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 2018 June Q6
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$\begin{aligned} a _ { 1 } & = 4
a _ { n + 1 } & = \frac { a _ { n } } { a _ { n } + 1 } , \quad n \geqslant 1 , n \in \mathbb { N } \end{aligned}$$
  1. Find the values of \(a _ { 2 } , a _ { 3 }\) and \(a _ { 4 }\) Write your answers as simplified fractions. Given that $$a _ { n } = \frac { 4 } { p n + q } , \text { where } p \text { and } q \text { are constants }$$
  2. state the value of \(p\) and the value of \(q\).
  3. Hence calculate the value of \(N\) such that \(a _ { N } = \frac { 4 } { 321 }\)
Edexcel C1 2018 June Q7
  1. The equation \(20 x ^ { 2 } = 4 k x - 13 k x ^ { 2 } + 2\), where \(k\) is a constant, has no real roots.
    1. Show that \(k\) satisfies the inequality
    $$2 k ^ { 2 } + 13 k + 20 < 0$$
  2. Find the set of possible values for \(k\).
Edexcel C1 2018 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-20_1063_1319_251_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the straight line \(l _ { 1 }\) with equation \(4 y = 5 x + 12\)
  1. State the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the point \(E ( 12,5 )\), as shown in Figure 2.
  2. Find the equation of \(l _ { 2 }\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined. The line \(l _ { 2 }\) cuts the \(x\)-axis at the point \(C\) and the \(y\)-axis at the point \(B\).
  3. Find the coordinates of
    1. the point \(B\),
    2. the point \(C\). The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(A\).
      The point \(D\) lies on \(l _ { 1 }\) such that \(A B C D\) is a parallelogram, as shown in Figure 2.
  4. Find the area of \(A B C D\).
Edexcel C1 2018 June Q9
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where
$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point \(P ( 1,20 )\) lies on \(C\),
  1. find \(\mathrm { f } ( x )\), simplifying each term.
  2. Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where \(A\) is a constant to be found.
  3. Sketch the graph of \(C\). Show clearly the coordinates of the points where \(C\) cuts or meets the \(x\)-axis and where \(C\) cuts the \(y\)-axis.
Edexcel C1 2018 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-28_643_1171_260_518} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x + \frac { 27 } { x } - 12 , \quad x > 0$$ The point \(A\) lies on \(C\) and has coordinates \(\left( 3 , - \frac { 3 } { 2 } \right)\).
  1. Show that the equation of the normal to \(C\) at \(A\) can be written as \(10 y = 4 x - 27\) 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 Q1
  1. Solve the inequality
    \(10 + x ^ { 2 } > x ( x - 2 )\).
    (3)
\begin{center} \begin{tabular}{|l|l|} \hline
Edexcel C1 Q2
2. Find \(\int \left( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } + \sqrt [ 3 ] { x } \right) \mathrm { d } x\) & Leave blank
\hline \end{tabular} \end{center}
Edexcel C1 Q3
Find the value of
  1. \(81 ^ { \frac { 1 } { 2 } }\),
  2. \(81 ^ { \frac { 3 } { 4 } }\),
  3. \(81 ^ { - \frac { 3 } { 4 } }\).
Edexcel C1 Q5
5. (a) Show that eliminating \(y\) from the equations $$\begin{gathered} 2 x + y = 8
3 x ^ { 2 } + x y = 1 \end{gathered}$$ produces the equation $$x ^ { 2 } + 8 x - 1 = 0$$ (b) Hence solve the simultaneous equations $$\begin{gathered} 2 x + y = 8
3 x ^ { 2 } + x y = 1 \end{gathered}$$ giving your answers in the form \(a + b \sqrt { } 17\), where \(a\) and \(b\) are integers.
5. continuedLeave blank
Edexcel C1 Q6
6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { } x } , \quad x > 0$$
  1. Show that \(\mathrm { f } ( x )\) can be written in the form \(P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }\), stating the values of the constants \(P , Q\) and \(R\).
