Questions C2 (1550 questions)

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AQA C2 2014 June Q8
11 marks Standard +0.3
8 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 5 terms of the series is 575 .
  1. Show that \(a + 2 d = 115\).
  2. Given also that the 10th term of the series is 87, find the value of \(d\).
  3. The \(n\)th term of the series is \(u _ { n }\). Given that \(u _ { k } > 0\) and \(u _ { k + 1 } < 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
    [0pt] [5 marks]
AQA C2 2014 June Q9
15 marks Moderate -0.3
9 A curve has equation \(y = 3 \times 12 ^ { x }\).
  1. The point ( \(k , 6\) ) lies on the curve \(y = 3 \times 12 ^ { x }\). Use logarithms to find the value of \(k\), giving your answer to three significant figures.
  2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x\), giving your answer to two significant figures.
  3. The curve \(y = 3 \times 12 ^ { x }\) is translated by the vector \(\left[ \begin{array} { l } 1 \\ p \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the curve \(y = \mathrm { f } ( x )\) passes through the origin ( 0,0 ), find the value of the constant \(p\).
  4. The curve with equation \(y = 2 ^ { 2 - x }\) intersects the curve \(y = 3 \times 12 ^ { x }\) at the point \(T\). Show that the \(x\)-coordinate of \(T\) can be written in the form \(\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }\), where \(q\) is an integer. State the value of \(q\).
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-20_2288_1707_221_153}
AQA C2 2015 June Q1
4 marks Moderate -0.8
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm . \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-02_378_451_648_790} The angle \(A O B\) is \(\theta\) radians and the area of the sector is \(15 \mathrm {~cm} ^ { 2 }\).
Find the perimeter of the sector.
[0pt] [4 marks]
AQA C2 2015 June Q2
7 marks Moderate -0.3
2 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(72 ^ { \circ }\) and the size of angle \(A B C\) is \(48 ^ { \circ }\). The length of \(B C\) is 20 cm .
  1. Show that the length of \(A C\) is 15.6 cm , correct to three significant figures.
  2. The midpoint of \(B C\) is \(M\). Calculate the length of \(A M\), giving your answer, in cm , to three significant figures.
    [0pt] [4 marks]
AQA C2 2015 June Q3
7 marks Moderate -0.8
3 The first term of an infinite geometric series is 48 . The common ratio of the series is 0.6 .
  1. Find the third term of the series.
  2. Find the sum to infinity of the series.
  3. The \(n\)th term of the series is \(u _ { n }\). Find the value of \(\sum _ { n = 4 } ^ { \infty } u _ { n }\).
AQA C2 2015 June Q4
10 marks Moderate -0.3
4 A curve is defined for \(x > 0\). The gradient of the curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x ^ { 2 } } - \frac { x } { 4 }$$
  1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. The curve has a stationary point \(M\) whose \(y\)-coordinate is \(\frac { 5 } { 2 }\).
    1. Find the \(x\)-coordinate of \(M\).
    2. Use your answers to parts (a) and (b)(i) to show that \(M\) is a maximum point.
    3. Find the equation of the curve.
AQA C2 2015 June Q5
6 marks Standard +0.3
5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by \(u _ { n + 1 } = p u _ { n } + q\), where \(p\) and \(q\) are constants.
The second term of the sequence is 160 . The third term of the sequence is 132 .
The limit of \(u _ { n }\) as \(n\) tends to infinity is 20 .
  1. Find the value of \(p\) and the value of \(q\).
  2. Hence find the value of the first term of the sequence.
AQA C2 2015 June Q6
10 marks Moderate -0.3
6
  1. Solve the equation \(\sin ( x + 0.7 ) = 0.6\) in the interval \(- \pi < x < \pi\), giving your answers in radians to two significant figures.
  2. It is given that \(5 \cos ^ { 2 } \theta - \cos \theta = \sin ^ { 2 } \theta\).
    1. By forming and solving a suitable quadratic equation, find the possible values of \(\cos \theta\).
    2. Hence show that a possible value of \(\tan \theta\) is \(2 \sqrt { 2 }\).
AQA C2 2015 June Q7
14 marks Moderate -0.3
7 The diagram shows a sketch of two curves. \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-14_448_527_370_762} The equations of the two curves are \(y = 1 + \sqrt { x }\) and \(y = 4 ^ { \frac { x } { 9 } }\).
The curves meet at the points \(P ( 0,1 )\) and \(Q ( 9,4 )\).
    1. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x }\) onto the graph of \(y = 1 + \sqrt { x }\).
    2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { \frac { x } { 9 } }\).
    1. Given that \(\int _ { 0 } ^ { 9 } \sqrt { x } \mathrm {~d} x = 18\), find the value of \(\int _ { 0 } ^ { 9 } ( 1 + \sqrt { x } ) \mathrm { d } x\).
