Questions C1 (1442 questions)

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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 \(k \sqrt { 2 }\).
Edexcel C1 Q2
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
Edexcel C1 Q3
  1. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant.
Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$
  1. find the value of \(a\),
  2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).
Edexcel C1 Q4
4. (a) Find in exact form the coordinates of the points where the curve \(y = x ^ { 2 } - 4 x + 2\) crosses the \(x\)-axis.
(b) Find the value of the constant \(k\) for which the straight line \(y = 2 x + k\) is a tangent to the curve \(y = x ^ { 2 } - 4 x + 2\).
Edexcel C1 Q5
5. The curve \(C\) with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).
  1. Sketch the curve \(C\), showing the coordinates of \(A\) and \(B\).
  2. Show that the tangent to \(C\) at \(A\) has the equation $$x + y = 2 .$$
Edexcel C1 Q6
6. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$
  1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
  2. State the maximum value of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
  4. Sketch the curve \(y = \mathrm { f } ( x )\).
Edexcel C1 Q7
7. (a) An arithmetic series has a common difference of 7 . Given that the sum of the first 20 terms of the series is 530 , find
  1. the first term of the series,
  2. the smallest positive term of the series.
    (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$ where \(k\) is a positive constant.
    Given that \(u _ { 2 } = 2 u _ { 1 }\),
  3. find the value of \(k\),
  4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).
Edexcel C1 Q8
8. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).
  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 \(B\) and is perpendicular to \(l _ { 1 }\).
  2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
  3. show that triangle \(A B C\) is isosceles.
Edexcel C1 Q9
9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).
  1. State the gradient of \(C\) at \(P\).
  2. Find the \(x\)-coordinate of \(P\).
  3. Find an equation for \(C\).
  4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.
Edexcel C1 Q1
  1. Evaluate
$$\sum _ { r = 1 } ^ { 30 } ( 3 r + 4 ) .$$
Edexcel C1 Q2
  1. (a) Express \(x ^ { 2 } + 6 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
    (b) State the coordinates of the minimum point of the curve \(y = x ^ { 2 } + 6 x + 7\).
  2. The straight line \(l _ { 1 }\) has the equation \(3 x - y = 0\).
The straight line \(l _ { 2 }\) has the equation \(x + 2 y - 4 = 0\).
(a) Sketch \(l _ { 1 }\) and \(l _ { 2 }\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes.
(b) Find, as exact fractions, the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.