectangle,
  3. find the \(y\) coordinate of \(K\).
Edexcel C1 2014 June Q8
8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { - \frac { 1 } { 2 } } + x \sqrt { } x , \quad x > 0$$ Given that \(y = 37\) at \(x = 4\), find \(y\) in terms of \(x\), giving each term in its simplest form.
Edexcel C1 2014 June Q9
9. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 2 } + 8\) The line \(L\) has equation \(y = 3 x + k\), where \(k\) is a positive constant.
  1. Sketch \(C\) and \(L\) on separate diagrams, showing the coordinates of the points at which \(C\) and \(L\) cut the axes. Given that line \(L\) is a tangent to \(C\),
  2. find the value of \(k\).
Edexcel C1 2014 June Q10
  1. Xin has been given a 14 day training schedule by her coach.
Xin will run for \(A\) minutes on day 1 , where \(A\) is a constant.
She will then increase her running time by ( \(d + 1\) ) minutes each day, where \(d\) is a constant.
  1. Show that on day 14 , Xin will run for $$( A + 13 d + 13 ) \text { minutes. }$$ Yi has also been given a 14 day training schedule by her coach.
    Yi will run for \(( A - 13 )\) minutes on day 1 .
    She will then increase her running time by ( \(2 d - 1\) ) minutes each day.
    Given that Yi and Xin will run for the same length of time on day 14,
  2. find the value of \(d\). Given that Xin runs for a total time of 784 minutes over the 14 days,
  3. find the value of \(A\).
Edexcel C1 2014 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-17_700_1556_276_201} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A sketch of part of the curve \(C\) with equation $$y = 20 - 4 x - \frac { 18 } { x } , \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has an \(x\) coordinate equal to 2
  1. Show that the equation of the normal to \(C\) at \(A\) is \(y = - 2 x + 7\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3 .
  2. Use algebra to find the coordinates of \(B\).
Edexcel C1 2015 June Q1
Simplify
  1. \(( 2 \sqrt { } 5 ) ^ { 2 }\)
  2. \(\frac { \sqrt { } 2 } { 2 \sqrt { } 5 - 3 \sqrt { } 2 }\) giving your answer in the form \(a + \sqrt { } b\), where \(a\) and \(b\) are integers.
Edexcel C1 2015 June Q3
Given that \(y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0\), find in their simplest form
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. \(\int y \mathrm {~d} x\)
Edexcel C1 2015 June Q5
  1. The equation
$$( p - 1 ) x ^ { 2 } + 4 x + ( p - 5 ) = 0 , \text { where } p \text { is a constant }$$ has no real roots.
  1. Show that \(p\) satisfies \(p ^ { 2 } - 6 p + 1 > 0\)
  2. Hence find the set of possible values of \(p\).
Edexcel C1 2015 June Q6
  1. The curve \(C\) has equation
$$y = \frac { \left( x ^ { 2 } + 4 \right) ( x - 3 ) } { 2 x } , \quad x \neq 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
  2. Find an equation of the tangent to \(C\) at the point where \(x = - 1\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2015 June Q7
  1. Given that \(y = 2 ^ { x }\),
    1. express \(4 ^ { x }\) in terms of \(y\).
    2. Hence, or otherwise, solve
    $$8 \left( 4 ^ { x } \right) - 9 \left( 2 ^ { x } \right) + 1 = 0$$
Edexcel C1 2015 June Q8
  1. (a) Factorise completely \(9 x - 4 x ^ { 3 }\)
    (b) Sketch the curve \(C\) with equation
$$y = 9 x - 4 x ^ { 3 }$$ Show on your sketch the coordinates at which the curve meets the \(x\)-axis. The points \(A\) and \(B\) lie on \(C\) and have \(x\) coordinates of - 2 and 1 respectively.
(c) Show that the length of \(A B\) is \(k \sqrt { } 10\) where \(k\) is a constant to be found.
Edexcel C1 2015 June Q9
Jess started work 20 years ago. In year 1 her annual salary was \(\pounds 17000\). Her annual salary increased by \(\pounds 1500\) each year, so that her annual salary in year 2 was \(\pounds 18500\), in year 3 it was \(\pounds 20000\) and so on, forming an arithmetic sequence. This continued until she reached her maximum annual salary of \(\pounds 32000\) in year \(k\). Her annual salary then remained at \(\pounds 32000\).
  1. Find the value of the constant \(k\).
  2. Calculate the total amount that Jess has earned in the 20 years.