Questions

The function

$F(x)$ , where

$x$ is a non-negative real number, is the result of subtracting the integer part of

$x$ from

$x$ . For example,

$F(3)=0$ ,

$F(5.43)=0.43$ ,

$F(\pi)=0.14159265\ldots$ . Find an expression for:

$\tfrac{1}{3} k^{3/2} - F\!\left( \tfrac{1}{3} k^{3/2} \right)$
B)

$\tfrac{1}{3} k^{3/2} - \sum_{n=0}^{k} n\!\left( \sqrt{n+1} - \sqrt{n} \right)$
C)

$\tfrac{1}{3} k^{3/2} + \sum_{n=1}^{k-1} n\!\left( \sqrt{n+1} - \sqrt{n} \right)$
D)

$\tfrac{1}{3} k^{3/2} - \sum_{n=1}^{k-1} n\!\left( \sqrt{n+1} + \sqrt{n} \right)$
E)

$\tfrac{1}{3} k^{3/2} + \left( \sum_{n=0}^{k-1} \sqrt{n} \right) + (k-1)\sqrt{k}$
F)

$\tfrac{1}{3} k^{3/2} + \left( \sum_{n=0}^{k-1} \sqrt{n} \right) + (1-k)\sqrt{k}$

Question 2

$f(x) = 1! \cdot x^{2! \cdot x^{3! \cdot x^{4! \cdot x^{5! \cdot x^{6! \cdot x}}}}}$

$0$

$\tfrac{1}{8}$

$\tfrac{1}{4}$

$\tfrac{1}{2}$

$1$

$2$

$1000000$

Question 3

The function

$f(x)$ has the property that

$f(x) = f(6-x)$ for all real

$x$ . Given that:

$\left( \int_{2}^{3} f(x)\, dx \right)^{2} + \left( \int_{4}^{6} f(x)\, dx \right)^{2} + \left( \int_{2}^{4} f(x)\, dx \right)\left( \int_{0}^{2} f(x)\, dx \right) - \int_{6}^{3} 3f(x)\, dx = -2$

$-10$

$-5$

$-3$

$0$

$3$

$5$

$10$

$\tfrac{27}{2}$

Question 4

no solutions
one solution
two solutions
three solutions

Question 5

$a = 0$

$a = \pm 1$

$a = \pm \tfrac{1}{\sqrt{2}} \text{ or } a = 0$

$a = \pm \tfrac{1}{\sqrt{2}}$

Question 6

has

$x = 2$ as a solution
has no real solutions
has an odd number of real solutions
has twenty real solutions

Question 7

A triangle

$ABC$ is to be drawn with

$AB = 10 \,\text{cm}$ ,

$BC = 7 \,\text{cm}$ and the angle at

$A$ equal to

$\theta$ , where

$\theta$ is a certain specified angle.

Of the two possible triangles that could be drawn, the larger triangle has three times the area of the smaller one.

$\tfrac{5}{7}$

$\tfrac{151}{200}$

$\tfrac{2\sqrt{2}}{5}$

$\tfrac{\sqrt{17}}{5}$

$\tfrac{\sqrt{51}}{8}$

$\tfrac{\sqrt{34}}{8}$

Question 8

The graph of

$y = \tan(\cos(\sin(x)))$ has a period of

$P$ and a maximum value of

$M$ . Which row is correct?

$P = 1, \; M < 1$

$P = 1, \; M > 1$

$P = \tfrac{\pi}{2}, \; M < 1$

$P = \tfrac{\pi}{2}, \; M > 1$

$P = \pi, \; M < 1$

$P = \pi, \; M > 1$

$P = 2\pi, \; M < 1$

$P = 2\pi, \; M > 1$

Question 9

Find the fraction of the interval

$0 \leq \theta \leq 2\pi$ for which the inequality

$\left( \cos\theta + \sin\theta \right) \left( \tfrac{\sqrt{3}}{2} + \cos\tfrac{1}{2}\theta \right) \left( \cos^{2}\theta - 1 \right) \leq 0$

$\tfrac{1}{4}$

$\tfrac{5}{12}$

$\tfrac{1}{2}$

$\tfrac{2}{3}$

$\tfrac{3}{4}$

$\tfrac{5}{6}$

Question 10

Let

$\lfloor x \rfloor$ denote the largest whole number that is less than or equal to

$x$ . For example,

$\lfloor -\pi \rfloor = -4$ .

and

$f(x) = 0$ otherwise. Note that, for example,

$f\!\left(\tfrac{1}{2}\right) = \tfrac{4}{3}$ . The value of

$\int_{0}^{2} f(x)\, dx$ is