Exam-Style Questions.

1.	IB Analysis and Approaches

Find:

$$ \int^{16}_9 \frac{6-2\sqrt{x}}{\sqrt{x}} dx $$

by first writing the algebraic fraction in the form \(ax^b+c\).

Worked Solution

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2.	IB Analysis and Approaches

(a) Find \(\int (4x+5) dx\).

(b) Given \(f'(x) = 4x+5\) find \(f(x)\) if \(f(3.4) = 10.12\).

Worked Solution

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3.	IB Analysis and Approaches

A particle moves in a straight line. During the first nine seconds the velocity, \(v\) ms^-1 of the particle at time \(t\) seconds is given by:

$$ v(t) = t \cos(t+5) $$

The following diagram shows the graph of v:

(a) Find the maximum value of \(v\).

(b) Find the acceleration of the particle when t = 6.

(c) Find the total distance travelled by the particle.

Worked Solution

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4.	IB Analysis and Approaches

Consider the region where \(0 \lt x \lt 2\pi \) and \(\sin{x} \gt \cos{2x} \)

Find the area enclosed by the graphs of \(y=\sin{x} \) and \(y= \cos{2x} \)

Worked Solution

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5.	IB Analysis and Approaches

The function \(f\) is defined by \(f(x) = 8 - 5 \sin{x} \), for \( x \ge 0 \).

The diagram shows part of the graph of \(y = f (x) \).

The shaded region is enclosed by the graph of \(y=f(x)\) and the x-axis for the first three periods of the function.

(a) Find the exact value of the x-coordinate of the right side of the shaded region.

(b) Show that the area of the shaded region is \( 48 \pi \).

A hemisphere has a total surface area in square centimetres equal to the shaded area in the previous diagram.

(c) Find the radius of the hemisphere.

Worked Solution

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6.	IB Analysis and Approaches

A particle moves in a straight line such that its velocity, \(v\) ms^-1, at time \(t\) seconds is given by:

$$ v(t)=10e^{-\frac{t}{7}} \sin\left(\frac{t}{2}\right) $$

for \( 0 \le t \le 4 \pi\). The graph of \(v\) is shown in the following diagram.

Let \(t_1 \) be the first time when the particle's acceleration is zero.

(a) Find the value of \(t_1\).

(b) Find the distance travelled by the particle between \(t = t_1 \), and t = \(4 \pi \).

Worked Solution

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7.	IB Standard

The diagram shows part of the graph of \(f(x)=Ae^{kx}+2\).

The y-intercept is at \((0,10)\).

(a) Show that \(A=8\).

(b) Given that \(f(8)=3.62\) (correct to 3 significant figures), find the value of \(k\).

(c) (i) Using your value of \(k\), find \(f'(x)\).

    (ii) Hence, explain why \(f\) is a decreasing function.

    (iii) Find the equation of the horizontal asymptote of the graph \(f\).

Let \(g(x)=-x^2+7x+5\)

(d) Find the area enclosed by the graphs of \(f\) and \(g\).

Worked Solution

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8.	IB Standard

Consider the graph of the function \(f(x)=x^2+2\).

(a) Find the area between the graph of \(f\) and the x-axis for \(2\le x \le 3\).

(b) If the area described above is rotated 360^o around the x-axis find the volume of the solid formed.

Worked Solution

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9.	IB Standard

Make a sketch of a graph showing the velocity (in \(ms^{-1}\)) against time of a particle travelling for six seconds according to the equation:

$$v=e^{\sin t}-1$$

(a) Find the point at which the graph crosses the \(t\) axis.

(b) How far does the particle travel during these first six seconds?

Worked Solution

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10.	IB Standard

Find the value of \(a\) if \(\pi \lt a \lt 2\pi\) and:

$$ \int_\pi^a sin3x dx = -\frac13$$

Worked Solution

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11.	IB Standard

This graph represents the function \(f:x\to a \cos x, a\in \mathbf N\)

(a) Find the value of \(a\).

(b) Find the area of the shaded region.

Worked Solution

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12.	IB Analysis and Approaches

Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \).

(a) Find an expression for the first derivative \(f'(x) \).

The graph of \(f\) has exactly one maximum point at P.

(b) Find the x-coordinate of P.

The graph of \(f\) has exactly one point of inflection at Q.

(c) Find the x-coordinate of Q.

(d) The region enclosed by the graph of \(f\), the x-axis, and the vertical lines through P and Q has an area of one square unit, find the value of \(k\).

Worked Solution

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