# Exam-Style Questions.

## Problems adapted from questions set for previous Mathematics exams.

### 1.

IB Standard

The acceleration, $$a$$ ms-2 , of an object at time $$t$$ seconds is given by

$$a=\frac1t+4sin3t, (t\ge1)$$

The object is at rest when $$t=1$$.

Find the velocity of the object when $$t=7$$.

### 2.

IB Analysis and Approaches

Given that $$\frac{dy}{dx} = \sin(x + \frac{\pi}{3})$$ and $$y = 5$$ when $$x = \frac{8\pi}{3}$$, find $$y$$ in terms of $$x$$.

### 3.

A-Level

(a) Using a suitable substitution, or otherwise, find

$$\int \frac{x}{(3x^2 - 5)^2} dx$$

(b) Solve the differential equation below giving your answer in the form $$y = f(x)$$. It is given that given that y = $$\frac{1}{2}$$ when x = 0.

$$\frac{dy}{dx} = \frac{2xy^3}{(3x^2 - 5)^2}$$

### 4.

IB Analysis and Approaches

Find $$f(x)$$ if:

$$f'(x) = \frac{12x}{\sqrt{3x^2+4}}$$

given that $$f(0) = 10$$

### 5.

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 $$k$$.

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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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