# Exam-Style Questions.

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

### 1.

IB Studies

A function is given as $$f(x)=3x^2-6x+4+\frac3x,-2\le x \le 4, x\ne 0$$.

(a) Find the derivative of the function.

(b) Find the coordinates of the local minimum point of $$f(x)$$ in the given domain using your calculator.

### 2.

IB Studies

Consider the graph of the function $$f(x)=7-3x^2-x^3$$

(a) Label the local maximum as A on the graph.

(b) Label the local minimum as B on the graph.

(c) Write down the interval where $$f(x)>5$$.

(d) Draw the tangent to the curve at $$x=-3$$ on the graph.

(e) Write down the equation of the tangent at $$x=-3$$.

### 3.

IB Standard

Consider the graph of $$f(x)=a\sin(b(x+c))+12$$, for $$0\le x\le 24$$.

The graph has a maximum at (8, 22) and a minimum at (18, 2).

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

(b) Find the value of $$b$$.

(c) Find the value of $$c$$.

(d) Solve $$f(x)=5$$.

### 4.

A-Level

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

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

(b) Find the following integral:

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

### 5.

IB Standard

The diagram shows part of the graph of $$y=asinbx+c$$ with a minimum at $$(-2.5,-2)$$ and a maximum at $$(2.5,4)$$.

(a) Find $$a$$.

(b) Find $$b$$.

(c) Find $$c$$.

### 6.

IB Analysis and Approaches

Consider the cubic function $$f(x)=\frac{1}{6}x^3-2x^2+6x-2$$

(a) Find $$f'(x)$$

The graph of $$f$$ has horizontal tangents at the points where $$x = a$$ and $$x = b$$ where $$a < b$$.

(b) Find the value of $$a$$ and the value of $$b$$

(c) Sketch the graph of $$y = f'(x)$$.

(d) Hence explain why the graph of $$f$$ has a local maximum point at $$x = a$$.

(e) Find $$f''(b)$$.

(f) Hence, use your answer to part (e) to show that the graph of $$f$$ has a local minimum point at $$x = b$$.

(g) Find the coordinates of the point where the normal to the graph of $$f$$ at $$x = a$$ and the tangent to the graph of $$f$$ at $$x = b$$ intersect.

### 7.

IB Studies

Consider the function $$f(x)=x^3-9x+2$$.

(a) Sketch the graph of $$y=f(x)$$ for $$-4\le x\le 4$$ and $$-14\le y\le 14$$ showing clearly the axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit on the x-axis, and a scale of 1 cm to represent 2 units on the y-axis.

(b) Find the value of $$f(-1)$$.

(c) Write down the coordinates of the y-intercept of the graph of $$f(x)$$.

(d) Find $$f'(x)$$.

(e) Find $$f'(-1)$$

(f) Explain what $$f'(-1)$$ represents.

(g) Find the equation of the tangent to the graph of $$f(x)$$ at the point where x is –1.

R and S are points on the curve such that the tangents to the curve at these points are horizontal. The x-coordinate of R is $$a$$ , and the x-coordinate of S is $$b$$ , $$b \gt a$$.

(h) Write down the value of $$a$$ ;

(i) Write down the value of $$b$$.

(j) Describe the behaviour of $$f(x)$$ for $$a \lt x \lt b$$.

### 8.

IB Analysis and Approaches

The displacement, in millimetres, of a particle from an origin, O, at time t seconds, is given by $$s(t) = t^3 cos t + 5t sin t$$ where $$0 \le t \le 5$$ .

(a) Find the maximum distance of the particle from O.

(b) Find the acceleration of the particle at the instant it first changes direction.

### 9.

IB Standard

The function $$f$$ is defined as follows:

$$f(x)=\frac{122}{1+60e^{-0.3x}}$$

(a) Calculate $$f(0)$$.

(b) Find a value of $$x$$ for which $$f(x)=85$$

(c) Find the range of $$f$$.

(d) Show that:

$$f'(x)=\frac{2196e^{-0.3x}}{(1+60e^{-0.3x})^2}$$

(e) Find the maximum rate of change of $$f$$.

### 10.

IB Standard

A particle P moves along a straight line. The velocity $$v$$ in metres per second of P after $$t$$ seconds is given by $$v(t) = 3\sin{t} - 8t^{\cos{t}}, 0 \le t \le 7$$.

(a) Find the initial velocity of P.

(b) Find the maximum speed of P.

(c) Write down the number of times that the acceleration of P is 0 ms-2.

(d) Find the acceleration of P at a time of 5 seconds.

(e) Find the total distance travelled by P.

### 11.

IB Analysis and Approaches

Consider a function $$f$$, such that $$f(x)=7.2\sin(\frac{\pi}{6}x + 2) + b$$ where $$0\le x \le 12$$

(a) Find the period of $$f$$.

The function f has a local maximum at the point (11.18,10.3) , and a local minimum at (5.18.-4.1).

(b) Find the value of b.

(c) Hence, find the value of $$f(7)$$.

A second function $$g$$ is given by $$g(x)=a\sin(\frac{2\pi}{7}(x -4)) + c$$ where $$0 \le x \le 10$$

The function $$g$$ passes through the points (2.25,-3) and (5.75,7).

(d) Find the value of $$a$$ and the value of $$c$$.

(e) Find the value of x for which the functions have the greatest difference.

### 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 $$k$$.

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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