Exam-Style Questions.

1.	IB Standard

The graph of \(f(x)=8-x^2\) crosses the x-axis at the points A and B.

(a) Find the x-coordinate of A and of B.

(b) The region enclosed by the graph of \(f\) and the x-axis is revolved 360^o about the x-axis. Find the volume of the solid formed.

Worked Solution

---

2.	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 \(t=7\).

Worked Solution

---

3.	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

---

4.	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 \(x\).

Worked Solution

---

5.	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

---

6.	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.

Worked Solution

---

7.	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

---

8.

A-Level

(a) Express the algebraic fraction

$$ \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} $$

in the form

$$A + \frac{B}{5-x} + \frac{C}{1-2x} $$

where \(A\), \(B\) and \(C\) are integers.

(b) Hence show that the following integral equates to 3.03 correct to three significant figures.

$$ \int^{0.25}_0 \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} dx $$

Worked Solution

---

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

---

10.	IB Standard

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

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

Worked Solution

---

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

---

12.	IB Standard

Find \(f(x)\) if \(f'(x)=6 \sin2x\) and the graph of \(f(x)\) passes through the point \((\frac{\pi}{3},11)\).

Worked Solution

---

13.

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.)

 

 

 

Worked Solution

---

14.	IB Standard

The following diagram shows the graph of \(f(x) = \cos(e^x) \; \text{for} \; 0 \le x \le 0.5\).

(a) Find the x-intercept of the graph of \(f(x)\).

The region enclosed by the graph of \(f(x)\), the y-axis and the x-axis is rotated 360° about the x-axis.

(b) Find the volume of the solid formed.

Worked Solution

---