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 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 \(28 \frac { 1 } { 8 }\).
Edexcel C1 Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e8f154b-c232-49ee-a798-f61ff08ca0b9-4_663_1113_950_402} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the parallelogram \(A B C D\).
The points \(A\) and \(B\) have coordinates \(( - 1,3 )\) and \(( 3,4 )\) respectively and lie on the straight line \(l _ { 1 }\).
  1. Find an equation for \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The points \(C\) and \(D\) lie on the straight line \(l _ { 2 }\) which has the equation \(x - 4 y - 21 = 0\).
  2. Show that the distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(k \sqrt { 17 }\), where \(k\) is an integer to be found.
  3. Find the area of parallelogram \(A B C D\).
Edexcel C1 Q1
  1. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively.
Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\).
Give your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers.
Edexcel C1 Q2
2. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).
Edexcel C1 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01488c70-db95-43cb-9216-23d7dbaaf9fe-2_549_944_708_347} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum at \(( - 3,4 )\) and a minimum at \(( 1 , - 2 )\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations
  1. \(y = 2 \mathrm { f } ( x )\),
  2. \(y = - \mathrm { f } ( x )\).
Edexcel C1 Q4
4. (a) Solve the inequality $$4 ( x - 2 ) < 2 x + 5$$ (b) Find the value of \(y\) such that $$4 ^ { y + 1 } = 8 ^ { 2 y - 1 } .$$
Edexcel C1 Q5
  1. A sequence of terms \(\left\{ t _ { n } \right\}\) is defined for \(n \geq 1\) by the recurrence relation
$$t _ { n + 1 } = k t _ { n } - 7 , \quad t _ { 1 } = 3$$ where \(k\) is a constant.
  1. Find expressions for \(t _ { 2 }\) and \(t _ { 3 }\) in terms of \(k\). Given that \(t _ { 3 } = 13\),
  2. find the possible values of \(k\).
Edexcel C1 Q6
6. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2. Find an equation for the tangent to the curve at \(A\).
Edexcel C1 Q7
7. As part of a new training programme, Habib decides to do sit-ups every day. He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
Edexcel C1 Q8
8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),
  1. find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
  2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
  3. find the rate at which the area of the stain is increasing when \(t = 15\).
Edexcel C1 Q9
9. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).
  1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
  2. Sketch \(l\) and \(C\) on the same set of axes.
  3. Find the coordinates of the points where \(I\) and \(C\) intersect.