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 Q8
8. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
  1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
  2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\). END
Edexcel C1 Q1
1. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
  1. Use integration to find \(y\) in terms of \(x\).
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
Edexcel C1 Q2
2. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r ) .$$
  1. Write down the first two terms of the series.
  2. Find the common difference of the series. Given that \(n = 50\),
  3. find the sum of the series.
Edexcel C1 Q3
3. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\).
  2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).
Edexcel C1 Q4
4. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c ,$$ where \(c\) is a constant.
(b) Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).
Edexcel C1 Q5
5. Find the set of values for \(x\) for which
  1. \(6 x - 7 < 2 x + 3\),
  2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
  3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
Edexcel C1 Q6
6. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),
  1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
Edexcel C1 Q7
7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),
  1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
  2. show that the population of fish dies out during the sixth year.
  3. Find the value of \(d\) which would leave the population each year unchanged.
Edexcel C1 Q8
8. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
  2. Find the \(x\)-coordinate of \(Q\).
  3. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. This tangent intersects the coordinate axes at the points \(R\) and \(S\).
  4. Find the length of \(R S\), giving your answer as a surd.
Edexcel C1 Q9
9. The points \(A ( - 1 , - 2 ) , B ( 7,2 )\) and \(C ( k , 4 )\), where \(k\) is a constant, are the vertices of \(\triangle A B C\). Angle \(A B C\) is a right angle.
  1. Find the gradient of \(A B\).
  2. Calculate the value of \(k\).
  3. Show that the length of \(A B\) may be written in the form \(p \sqrt { 5 }\), where \(p\) is an integer to be found.
  4. Find the exact value of the area of \(\triangle A B C\).
  5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 Q1
  1. (a) Solve the inequality
$$3 x - 8 > x + 13$$ (b) Solve the inequality $$x ^ { 2 } - 5 x - 14 > 0$$
Edexcel C1 Q2
  1. Given that \(2 ^ { x } = \frac { 1 } { \sqrt { 2 } }\) and \(2 ^ { y } = 4 \sqrt { } 2\),
    1. find the exact value of \(x\) and the exact value of \(y\),
    2. calculate the exact value of \(2 ^ { y - x }\).
    3. (a) Prove, by completing the square, that the roots of the equation \(x ^ { 2 } + 2 k x + c = 0\), where \(k\) and \(c\) are constants, are \(- k \pm v ^ { 2 } - c\) ).
    The equation \(x ^ { 2 } + 2 k x \pm 81 = 0\) has equal roots.
  2. Find the possible values of \(k\).
Edexcel C1 Q4
4. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \(( 280 + x )\) phones will be sold in the second month, \(( 280 + 2 x )\) in the third month, and so on. Using this model with \(x = 5\), calculate
    1. the number of phones sold in the 36th month,
    2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
      Using the same model,
  2. find the least value of \(x\) required to achieve this target.
Edexcel C1 Q5
5. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{3d5491ef-91cf-4ba9-b00b-cd326d51e6e1-3_686_1066_276_559}
Figure 1 shows the curve with equation \(y ^ { 2 } = 4 ( x - 2 )\) and the line with equation \(2 x - 3 y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).
  1. Write down the coordinates of \(A\).
  2. Find, using algebra, the coordinates of \(P\) and \(Q\).
  3. Show that \(\angle P A Q\) is a right angle. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{3d5491ef-91cf-4ba9-b00b-cd326d51e6e1-4_646_1043_291_336}
    \end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.
Edexcel C1 Q7
7. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that \(C\) passes through the point \(P ( 2,6 )\),
  1. find \(y\) in terms of \(x\).
  2. Verify that \(C\) passes through the point \(( 4,0 )\).
  3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
  4. Calculate the exact \(x\)-coordinate of \(Q\). END
Edexcel C1 Q1
  1. (a) Given that \(8 = 2 ^ { k }\), write down the value of \(k\).
    (b) Given that \(4 ^ { x } = 8 ^ { 2 - x }\), find the value of \(x\).
