Mathematics

Perimeter = 36 cm



What is the area of an equilateral triangle that has a perimeter of 36 centimeters? Round to the nearest square centimeter.



A. 15 square centimeters

B. 25 square centimeters

C. 62 square centimeters

D. 72 square centimeters



Heron's formula: Area = [tex]$\sqrt{s(s-\alpha)(s-b)(s-c)}$[/tex]

The mean test scores with standard deviations of four English classes are given.



| Class | Mean | Standard Deviation |

|------------|------|--------------------|

| Mrs. Jones | 89 | 1.9 |

| Mrs. Rijo | 82 | 1.4 |

| Mr. Phan | 73 | 3.4 |

| Mrs. Scott | 90 | 6.1 |



Which statement is most likely to be true?



A. The scores of Mrs. Scott's class are the closest to the class mean.

B. The scores of Mr. Phan's class are the closest to the class mean.

C. The scores of Mrs. Rijo's class are the closest to the class mean.

Find the product.

[tex]$5^{56} \times 5^{22} \times 5^{96}= \square$[/tex]

What are the center and the radius of the circle [tex]$2 x^2+y^2+4 x-4 y-3=0$[/tex]?

A. $(-1,2)$ and 3

B. $(1,2)$ and 3

C. $(1,-2)$ and 9

D. $(-1,2)$ and 9



If [tex]$P=(2,5)$[/tex] and [tex]$Q=(5,1)$[/tex], then what is the position vector of [tex]$Q$[/tex] relative to [tex]$P$[/tex]?

A. [tex]$-3 i+4 j$[/tex]

B. [tex]$3 i-4 j$[/tex]

C. [tex]$4 i-3 j$[/tex]

D. [tex]$-4 i+3 j$[/tex]



What is the solution set of the system [tex]$\left\{\begin{array}{l}x-z=y-1 \ 3 x-2 y=z-1 \ 4 x-3 y+2=2 z\end{array}\right.$[/tex] ?

A. [tex]$\varnothing$[/tex]

B. [tex]$\left\{(1-k, 2(1-k), k): k \in R \right\}$[/tex]

C. [tex]$\left\{(1-k, 2 k, k): k \in R \right\}$[/tex]

D. [tex]$\left\{(0,0,1)\right\}$[/tex]



If [tex]$T$[/tex] is a translation that takes [tex]$(1,2)$[/tex] to [tex]$(3,4)$[/tex], then which one of the following is not true?

A. [tex]$T(-3,2)=(1,4)$[/tex]

B. [tex]$T(0,0)=(2,2)$[/tex]

C. [tex]$T^{-1}(0,0)=(-2,-2)$[/tex]

D. [tex]$T(5,6)=(7,8)$[/tex]



Let [tex]$A$[/tex] be a [tex]$3 \times 3$[/tex] matrix invertible matrix and [tex]$B$[/tex] be any [tex]$3 \times 3$[/tex] matrix. If [tex]$\operatorname{det}(A)=a$[/tex] and [tex]$\operatorname{det}(B)=b$[/tex]. then which one of the following is not true?

A. [tex]$\operatorname{det}(\operatorname{adj}(A))=(\operatorname{det}(A))^2$[/tex]

B. If [tex]$b=0$[/tex], then [tex]$B$[/tex] is singular

C. [tex]$\operatorname{det}\left(k A^t\right)=k^3 \operatorname{det}(A)$[/tex] for any [tex]$k \in R$[/tex]

D. [tex]$\operatorname{det}\left(2(A B)^{-1}\right)=8 a b$[/tex]

$1 \cdot \frac{8}{14}+\frac{6}{14}+\frac{8}{14}$

What is the net (take-home) pay for an individual with expenses shown who has a gross income of $38,000?



Tax Table



| Expenses | Amount Taxed |

| ------------------ | ------------- |

| FICA | $2,907.00 |

| State Income Tax | $1,178.00 |

| Federal Income Tax | $3.990 .00 |

| Medical Insurance | $1,500.00 |



$[?]

