Questions — Edexcel (10514 questions)

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Edexcel C1 Q5
6 marks Moderate -0.3
  1. A sequence of terms \(\left\{ t _ { n } \right\}\) is defined for \(n \geq 1\) by the recurrence relation
$$t _ { n + 1 } = k t _ { n } - 7 , \quad t _ { 1 } = 3$$ where \(k\) is a constant.
  1. Find expressions for \(t _ { 2 }\) and \(t _ { 3 }\) in terms of \(k\). Given that \(t _ { 3 } = 13\),
  2. find the possible values of \(k\).
Edexcel C1 Q6
7 marks Easy -1.2
6. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2. Find an equation for the tangent to the curve at \(A\).
Edexcel C1 Q7
8 marks Easy -1.2
7. As part of a new training programme, Habib decides to do sit-ups every day. He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
Edexcel C1 Q8
11 marks Moderate -0.8
8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),
  1. find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
  2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
  3. find the rate at which the area of the stain is increasing when \(t = 15\).
Edexcel C1 Q9
11 marks Moderate -0.8
9. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).
  1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
  2. Sketch \(l\) and \(C\) on the same set of axes.
  3. Find the coordinates of the points where \(I\) and \(C\) intersect.
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 C2 Q1
6 marks Moderate -0.8
  1. Given that \(p = \log _ { q } 16\), express in terms of \(p\),
    1. \(\log _ { q } 2\),
    2. \(\log _ { q } ( 8 q )\).
      [0pt] [P2 January 2002 Question 2]
    3. \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c\), where \(c\) is a constant.
    Given that \(\mathrm { f } ( 4 ) = 0\),
    1. find the value of \(c\),
    2. factorise \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
    3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(\mathrm { f } ( x ) = 0\).
Edexcel C2 Q3
8 marks Moderate -0.3
3. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]
Edexcel C2 Q4
9 marks Challenging +1.2
4. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by \(P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }\), where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,
  1. calculate, to 4 decimal places, the value of \(a\),
  2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
  3. With reference to this model, give a reason why the population of deer cannot exceed 2000.
Edexcel C2 Q5
11 marks Standard +0.3
5.
  1. Given that \(( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = A + B x ^ { 2 } + C x ^ { 4 }\), find the values of the constants \(A , B\) and \(C\).
  2. Using the substitution \(y = x ^ { 2 }\) and your answers to part (a), solve, $$( 2 + x ) ^ { 5 } + ( 2 - x ) ^ { 5 } = 349$$
Edexcel C2 Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1033051d-18bf-4734-a556-4c8e1c789992-3_842_963_280_392}
\end{figure} Fig. 1 shows a gardener's design for the shape of a flower bed with perimeter \(A B C D . A D\) is an arc of a circle with centre \(O\) and radius \(5 \mathrm {~m} . B C\) is an arc of a circle with centre \(O\) and radius \(7 \mathrm {~m} . O A B\) and \(O D C\) are straight lines and the size of \(\angle A O D\) is \(\theta\) radians.
  1. Find, in terms of \(\theta\), an expression for the area of the flower bed. Given that the area of the flower bed is \(15 \mathrm {~m} ^ { 2 }\),
  2. show that \(\theta = 1.25\),
  3. calculate, in m , the perimeter of the flower bed. The gardener now decides to replace arc \(A D\) with the straight line \(A D\).
  4. Find, to the nearest cm , the reduction in the perimeter of the flower bed.
Edexcel C2 Q7
12 marks Moderate -0.8
7. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The second and fourth terms of the series are 3 and 1.08 respectively.
    Given that all terms in the series are positive, find
  2. the value of \(r\) and the value of \(a\),
  3. the sum to infinity of the series. \includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299} Fig. 2 shows part of the curve with equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\). The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).
    1. Show that the equation of the curve may be written as \(y = x ( x - 3 ) ^ { 2 }\), and hence write down the coordinates of \(A\).
    2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
    3. Find the area of \(R\).
Edexcel C2 Q1
6 marks Moderate -0.3
  1. Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \(( 1 + 3 x ) ^ { n }\), where \(n > 2\). Given that the coefficient of \(x ^ { 3 }\) in this expansion is ten times the coefficient of \(x ^ { 2 }\),
  2. find the value of \(n\),
  3. find the coefficient of \(x ^ { 4 }\) in the expansion.
