# 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

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

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

### 3.

A-Level

(a) Find $$\frac{dy}{dx}$$ when:

$$y= (7-5x^2)^{ \frac12 }$$

(b) Find the following integral:

$$\int (1 - cos3x) dx$$

### 4.

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.

### 5.

IB Applications and Interpretation

The following diagram shows part of the graph of:

$$f (x) = \frac{(9 + 4x) (6 - x)}{10}, x \in \mathbf R$$

The shaded region A is bounded by the x-axis, y-axis and the graph of $$f$$.

(a) Write down an integral for the area of region A.

(b) Find the area of region A.

The three points A (0 , 0) , B (6 , 9) and C (p , 0) define the vertices of a triangle.

(c) Find the value of p, the x-coordinate of C , such that the area of the triangle is equal to the area of region A.

### 6.

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 360o around the x-axis find the volume of the solid formed.

### 7.

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.

### 8.

A-Level

The diagram shows a sketch of the curve C with equation:

$$y=2x^{\frac32}-4x+3$$

The point P with coordinates (4, 3) lies on C.

The line L is the tangent to C at the point P.

The region A, shown shaded, is bounded by the curve C, the line L and the y-axis.

Find the area of A making your method clear.

Note that solutions based entirely on graphical or numerical methods are not acceptable.)

### 9.

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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