Round to the nearest whole dollar.

Health insurance coverage for an employee and their children is $6,240 per year. The employer pays 70% of that cost. How much is taken from the employee's biweekly paycheck?



Biweekly Insurance Cost = $[?]

What is the employee's yearly contribution to the U.S. Medicare tax on a salary of $[tex]$24,000$[/tex]?



| FICA Taxes |

| --- | --- |

| | Social Security | Medicare |

| Total Due: | [tex]$12.40 \%$[/tex] | [tex]$2.90 \%$[/tex] |

| Employer's Responsibility | [tex]$6.20 \%$[/tex] | [tex]$1.45 \%$[/tex] |

| Employee's Responsibility | [tex]$6.20 \%$[/tex] | [tex]$1.45 \%$[/tex] |

Which set of numbers can represent the side lengths, in centimeters, of a right triangle?

A. 8, 12, 15

B. 10, 24, 26

C. 12, 20, 25

D. 15, 18, 20

What is the order of this matrix?



$\left[\begin{array}{ccc}2 & -1 & 0 \ 4 & 1 & -2 \ 3 & -5 & 1 \ 3 & 6 & 1 \ 3 & 1 & 0\end{array}\right]$



A. $2 \times 3$

B. $3 \times 5$

C. $5 \times 3$

D. $3 \times 2$

In which triangle is the value of $x$ equal to $\tan ^{-1}\left(\frac{3.1}{5.2}\right)$? (Images may not be drawn to scale.)

The expression $\frac{\left(27 y^{-2}\right)^{\frac{2}{3}}}{y^{-\frac{1}{3}}}$ can be simplified and written without negative exponents as $\square$

Select all the correct answers.



Which expressions are equivalent to the given expression?

$(-\sqrt{9}+\sqrt{4})-(2 \sqrt{578}+\sqrt{-84})$



-3-2 i-2(24)+8 i

-51-6 i

-3+2 i +2(24)+8 i

-51+61

45+10 i

-3+2 i-2(24)-8 i

The stem-and-leaf plot shows the number of pages each student in a class read the previous evening.



0 | 0 0 5 8

1 | 2 3 5 8 8 9

2 | 2 4 6 7 7 7

3 | 3 5 6

4 | 2 4 6

5 | 7



Which statement is true about the data set?



A. The value of the first quartile is 13.

B. Its median is greater than its mode.

C. The data is symmetric.

D. It has a range of 52 pages.

1. Solve for x in log_2(log_3x) = 1.
   
   5. How many integer solutions does the equation x^2 - 2x - 8 = 0 have?

Solve the equation:
\( \left(\frac{28}{38}\right)^{\frac{x-1}{2}} = \sqrt[12]{\frac{32}{28}} \)

1. A random sample of size 16 has a mean of 53 and the sum of the squares of the deviations taken from the mean is 150. Can this sample be regarded as taken from the population having a mean of 56? Obtain 95% and 99% confidence limits of the mean of the population.

1] x = sin(θ), y = sin(3θ) when θ = π/2.

A2. Mrs. Arora bought two houses for ₹75,85,900 and ₹1,15,25,000, respectively. She spent ₹15,54,600 on renovation. How much money did she spend in total?

A3. In a city, there are three shopping malls. In the month of December, the malls did a trade of ₹2,74,23,505, ₹1,89,75,456 and ₹2,15,46,643, respectively.

(a) What was the total trade done by the shopping malls in the month of December?

(b) If the total trade for the month of January is ₹5,86,48,750, in which month was more trade done and by how much?

True or False. Identify the statement if it is true or false.

1. A sequence is always infinite.
2. In an arithmetic sequence, the same number is added to each term to get the next.
3. A geometric sequence adds the same value to get the next term.
4. The Fibonacci sequence starts with 0 and 1.
5. All sequences have a visible pattern.

Stan guessed on all 10 questions of a multiple-choice quiz. Each question has 4 answer choices. What is the probability that he got at least 2 questions correct? Round the answer to the nearest thousandth.



