CAIE P1 (Pure Mathematics 1) 2021 June

1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { x ^ { 4 } } + 32 x ^ { 3 }\). It is given that the curve passes through the point \(\left( \frac { 1 } { 2 } , 4 \right)\). Find the equation of the curve.

2 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410 . Find the 60th term of the progression.

3

1. Find the first three terms in the expansion of \(( 3 - 2 x ) ^ { 5 }\) in ascending powers of \(x\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + x ) ^ { 2 } ( 3 - 2 x ) ^ { 5 }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-05_677_1591_260_278} The diagram shows part of the graph of \(y = a \tan ( x - b ) + c\).
Given that \(0 < b < \pi\), state the values of the constants \(a , b\) and \(c\).

5 The fifth, sixth and seventh terms of a geometric progression are \(8 k , - 12\) and \(2 k\) respectively. Given that \(k\) is negative, find the sum to infinity of the progression.

6 The equation of a curve is \(y = ( 2 k - 3 ) x ^ { 2 } - k x - ( k - 2 )\), where \(k\) is a constant. The line \(y = 3 x - 4\) is a tangent to the curve. Find the value of \(k\).

7

1. Prove the identity \(\frac { 1 - 2 \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \equiv 1 - \tan ^ { 2 } \theta\).
2. Hence solve the equation \(\frac { 1 - 2 \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } = 2 \tan ^ { 4 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8
\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre \(C\). The boundary of the plate consists of two \(\operatorname { arcs } P S\) and \(Q R\) of the original circle and two semicircles with \(P Q\) and \(R S\) as diameters. The radius of the circle with centre \(C\) is 4 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.

1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
2. Find the exact perimeter of the plate.
3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).

9 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } - 4 \text { for } x \geqslant 2 ,
& \mathrm {~g} ( x ) = a x + 2 \text { for } x \in \mathbb { R } , \end{aligned}$$ where \(a\) is a constant.

1. State the range of f.
2. Find \(\mathrm { f } ^ { - 1 } ( x )\).
3. Given that \(a = - \frac { 5 } { 3 }\), solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\).
4. Given instead that \(\operatorname { ggf } ^ { - 1 } ( 12 ) = 62\), find the possible values of \(a\).

10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).

1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).

11 The equation of a curve is \(y = 2 \sqrt { 3 x + 4 } - x\).

1. Find the equation of the normal to the curve at the point (4,4), giving your answer in the form \(y = m x + c\).
2. Find the coordinates of the stationary point.
3. Determine the nature of the stationary point.
4. Find the exact area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.