CAIE P1 (Pure Mathematics 1) 2021 June

1

1. Express \(16 x ^ { 2 } - 24 x + 10\) in the form \(( 4 x + a ) ^ { 2 } + b\).
2. It is given that the equation \(16 x ^ { 2 } - 24 x + 10 = k\), where \(k\) is a constant, has exactly one root. Find the value of this root.

2

1. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 2 \mathrm { f } ( x - 1 )\).
   Describe fully the two single transformations which have been combined to give the resulting transformation.
2. The curve \(y = \sin 2 x - 5 x\) is reflected in the \(y\)-axis and then stretched by scale factor \(\frac { 1 } { 3 }\) in the \(x\)-direction. Write down the equation of the transformed curve.

3 The equation of a curve is \(y = ( x - 3 ) \sqrt { x + 1 } + 3\). The following points lie on the curve. Non-exact values are rounded to 4 decimal places. $$A ( 2 , k ) \quad B ( 2.9,2.8025 ) \quad C ( 2.99,2.9800 ) \quad D ( 2.999,2.9980 ) \quad E ( 3,3 )$$

1. Find \(k\), giving your answer correct to 4 decimal places.
2. Find the gradient of \(A E\), giving your answer correct to 4 decimal places.
   The gradients of \(B E , C E\) and \(D E\), rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively.
3. State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point \(E\).

4 The coefficient of \(x\) in the expansion of \(\left( 4 x + \frac { 10 } { x } \right) ^ { 3 }\) is \(p\). The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { k } { x ^ { 2 } } \right) ^ { 5 }\) is \(q\). Given that \(p = 6 q\), find the possible values of \(k\).

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

1. Find and simplify an expression for \(\mathrm { ff } ( x )\).
2. Solve the equation \(\mathrm { ff } ( x ) = 34 x ^ { 2 } + 19\).

6 Points \(A\) and \(B\) have coordinates \(( 8,3 )\) and \(( p , q )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(y = - 2 x + 4\). Find the values of \(p\) and \(q\).

7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).

1. Show that \(l\) is the tangent to the circle at \(A\).
2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).

8 The first, second and third terms of an arithmetic progression are \(a , \frac { 3 } { 2 } a\) and \(b\) respectively, where \(a\) and \(b\) are positive constants. The first, second and third terms of a geometric progression are \(a , 18\) and \(b + 3\) respectively.

1. Find the values of \(a\) and \(b\).
2. Find the sum of the first 20 terms of the arithmetic progression.

9
\includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-10_540_1113_260_516} The diagram shows part of the curve with equation \(y ^ { 2 } = x - 2\) and the lines \(x = 5\) and \(y = 1\). The shaded region enclosed by the curve and the lines is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained.

10

1. Prove the identity \(\frac { 1 + \sin x } { 1 - \sin x } - \frac { 1 - \sin x } { 1 + \sin x } \equiv \frac { 4 \tan x } { \cos x }\).
2. Hence solve the equation \(\frac { 1 + \sin x } { 1 - \sin x } - \frac { 1 - \sin x } { 1 + \sin x } = 8 \tan x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).

1. Find the value of \(k\).
   ................................................................................................................................................. . .
2. Find the equation of the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).

12
\includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-16_598_609_264_769} The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm , held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are \(A , B , C , D\), \(E\) and \(F\). Points \(P\) and \(Q\) are situated where straight sections of the rope meet the pipe with centre \(A\).

1. Show that angle \(P A Q = \frac { 1 } { 3 } \pi\) radians.
2. Find the length of the rope.
3. Find the area of the hexagon \(A B C D E F\), giving your answer in terms of \(\sqrt { 3 }\).
4. Find the area of the complete region enclosed by the rope.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.