[tex]\begin{array}{l}

P(k \text { successes })={ }_n C_k p^k(1-p)^{n-k} \

{ }_n C_k=\frac{n!}{(n-k)!\cdot k!}

\end{array}[/tex]

Simplify the ratio to its simplest form: 15 minutes : 1 hour 30 minutes

Use set-builder notation and the roster method to represent the following set.

The set of all digits less than 4

Use set-builder notation to represent the set. Choose the correct answer below.

A. {x | x is a digit less than or equal to 4}

B. {x | x is a digit less than 4}

C. {0, 1, 2, 3}

D. {4, 5, 6, 7, 8, 9}

Simplify the expression using properties of exponents.



[tex]\frac{2 x^4 y^8}{8 x^2 y^4}[/tex]



A. [tex]$4 x^2 y^4$[/tex]

B. [tex]$\frac{x^2 y^4}{4}$[/tex]

C. [tex]$2 x^2 y^4$[/tex]

D. [tex]$\frac{x y^2}{2}$[/tex]

Which polynomial correctly combines the like terms and expresses the given polynomial in standard form?



[tex]$9 x y^3-4 y^4-10 x^2 y^2+x^3 y+3 x^4+2 x^2 y^2-9 y^4$[/tex]



A. [tex]$-13 y^4+3 x^4-8 x^2 y^2+x^3 y+9 x y^3$[/tex]

B. [tex]$-13 y^4+x^3 y-8 x^2 y^2+9 x y^3+3 x^4$[/tex]

C. [tex]$3 x^4-8 x^2 y^2+x^3 y+9 x y^3-13 y^4$[/tex]

D. [tex]$3 x^4+x^3 y-8 x^2 y^2+9 x y^3-13 y^4$[/tex]

Which equation is equivalent to $16^{2 p}=32^{p+3}$?



A. $8^{4 p}=8^{4 p+3}$

B. $8^{4 p}=8^{4 p+12}$

C. $2^{8 p}=2^{5 p+15}$

D. $2^{8 p}=2^{5 p+3}$

For a standard normal distribution, which of the following expressions must always be equal to 1?



A. [tex]P(z \leq-a)-P(-a \leq z \leq a)-P(z \geq a)[/tex]

B. [tex]P(z \leq-a)+P(-a \leq z \leq a)+P(z \geq a)[/tex]

C. [tex]P(z \leq-a)-P(-a \leq z \leq a)+P(z \geq a)[/tex]

D. [tex]P(z \leq-a)+P(-a \leq z \leq a)-P(z \geq a)[/tex]

Replace the question mark by $<,>$, or $=$, whichever is correct.



$\frac{17}{5} \square 3.4$

Simplify the expression below if [tex]x\ \textgreater \ 0[/tex] and [tex]y\ \textgreater \ 0[/tex].



[tex]\frac{x \sqrt{y}}{y \sqrt{x}}[/tex]



A. [tex]\frac{\sqrt{y}}{x y}[/tex]

B. [tex]\frac{\sqrt{y}}{y}[/tex]

C. [tex]\frac{\sqrt{x y}}{y}[/tex]

D. [tex]\frac{x \sqrt{x y}}{y}[/tex]

Is [tex]$A = \{3, 4, 5, 6\}$[/tex], [tex]$B = \{6, 7, 8, 10\}$[/tex] and [tex]$R$[/tex] is a relation 'is half of' then find the elements of [tex]$R$[/tex].

$\frac{\sum_{i=1}^{4.5} 5_i}{1=1}$

For a standard normal distribution, which of the following expressions must always be equal to 1?



A. [tex]$P(z \leq -a)-P(-a \leq z \leq a)-P(z \geq a)$[/tex]

B. [tex]$P(z \leq -a)+P(-a \leq z \leq a)+P(z \geq a)$[/tex]

C. [tex]$P(z \leq -a)-P(-a \leq z \leq a)+P(z \geq a)$[/tex]

D. [tex]$P(z \leq -a)+P(-a \leq z \leq a)-P(z \geq a)$[/tex]

Replace the question mark by $<,>$, or $=$, whichever is correct.



