Questions — Edexcel C2 (476 questions)

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Edexcel C2 Q1
  1. Evaluate
$$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$
Edexcel C2 Q2
2. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
Edexcel C2 Q3
3. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\quad \log _ { 2 } 0.3\)
Edexcel C2 Q4
4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.
  1. Find the value of \(k\). Using this value of \(k\),
  2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375,
  3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$
Edexcel C2 Q5
  1. (a) Write down the exact value of \(\cos \frac { \pi } { 6 }\).
The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(b) Use the trapezium rule with three equally-spaced ordinates to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(c) Using your answer to part (b), find an estimate for the area of \(S\).
Edexcel C2 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{288b99b5-1198-4463-baed-f0a4bf03e485-3_335_890_246_456} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows triangle \(A B C\) in which \(A C = 8 \mathrm {~cm}\) and \(\angle B A C = \angle B C A = 30 ^ { \circ }\).
  1. Find the area of triangle \(A B C\) in the form \(k \sqrt { 3 }\). The point \(M\) is the mid-point of \(A C\) and the points \(N\) and \(O\) lie on \(A B\) and \(B C\) such that \(M N\) and \(M O\) are arcs of circles with centres \(A\) and \(C\) respectively.
  2. Show that the area of the shaded region \(B N M O\) is \(\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }\).
Edexcel C2 Q7
7. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0 ,$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),
  1. find the value of \(k\),
  2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
  3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).
Edexcel C2 Q8
8. Amy plans to join a savings scheme in which she will pay in \(\pounds 500\) at the start of each year. One scheme that she is considering pays 6\% interest on the amount in the account at the end of each year. For this scheme,
  1. find the amount of interest paid into the account at the end of the second year,
  2. show that after interest is paid at the end of the eighth year, the amount in the account will be \(\pounds 5246\) to the nearest pound. Another scheme that she is considering pays \(0.5 \%\) interest on the amount in the account at the end of each month.
  3. Find, to the nearest pound, how much more or less will be in the account at the end of the eighth year under this scheme.
Edexcel C2 Q9
9. The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where \(k\) is a constant.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) the remainder is \(r\).
When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) the remainder is \(3 r\).
  1. Find the value of \(k\).
  2. Find the value of \(r\).
  3. Show that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\).
  4. Show that there is only one real solution to the equation \(\mathrm { f } ( x ) = 0\). END
Edexcel C2 Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-2_383_707_246_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\).
Find, to 3 significant figures,
  1. the length \(B C\),
  2. the area of triangle \(A B C\).
Edexcel C2 Q2
2. Show that $$\int _ { 2 } ^ { 3 } \left( 6 \sqrt { x } - \frac { 4 } { \sqrt { x } } \right) \mathrm { d } x = k \sqrt { 3 } ,$$ where \(k\) is an integer to be found.
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-2_439_848_1560_461} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).
  1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
  2. find an estimate for the volume of the support.
Edexcel C2 Q4
4. (a) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(b) Hence expand ( \(\left. 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.
Edexcel C2 Q5
5. (a) Given that $$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0 .$$ (b) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0 .$$
Edexcel C2 Q6
  1. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for
    1. \(3 ^ { x + 1 }\),
    2. \(3 ^ { 2 x - 1 }\).
      (b) Hence, or otherwise, solve the equation
    $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$ giving non-exact answers to 2 decimal places.
Edexcel C2 Q7
7. The circle \(C\) has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).
  1. Find the length of the diameter of \(C\).
  2. Find an equation for \(C\).
  3. Show that the line \(y = 2 x - 3\) is a tangent to \(C\) and find the coordinates of the point of contact.
Edexcel C2 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-4_536_883_248_486} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
  1. Find the coordinates of the minimum point of the curve.
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).
Edexcel C2 Q9
9. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
  3. find the common ratio of the series,
  4. find the sum of the first eight terms of the series.
Edexcel C2 Q1
  1. Evaluate
$$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } \mathrm {~d} x .$$
Edexcel C2 Q2
2. $$f ( x ) = x ^ { 3 } + k x - 20 .$$ Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),
  1. find the value of the constant \(k\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\).
Edexcel C2 Q3
3. (a) Given that $$5 \cos \theta - 2 \sin \theta = 0 ,$$ show that \(\tan \theta = 2.5\)
(b) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.
Edexcel C2 Q4
4. Solve each equation, giving your answers to an appropriate degree of accuracy.
  1. \(3 ^ { x - 2 } = 5\)
  2. \(\quad \log _ { 2 } ( 6 - y ) = 3 - \log _ { 2 } y\)
Edexcel C2 Q5
5. A geometric series has third term 36 and fourth term 27. Find
  1. the common ratio of the series,
  2. the fifth term of the series,
  3. the sum to infinity of the series.
Edexcel C2 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{857bf144-b03e-4b46-b043-1119b30f9e78-3_572_954_246_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0\).
  1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.
    \(x\)23456
    \(y\)2.896.36
    The shaded area is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).
  2. Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
  3. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.
Edexcel C2 Q7
7. $$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$
  1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  2. Determine whether each stationary point is a maximum or minimum point.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions.