Questions — CAIE P2 (699 questions)

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CAIE P2 2011 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{55794ceb-2d52-459c-8724-6a6a29ab159a-2_705_737_591_703} The diagram shows the part of the curve \(y = \frac { 1 } { 2 } \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinates of the points on this part of the curve at which the gradient is 4 .
CAIE P2 2011 November Q4
4 Solve the equation \(3 ^ { 2 x } - 7 \left( 3 ^ { x } \right) + 10 = 0\), giving your answers correct to 3 significant figures.
CAIE P2 2011 November Q5
5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .
  1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
  2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).
CAIE P2 2011 November Q6
6
  1. Verify by calculation that the cubic equation $$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$ has a root that lies between \(x = 0.7\) and \(x = 0.8\).
  2. Show that this root also satisfies an equation of the form $$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$ where the values of \(a\) and \(b\) are to be found.
  3. With these values of \(a\) and \(b\), use the iterative formula $$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 November Q7
7 The parametric equations of a curve are $$x = \mathrm { e } ^ { 3 t } , \quad y = t ^ { 2 } \mathrm { e } ^ { t } + 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ( t + 2 ) } { 3 \mathrm { e } ^ { 2 t } }\).
  2. Show that the tangent to the curve at the point \(( 1,3 )\) is parallel to the \(x\)-axis.
  3. Find the exact coordinates of the other point on the curve at which the tangent is parallel to the \(x\)-axis.
CAIE P2 2011 November Q8
8
  1. By first expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( 2 \cos ^ { 3 } x - \cos x \right) d x = \frac { 5 } { 12 }$$
CAIE P2 2011 November Q1
1 Solve the inequality \(| x + 2 | > \left| \frac { 1 } { 2 } x - 2 \right|\).
CAIE P2 2011 November Q2
2 Use logarithms to solve the equation \(4 ^ { x + 1 } = 5 ^ { 2 x - 3 }\), giving your answer correct to 3 significant figures.
CAIE P2 2011 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{322eb555-d40a-460c-8c71-5780f5772bcd-2_535_1041_573_552} The diagram shows the curve \(y = x - 2 \ln x\) and its minimum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 2 } ^ { 5 } ( x - 2 \ln x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
CAIE P2 2011 November Q4
4 Find the exact value of the positive constant \(k\) for which $$\int _ { 0 } ^ { k } e ^ { 4 x } d x = \int _ { 0 } ^ { 2 k } e ^ { x } d x$$
CAIE P2 2011 November Q5
5
  1. By sketching a suitable pair of graphs, show that the equation $$\frac { 1 } { x } = \sin x$$ where \(x\) is in radians, has only one root for \(0 < x \leqslant \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between \(x = 1.1\) and \(x = 1.2\).
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 November Q6
6 The parametric equations of a curve are $$x = 1 + 2 \sin ^ { 2 } \theta , \quad y = 4 \tan \theta$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sin \theta \cos ^ { 3 } \theta }\).
  2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\), giving your answer in the form \(y = m x + c\).
CAIE P2 2011 November Q7
7 The polynomial \(a x ^ { 3 } - 3 x ^ { 2 } - 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). 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 12 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2011 November Q8
8
  1. Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \cos \theta - 3 \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  3. Write down the least value of \(15 \cos \theta - 9 \sin \theta\) as \(\theta\) varies.
CAIE P2 2011 November Q1
1 Find the gradient of the curve \(y = \ln ( 5 x + 1 )\) at the point where \(x = 4\).
CAIE P2 2011 November Q2
2 Solve the inequality \(| 2 x - 3 | \leqslant | 3 x |\).
CAIE P2 2011 November Q3
3 Solve the equation \(2 \ln ( x + 3 ) - \ln x = \ln ( 2 x - 2 )\).
CAIE P2 2011 November Q4
4
  1. Express \(\cos ^ { 2 } x\) in terms of \(\cos 2 x\).
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( \cos ^ { 2 } x + \sin 2 x \right) \mathrm { d } x = \frac { 1 } { 8 } \sqrt { } 3 + \frac { 1 } { 12 } \pi + \frac { 1 } { 4 }$$
CAIE P2 2011 November Q5
5 Solve the equation \(5 \sec ^ { 2 } 2 \theta = \tan 2 \theta + 9\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P2 2011 November Q6
6
  1. The polynomial \(x ^ { 4 } + a x ^ { 3 } - x ^ { 2 } + b x + 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\). Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 2\).
CAIE P2 2011 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{e82fee05-0c55-4fe2-b781-e5e82186c153-2_608_999_1430_571} The diagram shows the curve \(y = ( x - 4 ) \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The curve has a gradient of 3 at the point \(P\).
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }$$
  2. Verify that the equation in part (i) has a root between \(x = 3.1\) and \(x = 3.3\).
  3. Use the iterative formula \(x _ { n + 1 } = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2011 November Q8
8 The equation of a curve is \(2 x ^ { 2 } - 3 x - 3 y + y ^ { 2 } = 6\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 3 } { 3 - 2 y }\).
  2. Find the coordinates of the two points on the curve at which the gradient is - 1 .
CAIE P2 2012 November Q1
1 Solve the inequality \(| x - 2 | \geqslant | x + 5 |\).
CAIE P2 2012 November Q2
2 Use logarithms to solve the equation \(5 ^ { x } = 3 ^ { 2 x - 1 }\), giving your answer correct to 3 significant figures.
CAIE P2 2012 November Q3
3 Solve the equation $$2 \cos 2 \theta = 4 \cos \theta - 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).