Questions — Edexcel C1 (574 questions)

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Edexcel C1 Q10
13 marks Standard +0.3
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
4 marks Easy -1.2
  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
4 marks Easy -1.8
2. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).
Edexcel C1 Q3
6 marks Moderate -0.8
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
6 marks Easy -1.8
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
6 marks Moderate -0.3
  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
7 marks Easy -1.2
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
8 marks Easy -1.2
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
11 marks Moderate -0.8
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
11 marks Moderate -0.8
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.
Edexcel C1 Q10
12 marks Moderate -0.3
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0 ,$$ and that the point \(A\) on \(C\) has coordinates (2, 6),
  1. find an equation for \(C\),
  2. find an equation for the tangent to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers,
  3. show that the line \(y = x + 3\) is also a tangent to \(C\).
Edexcel C1 2006 January Q6
9 marks Moderate -0.8
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = 2 \mathrm { f } ( x )\),
  3. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). On each diagram show clearly the coordinates of all the points where the curve meets the axes.
Edexcel C1 Q1
Easy -1.3
  1. (a) Write down the value of \(8 ^ { \frac { 1 } { 3 } }\).
    (b) Find the value of \(8 ^ { - \frac { 2 } { 3 } }\).
  2. Given that \(y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0\),
    (a) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
    (b) find \(\int y \mathrm {~d} x\).
Edexcel C1 Q2
Moderate -0.8
2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1$$
  1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Write down the value of \(u _ { 20 }\).
Edexcel C1 Q4
Moderate -0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-006_689_920_292_511}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the origin \(O\) and through the point \(( 6,0 )\). The maximum point on the curve is \(( 3,5 )\). On separate diagrams, sketch the curve with equation
  1. \(y = 3 \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( x + 2 )\). On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.
Edexcel C1 Q5
Moderate -0.5
5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$
Edexcel C1 Q8
Moderate -0.8
8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
  2. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
  3. calculate the exact area of \(\triangle O C P\). \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}
Edexcel C1 Q9
Moderate -0.8
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 $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence. He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
  2. Find the amount Sean repays in the 21st month. Over the \(n\) months, he repays a total of \(\pounds 5000\).
  3. Form an equation in \(n\), and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
  4. Solve the equation in part (c).
  5. State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
Edexcel C1 Q1
3 marks Easy -1.8
  1. Write down the value of \(16^{-1}\). [1]
  2. Find the value of \(16^{-\frac{1}{2}}\). [2]
Edexcel C1 Q2
8 marks Easy -1.8
  1. Given that \(y = 5x^3 + 7x + 3\), find
    1. \(\frac{dy}{dx}\), [3]
    2. \(\frac{d^2y}{dx^2}\). [1]
  2. Find \(\int \left(1 + 3\sqrt{x} - \frac{1}{x^2}\right) dx\). [4]
Edexcel C1 Q3
4 marks Moderate -0.8
Given that the equation \(kx^2 + 12x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\). [4]
Edexcel C1 Q4
6 marks Moderate -0.3
Solve the simultaneous equations $$x + y = 2$$ $$x^2 + 2y = 12.$$ [6]
Edexcel C1 Q5
6 marks Easy -1.2
The \(r\)th term of an arithmetic series is \((2r - 5)\).
  1. Write down the first three terms of this series. [2]
  2. State the value of the common difference. [1]
  3. Show that \(\sum_{r=1}^n (2r - 5) = n(n - 4)\). [3]
Edexcel C1 Q6
6 marks Moderate -0.8
\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve crosses the \(x\)-axis at the points \((2, 0)\) and \((4, 0)\). The minimum point on the curve is \(P(3, -2)\). In separate diagrams sketch the curve with equation
  1. \(y = -f(x)\), [3]
  2. \(y = f(2x)\). [3]
On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.
Edexcel C1 Q7
10 marks Moderate -0.8
The curve \(C\) has equation \(y = 4x^2 + \frac{5-x}{x}\), \(x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate \(1\).
  1. Show that the value of \(\frac{dy}{dx}\) at \(P\) is \(3\). [5]
  2. Find an equation of the tangent to \(C\) at \(P\). [3]
This tangent meets the \(x\)-axis at the point \((k, 0)\).
  1. Find the value of \(k\). [2]