Questions C1 (1562 questions)

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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.
AQA C1 2014 June Q7
14 marks Moderate -0.5
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    1. Write down the coordinates of \(C\).
    2. Show that the circle has radius \(n \sqrt { 5 }\), where \(n\) is an integer.
  3. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(x + p y = q\), where \(p\) and \(q\) are integers.
  4. The point \(B\) lies on the tangent to the circle at \(A\) and the length of \(B C\) is 6. Find the length of \(A B\).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}
AQA C1 2015 June Q4
11 marks Standard +0.3
  1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
  3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
  4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
    [0pt] [2 marks]
OCR C1 2007 January Q9
12 marks Moderate -0.3
  1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2009 June Q9
8 marks Moderate -0.8
  1. Calculate the length of \(A B\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the equation of the line through \(( 1,3 )\) which is parallel to \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2009 January Q11
14 marks Moderate -0.3
  1. Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
  2. Find the coordinates of the points of intersection of this circle with the \(y\)-axis. Give your answer in the form \(a \pm \sqrt { b }\).
  3. Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.
OCR MEI C1 Q10
12 marks Moderate -0.3
  1. Write down the equations of the circles A and B .
  2. Find the \(x\) coordinates of the points where the two curves intersect.
  3. Find the \(y\) coordinates of these points, giving your answers in surd form.
OCR C1 Q9
10 marks Standard +0.3
  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 \(O P = O R\).
OCR MEI C1 2007 January Q11
12 marks Moderate -0.3
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 2009 January Q13
11 marks Moderate -0.3
13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q4
12 marks Moderate -0.8
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\) (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
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.
AQA C1 2007 January Q1
11 marks Moderate -0.8
1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.
    1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
  3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)
AQA C1 2007 January Q2
11 marks Moderate -0.3
2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).
    1. Find the gradient of \(A B\).
    2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
  2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
  3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).
AQA C1 2007 January Q3
8 marks Moderate -0.8
3
  1. Express \(\frac { \sqrt { 5 } + 3 } { \sqrt { 5 } - 2 }\) in the form \(p \sqrt { 5 } + q\), where \(p\) and \(q\) are integers.
    1. Express \(\sqrt { 45 }\) in the form \(n \sqrt { 5 }\), where \(n\) is an integer.
    2. Solve the equation $$x \sqrt { 20 } = 7 \sqrt { 5 } - \sqrt { 45 }$$ giving your answer in its simplest form.
AQA C1 2007 January Q4
14 marks Moderate -0.8
4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0\).
  1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
  3. Show that the circle does not intersect the \(x\)-axis.
  4. The line with equation \(x + y = 4\) intersects the circle at the points \(P\) and \(Q\).
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } + 3 x - 10 = 0$$
    2. Given that \(P\) has coordinates (2,2), find the coordinates of \(Q\).
    3. Hence find the coordinates of the midpoint of \(P Q\).
AQA C1 2007 January Q5
10 marks Moderate -0.5
5 The diagram shows an open-topped water tank with a horizontal rectangular base and four vertical faces. The base has width \(x\) metres and length \(2 x\) metres, and the height of the tank is \(h\) metres. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-4_403_410_477_792} The combined internal surface area of the base and four vertical faces is \(54 \mathrm {~m} ^ { 2 }\).
    1. Show that \(x ^ { 2 } + 3 x h = 27\).
    2. Hence express \(h\) in terms of \(x\).
    3. Hence show that the volume of water, \(V \mathrm {~m} ^ { 3 }\), that the tank can hold when full is given by $$V = 18 x - \frac { 2 x ^ { 3 } } { 3 }$$
    1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
    2. Verify that \(V\) has a stationary value when \(x = 3\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 3\).
    (2 marks)
AQA C1 2007 January Q6
14 marks Moderate -0.8
6 The curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-5_428_563_372_740} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and cuts the \(y\)-axis at the point \(B\).
    1. State the coordinates of the point \(B\) and hence find the area of the triangle \(A O B\), where \(O\) is the origin.
    2. Find \(\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x\).
    3. Hence find the area of the shaded region bounded by the curve and the line \(A B\).
    1. Find the gradient of the curve with equation \(y = 3 x ^ { 5 } + 2 x + 5\) at the point \(A ( - 1,0 )\).
    2. Hence find an equation of the tangent to the curve at the point \(A\).
AQA C1 2007 January Q7
7 marks Moderate -0.3
7 The quadratic equation \(( k + 1 ) x ^ { 2 } + 12 x + ( k - 4 ) = 0\) has real roots.
  1. Show that \(k ^ { 2 } - 3 k - 40 \leqslant 0\).
  2. Hence find the possible values of \(k\).