CAIE P1 (Pure Mathematics 1) 2019 March

The coefficient of \(x^3\) in the expansion of \((1 - px)^5\) is \(-2160\). Find the value of the constant \(p\). [3]

A curve with equation \(y = f(x)\) passes through the points \((0, 2)\) and \((3, -1)\). It is given that \(f'(x) = kx^2 - 2x\), where \(k\) is a constant. Find the value of \(k\). [5]

\includegraphics{figure_3} In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]

A curve has equation \(y = (2x - 1)^{-1} + 2x\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]

Two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), are such that $$\mathbf{u} = \begin{pmatrix} q \\ 1 \\ 6 \end{pmatrix} \quad \text{and} \quad \mathbf{v} = \begin{pmatrix} 8 \\ q - 1 \\ q^2 - 7 \end{pmatrix},$$ where \(q\) is a constant.

1. Find the values of \(q\) for which \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\). [3]
2. Find the angle between \(\mathbf{u}\) and \(\mathbf{v}\) when \(q = 0\). [4]

1. The first and second terms of a geometric progression are \(p\) and \(2p\) respectively, where \(p\) is a positive constant. The sum of the first \(n\) terms is greater than \(1000p\). Show that \(2^n > 1001\). [2]
2. In another case, \(p\) and \(2p\) are the first and second terms respectively of an arithmetic progression. The \(n\)th term is \(336\) and the sum of the first \(n\) terms is \(7224\). Write down two equations in \(n\) and \(p\) and hence find the values of \(n\) and \(p\). [5]

1. Solve the equation \(3\sin^2 2\theta + 8\cos 2\theta = 0\) for \(0° < \theta < 180°\). [5]
2. \includegraphics{figure_7b} The diagram shows part of the graph of \(y = a + \tan bx\), where \(x\) is measured in radians and \(a\) and \(b\) are constants. The curve intersects the \(x\)-axis at \(\left(-\frac{1}{6}\pi, 0\right)\) and the \(y\)-axis at \((0, \sqrt{3})\). Find the values of \(a\) and \(b\). [3]

1. Express \(x^2 - 4x + 7\) in the form \((x + a)^2 + b\). [2]

The function \(f\) is defined by \(f(x) = x^2 - 4x + 7\) for \(x < k\), where \(k\) is a constant.

1. State the largest value of \(k\) for which \(f\) is a decreasing function. [1]

The value of \(k\) is now given to be \(1\).

1. Find an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\). [3]
2. The function \(g\) is defined by \(g(x) = \frac{2}{x-1}\) for \(x > 1\). Find an expression for \(gf(x)\) and state the range of \(gf\). [4]

\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \sqrt{x^3 + x^2}\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [4]
2. \(P\) is the point on the curve with \(x\)-coordinate \(3\). Find the \(y\)-coordinate of the point where the normal to the curve at \(P\) crosses the \(y\)-axis. [6]

\includegraphics{figure_10} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}}\).

1. The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(AB\). [6]
2. The tangent to the curve at a point \(T\) is parallel to \(AB\). Find the coordinates of \(T\). [3]
3. Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(AB\). [3]