Questions C2 (1410 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel C2 Q2
2. The first three terms of a geometric series are ( \(p - 1\) ), 2 and ( \(2 p + 5\) ) respectively, where \(p\) is a constant. Find the two possible values of \(p\).
Edexcel C2 Q3
3. Find the area of the finite region enclosed by the curve \(y = 5 x - x ^ { 2 }\) and the \(x\)-axis.
Edexcel C2 Q4
4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta ,$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).
Edexcel C2 Q5
5. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6 ,$$ and that \(x = - 3\) is a solution to the equation \(\mathrm { f } ( x ) = 0\),
  1. find the value of the constant \(p\),
  2. show that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) there is a remainder of 50 ,
  3. find the other solutions to the equation \(\mathrm { f } ( x ) = 0\), giving your answers to 2 decimal places.
Edexcel C2 Q6
6. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0 .$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
  3. Sketch C. Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
  4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).
Edexcel C2 Q7
7. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 } ,$$ where \(n\) is an integer and \(n > 1\),
  1. show that \(n = 16\) and find the value of \(a\),
  2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.
Edexcel C2 Q8
8. (a) Given that $$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x ,$$ show that $$y = 2 x + 1 .$$ (b) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x
& 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$
Edexcel C2 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9215e382-406c-41a3-8907-f465b134dd87-4_499_1137_954_319} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a tray made from sheet metal.
The horizontal base is a rectangle measuring \(8 x \mathrm {~cm}\) by \(y \mathrm {~cm}\) and the two vertical sides are trapezia of height \(x \mathrm {~cm}\) with parallel edges of length \(8 x \mathrm {~cm}\) and \(10 x \mathrm {~cm}\). The remaining two sides are rectangles inclined at \(45 ^ { \circ }\) to the horizontal. Given that the capacity of the tray is \(900 \mathrm {~cm} ^ { 3 }\),
  1. find an expression for \(y\) in terms of \(x\),
  2. show that the area of metal used to make the tray, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 18 x ^ { 2 } + \frac { 200 ( 4 + \sqrt { 2 } ) } { x } ,$$
  3. find to 3 significant figures, the value of \(x\) for which \(A\) is stationary,
  4. find the minimum value of \(A\) and show that it is a minimum.
Edexcel C2 Q1
  1. During one day, a biological culure is allowed to grow under controlled conditions.
At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times ( 1.06 ) ^ { t } .$$ Using this model,
  1. find the number of bacteria present at 11 a.m.,
  2. find, to the nearest minute, the time when the initial number of bacteria will have doubled.
Edexcel C2 Q2
2. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{824da843-3ea1-4a83-9170-d0082b2e8d1c-2_476_880_1254_539} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 1\).
  1. Use the trapezium rule with four intervals of equal width to find an estimate for the area of the shaded region.
  2. State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.
Edexcel C2 Q4
4. The first three terms in the expansion in descending powers of \(x\) of $$\left( x + \frac { k } { x ^ { 2 } } \right) ^ { 15 } ,$$ where \(k\) is a constant, are $$x ^ { 15 } + 30 x ^ { 12 } + A x ^ { 9 } .$$
  1. Find the values of \(k\) and \(A\).
  2. Find the value of the term independent of \(x\) in the expansion.
Edexcel C2 Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{824da843-3ea1-4a83-9170-d0082b2e8d1c-3_458_862_906_511} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).
  1. Show that \(a = 8\).
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.
Edexcel C2 Q6
6. $$f ( x ) = \cos 2 x , \quad 0 \leq x \leq \pi .$$
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
  3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).
Edexcel C2 Q7
7. The points \(P\) and \(Q\) have coordinates \(( - 2,6 )\) and \(( 4 , - 1 )\) respectively. Given that \(P Q\) is a diameter of circle \(C\),
  1. find the coordinates of the centre of \(C\),
  2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0 .$$ The point \(R\) has coordinates (2, 7).
  3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.
Edexcel C2 Q8
8. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.
  1. Find the common ratio of the series.
  2. Show that the first term of the series is \(\log _ { 3 } 2\).
  3. Find, to 3 significant figures, the sum of the first six terms of the series.
Edexcel C2 Q9
9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).
  1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
  4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).
Edexcel C2 Q1
  1. Evaluate
$$\int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) d x .$$
Edexcel C2 Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-2_510_842_534_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \sqrt { 4 x - 1 }\). Use the trapezium rule with five equally-spaced ordinates to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Edexcel C2 Q3
3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
  2. \(\log _ { 2 } ( \sqrt { x } )\).
    (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
Edexcel C2 Q4
4. $$f ( x ) = 2 - x - x ^ { 3 }$$
  1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
  2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
  3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.
Edexcel C2 Q5
5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,
  1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
  2. show that \(Q R = 2 \sqrt { 6 }\),
  3. find \(\angle P Q R\) in degrees to 1 decimal place.
Edexcel C2 Q6
6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,
  1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
Edexcel C2 Q7
7. (a) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < 2 \pi\) for which $$\tan \left( x + \frac { \pi } { 4 } \right) = 3 .$$ (b) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y .$$
Edexcel C2 Q8
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.