Questions C2 (1410 questions)

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Edexcel C2 2005 January Q2
The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
  2. find an equation for \(C\).
Edexcel C2 2009 January Q2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-03_870_1027_205_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = ( 1 + x ) ( 4 - x )\).
The curve intersects the \(x\)-axis at \(x = - 1\) and \(x = 4\). The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis. Use calculus to find the exact area of \(R\).
Edexcel C2 2005 June Q2
Solve
  1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
  2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
Edexcel C2 2005 June Q3
  1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
  2. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.
Edexcel C2 2005 June Q4
  1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
  2. find the value of \(p\) and the value of \(q\).
Edexcel C2 2006 June Q2
Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
Edexcel C2 2006 June Q3
  1. Write down the value of \(\log _ { 6 } 36\).
  2. Express \(2 \log _ { a } 3 + \log _ { a } 11\) as a single logarithm to base \(a\).
Edexcel C2 2006 June Q4
$$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 2007 June Q2
$$f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } - 16 x + 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ). Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  2. factorise \(\mathrm { f } ( x )\) completely.
Edexcel C2 Specimen Q2
The circle \(C\) has centre \(( 3,4 )\) and passes through the point \(( 8 , - 8 )\). Find an equation for C
Edexcel C2 Specimen Q3
The trapezium rule, with the table below, was used to estimate the area between the curve \(y = \sqrt { x ^ { 3 } + 1 }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis.
\(x\)11.522.53
\(y\)1.4142.0923.000
  1. Calculate, to 3 decimal places, the values of \(y\) for \(x = 2.5\) and \(x = 3\).
  2. Use the values from the table and your answers to part (a) to find an estimate, to 2 decimal places, for this area.
OCR C2 2005 January Q1
1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).
OCR C2 2005 January Q2
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = \frac { 1 } { 1 - u _ { n } } \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 2 } , u _ { 3 } , u _ { 4 }\) and \(u _ { 5 }\).
  2. Deduce the value of \(u _ { 200 }\), showing your reasoning.
OCR C2 2005 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_488_604_895_769} A landmark \(L\) is observed by a surveyor from three points \(A , B\) and \(C\) on a straight horizontal road, where \(A B = B C = 200 \mathrm {~m}\). Angles \(L A B\) and \(L B A\) are \(65 ^ { \circ }\) and \(80 ^ { \circ }\) respectively (see diagram). Calculate
  1. the shortest distance from \(L\) to the road,
  2. the distance \(L C\).
OCR C2 2005 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_547_511_1813_817} The diagram shows a sketch of parts of the curves \(y = \frac { 16 } { x ^ { 2 } }\) and \(y = 17 - x ^ { 2 }\).
  1. Verify that these curves intersect at the points \(( 1,16 )\) and \(( 4,1 )\).
  2. Calculate the exact area of the shaded region between the curves.
OCR C2 2005 January Q5
5
  1. Prove that the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ can be expressed in the form $$2 \cos ^ { 2 } \theta + \cos \theta - 1 = 0$$
  2. Hence solve the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2005 January Q6
6
  1. Find \(\int x \left( x ^ { 2 } + 2 \right) \mathrm { d } x\).
    1. Find \(\int \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).
    2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } x }\). Find the equation of the curve, given that it passes through the point \(( 4,0 )\).
OCR C2 2005 January Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).
  1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
  2. Find the exact area of the shaded region.
OCR C2 2005 January Q8
8
  1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
    (a) \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
    (b) \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
  2. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$
OCR C2 2005 January Q9
9 A geometric progression has first term \(a\), where \(a \neq 0\), and common ratio \(r\), where \(r \neq 1\). The difference between the fourth term and the first term is equal to four times the difference between the third term and the second term.
  1. Show that \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1 = 0\).
  2. Show that \(r - 1\) is a factor of \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\). Hence factorise \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\).
  3. Hence find the two possible values for the ratio of the geometric progression. Give your answers in an exact form.
  4. For the value of \(r\) for which the progression is convergent, prove that the sum to infinity is \(\frac { 1 } { 2 } a ( 1 + \sqrt { } 5 )\).
OCR C2 2006 January Q1
1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .
  1. Find the first term and the common difference.
  2. Show that the sum of the first 29 terms is zero.
OCR C2 2006 January Q2
2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate
  1. the area of the triangle,
  2. the length of \(C A\),
  3. the size of angle \(C\).
OCR C2 2006 January Q3
3
  1. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$
OCR C2 2006 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The angle \(A O B\) is 1.8 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. It is given that \(O A = O B = 20 \mathrm {~cm}\) and \(O C = O D = 15 \mathrm {~cm}\). The shaded region is bounded by the arcs \(A B\) and \(C D\) and by the lines \(C A\) and \(D B\).
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
OCR C2 2006 January Q5
5 In a geometric progression, the first term is 5 and the second term is 4.8 .
  1. Show that the sum to infinity is 125 .
  2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).