-8 ? -6

This table displays a scenario.





What can be determined from the table? Check all that apply.



* The independent variable is the number of gallons.

* Liters is a function of Gallons.

* The equation [tex]$l=3.79 g$[/tex] represents the table.

* As the number of gallons increases, the number of liters increases.

* This is a function because every input has exactly one output.

6.3 A school is planning to build a mini-school hall. The cost is calculated to be R2 000 000 in ten years' time. On 1 January 2025, an initial deposit of R650 197,00 is made into the school hall project savings account. Interest is earned at 6,1% p.a. compounded monthly for the first 5 years. On 1 January 2030, another amount, of Rx, is deposited into the savings account. The interest rate for the last 5 years is 7,47% p.a. compounded quarterly. Determine the amount that was deposited into the savings account on 1 January 2030, i.e. calculate the value of x. ​

Simplify the expression $\frac{\left(x^{25}\right)^{-6}}{\left(x^{-3}\right)^{45}}$. The power of $x$ in the simplified expression is $\square$

Find the quotient.



$\frac{2 x-3}{x} \div \frac{7}{x^3}$



A. $\frac{7}{x(2 x-3)}$

B. $\frac{7 x}{2 x-3}$

C. $\frac{2 x-3}{7 x}$

D. $\frac{x(2 x-3)}{7}$

$\frac{\sum_{i=1}^{4.5} 5 i}{}$

Jace ordered a banner in the shape of a parallelogram from a print shop.



The print shop charges $1.10 per square foot for banners of any shape and size. What is the approximate cost of the banner before tax?



A. $41.95

B. $46.14

C. $83.90

D. $92.30



Heron's formula: Area = [tex]$\sqrt{s(s-\alpha)(s-b)(s-c)}$[/tex]

Fill in the blank with the appropriate symbol, [tex]$\in$[/tex] or [tex]$\notin$[/tex]. [tex]$\sqrt{2} \ldots\{ x \mid x$[/tex] is a rational number [tex]\}[/tex]

Simplify the expression below.



$\frac{\sqrt[3]{4}}{\sqrt{2}}$



A. $\sqrt{2}$

B. $\sqrt[3]{2}$

C. $\sqrt[6]{2}$

D. $\sqrt[7]{2}$

Select the correct answer.

Which statement best describes the zeros of the function [tex]h(x)=(x+9)(x^2-10 x+25)[/tex] ?

A. The function has three complex zeros.

B. The function has three distinct real zeros.

C. The function has one real zero and two complex zeros.

D. The function has two distinct real zeros.

Fiona recorded the number of miles she biked each day last week as shown:



[tex]$4,7,4,10,5$[/tex]



The mean is given by [tex]$\mu=6$[/tex]. Which equation shows the variance for the number of miles Fiona biked last week?



A. [tex]$s^2=\frac{(4-6)^2+(7-6)^2+(4-6)^2+(10-6)^2+(5-6)^2}{6}$[/tex]

B. [tex]$\sigma=\sqrt{\frac{(4-6)^2+(7-6)^2+(4-6)^2+(10-6)^2+(5-6)^2}{5}}$[/tex]

C. [tex]$s=\sqrt{\frac{(4-6)^2+(7-6)^2+(4-6)^2+(10-6)^2+(5-6)^2}{4}}$[/tex]

Fill in the blank with the appropriate symbol, $\in$ or $\notin$.



$\frac{6}{7}-\{x \mid x$ is an irrational number $\}$



$\frac{6}{7}$ $\square$ $\{ x \mid x$ is an irrational number $\}$

Simplify the expression using properties of exponents.



