Exam-Style Questions.

1.	IB Analysis and Approaches

Consider the functions \(f(x)=\sin(x) \text{ and } g(x) = \tan(x+\frac{\pi}{2})\) in the region where \(\frac{\pi}{2}\le x \le \pi\)

The graphs \(y=f (x)\) and \(y = g(x)\) intersect at a point A whose x-coordinate is \(a\).

(a) Show that \(\sin^2{a}=-\cos{a}\).

(b) Hence, show that the tangents to the curves at A intersect at right angles.

(c) Find the value of \(\cos(a)\). Give your answer in the form \( \dfrac{a + \sqrt{b}}{c} \).

Worked Solution

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2.	IB Analysis and Approaches

The curve \( y=\sin(\sqrt{x}) \text{ where } x \ge 0 \) intersects the x axis at the points \(x_0, x_1, x_2, x_3, x_4, ... \) and \(x_0 = 0\).

(a) Find \(x_1\), the first x-intercept of the function to the right of the origin. Give your answer in terms of \(\pi\).

(b) Find an expression for the nth x-intercept of the curve, in terms of \(\pi\).

(c) By using an appropriate substitution, show that:

$$ \int \sin(\sqrt{x}) \; dx = 2\sin(\sqrt{x}) - 2 \sqrt{x} \cos(\sqrt{x})$$

The area of the region bounded by the curve and the x-axis is denoted by \(R_n\) where:

$$ R_n = \int^{x_{n+1}}_{x_n} y \; dx$$

(d) Calculate the area of region \(R_n\) giving your answer in terms of \(\pi\).

(e) Hence, show that the areas of the regions bounded by the curve and the x-axis, form an arithmetic sequence.

Worked Solution

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3.	IB Analysis and Approaches

A function \( f \) is defined by \( f(x) = \arccos\left(\frac{x^2}{1-x^4}\right) \), \( x \in \mathbb{R} \), \( 0.785 \lt |x| \lt 5\).

(a) Show that \( f \) is an even function.

(b) Find the equations of the horizontal asymptote of \( y = f(x) \).

(c) Find \( f'(x) \) for \( x \in \mathbb{R} \), \( x \neq \pm1 \).

(d) Determine whether \( f \) is increasing or decreasing for \( x > 1 \).

(e) Find an expression for \( f^{-1}(x) \), for \( x \in \mathbb{R} \), \( x \neq \pm1\)

Worked Solution

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