Questions — Edexcel (9685 questions)

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Edexcel C1 Q9
12 marks Moderate -0.8
9. (a) Prove that the sum of the first \(n\) terms of an arithmetic series with first term \(a\) and common difference \(d\) is given by $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A novelist begins writing a new book. She plans to write 16 pages during the first week, 18 during the second and so on, with the number of pages increasing by 2 each week. Find, according to her plan,
(b) how many pages she will write in the fifth week,
(c) the total number of pages she will write in the first five weeks.
(d) Using algebra, find how long it will take her to write the book if it has 250 pages.
Edexcel C1 Q10
14 marks Moderate -0.3
10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x + 2 ) ^ { 3 }$$
  1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
  2. Find f \({ } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
  3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
  4. Show that \(m\) has the equation \(y = 3 x + 8\).
Edexcel C1 Q1
3 marks Easy -1.2
  1. Express \(\sqrt { 50 } + 3 \sqrt { 8 }\) in the form \(k \sqrt { 2 }\).
  2. Differentiate with respect to \(x\)
$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$
Edexcel C1 Q3
4 marks Moderate -0.8
  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
6 marks Moderate -0.8
  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
7 marks Moderate -0.3
  1. Solve the simultaneous equations
$$\begin{aligned} & x + y = 2 \\ & 3 x ^ { 2 } - 2 x + y ^ { 2 } = 2 \end{aligned}$$
Edexcel C1 Q6
7 marks Moderate -0.8
  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
10 marks Moderate -0.3
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
10 marks Moderate -0.3
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
11 marks Moderate -0.3
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
13 marks Moderate -0.3
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
4 marks Easy -1.3
  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
5 marks Moderate -0.8
  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
6 marks Moderate -0.8
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
7 marks Moderate -0.8
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
8 marks Moderate -0.8
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
9 marks Moderate -0.3
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
10 marks Moderate -0.8
  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
12 marks Standard +0.3
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
3 marks Moderate -0.3
  1. Find in exact form the real solutions of the equation
$$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$
Edexcel C1 Q3
4 marks Easy -1.3
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
5 marks Standard +0.3
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
6 marks Moderate -0.8
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
8 marks Moderate -0.8
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
10 marks Moderate -0.3
  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 \(B\).