$\frac{10 a^{-8} b^{-2}}{4 a^3 b^5}$



A. $\frac{5 a^5 b^3}{2}$

B. $\frac{5 a^{11} b^7}{2}$

C. $\frac{5}{2 a^{11} b^7}$

D. $\frac{5 b^3}{2 a^{11}}$

[tex]\begin{array}{l}\sum_{i=1}^{4.5} 5 i\end{array}[/tex]

Simplify the expression below.



$\sqrt{\frac{3}{15 x}}$



A. $\sqrt{3}$

B. $\frac{\sqrt{3}}{15 x}$

C. $\frac{\sqrt{5}}{5}$

D. $\frac{\sqrt{5 x}}{5 x}$

Simplify the expression below.

$(4 x)^3$

A. $4 x^3$

B. $7 x^3$

C. $12 x^3$

D. $64 x^3

Jason rolls a fair number cube labeled 1 through 6, and then he flips a coin. What is the probability that he rolls a 3 and flips a head?

Use the binomial theorem to find the value [tex]$\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6$[/tex]

For what value of a does [tex]\left(\frac{1}{7}\right)^{3 a+3}=343^{a-1}[/tex] ?

A. -1

B. 0

C. 1

D. no solution

The times of the runners in a marathon are normally distributed, with a mean of 3 hours and 50 minutes and a standard deviation of 30 minutes. What is the probability that a randomly selected runner has a time less than or equal to 3 hours and 20 minutes? Use the portion of the standard normal table to help answer the question.



| z | Probability |

|---|---|

| 0.00 | 0.5000 |

| 0.50 | 0.6915 |

| 1.00 | 0.8413 |

| 2.00 | 0.9772 |

| 3.00 | 0.9987 |



A. 34%

B. 32%

C. 84%

D. 16%

There were 3 apple, 1 mixed fruit, 2 grape, and 4 tropical fruit juice boxes in the cooler at the picnic. What is the probability that, when Jill reaches into the cooler to grab two juice boxes without replacing them, she grabs two that are grape?



$\frac{1}{100}$

$\frac{1}{50}$

$\frac{1}{45}$

$\frac{1}{25}$

For what value of $b$ does $\left(\frac{1}{12}\right)^{-2 b} \cdot 12^{-2 b+2}=12$?

Which is a factor of $x^2+8 x-48$?



A. $(x-6)$

B. $(x+4)$

C. $(x-16)$

D. $(x+12)$

What is the probability of one of the coins landing on tails and two of them landing on heads?



A.$\frac{1}{4}$

B.$\frac{3}{8}$

C.$\frac{1}{2}$

What is the multiplicative inverse of 10? (Type an integer or a simplified fraction.)

The mean of a set of credit scores is [tex]$\mu=690$[/tex] and [tex]$\sigma=14$[/tex]. Which statement must be true about [tex]$z {\theta \rho 4}$[/tex]?



A. [tex]$Z {694}$[/tex] is between 1 and 2 standard deviations of the mean.

B. [tex]$z_{694}$[/tex] is more than 3 standard deviations of the mean.

C. [tex]$Z_{694}$[/tex] is between 2 and 3 standard deviations of the mean.

D. [tex]$Z _{694}$[/tex] is within 1 standard deviation of the mean.

If $3^{2 x+1}=3^{x+5}$, what is the value of $x$?

Find the multiplicative inverse (or reciprocal) of -3.



What is the multiplicative inverse of -3?

$\square$ (Type an integer or a simplified fraction.)

Select the correct answer.



Which expression is equivalent to $\frac{(x-1)^2}{x^2-x-12} \cdot \frac{x^2+x-6}{x^2-6 x+5}$ if no denominator equals zero?

A. $\frac{x^2-3 x+2}{x^2-9 x+20}$

B. $\frac{x^2+x-2}{x^2-x-20}$

C. $\frac{x^2-3 x+2}{x^2-x-20}$

D. $\frac{x^2+3 x+2}{x^2+x-20}$

EXERCISE 2.2



1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.



