CAIE P1 (Pure Mathematics 1) 2015 November

1 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + 2 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto 4 x - 12 , \quad x \in \mathbb { R } . \end{aligned}$$ Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \operatorname { gf } ( x )\).

2 In the expansion of \(( x + 2 k ) ^ { 7 }\), where \(k\) is a non-zero constant, the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal. Find the value of \(k\).

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends \(A B E\) and \(D C F\) are identical isosceles triangles. Angle \(A B E =\) angle \(B A E = 30 ^ { \circ }\). The length of \(A D\) is 40 cm . The tank is fixed in position with the open top \(A B C D\) horizontal. Water is poured into the tank at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). The depth of water, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\) (see Fig. 2).

1. Show that, when the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by \(V = ( 40 \sqrt { } 3 ) h ^ { 2 }\).
2. Find the rate at which \(h\) is increasing when \(h = 5\).

4

1. Prove the identity \(\left( \frac { 1 } { \sin x } - \frac { 1 } { \tan x } \right) ^ { 2 } \equiv \frac { 1 - \cos x } { 1 + \cos x }\).
2. Hence solve the equation \(\left( \frac { 1 } { \sin x } - \frac { 1 } { \tan x } \right) ^ { 2 } = \frac { 2 } { 5 }\) for \(0 \leqslant x \leqslant 2 \pi\).

5
\includegraphics[max width=\textwidth, alt={}, center]{9cdb00a6-1e86-4185-bb73-ed3ecab981ba-3_560_506_258_822} The diagram shows a metal plate \(O A B C\), consisting of a right-angled triangle \(O A B\) and a sector \(O B C\) of a circle with centre \(O\). Angle \(A O B = 0.6\) radians, \(O A = 6 \mathrm {~cm}\) and \(O A\) is perpendicular to \(O C\).

1. Show that the length of \(O B\) is 7.270 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

6 Points \(A , B\) and \(C\) have coordinates \(A ( - 3,7 ) , B ( 5,1 )\) and \(C ( - 1 , k )\), where \(k\) is a constant.

1. Given that \(A B = B C\), calculate the possible values of \(k\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at \(D\).
2. Calculate the coordinates of \(D\).

7 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 0
2
- 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
5
- 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 3
p
q \end{array} \right)$$

1. In the case where \(A B C\) is a straight line, find the values of \(p\) and \(q\).
2. In the case where angle \(B A C\) is \(90 ^ { \circ }\), express \(q\) in terms of \(p\).
3. In the case where \(p = 3\) and the lengths of \(A B\) and \(A C\) are equal, find the possible values of \(q\).

8 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. In the case where \(a = 6\) and \(b = - 8\), find the range of f .
2. In the case where \(a = 5\), the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots, then \(a ^ { 2 } < 4 ( b - a )\).

9 The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 2,10 )\) and it is given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 12 } { x ^ { 3 } }\).

1. Find \(\mathrm { f } ( x )\).
2. Find the coordinates of the other stationary point.
3. Find the nature of each of the stationary points.

10
\includegraphics[max width=\textwidth, alt={}, center]{9cdb00a6-1e86-4185-bb73-ed3ecab981ba-4_634_937_696_603} The diagram shows part of the curve \(y = \sqrt { } \left( 9 - 2 x ^ { 2 } \right)\). The point \(P ( 2,1 )\) lies on the curve and the normal to the curve at \(P\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Show that \(B\) is the mid-point of \(A P\). The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\).
2. Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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