Edexcel C2 Q2
7 marks Moderate -0.5
2. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), the remainder is 14 . When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .
  1. Find the value of \(a\) and the value of \(b\).
  2. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
    [0pt] [P3 June 2002 Question 1]
Edexcel C2 Q3
8 marks Moderate -0.8
3. 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 \(y = \mathrm { f } ( x )\).
  3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
    [0pt] [P1 June 2002 Question 3]
Edexcel C2 Q4
10 marks Moderate -0.8
4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$
  1. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\) axis.
  2. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
  3. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\) [0pt] [P1 June 2002 Question 5]
Edexcel C2 Q5
10 marks Standard +0.3
5.
  1. Given that \(3 + 2 \log _ { 2 } x = \log _ { 2 } y\), show that \(y = 8 x ^ { 2 }\).
  2. Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$
  3. Show that \(\log _ { 2 } \alpha = - 2\).
  4. Calculate \(\log _ { 2 } \beta\), giving your answer to 3 significant figures.
Edexcel C2 Q6
11 marks Standard +0.3
6. Given that \(\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0\),
  1. find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 5\).
  2. Show that \(\mathrm { f } ( x )\) may be written in the form \(A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C\), where \(A , B\) and \(C\) are constants to be found.
  3. Hence evaluate \(\int _ { 1 } ^ { 2 } f ( x ) d x\).
Edexcel C2 Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ba5cb933-dedd-4ad9-9e66-49636870b3de-3_739_1272_826_328}
\end{figure} Fig. 1 shows the cross-section \(A B C D\) of a chocolate bar, where \(A B , C D\) and \(A D\) are straight lines and \(M\) is the mid-point of \(A D\). The length \(A D\) is 28 mm , and \(B C\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B , C\) and \(D\) have coordinates (7,24), (21,24) and (28,0) respectively.
  1. Show that the length of \(B M\) is 25 mm .
  2. Show that, to 3 significant figures, \(\angle B M C = 0.568\) radians.
  3. Hence calculate, in \(\mathrm { mm } ^ { 2 }\), the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
  4. calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the volume of the bar.
Edexcel C2 Q8
13 marks Moderate -0.8
8.
  1. An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
  2. Find the value of \(d\). Using your value of \(d\),
  3. find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be \(\pounds 54000\),
  4. find the predicted profit for the year 2011.
Edexcel C2 Q2
7 marks Moderate -0.3
2.
  1. Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
  2. Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
Edexcel C2 Q3
7 marks Easy -1.2
3. Every \(\pounds 1\) of money invested in a savings scheme continuously gains interest at a rate of \(4 \%\) per year. Hence, after \(x\) years, the total value of an initial \(\pounds 1\) investment is \(\pounds y\), where $$y = 1.04 ^ { x }$$
  1. Sketch the graph of \(y = 1.04 ^ { x } , x \geq 0\).
  2. Calculate, to the nearest \(\pounds\), the total value of an initial \(\pounds 800\) investment after 10 years.
  3. Use logarithms to find the number of years it takes to double the total value of any initial investment.
Edexcel C2 Q4
7 marks Moderate -0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-2_554_561_1635_762}
\end{figure} Fig. 1 shows the sector \(A O B\) of a circle, with centre \(O\) and radius 6.5 cm , and \(\angle A O B = 0.8\) radians.
  1. Calculate, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(A O B\).
  2. Show that the length of the chord \(A B\) is 5.06 cm , to 3 significant figures. The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
  3. Calculate, in cm , the perimeter of \(R\).
Edexcel C2 Q5
8 marks Moderate -0.3
5.
  1. Write down the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 1 + a x ) ^ { n } , n > 2 .$$ Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 ,
  2. calculate the value of \(n\) and the value of \(a\),
  3. find the coefficient of \(x ^ { 3 }\).
    [0pt] [P2 November 2003 Question 3]
Edexcel C2 Q6
12 marks Standard +0.3
6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.
  1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
  2. Find the value of \(r\) for which \(A\) is a minimum.
  3. Prove that this value of \(r\) gives a minimum value of \(A\).
  4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.