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Test 1
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120:00
June Exam - Test 1 [100 Marks]
Topics Covered: Whole Numbers & Finance, Integers & Rational Numbers, Exponents & Scientific Notation, Numeric & Geometric Patterns, and Functions & Relationships.
Question 1: Whole Numbers & Finance [20 Marks]
1.1
Write down the prime factors of \(144\) using product-of-primes index notation. (3 Marks)
1.2
Determine the highest common factor (HCF) of the numbers \(12, 18\) and \(30\) using prime factorisation. (2 Marks)
1.3
Determine the lowest common multiple (LCM) of the numbers \(12, 18\) and \(30\) using prime factorisation. (2 Marks)
1.4
Divide \( \text{R}1\,500 \) in the ratio \( 2 : 3 \). (3 Marks)
1.5
**Rate:** A vehicle travels a distance of \( 180\text{ km} \) in exactly \( 3\text{ hours} \). Calculate its speed in \(\text{km/h}\). (2 Marks)
1.6
**Hire Purchase:** A washing machine costs \( \text{R}8\,000 \). Under a hire purchase agreement, you pay a \( 10\% \) deposit, and simple interest is charged on the balance at a rate of \( 15\% \) per annum for \( 2 \text{ years} \). Calculate your monthly installment. (5 Marks)
1.7
**Exchange Rates:** Convert \( \text{R}3\,800 \) into United States Dollars (USD) if the current exchange rate is \( \$1\text{ USD} = \text{R}19,00 \). (3 Marks)
Question 2: Integers & Rational Numbers [20 Marks]
2.1
Calculate using order of operations (BODMAS): \( -15 + (-3) \times (-4) \div (-2) \). (3 Marks)
2.2
Evaluate the radical calculation: \( \sqrt{64} + \sqrt[3]{-125} \). (3 Marks)
2.3
Calculate the decimal product without a calculator: \( 0,3 \times 0,04 \). (2 Marks)
2.4
Use the reciprocal relationship to divide common fractions: \( \frac{3}{4} \div \frac{9}{8} \). (4 Marks)
2.5
Evaluate the powers of rational numbers: \( \sqrt{\frac{25}{36}} + \left(\frac{1}{2}\right)^3 \). (4 Marks)
2.6
Increase the amount \( \text{R}400 \) by \( 15\% \). (4 Marks)
Question 3: Exponents & Scientific Notation [20 Marks]
3.1
Simplify using exponent laws: \( x^3 \cdot x^4 \). (2 Marks)
3.2
Simplify: \( \frac{y^7}{y^2} \). (2 Marks)
3.3
Simplify the power of a product fully: \( (a^2b^3)^3 \). (2 Marks)
3.4
Determine the numerical value of: \( 5x^0 + (5x)^0 \). (2 Marks)
3.5
Evaluate and contrast the difference between \( (-2)^4 \) and \( -2^4 \). (2 Marks)
3.6
Simplify fully using base conversion: \( \frac{3^5 \cdot 9}{27} \). (3 Marks)
3.7
Express the following large number in scientific notation: \( 45\,000 \). (3 Marks)
3.8
Express the following number in standard decimal form: \( 3,2 \times 10^4 \). (3 Marks)
Question 4: Numeric & Geometric Patterns [20 Marks]
4.1
Consider the arithmetic number pattern: \( 4, 7, 10, 13, \ldots \)
Determine the common difference and write down the next three terms. (3 Marks)
Determine the common difference and write down the next three terms. (3 Marks)
4.2
Determine the general term formula \( T_n \) for the arithmetic sequence in **4.1**. (2 Marks)
4.3
Calculate the 100th term (\( T_{100} \)) of the sequence in **4.1** using your formula. (2 Marks)
4.4
Consider the geometric sequence: \( 2, 6, 18, \ldots \)
Find the common ratio and calculate the next term. (3 Marks)
Find the common ratio and calculate the next term. (3 Marks)
4.5
A decreasing arithmetic pattern is given by: \( 20, 17, 14, 11, \ldots \)
Find the general term formula \( T_n \) for this pattern. (2 Marks)
Find the general term formula \( T_n \) for this pattern. (2 Marks)
4.6
Calculate the 25th term \( T_{25} \) of the decreasing sequence in **4.5**. (2 Marks)
4.7
A pattern of matchstick squares has \(4\) matchsticks in the 1st figure (1 square), \(7\) in the 2nd (2 squares), and \(10\) in the 3rd (3 squares). Determine the general formula for the number of matchsticks, and find the number needed for the 4th figure. (4 Marks)
Question 5: Functions & Relationships [20 Marks]
5.1
Calculate the outputs for a spider diagram with inputs \( \{ 2, 4, 6 \} \) under the rule \( y = 3x - 1 \). (4 Marks)
5.2
Complete the output results in the linear table for \( y = -3x + 1 \):
(4 Marks)
| Input (\(x\)) | -2 | 0 | 3 | 5 |
|---|---|---|---|---|
| Output (\(y\)) | \(a\) | \(b\) | \(c\) | \(d\) |
5.3
Determine the linear relationship formula from the table: Input \(x \in \{1, 2, 3\}\) resulting in outputs \(y \in \{2, 5, 8\}\). (5 Marks)
5.4
Complete the outputs for the table representing \( y = 2x - 4 \):
(4 Marks)
| Input (\(x\)) | -1 | 1 | 3 | 5 |
|---|---|---|---|---|
| Output (\(y\)) | \(e\) | \(f\) | \(g\) | \(h\) |
5.5
Determine the linear relationship formula from the table: Input \(x \in \{1, 2, 3\}\) resulting in outputs \(y \in \{5, 7, 9\}\). (2 Marks)
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Test 2
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120:00
June Exam - Test 2 [100 Marks]
Topics Covered: Whole Numbers & Finance, Integers & Rational Numbers, Exponents & Scientific Notation, Numeric & Geometric Patterns, and Functions & Relationships.
