Questions — Edexcel C1 (490 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 Q6
6. (a) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 4 k x - k = 0$$ (b) Hence find the set of values of \(k\) for which the equation has no real roots.
Edexcel C1 Q7
7. (a) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(b) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(c) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).
Edexcel C1 Q8
8. The points \(P\) and \(Q\) have coordinates \(( 7,4 )\) and \(( 9,7 )\) respectively.
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The straight line \(m\) has gradient 8 and passes through the origin, \(O\).
  2. Write down an equation for \(m\). The lines \(l\) and \(m\) intersect at the point \(R\).
  3. Show that \(O P = O R\).
Edexcel C1 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e1a283dd-c30a-45ee-af0c-429791036753-4_549_721_251_390} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$f ^ { \prime } ( x ) = 6 - 4 x - 3 x ^ { 2 } ,$$
  1. find an expression for \(y\) in terms of \(x\),
  2. show that \(A B = k \sqrt { 7 }\), where \(k\) is an integer to be found.
Edexcel C1 Q10
10. A curve has the equation \(y = x + \frac { 3 } { x } , x \neq 0\). The point \(P\) on the curve has \(x\)-coordinate 1 .
  1. Show that the gradient of the curve at \(P\) is - 2 .
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(y = m x + c\).
  3. Find the coordinates of the point where the normal to the curve at \(P\) intersects the curve again.
Edexcel C1 Q1
1. $$f ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }$$ Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.
Edexcel C1 Q2
2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).
Edexcel C1 Q3
3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 ^ { n } + k n ,$$ where \(k\) is a constant. Given that \(u _ { 1 } = u _ { 3 }\),
  1. find the value of \(k\),
  2. find the value of \(u _ { 5 }\).
Edexcel C1 Q4
4. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1 ,$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).
Edexcel C1 Q5
5. $$f ( x ) = 4 x - 3 x ^ { 2 } - x ^ { 3 }$$
  1. Fully factorise \(4 x - 3 x ^ { 2 } - x ^ { 3 }\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
Edexcel C1 Q6
6. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\). Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).
Edexcel C1 Q7
7. (a) Given that \(y = 2 ^ { x }\), find expressions in terms of \(y\) for
  1. \(2 ^ { x + 2 }\),
  2. \(2 ^ { 3 - x }\).
    (b) Show that using the substitution \(y = 2 ^ { x }\), the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$ can be rewritten as $$4 y ^ { 2 } - 33 y + 8 = 0$$ (c) Hence solve the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$
Edexcel C1 Q8
  1. Given that
$$y = 2 x ^ { \frac { 3 } { 2 } } - 1$$
  1. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
  2. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 y = k$$ where \(k\) is an integer to be found,
  3. find $$\int y ^ { 2 } \mathrm {~d} x$$
Edexcel C1 Q9
  1. The second and fifth terms of an arithmetic series are 26 and 41 repectively.
    1. Show that the common difference of the series is 5 .
    2. Find the 12th term of the series.
    Another arithmetic series has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two series are equal,
  2. find the value of \(n\).
Edexcel C1 Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ddc2483c-fc21-4d6f-9e5b-7c48339dbc88-4_647_775_879_475} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).
  1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
  2. Find an equation for the tangent to the curve at \(P\).
  3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
  4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).
Edexcel C1 Q2
2. Find the set of values of \(x\) for which $$( x - 1 ) ( x - 2 ) < 20$$
Edexcel C1 Q3
  1. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point (8, 7).
Given that $$\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { \frac { 1 } { 3 } } - 5$$ find \(\mathrm { f } ( x )\).
Edexcel C1 Q4
4. (a) Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
(b) Find the value of \(x\) such that $$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
Edexcel C1 Q5
5. Given that $$y = x + 5 + \frac { 3 } { \sqrt { x } }$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find \(\int y \mathrm {~d} x\).
Edexcel C1 Q6
6. $$f ( x ) = x ^ { \frac { 3 } { 2 } } - 8 x ^ { - \frac { 1 } { 2 } } .$$
  1. Evaluate f(3), giving your answer in its simplest form with a rational denominator.
  2. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(k \sqrt { 2 }\).
Edexcel C1 Q7
7. The straight line \(l _ { 1 }\) has gradient 2 and passes through the point with coordinates \(( 4 , - 5 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) is perpendicular to the line with equation \(3 x - y = 4\) and passes through the point with coordinates \(( 3,0 )\).
  2. Find an equation for \(l _ { 2 }\).
  3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
Edexcel C1 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d05cfae5-1d1d-4c90-80df-2975b9481c82-3_522_844_1235_379} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).
  1. Write down the number of solutions that exist for the equation
    1. \(\mathrm { f } ( x ) = 1\),
    2. \(\mathrm { f } ( x ) = - x\).
  2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
    1. \(\quad y = \mathrm { f } ( x - 2 )\),
    2. \(y = \mathrm { f } ( 2 x )\).
Edexcel C1 Q9
9. (a) Prove that the sum of the first \(n\) terms of an arithmetic series with first term \(a\) and common difference \(d\) is given by $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A novelist begins writing a new book. She plans to write 16 pages during the first week, 18 during the second and so on, with the number of pages increasing by 2 each week. Find, according to her plan,
(b) how many pages she will write in the fifth week,
(c) the total number of pages she will write in the first five weeks.
(d) Using algebra, find how long it will take her to write the book if it has 250 pages.
Edexcel C1 Q10
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x + 2 ) ^ { 3 }$$
  1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
  2. Find f \({ } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
  3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
  4. Show that \(m\) has the equation \(y = 3 x + 8\).
Edexcel C1 Q1
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(k \sqrt { 2 }\).
  2. Differentiate with respect to \(x\)
$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$