Questions C1 (1562 questions)

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Edexcel C1 Q4
9 marks Moderate -0.8
  1. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0$$
  1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
  2. Using integration, find \(\mathrm { f } ( x )\).
Edexcel C1 Q5
10 marks Moderate -0.8
5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  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 the origin and has gradient - 4 .
  2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
  3. Find the exact coordinates of the mid-point of \(A C\).
Edexcel C1 Q6
14 marks Easy -1.2
6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$
  1. Write down the maximum value of \(\mathrm { f } ( x )\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
  3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
  4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
  5. Find the value of \(k\).
Edexcel C1 Q7
7 marks Moderate -0.8
7. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 Q8
5 marks Moderate -0.8
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
7 marks Easy -1.2
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
6 marks Easy -1.2
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
6 marks Easy -1.2
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
7 marks Moderate -0.8
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
7 marks Moderate -0.3
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
5 marks Moderate -0.8
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
8 marks Moderate -0.5
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
11 marks Moderate -0.8
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
12 marks Standard +0.3
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
5 marks Easy -1.2
  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. \(f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0\).
    4. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
    5. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
    6. The sum of an arithmetic series is \(\sum _ { r = 1 } ^ { n } ( 80 - 3 r )\).
    7. Write down the first two terms of the series.
    8. Find the common difference of the series.
    Given that \(n = 50\),
  2. find the sum of the series.
Edexcel C1 Q4
7 marks Moderate -0.5
4. Find the set of values for \(x\) for which
  1. \(6 x - 7 < 2 x + 3\),
  2. \(\quad 2 x ^ { 2 } - 11 x + 5 < 0\),
  3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
Edexcel C1 Q5
7 marks Moderate -0.8
5. The equation \(x ^ { 2 } + 5 k x + 2 k = 0\), where \(k\) is a constant, has real roots.
  1. Prove that \(k ( 25 k - 8 ) \geq 0\).
  2. Hence find the set of possible values of \(k\).
  3. Write down the values of \(k\) for which the equation \(x ^ { 2 } + 5 k x + 2 k = 0\) has equal roots.
Edexcel C1 Q6
8 marks Moderate -0.3
6. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation \(\quad u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000\).
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 Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{813612f1-92c8-456d-84a2-aa6bb91b8a6a-3_689_1077_927_484}
\end{figure} Fig. 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.
Edexcel C1 Q8
12 marks Standard +0.3
8. 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 Q9
14 marks Standard +0.3
9. 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 \(Q\).
Edexcel C1 Q1
7 marks Moderate -0.8
  1. (a) Solve the equation \(4 x ^ { 2 } + 12 x = 0\).
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 2 } + 12 x + c\), 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 Q2
7 marks Standard +0.3
2. A sequence is defined by the recurrence relation \(u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , n = 1,2,3 , \ldots\), where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
  2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
    1. calculate the value of \(a\),
    2. write down the value of \(u _ { 5 }\).
Edexcel C1 Q3
7 marks Moderate -0.8
3. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    [0pt] [P1 June 2003 Question 8*]
Edexcel C1 Q4
8 marks Moderate -0.8
4. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,
  1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
  2. form a quadratic inequality in \(x\).
  3. by solving your inequalities, find the set of possible values of \(x\).