Question 1: Whole Numbers & Finance [20 Marks]
1.1
Write down the prime factors of \(180\) using product-of-primes index notation. (3 Marks)
1.2
Determine the highest common factor (HCF) of the numbers \(24\) and \(36\). (2 Marks)
1.3
Determine the lowest common multiple (LCM) of the numbers \(24\) and \(36\). (2 Marks)
1.4
Divide \( \text{R}800 \) in the ratio \( 3 : 5 \). (3 Marks)
1.5
If \(4\) mathematics notebooks cost \( \text{R}120 \), calculate the cost of \(6\) of the same notebooks. (2 Marks)
1.6
**Rate:** A high-speed train travels \( 350\text{ km} \) in \( 5 \) hours. Calculate its average speed in \(\text{km/h}\). (2 Marks)
1.7
**Compound Interest:** An investment of \( \text{R}8\,000 \) is placed at a compound interest rate of \( 6\% \) per annum for \( 2 \) years. Calculate the final accumulated amount. (6 Marks)
Question 2: Integers & Rational Numbers [20 Marks]
2.1
Calculate the simplified value of: \( -10 - (-15) + (-6) \). (3 Marks)
2.2
Evaluate the quotient calculation: \( -8 \times 3 \div (-6) \). (3 Marks)
2.3
Calculate the root expression: \( \sqrt{100 - 36} + (-2)^3 \). (3 Marks)
2.4
Calculate without a calculator: \( 0,4 \times 0,05 \). (3 Marks)
2.5
Divide fractions and simplify: \( \frac{2}{3} \div \frac{4}{9} \). (4 Marks)
2.6
Increase the amount \( \text{R}600 \) by \( 12\% \). (4 Marks)
Question 3: Exponents & Scientific Notation [20 Marks]
3.1
Simplify using exponent laws: \( y^5 \cdot y^2 \). (2 Marks)
3.2
Simplify: \( \frac{a^{12}}{a^4} \). (2 Marks)
3.3
Simplify: \( (x^3y^2)^3 \). (2 Marks)
3.4
Evaluate: \( 4y^0 + (4y)^0 \). (2 Marks)
3.5
Evaluate both \( (-3)^2 \) and \( -3^2 \), showing clearly how they differ. (2 Marks)
3.6
Simplify fully using base conversion: \( \frac{2^6 \cdot 4}{8} \). (3 Marks)
3.7
Express \( 580\,000 \) in scientific notation. (3 Marks)
3.8
Express \( 0,000042 \) in scientific notation. (4 Marks)
Question 4: Numeric & Geometric Patterns [20 Marks]
4.1
Consider the pattern: \( 6, 11, 16, 21, \ldots \). Find the common difference and the next three terms. (3 Marks)
4.2
Find the general term formula \( T_n \) for the pattern in **4.1**. (2 Marks)
4.3
Calculate the 80th term \( T_{80} \) of the pattern in **4.1**. (2 Marks)
4.4
Consider the geometric sequence: \( 3, 9, 27, \ldots \). Determine the common ratio and find the next term. (3 Marks)
4.5
Find the general term formula \( T_n \) for the decreasing pattern: \( 30, 26, 22, 18, \ldots \). (2 Marks)
4.6
Calculate the 40th term \( T_{40} \) of the decreasing pattern in **4.5**. (3 Marks)
4.7
A matchstick pattern has squares: Figure 1 (1 square) has \(4\) matches, Figure 2 (2 squares) has \(7\) matches, Figure 3 (3 squares) has \(10\) matches. Find the general formula and determine the matches needed for the 5th figure. (5 Marks)
Question 5: Functions & Relationships [20 Marks]
5.1
Calculate the outputs for a spider diagram with inputs \( \{ 1, 3, 5 \} \) under the rule \( y = 4x - 2 \). (3 Marks)
5.2
Complete the output results in the linear table for \( y = -2x + 5 \):
(4 Marks)
| Input (\(x\)) | -2 | 0 | 3 | 5 |
|---|---|---|---|---|
| Output (\(y\)) | \(a\) | \(b\) | \(c\) | \(d\) |
5.3
Determine the linear relationship formula from the table: Input \(x \in \{1, 2, 3\}\) resulting in outputs \(y \in \{3, 5, 7\}\). (5 Marks)
5.4
Complete the outputs for the table representing \( y = 3x - 1 \):
(5 Marks)
| Input (\(x\)) | -1 | 1 | 3 | 5 |
