Questions P2 (867 questions)

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CAIE P2 2014 November Q3
7 marks Standard +0.3
3 A curve has equation $$3 \ln x + 6 x y + y ^ { 2 } = 16$$ Find the equation of the normal to the curve at the point \(( 1,2 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
CAIE P2 2014 November Q4
8 marks Moderate -0.8
4
  1. Find the value of \(x\) satisfying the equation \(2 \ln ( x - 4 ) - \ln x = \ln 2\).
  2. Use logarithms to find the smallest integer satisfying the inequality $$1.4 ^ { y } > 10 ^ { 10 }$$
CAIE P2 2014 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-2_554_689_1354_726} The diagram shows part of the curve $$y = 2 \cos x - \cos 2 x$$ and its maximum point \(M\). The shaded region is bounded by the curve, the axes and the line through \(M\) parallel to the \(y\)-axis.
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. Find the exact value of the area of the shaded region.
CAIE P2 2014 November Q6
8 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-3_597_931_260_607} The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 3 x ^ { 2 } - 25 x + 48 .$$ The diagram shows the curve \(y = \mathrm { p } ( x )\) which crosses the \(x\)-axis at ( \(\alpha , 0\) ) and ( 3,0 ).
  1. Divide \(\mathrm { p } ( x )\) by a suitable linear factor and hence show that \(\alpha\) is a root of the equation \(x = \sqrt [ 3 ] { } ( 16 - 3 x )\).
  2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 3 ] { } \left( 16 - 3 x _ { n } \right)\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2014 November Q7
11 marks Standard +0.3
7
  1. Express \(5 \cos \theta - 12 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(5 \cos \theta - 12 \sin \theta = 8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. Find the greatest possible value of $$7 + 5 \cos \frac { 1 } { 2 } \phi - 12 \sin \frac { 1 } { 2 } \phi$$ as \(\phi\) varies, and determine the smallest positive value of \(\phi\) for which this greatest value occurs.
    [0pt] [4]
CAIE P2 2014 November Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{293e1e27-77e9-4b19-a152-96d71b75346e-2_654_693_532_724} 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 ( \(0.75,1.70\) ) and ( \(1.53,2.18\) ), as shown in the diagram. Find the values of \(a\) and \(b\) correct to 2 decimal places.
CAIE P2 2014 November Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{293e1e27-77e9-4b19-a152-96d71b75346e-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
  2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
  3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2015 November Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation $$5 ^ { x + 3 } = 7 ^ { x - 1 }$$ giving the answer correct to 3 significant figures.
CAIE P2 2015 November Q2
5 marks Standard +0.3
2 A curve has equation $$y = \frac { 3 x + 1 } { x - 5 }$$ Find the coordinates of the points on the curve at which the gradient is - 4 .
CAIE P2 2015 November Q3
7 marks Moderate -0.3
3
  1. Express \(8 \sin \theta + 15 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$8 \sin \theta + 15 \cos \theta = 6$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2015 November Q4
7 marks Moderate -0.3
4
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
  2. Verify by calculation that \(4.5 < \alpha < 5.0\).
  3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2015 November Q5
7 marks Moderate -0.3
5
  1. Find \(\int \left( \tan ^ { 2 } x + \sin 2 x \right) \mathrm { d } x\).
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } 3 \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x\).
CAIE P2 2015 November Q6
9 marks Standard +0.3
6
  1. Find the quotient and remainder when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + 12 x + 6$$ is divided by ( \(x ^ { 2 } - x + 4\) ).
  2. It is given that, when $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q$$ is divided by ( \(x ^ { 2 } - x + 4\) ), the remainder is zero. Find the values of the constants \(p\) and \(q\).
  3. When \(p\) and \(q\) have these values, show that there is exactly one real value of \(x\) satisfying the equation $$x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } + p x + q = 0$$ and state what that value is.
CAIE P2 2015 November Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{250b4df9-2646-4246-bb6d-2be92bf29598-3_553_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
  2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
  3. Find the value of the gradient of the curve at \(B\).
CAIE P2 2015 November Q1
5 marks Moderate -0.8
1
  1. Solve the equation \(| 3 x - 2 | = 5\).
  2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.
CAIE P2 2015 November Q2
5 marks Standard +0.3
2 The sequence of values given by the iterative formula $$x _ { n + 1 } = 2 + \frac { 4 } { x _ { n } ^ { 2 } + 2 x _ { n } + 4 }$$ with initial value \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Determine the value of \(\alpha\) correct to 3 decimal places, giving the result of each iteration to 5 decimal places.
  2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P2 2015 November Q3
6 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-2_456_725_1082_712} 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 ( \(0.22,3.96\) ) and ( \(1.32,2.43\) ), as shown in the diagram. Find the values of \(K\) and \(m\) correct to 2 significant figures.
CAIE P2 2015 November Q4
7 marks Standard +0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + 11 x ^ { 2 } + a x + a$$ where \(a\) is a constant. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Use the factor theorem to show that \(a = - 4\).
  2. When \(a = - 4\),
    1. factorise \(\mathrm { p } ( x )\) completely,
    2. solve the equation \(6 \sec ^ { 3 } \theta + 11 \sec ^ { 2 } \theta + a \sec \theta + a = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P2 2015 November Q5
8 marks Moderate -0.3
5 Find the \(x\)-coordinates of the stationary points of the following curves:
  1. \(y = 4 x \mathrm { e } ^ { - 3 x }\);
  2. \(y = \frac { 4 x ^ { 2 } } { x + 1 }\).
CAIE P2 2015 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_453_650_258_744} The diagram shows the curve with parametric equations $$x = 3 \cos t , \quad y = 2 \cos \left( t - \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant t < 2 \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 } ( \sqrt { } 3 - \cot t )\).
  2. Find the equation of the tangent to the curve at the point where the curve crosses the positive \(y\)-axis. Give the answer in the form \(y = m x + c\).
CAIE P2 2015 November Q7
10 marks Standard +0.8
7
  1. Show that the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 2 } x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x\) is \(\frac { 1 } { 6 } \pi + \frac { 9 } { 8 } \sqrt { } 3\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_444_495_1523_865} The diagram shows the curve \(y = \cos x + \frac { 1 } { \cos x }\) for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). The shaded region is bounded by the curve and the lines \(x = 0 , x = \frac { 1 } { 3 } \pi\) and \(y = 0\). Find the exact volume of the solid obtained when the shaded region is rotated completely about the \(x\)-axis.
CAIE P2 2015 November Q1
5 marks Moderate -0.8
1 Find the exact value of \(\int _ { - 1 } ^ { 35 } \frac { 3 } { 2 x + 5 } \mathrm {~d} x\), giving the answer in the form \(\ln k\).
CAIE P2 2015 November Q2
5 marks Standard +0.3
2
  1. Solve the equation \(| 2 x + 3 | = | x + 8 |\).
  2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 1 } + 3 \right| = \left| 2 ^ { y } + 8 \right|\). Give the answer correct to 3 significant figures.
CAIE P2 2015 November Q3
6 marks Moderate -0.3
3 The parametric equations of a curve are $$x = ( t + 1 ) \mathrm { e } ^ { t } , \quad y = 6 ( t + 4 ) ^ { \frac { 1 } { 2 } }$$ Find the equation of the tangent to the curve when \(t = 0\), giving the answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
CAIE P2 2015 November Q4
7 marks Standard +0.3
4
  1. Find the quotient when \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1\) is divided by ( \(x - 2\) ), and show that the remainder is 39 .
  2. Hence show that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 40 = 0\) has exactly one real root.