Questions — Edexcel C1 (490 questions)

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Edexcel C1 Q3
  1. A sequence is defined by the recurrence relation
$$u _ { n + 1 } = u _ { n } - 2 , \quad n > 0 , \quad u _ { 1 } = 50 .$$
  1. Write down the first four terms of the sequence.
  2. Evaluate $$\sum _ { r = 1 } ^ { 20 } u _ { r } .$$
Edexcel C1 Q4
  1. (a) Find the value of the constant \(k\) such that the equation
$$x ^ { 2 } - 6 x + k = 0$$ has equal roots.
(b) Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0$$
Edexcel C1 Q5
  1. Solve the simultaneous equations
$$\begin{aligned} & x + y = 2
& 3 x ^ { 2 } - 2 x + y ^ { 2 } = 2 \end{aligned}$$
Edexcel C1 Q6
  1. Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { x } - x ^ { 2 }$$ and that \(y = \frac { 2 } { 3 }\) when \(x = 1\), find the value of \(y\) when \(x = 4\).
Edexcel C1 Q7
7. The first three terms of an arithmetic series are \(( 12 - p ) , 2 p\) and \(( 4 p - 5 )\) respectively, where \(p\) is a constant.
  1. Find the value of \(p\).
  2. Show that the sixth term of the series is 50 .
  3. Find the sum of the first 15 terms of the series.
  4. Find how many terms of the series have a value of less than 400.
Edexcel C1 Q8
8. $$f ( x ) = 2 x ^ { 2 } + 3 x - 2$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
  4. Find the values of \(a\), \(b\) and \(c\).
Edexcel C1 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a976970c-32a7-4808-9c82-4b71a539c875-4_611_828_251_392} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) with the equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 4 x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). The line \(l\) is the tangent to \(C\) at \(O\).
  2. Find an equation for \(l\).
  3. Find the coordinates of the point where \(l\) intersects \(C\) again.
Edexcel C1 Q10
10. The straight line \(l _ { 1 }\) has equation \(2 x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). The straight line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the point \(B ( - 6,6 )\).
  2. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\). The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(C\).
  3. Find the coordinates of \(C\). The point \(D\) lies on \(l _ { 1 }\) and is such that \(C D\) is perpendicular to \(l _ { 1 }\).
  4. Show that \(D\) has coordinates \(( 5,4 )\).
  5. Find the area of triangle \(A C D\).
Edexcel C1 Q1
  1. (a) Express \(\frac { 18 } { \sqrt { 3 } }\) in the form \(k \sqrt { 3 }\).
    (b) Express \(( 1 - \sqrt { 3 } ) ( 4 - 2 \sqrt { 3 } )\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
  2. Solve the equation
$$3 x - \frac { 5 } { x } = 2 .$$
Edexcel C1 Q3
  1. The straight line \(l\) has the equation \(x - 5 y = 7\).
The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).
Edexcel C1 Q4
4. A sequence of terms is defined by $$u _ { n } = 3 ^ { n } - 2 , \quad n \geq 1$$
  1. Write down the first four terms of the sequence. The same sequence can also be defined by the recurrence relation $$u _ { n + 1 } = a u _ { n } + b , \quad n \geq 1 , \quad u _ { 1 } = 1 ,$$ where \(a\) and \(b\) are constants.
  2. Find the values of \(a\) and \(b\).
Edexcel C1 Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91b8965d-5003-4a64-b863-fb6af956abd3-3_534_686_248_475} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 8 x - x ^ { \frac { 5 } { 2 } } , x \geq 0\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).
  1. Find the \(x\)-coordinate of \(A\).
  2. Find the gradient of the tangent to the curve at \(A\).
Edexcel C1 Q6
6. $$f ( x ) = 2 x ^ { 2 } - 4 x + 1$$
  1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\mathrm { f } ( x ) = a ( x + b ) ^ { 2 } + c .$$
  2. State the equation of the line of symmetry of the curve \(y = \mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 3\), giving your answers in exact form.
Edexcel C1 Q7
7. \(\quad \mathrm { f } ( x ) \equiv \frac { ( x - 4 ) ^ { 2 } } { 2 x ^ { \frac { 1 } { 2 } } } , x > 0\).
  1. Find the values of the constants \(A , B\) and \(C\) such that $$f ( x ) = A x ^ { \frac { 3 } { 2 } } + B x ^ { \frac { 1 } { 2 } } + C x ^ { - \frac { 1 } { 2 } } .$$
  2. Show that $$f ^ { \prime } ( x ) = \frac { ( 3 x + 4 ) ( x - 4 ) } { 4 x ^ { \frac { 3 } { 2 } } }$$
Edexcel C1 Q8
  1. (a) Describe fully the single transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( x - 1 )\).
    (b) Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac { 1 } { x - 1 }\).
    (c) Find the \(x\)-coordinates of any points where the graph of \(y = \frac { 1 } { x - 1 }\) intersects the graph of \(y = 2 + \frac { 1 } { x }\). Give your answers in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are rational.
  2. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month.
In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, so that sales of \(\pounds ( 1500 - x )\) and \(\pounds ( 1500 - 2 x )\) will be achieved in the second and third months respectively. Given that sales total \(\pounds 8100\) during the first six months, use the model to
Edexcel C1 Q10
10. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 4 x + k$$ where \(k\) is a constant. Given that \(C\) passes through the points \(( 0 , - 2 )\) and \(( 2,18 )\),
  1. show that \(k = 2\) and find an equation for \(C\),
  2. show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact.
Edexcel C1 Q1
  1. Find in exact form the real solutions of the equation
$$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$
Edexcel C1 Q3
3. (a) Solve the equation $$x ^ { \frac { 3 } { 2 } } = 27 .$$ (b) Express \(\left( 2 \frac { 1 } { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an exact fraction in its simplest form.
Edexcel C1 Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7cb02f85-96e6-42dd-908d-77973073b683-2_526_919_1297_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). Show that \(a = - 5\) and find the values of \(b\) and \(c\).
Edexcel C1 Q5
5. Given that $$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } }$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).
Edexcel C1 Q6
6. (a) Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
(b) Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$
Edexcel C1 Q7
  1. A curve has the equation \(y = \frac { x } { 2 } + 3 - \frac { 1 } { x } , x \neq 0\).
The point \(A\) on the curve has \(x\)-coordinate 2 .
  1. Find the gradient of the curve at \(A\).
  2. Show that the tangent to the curve at \(A\) has equation $$3 x - 4 y + 8 = 0$$ The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).
  3. Find the coordinates of \(B\).
Edexcel C1 Q8
8. The straight line \(l _ { 1 }\) has gradient \(\frac { 3 } { 2 }\) and passes through the point \(A ( 5,3 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) has the equation \(3 x - 4 y + 3 = 0\) and intersects \(l _ { 1 }\) at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find the coordinates of the mid-point of \(A B\).
  4. Show that the straight line parallel to \(l _ { 2 }\) which passes through the mid-point of \(A B\) also passes through the origin.
Edexcel C1 Q9
9. The third term of an arithmetic series is \(5 \frac { 1 } { 2 }\). The sum of the first four terms of the series is \(22 \frac { 3 } { 4 }\).
  1. Show that the first term of the series is \(6 \frac { 1 } { 4 }\) and find the common difference.
  2. Find the number of positive terms in the series.
  3. Hence, find the greatest value of the sum of the first \(n\) terms of the series.
Edexcel C1 Q10
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$ and that the point \(P ( 1,1 )\) lies on \(C\),
  1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = m x + c\),
  2. find an equation for \(C\),
  3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k \sqrt { 2 }\).