Questions — CAIE P2 (699 questions)

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CAIE P2 2013 June Q8
8
  1. Prove the identity $$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
  2. Hence solve the equation $$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2014 June Q1
1 Solve the inequality \(| 3 x - 2 | \geqslant | x + 4 |\).
CAIE P2 2014 June Q2
2 Find the gradient of each of the following curves at the point for which \(x = 0\).
  1. \(y = 3 \sin x + \tan 2 x\)
  2. \(y = \frac { 6 } { 1 + \mathrm { e } ^ { 2 x } }\)
CAIE P2 2014 June Q3
3
  1. Find the quotient when \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 15\) is divided by ( \(x ^ { 2 } - 4\) ), and confirm that the remainder is 7 .
  2. Hence solve the equation \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).
CAIE P2 2014 June Q4
4
  1. By sketching a suitable pair of graphs, show that the equation $$3 \ln x = 15 - x ^ { 3 }$$ has exactly one real root.
  2. Show by calculation that the root lies between 2.0 and 2.5.
  3. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 15 - 3 \ln x _ { n } \right)\) to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2014 June Q6
6
  1. Show that \(\int _ { 6 } ^ { 16 } \frac { 6 } { 2 x - 7 } \mathrm {~d} x = \ln 125\).
  2. Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 17 } \log _ { 10 } x d x$$ giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q7
7 The equation of a curve is $$2 x ^ { 2 } + 3 x y + y ^ { 2 } = 3$$
  1. Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  2. Show that the curve has no stationary points.
CAIE P2 2014 June Q1
1
  1. Solve the equation \(| x + 2 | = | x - 13 |\).
  2. Hence solve the equation \(\left| 3 ^ { y } + 2 \right| = \left| 3 ^ { y } - 13 \right|\), giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q2
2 Solve the equation \(3 \sin 2 \theta \tan \theta = 2\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2014 June Q3
3
  1. Find \(\int 4 \cos \left( \frac { 1 } { 3 } x + 2 \right) \mathrm { d } x\).
  2. Use the trapezium rule with three intervals to find an approximation to $$\int _ { 0 } ^ { 12 } \sqrt { } \left( 4 + x ^ { 2 } \right) \mathrm { d } x$$ giving your answer correct to 3 significant figures.
CAIE P2 2014 June Q4
4 The parametric equations of a curve are $$x = 2 \ln ( t + 1 ) , \quad y = 4 \mathrm { e } ^ { t }$$ Find the equation of the tangent to the curve at the point for which \(t = 0\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
CAIE P2 2014 June Q5
6 marks
5
\includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
[0pt] [6]
CAIE P2 2014 June Q6
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x + a$$ 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, find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
CAIE P2 2014 June Q7
7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 1 } { 2 } \mathrm { e } ^ { 3 x } + x ^ { 2 } \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.
  1. Show that \(a = \frac { 1 } { 3 } \ln \left( 61 - 2 a ^ { 3 } \right)\).
  2. Show by calculation that the value of \(a\) lies between 1.0 and 1.5.
  3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2014 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
  2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
CAIE P2 2014 June Q5
6 marks
5
\includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-2_583_597_1457_772} The variables \(x\) and \(y\) satisfy the equation \(y = K \left( 2 ^ { p x } \right)\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points ( \(1.35,1.87\) ) and ( \(3.35,3.81\) ), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places.
[0pt] [6]
CAIE P2 2014 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
  2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
CAIE P2 2015 June Q1
1
  1. Solve the equation \(| 3 x + 4 | = | 3 x - 11 |\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 2 ^ { y } + 4 \right| = \left| 3 \times 2 ^ { y } - 11 \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2015 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{595e38f4-c52e-4509-8b16-f08e30dec96b-2_456_716_529_712} The variables \(x\) and \(y\) satisfy the equation $$y = A \mathrm { e } ^ { p ( x - 1 ) } ,$$ where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 2,1.60 )\) and \(( 5,2.92 )\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 significant figures.
CAIE P2 2015 June Q3
3 The equation of a curve is $$y = 6 \sin x - 2 \cos 2 x$$ Find the equation of the tangent to the curve at the point \(\left( \frac { 1 } { 6 } \pi , 2 \right)\). Give the answer in the form \(y = m x + c\), where the values of \(m\) and \(c\) are correct to 3 significant figures.
CAIE P2 2015 June Q4
4 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\). It is also given that, when \(\mathrm { g } ( x )\) 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, find the greatest possible value of \(\mathrm { g } ( x ) - \mathrm { f } ( x )\) as \(x\) varies.
CAIE P2 2015 June Q5
5
  1. Given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 1 \right) \mathrm { d } x = 10\), show that the positive constant \(a\) satisfies the equation $$a = 2 \ln \left( \frac { 16 - a } { 6 } \right)$$
  2. Use the iterative formula \(a _ { n + 1 } = 2 \ln \left( \frac { 16 - a _ { n } } { 6 } \right)\) with \(a _ { 1 } = 2\) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2015 June Q6
6
  1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
  2. Hence
    (a) solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
    (b) find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).
CAIE P2 2015 June Q7
7 The equation of a curve is $$y ^ { 3 } + 4 x y = 16$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 y } { 3 y ^ { 2 } + 4 x }\).
  2. Show that the curve has no stationary points.
  3. Find the coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P2 2015 June Q1
1
  1. Use logarithms to solve the equation \(2 ^ { x } = 20 ^ { 5 }\), giving the answer correct to 3 significant figures.
  2. Hence determine the number of integers \(n\) satisfying $$20 ^ { - 5 } < 2 ^ { n } < 20 ^ { 5 }$$