CAIE P1 (Pure Mathematics 1) 2019 June

The function f is defined by \(\text{f}(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\).

1. Express \(x^2 - 4x + 8\) in the form \((x - a)^2 + b\). [2]
2. Hence find the set of values of \(x\) for which \(\text{f}(x) < 9\), giving your answer in exact form. [3]

1. In the binomial expansion of \(\left(2x - \frac{1}{2x}\right)^5\), the first three terms are \(32x^5 - 40x^3 + 20x\). Find the remaining three terms of the expansion. [3]
2. Hence find the coefficient of \(x\) in the expansion of \((1 + 4x^2)\left(2x - \frac{1}{2x}\right)^5\). [2]

\includegraphics{figure_3} The diagram shows triangle \(ABC\) which is right-angled at \(A\). Angle \(ABC = \frac{1}{4}\pi\) radians and \(AC = 8\) cm. The points \(D\) and \(E\) lie on \(BC\) and \(BA\) respectively. The sector \(ADE\) is part of a circle with centre \(A\) and is such that \(BDC\) is the tangent to the arc \(DE\) at \(D\).

1. Find the length of \(AD\). [3]
2. Find the area of the shaded region. [3]

The function f is defined by \(\text{f}(x) = \frac{48}{x - 1}\) for \(3 \leqslant x \leqslant 7\). The function g is defined by \(\text{g}(x) = 2x - 4\) for \(a \leqslant x \leqslant b\), where \(a\) and \(b\) are constants.

1. Find the greatest value of \(a\) and the least value of \(b\) which will permit the formation of the composite function gf. [2] It is now given that the conditions for the formation of gf are satisfied.
2. Find an expression for \(\text{gf}(x)\). [1]
3. Find an expression for \((\text{gf})^{-1}(x)\). [2]

Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of 13 kg. At the end of week 1 they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer A's weight loss in week 2 is 0.98 kg. It is given that his weekly weight loss follows an arithmetic progression.

1. Write down an expression for his total weight loss after \(x\) weeks. [1]
2. He reaches his 13 kg target during week \(n\). Use your answer to part (i) to find the value of \(n\). [2]

Boxer B's weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a geometric progression.

1. Calculate his total weight loss after 20 weeks and show that he can never reach his target. [4]

\includegraphics{figure_6} The diagram shows a solid figure \(ABCDEF\) in which the horizontal base \(ABC\) is a triangle right-angled at \(A\). The lengths of \(AB\) and \(AC\) are 8 units and 4 units respectively and \(M\) is the mid-point of \(AB\). The point \(D\) is 7 units vertically above \(A\). Triangle \(DEF\) lies in a horizontal plane with \(DE\), \(DF\) and \(FE\) parallel to \(AB\), \(AC\) and \(CB\) respectively and \(N\) is the mid-point of \(FE\). The lengths of \(DE\) and \(DF\) are 4 units and 2 units respectively. Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(\overrightarrow{AB}\), \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) respectively.

1. Find \(\overrightarrow{MF}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
2. Find \(\overrightarrow{FN}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1]
3. Find \(\overrightarrow{MN}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
4. Use a scalar product to find angle \(FMN\). [4]

The coordinates of two points \(A\) and \(B\) are \((1, 3)\) and \((9, -1)\) respectively and \(D\) is the mid-point of \(AB\). A point \(C\) has coordinates \((x, y)\), where \(x\) and \(y\) are variables.

1. State the coordinates of \(D\). [1]
2. It is given that \(CD^2 = 20\). Write down an equation relating \(x\) and \(y\). [1]
3. It is given that \(AC\) and \(BC\) are equal in length. Find an equation relating \(x\) and \(y\) and show that it can be simplified to \(y = 2x - 9\). [3]
4. Using the results from parts (ii) and (iii), and showing all necessary working, find the possible coordinates of \(C\). [4]

A curve is such that \(\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b\). The curve has stationary points at \((-1, 2)\) and \((3, k)\). Find the values of the constants \(a\), \(b\) and \(k\). [8]

\includegraphics{figure_9} The function f : \(x \mapsto p \sin^2 2x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \text{f}(x)\).

1. In terms of \(p\) and \(q\), state the range of f. [2]
2. State the number of solutions of the following equations.
   1. \(\text{f}(x) = p + q\) [1]
   2. \(\text{f}(x) = q\) [1]
   3. \(\text{f}(x) = \frac{1}{2}p + q\) [1]
3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\text{f}(x) = 4\), showing all necessary working. [5]

\includegraphics{figure_10} The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{3}}\) and the tangent to the curve at the point \(A\). The \(x\)-coordinate of \(A\) is 4.

1. Find the equation of the tangent to the curve at \(A\). [5]
2. Find, showing all necessary working, the area of the shaded region. [5]
3. A point is moving along the curve. At the point \(P\) the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of \(P\). [3]