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 Q2
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
Edexcel C1 Q3
  1. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant.
Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$
  1. find the value of \(a\),
  2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).
Edexcel C1 Q4
4. (a) Find in exact form the coordinates of the points where the curve \(y = x ^ { 2 } - 4 x + 2\) crosses the \(x\)-axis.
(b) Find the value of the constant \(k\) for which the straight line \(y = 2 x + k\) is a tangent to the curve \(y = x ^ { 2 } - 4 x + 2\).
Edexcel C1 Q5
5. The curve \(C\) with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).
  1. Sketch the curve \(C\), showing the coordinates of \(A\) and \(B\).
  2. Show that the tangent to \(C\) at \(A\) has the equation $$x + y = 2 .$$
Edexcel C1 Q6
6. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$
  1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
  2. State the maximum value of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
  4. Sketch the curve \(y = \mathrm { f } ( x )\).
Edexcel C1 Q7
7. (a) An arithmetic series has a common difference of 7 . Given that the sum of the first 20 terms of the series is 530 , find
  1. the first term of the series,
  2. the smallest positive term of the series.
    (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$ where \(k\) is a positive constant.
    Given that \(u _ { 2 } = 2 u _ { 1 }\),
  3. find the value of \(k\),
  4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).
Edexcel C1 Q8
8. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
  2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
  3. show that triangle \(A B C\) is isosceles.
Edexcel C1 Q9
9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).
  1. State the gradient of \(C\) at \(P\).
  2. Find the \(x\)-coordinate of \(P\).
  3. Find an equation for \(C\).
  4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.
Edexcel C1 Q1
  1. Evaluate
$$\sum _ { r = 1 } ^ { 30 } ( 3 r + 4 ) .$$
Edexcel C1 Q2
  1. (a) Express \(x ^ { 2 } + 6 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
    (b) State the coordinates of the minimum point of the curve \(y = x ^ { 2 } + 6 x + 7\).
  2. The straight line \(l _ { 1 }\) has the equation \(3 x - y = 0\).
The straight line \(l _ { 2 }\) has the equation \(x + 2 y - 4 = 0\).
(a) Sketch \(l _ { 1 }\) and \(l _ { 2 }\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes.
(b) Find, as exact fractions, the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
Edexcel C1 Q4
4. Find the pairs of values \(( x , y )\) which satisfy the simultaneous equations $$\begin{aligned} & 3 x ^ { 2 } + y ^ { 2 } = 21
& 5 x + y = 7 \end{aligned}$$
Edexcel C1 Q5
  1. (a) Sketch on the same diagram the graphs of \(y = ( x - 1 ) ^ { 2 } ( x - 5 )\) and \(y = 8 - 2 x\).
Label on your diagram the coordinates of any points where each graph meets the coordinate axes.
(b) Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$( x - 1 ) ^ { 2 } ( x - 5 ) = 8 - 2 x$$ (c) State the integer, \(n\), such that $$n < \alpha < n + 1 .$$
Edexcel C1 Q6
  1. The curve with equation \(y = x ^ { 2 } + 2 x\) passes through the origin, \(O\).
    1. Find an equation for the normal to the curve at \(O\).
    2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again.
    3. Given that
    $$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$
  2. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  3. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} ^ { 2 } }\),
  4. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0$$
Edexcel C1 Q8
  1. (a) Prove that the sum of the first \(n\) positive integers is given by
$$\frac { 1 } { 2 } n ( n + 1 ) .$$ (b) Hence, find the sum of
  1. the integers from 100 to 200 inclusive,
  2. the integers between 300 to 600 inclusive which are divisible by 3 .
Edexcel C1 Q9
9. (a) Express each of the following in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are rational.
  1. \(( 4 - 3 \sqrt { 2 } ) ^ { 2 }\)
  2. \(\frac { 1 } { 2 + \sqrt { 2 } }\)
    (b) (i) Solve the equation $$y ^ { 2 } + 8 = 9 y .$$
  3. Hence solve the equation $$x ^ { 3 } + 8 = 9 x ^ { \frac { 3 } { 2 } } .$$
Edexcel C1 Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{32aa549b-e8b8-4ce5-927e-103f7e846f28-4_524_821_1078_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The curve meets the \(x\)-axis at the origin and at the point \(A\).
Given that $$f ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } } ,$$
  1. find \(\mathrm { f } ( x )\),
  2. find the coordinates of \(A\). The point \(B\) on the curve has \(x\)-coordinate 2 .
  3. Find an equation for the tangent to the curve at \(B\) in the form \(y = m x + c\).
Edexcel C1 Q1
  1. The \(n\)th term of a sequence is defined by
$$u _ { n } = n ^ { 2 } - 6 n + 11 , \quad n \geq 1 .$$ Given that the \(k\) th term of the sequence is 38 , find the value of \(k\).
Edexcel C1 Q2
2. Find $$\int \left( 4 x ^ { 2 } - \sqrt { x } \right) \mathrm { d } x$$
Edexcel C1 Q3
  1. Find the integer \(n\) such that
$$4 \sqrt { 12 } - \sqrt { 75 } = \sqrt { n }$$
Edexcel C1 Q4
  1. (a) Evaluate
$$\left( 36 ^ { \frac { 1 } { 2 } } + 16 ^ { \frac { 1 } { 4 } } \right) ^ { \frac { 1 } { 3 } }$$ (b) Solve the equation $$3 x ^ { - \frac { 1 } { 2 } } - 4 = 0 .$$
Edexcel C1 Q5
  1. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { x ^ { 2 } } , \quad x \neq 0 .$$
  1. Using integration, find \(\mathrm { f } ( x )\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\) and write down the equations of its asymptotes.
Edexcel C1 Q6
6. \(f ( x ) = x ^ { 2 } - 10 x + 17\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the coordinates of the minimum point of the curve \(y = \mathrm { f } ( x )\).
  3. Deduce the coordinates of the minimum point of each of the following curves:
    1. \(\quad y = \mathrm { f } ( x ) + 4\),
    2. \(y = \mathrm { f } ( 2 x )\).
Edexcel C1 Q7
7. Given that the equation $$4 x ^ { 2 } - k x + k - 3 = 0$$ where \(k\) is a constant, has real roots,
  1. show that $$k ^ { 2 } - 16 k + 48 \geq 0 ,$$
  2. find the set of possible values of \(k\),
  3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value.
Edexcel C1 Q8
8. (a) The first and third terms of an arithmetic series are 3 and 27 respectively.
  1. Find the common difference of the series.
  2. Find the sum of the first 11 terms of the series.
    (b) Find the sum of the integers between 50 and 150 which are divisible by 8 .
Edexcel C1 Q9
9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).