CAIE P1 (Pure Mathematics 1) 2023 June

The equation of a curve is such that \(\frac{dy}{dx} = \frac{4}{(x-3)^2}\) for \(x > 3\). The curve passes through the point \((4, 5)\). Find the equation of the curve. [3]

The coefficient of \(x^4\) in the expansion of \((x + a)^6\) is \(p\) and the coefficient of \(x^2\) in the expansion of \((ax + 3)^4\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\). [4]

1. Express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\). [2]
2. Hence or otherwise find the set of values of \(p\) for which the equation \(4x^2 - 24x + p = 0\) has no real roots. [1]

Solve the equation \(8x^6 + 215x^3 - 27 = 0\). [3]

\includegraphics{figure_5} The diagram shows the curve with equation \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) for \(x > 0\). The curve meets the \(x\)-axis at the points \((0, 0)\) and \((4, 0)\). Find the area of the shaded region. [4]

\includegraphics{figure_6} The diagram shows a sector \(OAB\) of a circle with centre \(O\). Angle \(AOB = \theta\) radians and \(OP = AP = x\).

1. Show that the arc length \(AB\) is \(2x\theta \cos \theta\). [2]
2. Find the area of the shaded region \(APB\) in terms of \(x\) and \(\theta\). [4]

1. 1. By first expanding \((\cos \theta + \sin \theta)^2\), find the three solutions of the equation $$(\cos \theta + \sin \theta)^2 = 1$$ for \(0 \leqslant \theta \leqslant \pi\). [3]
   2. Hence verify that the only solutions of the equation \(\cos \theta + \sin \theta = 1\) for \(0 < \theta < \pi\) are \(0\) and \(\frac{1}{2}\pi\). [2]
2. Prove the identity \(\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} \equiv \frac{\cos \theta + \sin \theta - 1}{1 - 2\sin^2 \theta}\). [3]
3. Using the results of (a)(ii) and (b), solve the equation $$\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} = 2(\cos \theta + \sin \theta - 1)$$ for \(0 \leqslant \theta \leqslant \pi\). [3]

\includegraphics{figure_8} The diagram shows the graph of \(y = f(x)\) where the function \(f\) is defined by $$f(x) = 3 + 2\sin \frac{1}{4}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$

1. On the diagram above, sketch the graph of \(y = f^{-1}(x)\). [2]
2. Find an expression for \(f^{-1}(x)\). [2]
3. \includegraphics{figure_8c} The diagram above shows part of the graph of the function \(g(x) = 3 + 2\sin \frac{1}{4}x\) for \(-2\pi \leqslant x \leqslant 2\pi\). Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]
4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\) to the graph of \(y = f(x)\), making clear the order in which the transformations are applied. [6]

The second term of a geometric progression is 16 and the sum to infinity is 100.

1. Find the two possible values of the first term. [4]
2. Show that the \(n\)th term of one of the two possible geometric progressions is equal to \(4^{n-2}\) multiplied by the \(n\)th term of the other geometric progression. [4]

The equation of a circle is \((x - a)^2 + (y - 3)^2 = 20\). The line \(y = \frac{1}{2}x + 6\) is a tangent to the circle at the point \(P\).

1. Show that one possible value of \(a\) is 4 and find the other possible value. [5]
2. For \(a = 4\), find the equation of the normal to the circle at \(P\). [4]
3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b). [4]

The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.

1. Find \(\frac{dy}{dx}\). [2]
2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan^{-1}(2)\) with the positive \(x\)-axis. [4]