    2. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 9 } 4 ^ { \frac { x } { 9 } } \mathrm {~d} x\). Give your answer to one decimal place.
    3. Hence find an approximate value for the area of the shaded region bounded by the two curves and state, with an explanation, whether your approximation will be an overestimate or an underestimate of the true value for the area of the shaded region.
      [0pt] [3 marks]
AQA C2 2015 June Q8
5 marks Moderate -0.3
8 The point \(A\) lies on the curve with equation \(y = x ^ { \frac { 1 } { 2 } }\). The tangent to this curve at \(A\) is parallel to the line \(3 y - 2 x = 1\). Find an equation of this tangent at \(A\).
[0pt] [5 marks]
AQA C2 2015 June Q9
14 marks Moderate -0.3
9
  1. Use logarithms to solve the equation \(2 ^ { 3 x } = 5\), giving your value of \(x\) to three significant figures.
  2. Given that \(\log _ { a } k - \log _ { a } 2 = \frac { 2 } { 3 }\), express \(a\) in terms of \(k\).
    1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 3 }\) in ascending powers of \(x\).
    2. It is given that $$\log _ { 2 } \left[ ( 1 + 2 n ) ^ { 3 } - 8 n \right] = \log _ { 2 } ( 1 + 2 n ) + \log _ { 2 } \left[ 4 \left( 1 + n ^ { 2 } \right) \right]$$ By forming and solving a suitable quadratic equation, find the possible values of \(n\). [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-20_1581_1714_1126_153} \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-24_2488_1728_219_141}
AQA C2 2016 June Q1
5 marks Moderate -0.8
1
  1. Find \(\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x\), where \(a\) is a constant.
  2. Hence, given that \(\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16\), find the value of the constant \(a\).
    [0pt] [2 marks]
AQA C2 2016 June Q2
5 marks Moderate -0.8
2
  1. Sketch the graph of \(y = ( 0.2 ) ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
  2. Use logarithms to solve the equation \(( 0.2 ) ^ { x } = 4\), giving your answer to three significant figures.
  3. Describe the geometrical transformation that maps the graph of \(y = ( 0.2 ) ^ { x }\) onto the graph of \(y = 5 ^ { x }\).
    [0pt] [1 mark]
AQA C2 2016 June Q3
11 marks Standard +0.3
3 The diagram shows a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-06_512_867_354_589} The curve is defined for \(x > 0\) by the equation $$y = 6 x ^ { \frac { 1 } { 2 } } - x - 3$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(y\)-coordinate of the maximum point \(M\).
  3. Find an equation of the normal to the curve at the point \(P ( 4,5 )\).
  4. It is given that the normal to the curve at \(P\), when translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), passes through the point \(M\). Find the value of the constant \(k\).
    [0pt] [3 marks]
AQA C2 2016 June Q4
10 marks Moderate -0.3
4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 21 terms is 168 .
  1. Show that \(a + 10 d = 8\).
  2. The sum of the second term and the third term is 50 . The \(n\)th term of the series is \(u _ { n }\).
    1. Find the value of \(u _ { 12 }\).
    2. Find the value of \(\sum _ { n = 4 } ^ { 21 } u _ { n }\).
AQA C2 2016 June Q5
8 marks Moderate -0.8
5
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 2 } ^ { 11 } \sqrt { x ^ { 2 } + 9 } \mathrm {~d} x\). Give your answer to one decimal place.
  2. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x ^ { 2 } + 9 }\) onto the graph of :
    1. \(y = 5 + \sqrt { x ^ { 2 } + 9 }\);
    2. \(y = 3 \sqrt { x ^ { 2 } + 1 }\).
AQA C2 2016 June Q6
11 marks Standard +0.3
6 The diagram shows a triangle \(A B C\). The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.
  1. Show that \(\theta = 0.586\), correct to three significant figures.
  2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
  3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-14_467_677_1462_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
    [0pt] [6 marks]
AQA C2 2016 June Q7
8 marks Standard +0.3
7
  1. The expression \(( 1 - 2 x ) ^ { 5 }\) can be written in the form $$1 + p x + q x ^ { 2 } + r x ^ { 3 } + 80 x ^ { 4 } - 32 x ^ { 5 }$$ By using the binomial expansion, or otherwise, find the values of the coefficients \(p , q\) and \(r\).
  2. Find the value of the coefficient of \(x ^ { 10 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 } ( 2 + x ) ^ { 7 }\).
    [0pt] [5 marks]
AQA C2 2016 June Q8
9 marks Standard +0.3
8
    1. Given that \(4 \sin x + 5 \cos x = 0\), find the value of \(\tan x\).
    2. Hence solve the equation \(( 1 - \tan x ) ( 4 \sin x + 5 \cos x ) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your values of \(x\) to the nearest degree.
  2. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
    [0pt] [4 marks]
AQA C2 2016 June Q9
8 marks Standard +0.3
9
  1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
  2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.
    [0pt] [4 marks]
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.