(i) [tex]$x^2-2 x-8$[/tex]

(ii) [tex]$4 s^2-4 s+1$[/tex]

(iii) [tex]$6 x^2-3-7 x$[/tex]

Zohar is using scissors to cut a rectangle with a length of $5x-2$ and a width of $3x+1$ out of a larger piece of paper. Which expression can be used to find the perimeter of the rectangle and what is the perimeter if $x=4$?



A. $(5x-2)+(3x+1) ; 31$ centimeters

B. $(5x-2)+(3x+1) ; 36$ centimeters

C. $2(5x-2)+2(3x+1) ; 62$ centimeters

D. $2(5x-2)+2(3x+1) ; 70$ centimeters

Simplify the expression using properties of exponents.



$\frac{25 a^{-5} b^{-8}}{5 a^2 b^6}$



A. $5 a b^7$

B. $5 a^9 b^9$

C. $\frac{5}{a b^7}$

D. $\frac{5}{a^5 b^5}$

$\begin{array}{l}\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6 \ =x^6+6 x^5\left(\sqrt{x^2-1}\right)+15 x^4\left(\sqrt{x^2-1}\right)^2+20 x^3\left(\sqrt{x^2-1}\right)^3\end{array}$

Which formula is used to calculate the standard deviation of sample data?



$s=\frac{\left(x_1-\bar{x}\right)^2+\left(x_2-\bar{x}\right)^2+\ldots+\left(x_n-\bar{x}\right)^2}{n-1}$



$\sigma=\sqrt{\frac{\left(x_1-\mu\right)^2+\left(x_2-\mu\right)^2+\ldots+\left(x_N-\mu\right)^2}{N}}$



$s=\sqrt{\frac{\left(x_1-\bar{x}\right)^2+\left(x_2-\bar{x}\right)^2+\ldots+\left(x_n-\bar{x}\right)^2}{n-1}}$



$a^2=\frac{\left(x_1-\mu\right)^2+\left(x_2-\mu\right)^2+\ldots+\left(x_N-\mu\right)^2}{N}$

What is the probability of at least two coins landing on heads?



| | |

| :--- | :--- |

| HHHH | |

| HHHT | |

| HHTH | |

| HHTT | |

| HTHH | |

| HTHT | |

| HTTH | |

| HTTT | |

| THHH | |

| THHT | |

| THTH | |

| THTT | |

| TTHH | |

| TTHT | |

| TTTH | |

| TTTT | |

The polynomial $2 x^3-5 x^2+4 x-10$ is split into two groups, $2 x^3+4 x$ and $-5 x^2-10$. The GCFs of each group is then factored out.



What is the common binomial factor between the two groups after their GCFs have been factored out?

A. $2 x+5$

B. $2 x-5$

C. $x^2-2$

D. $x^2+2$

The second term of an arithmetic sequence is 24 and fifth term is 3, find the first termand the common difference?

For what value of a does [tex]\left(\frac{1}{9}\right)^{a+1}=81^{a+1} \cdot 27^{a-2}[/tex]?

Select the correct answer.



If no denominator equals zero, which expression is equivalent to $\frac{x^2+10 x+25}{x+5}-\frac{x^2-6}{x-5} ?$

A. $\frac{2 x^2-19}{x-5}$

B. $\frac{2 x^2-19}{x^3-25}$

C. $\frac{19}{x-5}$

D. $\frac{-19}{x-5}$

Which expression is equivalent to $\sqrt[3]{125 x^6 y^{15} z^3}$ ?



A. $5 x^3 y^3 z$

B. $5 x^2 y^5 z$

C. $25 x^2 y^3 z$

D. $25 x^2 y^5 z$

For what value of a does $9-\left(\frac{1}{27}\right)^{a+3}$ = $-\frac{11}{3}$?

Consider this product.



[tex]\frac{x^2-3 x-10}{x^2-6 x+5} \cdot \frac{x-2}{x-5}[/tex]



The simplest form of this product has a numerator of [ ] and a denominator of [ ]. The expression has an excluded value of x = [ ]

Simplify this expression. $6^7 \div 6^5$

If $\log {5 \sqrt{5}} 125=x$ and $\log {2 \sqrt{2}} 64=y$, what is the product of $x$ and $y$?