  2. Find f \({ } ^ { \prime } ( x )\).
  3. Show that the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) is parallel to the line with equation \(2 y = 11 x + 3\).
    (3)
    6. continuedLeave blank
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
Edexcel C1 Q7
7. (a) Factorise completely \(x ^ { 3 } - 4 x\).
(3)
(b) Sketch the curve with equation \(y = x ^ { 3 } - 4 x\), showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
(c) On a separate diagram, sketch the curve with equation \(y = ( x - 1 ) ^ { 3 } - 4 ( x - 1 ) ,\)
showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
\end{tabular} & Leave blank
\hline \end{tabular} \end{center}
\includegraphics[max width=\textwidth, alt={}]{6400bb0c-f199-45f2-a4b1-55534e2c63b0-11_2608_1924_141_75}
\begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
Edexcel C1 Q8
8. The straight line \(l _ { 1 }\) has equation \(y = 3 x - 6\).
The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point (6, 2).
  1. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    (3)
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\).
  2. Use algebra to find the coordinates of \(C\).
    (3)
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the \(x\)-axis at the points \(A\) and \(B\) respectively.
  3. Calculate the exact area of triangle \(A B C\).
    (4) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    \end{tabular} & Leave blank
    \hline \end{tabular} \end{center}
    8. continuedLeave blank
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
Edexcel C1 Q9
9. An arithmetic series has first term \(a\) and common difference \(d\).
  1. Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .\)
    (4)
    A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference \(d \mathrm {~cm}\).
    The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm .
    Find
  2. the length of the shortest side of the polygon,
    (5)
  3. the value of \(d\).
    (2) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    \end{tabular} & Leave blank
    \hline \end{tabular} \end{center}
    Leave blank
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
Edexcel C1 Q10
10. For the curve \(C\) with equation \(y = \mathrm { f } ( x )\), \(\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .\)
  1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    (2)
  2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2\) for all values of \(x\).
    (1)
    Given that the point \(P ( 2,4 )\) lies on \(C\),
  3. find \(y\) in terms of \(x\),
    (5)
  4. find an equation for the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
    (5)
    \end{tabular} & Leave blank
    \hline \end{tabular} \end{center}
    1. continued
OCR MEI C1 Q1
1 Expand \(( 2 x + 5 ) ( x - 1 ) ( x + 3 )\), simplifying your answer.
OCR MEI C1 Q2
2 Find the discriminant of \(3 x ^ { 2 } + 5 x + 2\). Hence state the number of distinct real roots of the equation \(3 x ^ { 2 } + 5 x + 2 = 0\).
OCR MEI C1 Q3
3 Make \(x\) the subject of the formula \(y = \frac { 1 - 2 x } { x + 3 }\).
OCR MEI C1 Q4
4 Factorise \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\). Hence prove that, when \(n\) is a positive integer, \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\) is always divisible by 6 .
OCR MEI C1 Q5
5 Express \(5 x ^ { 2 } + 20 x + 6\) in the form \(a ( x + b ) ^ { 2 } + c\).
OCR MEI C1 Q6
6 Rearrange the formula \(c = \sqrt { \frac { a + b } { 2 } }\) to make \(a\) the subject.
\(7 \quad\) Make \(a\) the subject of the formula \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 Q8
8 Prove that, when \(n\) is an integer, \(n ^ { 3 } - n\) is always even.
OCR MEI C1 Q9
9
  1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } + 6 x + 5\).
OCR MEI C1 Q10
10 Find the real roots of the equation \(x ^ { 4 } - 5 x ^ { 2 } - 36 = 0\) by considering it as a quadratic equation in \(x ^ { 2 }\).
OCR MEI C1 Q11
11 Solve the equation \(\frac { 3 x + 1 } { 2 x } = 4\).
OCR MEI C1 Q12
12 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.