  2. Given that \(( 2 + \sqrt { } 7 ) ( 4 - \sqrt { 7 } ) = a + b \sqrt { } 7\), where a and \(b\) are integers,
    (a) find the value of a and the value of \(b\).
Given that \(\frac { 2 + \sqrt { 7 } } { 4 + \sqrt { 7 } } = c + d \sqrt { 7 }\) where \(c\) and \(d\) are rational numbers,
(b) find the value of \(c\) and the value of \(d\).
Edexcel C1 Q3
3. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } } .$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find \(\int y \mathrm {~d} x\).
Edexcel C1 Q4
4. (a) By completing the square, find in terms of \(k\) the roots of the equation $$x ^ { 2 } + 2 k x - 7 = 0 .$$ (b) Prove that, for all values of \(k\), the roots of \(x ^ { 2 } + 2 k x - 7 = 0\) are real and different.
(c) Given that \(k = \sqrt { } 2\), find the exact roots of the equation.
Edexcel C1 Q5
5. The straight line \(l _ { 1 }\) has equation \(4 y + x = 0\). The straight line \(l _ { 2 }\) has equation \(y = 2 x - 3\).
  1. On the same axes, sketch the graphs of \(l _ { 1 }\) and \(l _ { 2 }\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).
  2. Calculate, as exact fractions, the coordinates of \(A\).
  3. Find an equation of the line through \(A\) which is perpendicular to \(l _ { 1 }\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 Q6
6. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays \(\pounds 500\). Her payments then increase by \(\pounds 50\) each year, so that she pays \(\pounds 550\) in the second year, \(\pounds 600\) in the third year, and so on.
  1. Find the amount that Anne will pay in the 40th year.
  2. Find the total amount that Anne will pay in over the 40 years. Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in \(\pounds 890\) and his payments then increase by \(\pounds d\) each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
  3. find the value of \(d\).
Edexcel C1 Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{beec3800-227a-45a1-87b8-0ea96e0e6270-4_661_746_283_721}
\end{figure} The points \(A\) and \(B\) have coordinates \(( 2 , - 3 )\) and \(( 8,5 )\) respectively, and \(A B\) is a chord of a circle with centre \(C\), as shown in Fig. 1.
  1. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
  2. Find an equation for the line through \(C\) and \(M\).
    (5) Given that the \(x\)-coordinate of \(C\) is 4 ,
  3. find the \(y\)-coordinate of \(C\),
    (2)
  4. show that the radius of the circle is \(\frac { 5 \sqrt { } 17 } { 4 }\).
    (4)
Edexcel C1 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{beec3800-227a-45a1-87b8-0ea96e0e6270-5_722_561_233_593} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2 x cm by x cm and height \(h \mathrm {~cm}\), as shown in Fig. 4. Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),
  1. express \(h\) in terms of \(x\),
  2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$
Edexcel C1 Q1
  1. (a) Given that \(8 = 2 ^ { k }\), write down the value of \(k\).
    (b) Given that \(4 ^ { x } = 8 ^ { 2 - x }\), find the value of \(x\).
  2. Given that \(( 2 + \sqrt { 7 } ) ( 4 - \sqrt { 7 } ) = a + b \sqrt { 7 }\), where a and \(b\) are integers,
    (a) find the value of a and the value of \(b\).
Given that \(\frac { 2 + \sqrt { 7 } } { 4 + \sqrt { 7 } } = c + d \sqrt { 7 }\) where \(c\) and \(d\) are rational numbers,
(b) find the value of \(c\) and the value of \(d\).
Edexcel C1 Q3
3. (a) Solve the inequality \(3 x - 8 > x + 13\).
(b) Solve the inequality \(x ^ { 2 } - 5 x - 14 > 0\).
Edexcel C1 Q4
4. (a) Prove, by completing the square, that the roots of the equation \(x ^ { 2 } + 2 k x + c = 0\), where \(k\) and \(c\) are constants, are \(- k \pm \sqrt { } \left( k ^ { 2 } - c \right)\). The equation \(x ^ { 2 } + 2 k x \pm 81 = 0\) has equal roots.
(b) Find the possible values of \(k\).