Questions C2 (1410 questions)

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Edexcel C2 Q2
  1. A geometric series has common ratio \(\frac { 1 } { 3 }\).
Given that the sum of the first four terms of the series is 200,
  1. find the first term of the series,
  2. find the sum to infinity of the series.
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ccac020a-c378-45db-80f4-c63b5c213e1d-2_513_775_945_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates ( \(- 4,0\) ),
  1. show that \(k = - 7\),
  2. find the coordinates of \(B\) and \(C\).
Edexcel C2 Q4
4. (a) (i) Sketch the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
(ii) Write down the coordinates of the turning points of the curve in this interval.
(b) Find all values of \(x\) in the interval \(- 180 \leq x \leq 180\) for which $$\sin ( x - 30 ) ^ { \circ } = 0.35$$ giving your answers to 1 decimal place.
Edexcel C2 Q5
5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0 .$$
Edexcel C2 Q6
  1. \(\quad \mathrm { f } ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0\).
    1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
    2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
    3. Determine whether the turning point is a maximum or minimum point.
    4. The points \(P , Q\) and \(R\) have coordinates \(( - 5,2 ) , ( - 3,8 )\) and \(( 9,4 )\) respectively.
    5. Show that \(\angle P Q R = 90 ^ { \circ }\).
    Given that \(P , Q\) and \(R\) all lie on circle \(C\),
  2. find the coordinates of the centre of \(C\),
  3. show that the equation of \(C\) can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.
Edexcel C2 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ccac020a-c378-45db-80f4-c63b5c213e1d-4_549_517_246_605} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(120 ^ { \circ }\) at the centre of the circle.
  1. Find the exact length of the major arc \(P Q\).
  2. Show that the perimeter of the shaded minor segment is given by \(k ( 2 \pi + 3 \sqrt { 3 } ) \mathrm { cm }\), where \(k\) is an integer to be found.
  3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle.
Edexcel C2 Q9
9. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).
  1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\).
  2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
  3. Find the percentage error in the estimate made in part (a).
Edexcel C2 Q1
  1. Expand \(( 3 - 2 x ) ^ { 4 }\) in ascending powers of \(x\) and simplify each coefficient.
Figure 1 Figure 1 shows triangle \(P Q R\) in which \(P Q = x , P R = 7 - x , Q R = x + 1\) and \(\angle P Q R = 60 ^ { \circ }\). Using the cosine rule, find the value of \(x\).
Edexcel C2 Q3
3. Find the coordinates of the stationary point of the curve with equation $$y = x + \frac { 4 } { x ^ { 2 } } .$$
Edexcel C2 Q4
  1. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which
$$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$
Edexcel C2 Q5
  1. (a) Sketch the curve \(y = 5 ^ { x - 1 }\), showing the coordinates of any points of intersection with the coordinate axes.
    (b) Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5 ^ { x - 1 }\) intersects
    1. the straight line \(y = 10\),
    2. the curve \(y = 2 ^ { x }\).
    $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1 .$$
Edexcel C2 Q7
7. (a) Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ (b) Evaluate \(\quad \sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)\).
Edexcel C2 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f824c38-ae19-4889-a2e8-05a3707e9b27-3_496_716_1407_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = 5 + x - x ^ { 2 }\) and the normal to the curve at the point \(P ( 1,5 )\).
  1. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
  3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).
Edexcel C2 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f824c38-ae19-4889-a2e8-05a3707e9b27-4_757_855_246_482} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$
  1. Find the coordinates of the centre and the radius of \(C\).
    \(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).
  2. Find the coordinates of \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(\theta\) at the centre of \(C\).
  3. Using the cosine rule, show that \(\cos \theta = \frac { 7 } { 25 }\).
  4. Find the area of the shaded minor segment bounded by \(C\) and the chord \(P Q\). END
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.