Questions — Edexcel (10514 questions)

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Edexcel C2 Q7
17 marks Moderate -0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-3_771_972_1322_557}
\end{figure} Fig. 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x\).
The curve crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).
  1. Factorise \(\mathrm { f } ( x )\) completely.
  2. Write down the \(x\)-coordinates of the points \(A\) and \(B\).
  3. Find the gradient of \(C\) at \(A\). The region \(R\) is bounded by \(C\) and the line \(O A\), and the region \(S\) is bounded by \(C\) and the line \(A B\).
  4. Use integration to find the area of the combined regions \(R\) and \(S\), shown shaded in Fig.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
    \end{figure} Fig. 3 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0\). The point \(P\), on \(C\), has \(x\)-coordinate 1 and the point \(Q\) is the minimum turning point of \(C\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find the coordinates of \(Q\).
    3. Show that \(P Q\) is parallel to the \(x\)-axis.
    4. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).
Edexcel C2 Q1
5 marks Moderate -0.8
  1. \(\quad \mathrm { f } ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).
Given that when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) there is a remainder of - 35 ,
  1. find the value of the constant \(k\),
  2. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(3 x - 2\) ).
Edexcel C2 Q2
5 marks Easy -1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-2_613_911_692_376} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).
Edexcel C2 Q3
6 marks Moderate -0.8
3. Giving your answers in terms of \(\pi\), solve the equation $$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).
Edexcel C2 Q4
7 marks Moderate -0.3
4.
  1. Expand \(( 1 + 3 x ) ^ { 8 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). You should simplify each coefficient in your expansion.
  2. Use your series, together with a suitable value of \(x\) which you should state, to estimate the value of (1.003) \({ } ^ { 8 }\), giving your answer to 8 significant figures.
Edexcel C2 Q5
7 marks Moderate -0.3
5. (a) Given that \(t = \log _ { 3 } x\), find expressions in terms of \(t\) for
  1. \(\log _ { 3 } x ^ { 2 }\),
  2. \(\log _ { 9 } x\).
    (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4 .$$
Edexcel C2 Q6
10 marks Moderate -0.8
  1. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).
    1. Find an equation for \(C\).
    2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
    3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-3_664_1016_1276_376} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the curve \(y = 2 x ^ { 2 } + 6 x + 7\) and the straight line \(y = 2 x + 13\).
Edexcel C2 Q8
11 marks Standard +0.3
8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),
  1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
  2. find the value of \(a\),
  3. show that \(S _ { 6 } = 728\).
Edexcel C2 Q9
13 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-4_661_915_932_431} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians. Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),
  1. show that the perimeter, \(P\) cm, of the design is given by $$P = 2 x + \frac { 100 } { x }$$
  2. Find the value of \(x\) for which \(P\) is a minimum.
  3. Show that \(P\) is a minimum for this value of \(x\).
  4. Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).
Edexcel C2 Q2
6 marks Moderate -0.5
  1. Given that
$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
Edexcel C2 Q3
6 marks Moderate -0.3
3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),
  1. find and simplify the first three terms,
  2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).
Edexcel C2 Q4
7 marks Moderate -0.3
4. Solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$
Edexcel C2 Q5
9 marks Moderate -0.3
The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).
  1. Find an equation for \(C\). The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
  2. Find the \(x\)-coordinates of \(A\) and \(B\).
  3. Show that \(A B = 2 \sqrt { 10 }\).
Edexcel C2 Q6
10 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e44996a-4635-46f6-bd45-7799a8c49463-3_589_894_248_397} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).
  1. Find the coordinates of the minimum point of the curve. The shaded region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
  2. Use the trapezium rule with three intervals of equal width to estimate the area of \(R\).
Edexcel C2 Q7
10 marks Moderate -0.3
7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes. Assuming that the times he takes to complete successive papers form a geometric sequence,
  1. find, to the nearest minute, how long he will take to complete the fifth paper,
  2. show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
  3. find the least number of papers he must work through if he is to complete a paper in less than one hour.
Edexcel C2 Q8
10 marks Standard +0.3
8. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm} , C D = 8 \mathrm {~cm}\), \(A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).
  1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
  2. Find the size of \(\angle B C D\) in degrees.
  3. Find the area of quadrilateral \(A B C D\).
Edexcel C2 Q9
13 marks Standard +0.3
9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16 .$$
  1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
Edexcel C2 Q1
4 marks Moderate -0.8
  1. Evaluate
$$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$
Edexcel C2 Q2
5 marks Moderate -0.3
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
6 marks Moderate -0.8
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
8 marks Moderate -0.3
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
9 marks Standard +0.3
  1. 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 \(x = \frac { \pi } { 3 }\).
  2. 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 \(x = \frac { \pi } { 3 }\).
  3. Using your answer to part (b), find an estimate for the area of \(S\).
Edexcel C2 Q6
9 marks Standard +0.3
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
10 marks Standard +0.3
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
12 marks Standard +0.3
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.