CAIE P1 (Pure Mathematics 1) 2018 November

1 Showing all necessary working, solve the equation \(4 x - 11 x ^ { \frac { 1 } { 2 } } + 6 = 0\).

2 A line has equation \(y = x + 1\) and a curve has equation \(y = x ^ { 2 } + b x + 5\). Find the set of values of the constant \(b\) for which the line meets the curve.

3 Two points \(A\) and \(B\) have coordinates ( \(3 a , - a\) ) and ( \(- a , 2 a\) ) respectively, where \(a\) is a positive constant.

1. Find the equation of the line through the origin parallel to \(A B\).
2. The length of the line \(A B\) is \(3 \frac { 1 } { 3 }\) units. Find the value of \(a\).

4 The first term of a series is 6 and the second term is 2 .

1. For the case where the series is an arithmetic progression, find the sum of the first 80 terms.
2. For the case where the series is a geometric progression, find the sum to infinity.

5

1. Show that the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ may be expressed as \(9 \cos ^ { 2 } \theta - 22 \cos \theta + 4 = 0\).
2. Hence solve the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 A curve has a stationary point at \(\left( 3,9 \frac { 1 } { 2 } \right)\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + a ^ { 2 } x\), where \(a\) is a non-zero constant.

1. Find the value of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-08_67_1569_461_328}
2. Find the equation of the curve.
3. Determine, showing all necessary working, the nature of the stationary point.

7
\includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-10_503_853_260_641} The diagram shows part of the curve with equation \(y = k \left( x ^ { 3 } - 7 x ^ { 2 } + 12 x \right)\) for some constant \(k\). The curve intersects the line \(y = x\) at the origin \(O\) and at the point \(A ( 2,2 )\).

1. Find the value of \(k\).
2. Verify that the curve meets the line \(y = x\) again when \(x = 5\).
3. Find, showing all necessary working, the area of the shaded region.

8
\includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-12_595_748_260_699} The diagram shows a solid figure \(O A B C D E F\) having a horizontal rectangular base \(O A B C\) with \(O A = 6\) units and \(A B = 3\) units. The vertical edges \(O F , A D\) and \(B E\) have lengths 6 units, 4 units and 4 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O F\) respectively.

1. Find \(\overrightarrow { D F }\).
2. Find the unit vector in the direction of \(\overrightarrow { E F }\).
3. Use a scalar product to find angle \(E F D\).

9
\includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-14_465_677_260_733} The diagram shows a triangle \(O A B\) in which angle \(A B O\) is a right angle, angle \(A O B = \frac { 1 } { 5 } \pi\) radians and \(A B = 5 \mathrm {~cm}\). The arc \(B C\) is part of a circle with centre \(A\) and meets \(O A\) at \(C\). The arc \(C D\) is part of a circle with centre \(O\) and meets \(O B\) at \(D\). Find the area of the shaded region.

10 A curve has equation \(y = \frac { 1 } { 2 } ( 4 x - 3 ) ^ { - 1 }\). The point \(A\) on the curve has coordinates \(\left( 1 , \frac { 1 } { 2 } \right)\).

1. (a) Find and simplify the equation of the normal through \(A\).
   (b) Find the \(x\)-coordinate of the point where this normal meets the curve again.
2. A point is moving along the curve in such a way that as it passes through \(A\) its \(x\)-coordinate is decreasing at the rate of 0.3 units per second. Find the rate of change of its \(y\)-coordinate at \(A\).

11

1. The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 1\) for \(x < a\), where \(a\) is a constant.
   1. State the greatest possible value of \(a\).
   2. It is given that \(a\) takes this greatest possible value. State the range of f and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. The function g is defined by \(\mathrm { g } ( x ) = ( x - 3 ) ^ { 2 }\) for \(x \geqslant 0\).
   1. Show that \(\operatorname { gg } ( 2 x )\) can be expressed in the form \(( 2 x - 3 ) ^ { 4 } + b ( 2 x - 3 ) ^ { 2 } + c\), where \(b\) and \(c\) are constants to be found.
   2. Hence expand \(\operatorname { gg } ( 2 x )\) completely, simplifying your answer.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.