Evaluate. [tex]$|8.8-18.8|$[/tex]

Consider the following polynomial inequality:

x(x+3)²(x-8) < 0

Step 2 of 2: Test each interval to find the solution of the polynomial inequality. Express your answer in interval notation.

Select the correct answer.



Which is the correct simplified form of the expression $\left(\frac{b^8}{a^3}\right)^{\frac{1}{3}}$ ?

A. $a b^{-2}$

B. $\frac{b^2}{a}$

C. $\frac{a^4}{b^3}$

D. $a^2 b^3$

1. Solve for x:
   
   $\frac{\frac{2x}{4} - \frac{3}{4}}{\frac{9x}{7} + \frac{4}{7}} = \frac{35}{68}$

The table represents the multiplication of two binomials.



\begin{tabular}{|c|c|c|}

\hline & $-2 x$ & 3 \

\hline $4 x$ & $A$ & $B$ \

\hline 1 & $C$ & $D$ \

\hline

\end{tabular}



Which letters from the table represent like terms?

A. A and B

B. B and C

C. A and D

D. B and D

Compare each pair of numbers. Put >, < or =.

1. (60, 40, 213) vs (98, 40, 213)
2. (222, 213) vs (212, 200)
3. (2, 30, 10,000) vs (2, 30, 13,000)
4. (8, 10, 10,203) vs (81, 010,203)

1. Find the product of the given fractions.
   (a) \( \frac{1}{2} \times \frac{1}{2} \)
   (b) \( \frac{2}{3} \times \frac{1}{2} \)
   (c) \( \frac{1}{4} \times \frac{1}{2} \)
   (d) \( \frac{1}{2} \times \frac{1}{3} \)
   
   42. Find the division of the given fraction.
   (a) \( \frac{1}{4} \div \frac{1}{2} \)
   (b) \( 2 \frac{1}{2} \div \frac{1}{2} \)
   (c) \( \frac{1}{2} \div \frac{11}{12} \)
   (d) \( 2 \frac{1}{2} \div 1 \frac{1}{2} \)

1. A hotel has x floors, with y apartments on each floor. How many apartments does the hotel have in total?
   
   5. My neighbour has x cats. Write an expression for their total number of:
   
   a. tails
   
   b. eyes
   
   c. legs

Q.29. If P = {x: x = n^2, n < 5, n ∈ N} and Q = {x: x = 2^{m-1}, m ≤ 5, m ∈ N}, then find (P - Q) × (P ∩ Q).

Q.30. Given A = {14, 15, 16, 17, 18}, let f: A → N be defined by f(n) = the smallest prime factor of n. Find the range of f.

Q.31. If f(x) = { 2x - 3, x ≥ 2; x + 2, x < 2 }, then

i. Find f(-3) and f(4.5).
ii. Find the domain of f.

Q.32. Evaluate the value of 3 cos(3π/2) + 5 sin(3π/2) - 7 cos(π) + 11 sin(2π).

Q.33. In a circle with a diameter of 60 cm, the length of a chord is 30 cm. Find the length of the minor arc subtended by this chord.

Q.34. In a quadrilateral, three angles are 65° 30', 92° 15' and 110° 45'. Find the measure of the fourth angle.

Q.35. If tan(θ) = 2/3 and tan(φ) = 3/4, then find the value of tan(θ + φ).

Q.36. Given that 8θ = π, prove that cos(θ) + cos(7θ) = 0.

Q.37. Prove that (cos(14°) - sin(14°)) / (cos(14°) + sin(14°)) = tan(31°).

Q.38. Let A = {x: x ∈ Z, x^4 = 16}, B = {y: y ∈ Z, y^3 + 8 = 0}. Find
a) A ∪ B
b) A ∩ B
c) A - B
d) B - A.