CAIE P1 (Pure Mathematics 1) 2017 March

1 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 k x + k = 0\) has distinct real roots.

2 In the expansion of \(\left( \frac { 1 } { a x } + 2 a x ^ { 2 } \right) ^ { 5 }\), the coefficient of \(x\) is 5 . Find the value of the constant \(a\).

3 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-04_489_465_258_840} The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is \(h \mathrm {~cm}\) the surface of the water is a square of side \(\frac { 1 } { 2 } h \mathrm {~cm}\).

1. Express the volume of water in the container in terms of \(h\).
   [0pt] [The volume of a pyramid having a base area \(A\) and vertical height \(h\) is \(\frac { 1 } { 3 } A h\).]
   Water is steadily dripping into the container at a constant rate of \(20 \mathrm {~cm} ^ { 3 }\) per minute.
2. Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm . \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-06_403_773_258_685} In the diagram, \(A B = A C = 8 \mathrm {~cm}\) and angle \(C A B = \frac { 2 } { 7 } \pi\) radians. The circular \(\operatorname { arc } B C\) has centre \(A\), the circular arc \(C D\) has centre \(B\) and \(A B D\) is a straight line.

1. Show that angle \(C B D = \frac { 9 } { 14 } \pi\) radians.
2. Find the perimeter of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-08_526_499_258_824} The diagram shows the graphs of \(y = \tan x\) and \(y = \cos x\) for \(0 \leqslant x \leqslant \pi\). The graphs intersect at points \(A\) and \(B\).

1. Find by calculation the \(x\)-coordinate of \(A\).
2. Find by calculation the coordinates of \(B\).

6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$

1. Use a scalar product to find angle \(O A B\).
2. Find the area of triangle \(O A B\).

7 The function f is defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   The first, second and third terms of a geometric progression are respectively \(\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )\) and \(k \mathrm { f } ^ { \prime \prime } ( 2 )\).
2. Find the value of the constant \(k\).

8 The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x ^ { 2 } + 3 \\ & \mathrm {~g} : x \mapsto 3 x + 2 \end{aligned}$$

1. Show that \(\operatorname { gf } ( x ) = 6 x ^ { 2 } + 11\) and obtain an unsimplified expression for \(\operatorname { fg } ( x )\).
2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and determine the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
3. Solve the equation \(\mathrm { gf } ( 2 x ) = \mathrm { fg } ( x )\).

9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).

1. Find the equation of the tangent to the curve at \(A\).
   The normal to the curve at \(A\) intersects the curve again at \(B\).
2. Find the coordinates of \(B\).
   The tangents at \(A\) and \(B\) intersect each other at \(C\).
3. Find the coordinates of \(C\).

10 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-18_611_531_262_808} The diagram shows the curve \(y = \mathrm { f } ( x )\) defined for \(x > 0\). The curve has a minimum point at \(A\) and crosses the \(x\)-axis at \(B\) and \(C\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - \frac { 2 } { x ^ { 3 } }\) and that the curve passes through the point \(\left( 4 , \frac { 189 } { 16 } \right)\).

1. Find the \(x\)-coordinate of \(A\).
2. Find \(\mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of \(B\) and \(C\).
4. Find, showing all necessary working, the area of the shaded region.
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