Questions — CAIE P2 (699 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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
CAIE P2 2010 June Q7
7 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) the remainder is 30 , and that when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) the remainder is 18 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, verify that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\) and hence factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2010 June Q8
8
  1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
  2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{af7c2b6e-1293-4744-8ea0-927fea5ab4ec-2_531_949_431_598} The diagram shows part of the curve \(y = x \mathrm { e } ^ { - x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 2 , x = 3\) and \(y = 0\).
  1. Use the trapezium rule with two intervals to estimate the area of \(R\), giving your answer correct to 2 decimal places.
  2. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of \(R\).
CAIE P2 2010 June Q3
3 Solve the inequality \(| 2 x - 1 | < | x + 4 |\).
CAIE P2 2011 June Q1
1 Solve the equation \(| 3 x + 4 | = | 2 x + 5 |\).
CAIE P2 2011 June Q2
2 A curve has parametric equations $$x = 3 t + \sin 2 t , \quad y = 4 + 2 \cos 2 t$$ Find the exact gradient of the curve at the point for which \(t = \frac { 1 } { 6 } \pi\).
CAIE P2 2011 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{d90dc270-b304-4b42-8e0e-37641b8a03b8-2_556_1113_680_516} The variables \(x\) and \(y\) satisfy the equation \(y = K x ^ { m }\), where \(K\) and \(m\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \(( 0,2.0 )\) and \(( 6,10.2 )\), as shown in the diagram. Find the values of \(K\) and \(m\), correct to 2 decimal places.
CAIE P2 2011 June Q4
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + a x + a$$ where \(a\) is a constant.
  1. Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. When \(a\) has the value found in part (i), find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
CAIE P2 2011 June Q5
5 Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\) in each of the following cases:
  1. \(y = x \ln ( x - 3 )\),
  2. \(y = \frac { x - 1 } { x + 1 }\).
CAIE P2 2011 June Q6
6
  1. Find \(\int 4 \mathrm { e } ^ { x } \left( 3 + \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
  2. Show that \(\int _ { - \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 4 } \pi } \left( 3 + 2 \tan ^ { 2 } \theta \right) \mathrm { d } \theta = \frac { 1 } { 2 } ( 8 + \pi )\).
CAIE P2 2011 June Q7
7
  1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 14 - x ^ { 2 }$$ has exactly two real roots.
  2. Show by calculation that the positive root lies between 1.2 and 1.3.
  3. Show that this root also satisfies the equation $$x = \frac { 1 } { 2 } \ln \left( 14 - x ^ { 2 } \right) .$$
  4. Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.
  5. Express \(4 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  6. Solve the equation \(4 \sin \theta - 6 \cos \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  7. Find the greatest and least possible values of \(( 4 \sin \theta - 6 \cos \theta ) ^ { 2 } + 8\) as \(\theta\) varies.
CAIE P2 2011 June Q1
1 Use logarithms to solve the equation \(3 ^ { x } = 2 ^ { x + 2 }\), giving your answer correct to 3 significant figures.
CAIE P2 2011 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{2c27f384-5289-4c1b-9199-6b4c6ac81e38-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
  2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).
CAIE P2 2011 June Q3
3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) defined by $$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { 1 } { 2 } \sqrt [ 3 ] { } \left( x _ { n } ^ { 2 } + 6 \right)$$ converges to the value \(\alpha\).
  1. Find the value of \(\alpha\) correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places.
  2. Find, in the form \(a x ^ { 3 } + b x ^ { 2 } + c = 0\), an equation of which \(\alpha\) is a root.
CAIE P2 2011 June Q4
4
  1. Find the value of \(\int _ { 0 } ^ { \frac { 2 } { 3 } \pi } \sin \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
  2. Find \(\int \mathrm { e } ^ { - x } \left( 1 + \mathrm { e } ^ { x } \right) \mathrm { d } x\).
CAIE P2 2011 June Q5
6 marks
5 A curve has equation \(x ^ { 2 } + 2 y ^ { 2 } + 5 x + 6 y = 10\). Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
[0pt] [6]
CAIE P2 2011 June Q6
6 The curve \(y = 4 x ^ { 2 } \ln x\) has one stationary point.
  1. Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P2 2011 June Q7
7 The cubic polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that, when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ), the remainder is 24 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2011 June Q8
8
  1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
  2. Hence
    (a) solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
    (b) find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).
CAIE P2 2011 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{ee420db2-bef4-4c2b-8dd2-c8f439dd561e-2_645_750_429_699} The diagram shows the curve \(y = \sqrt { } \left( 1 + x ^ { 3 } \right)\). Region \(A\) is bounded by the curve and the lines \(x = 0\), \(x = 2\) and \(y = 0\). Region \(B\) is bounded by the curve and the lines \(x = 0\) and \(y = 3\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 2 decimal places.
  2. Deduce an approximation to the area of region \(B\) and explain why this approximation underestimates the true area of region \(B\).
CAIE P2 2012 June Q1
1 Solve the equation \(\left| x ^ { 3 } - 14 \right| = 13\), showing all your working.
CAIE P2 2012 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-2_453_771_386_685} The variables \(x\) and \(y\) satisfy the equation \(y = A \left( b ^ { x } \right)\), where \(A\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0,2.14 )\) and \(( 5,4.49 )\), as shown in the diagram. Find the values of \(A\) and \(b\), correct to 1 decimal place.
CAIE P2 2012 June Q3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 3 x ^ { 2 } - 5 x + a + 4 ,$$ where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
  2. When \(a\) has this value,
    (a) factorise \(\mathrm { p } ( x )\) completely,
    (b) find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).
CAIE P2 2012 June Q4
4
  1. Given that \(35 + \sec ^ { 2 } \theta = 12 \tan \theta\), find the value of \(\tan \theta\).
  2. Hence, showing the use of an appropriate formula in each case, find the exact value of
    (a) \(\tan \left( \theta - 45 ^ { \circ } \right)\),
    (b) \(\tan 2 \theta\).
CAIE P2 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
  2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).