CAIE P1 (Pure Mathematics 1) 2016 November

Find the set of values of \(k\) for which the curve \(y = kx^2 - 3x\) and the line \(y = x - k\) do not meet. [3]

The coefficient of \(x^3\) in the expansion of \((1 - 3x)^6 + (1 + ax)^5\) is 100. Find the value of the constant \(a\). [4]

Showing all necessary working, solve the equation \(6\sin^2 x - 5\cos^2 x = 2\sin^2 x + \cos^2 x\) for \(0° \leq x \leq 360°\). [4]

The function \(f\) is such that \(f(x) = x^3 - 3x^2 - 9x + 2\) for \(x > n\), where \(n\) is an integer. It is given that \(f\) is an increasing function. Find the least possible value of \(n\). [4]

\includegraphics{figure_1} The diagram shows a major arc \(AB\) of a circle with centre \(O\) and radius 6 cm. Points \(C\) and \(D\) on \(OA\) and \(OB\) respectively are such that the line \(AB\) is a tangent at \(E\) to the arc \(CED\) of a smaller circle also with centre \(O\). Angle \(COD = 1.8\) radians.

1. Show that the radius of the arc \(CED\) is 3.73 cm, correct to 3 significant figures. [2]
2. Find the area of the shaded region. [4]

Three points, \(A\), \(B\) and \(C\), are such that \(B\) is the mid-point of \(AC\). The coordinates of \(A\) are \((2, m)\) and the coordinates of \(B\) are \((n, -6)\), where \(m\) and \(n\) are constants.

1. Find the coordinates of \(C\) in terms of \(m\) and \(n\). [2]

The line \(y = x + 1\) passes through \(C\) and is perpendicular to \(AB\).

1. Find the values of \(m\) and \(n\). [5]

\includegraphics{figure_2} The diagram shows a triangular pyramid \(ABCD\). It is given that $$\overrightarrow{AB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}, \quad \overrightarrow{AC} = \mathbf{i} - 2\mathbf{j} - \mathbf{k} \quad \text{and} \quad \overrightarrow{AD} = \mathbf{i} + 4\mathbf{j} - 7\mathbf{k}.$$

1. Verify, showing all necessary working, that each of the angles \(DAB\), \(DAC\) and \(CAB\) is \(90°\). [3]
2. Find the exact value of the area of the triangle \(ABC\), and hence find the exact value of the volume of the pyramid. [4]

[The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac{1}{3}Ah\).]

1. Express \(4x^2 + 12x + 10\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Functions \(f\) and \(g\) are both defined for \(x > 0\). It is given that \(f(x) = x^2 + 1\) and \(fg(x) = 4x^2 + 12x + 10\). Find \(g(x)\). [1]
3. Find \((fg)^{-1}(x)\) and give the domain of \((fg)^{-1}\). [4]

1. Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are \(6\) and \(6r\) respectively. The first and second terms of \(Q\) are \(12\) and \(-12r\) respectively. Find the value of the common sum to infinity. [3]
2. The first term of an arithmetic progression is \(\cos\theta\) and the second term is \(\cos\theta + \sin^2\theta\), where \(0 \leq \theta \leq \pi\). The sum of the first \(13\) terms is \(52\). Find the possible values of \(\theta\). [5]

A curve is such that \(\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-\frac{3}{2}}\), where \(a\) is a positive constant. The point \(A(a^2, 3)\) lies on the curve. Find, in terms of \(a\),

1. the equation of the tangent to the curve at \(A\), simplifying your answer, [3]
2. the equation of the curve. [4]

It is now given that \(B(16, 8)\) also lies on the curve.

1. Find the value of \(a\) and, using this value, find the distance \(AB\). [5]

A curve has equation \(y = (kx - 3)^{-1} + (kx - 3)\), where \(k\) is a non-zero constant.

1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point. Justify your answers. [7]

1. \includegraphics{figure_3} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360°\) about the \(x\)-axis. [5]