|---|---|---|---|---|
| Output (\(y\)) | \(e\) | \(f\) | \(g\) | \(h\) |
5.5
Determine the linear relationship formula from the table: Input \(x \in \{1, 2, 3\}\) resulting in outputs \(y \in \{4, 7, 10\}\). (3 Marks)
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Test 3
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120:00
June Exam - Test 3 [100 Marks]
Topics Covered: Whole Numbers & Finance, Integers & Rational Numbers, Exponents & Scientific Notation, Numeric & Geometric Patterns, and Functions & Relationships.
Question 1: Whole Numbers, Finance & Proportions [20 Marks]
1.1
**Inverse Proportion:** If \( 5 \) builders take \( 12 \) days to paint a school hall, calculate how many days it will take \( 6 \) builders to paint the same hall at the same working rate. (4 Marks)
1.2
**Exchange Rates:** A tourist exchanges \( \$500 \) USD into South African Rands (ZAR). The bank charges a flat \( 2\% \) commission fee on the transaction. If the exchange rate is \( \$1\text{ USD} = \text{R}18,50 \), calculate the final amount in ZAR the tourist receives. (4 Marks)
1.3
**Profit & Loss:** A merchant buys a box of goods for \( \text{R}1\,500 \) and spends \( \text{R}100 \) on transportation. If he sells the goods for \( \text{R}1\,920 \), calculate his percentage profit based on the total cost price. (4 Marks)
1.4
**Hire Purchase:** A computer is marked at \( \text{R}10\,000 \) cash. Under a hire purchase agreement, a student pays an \( 8\% \) deposit. Simple interest of \( 12\% \) per annum is charged on the outstanding balance over \( 18\text{ months} \). Calculate the monthly installment. (5 Marks)
1.5
**Compound Ratio:** Share \( \text{R}4\,800 \) among A, B, and C such that A receives twice as much as B, and B receives three times as much as C. Calculate each share. (3 Marks)
Question 2: Integers & Rational Numbers [20 Marks]
2.1
Simplify the complex integer calculation using BODMAS: \( \frac{(-3)^2 \times (-2)^3 - \sqrt{100 - 36}}{-4 + (-1)^5} \). (5 Marks)
2.2
Evaluate the rational operations without a calculator: \( (1,2 \times 0,3) \div \frac{3}{10} + \sqrt{\frac{16}{25}} \). (5 Marks)
2.3
**Multi-step Percentage:** Decrease the amount \( \text{R}800 \) by \( 10\% \), and then increase the new value by \( 10\% \). Determine the final amount and show why it is different from the original \( \text{R}800 \). (5 Marks)
2.4
Show by choosing suitable positive integers \(a\) and \(b\) that the following statement is **invalid**: \( \sqrt{a^2 + b^2} = a + b \). (5 Marks)
Question 3: Exponents & Scientific Notation [20 Marks]
3.1
Simplify the expression using the laws of exponents: \( \frac{(-x)^3 \cdot (-y)^4}{-(xy)^2} \). (4 Marks)
3.2
Simplify using base conversion and prime factorization: \( \frac{2^{a+2} \cdot 4^{a-1}}{8^a} \). (5 Marks)
3.3
Solve the following exponential equation for \(x\) by factoring out a common base factor: \( 2^{x+2} - 2^x = 24 \). (5 Marks)
3.4
Evaluate the following product and write the final result in standard scientific notation: \( \frac{6 \times 10^5 \times 4 \times 10^{-3}}{8 \times 10^2} \). (4 Marks)
3.5
Express the following product in standard scientific notation: \( 0,00000305 \times 10^2 \). (2 Marks)
Question 4: Numeric & Geometric Patterns [20 Marks]
4.1
Given the decreasing arithmetic number pattern: \( -2, -7, -12, -17, \ldots \)
Determine the common difference and find the general term formula \( T_n \). (5 Marks)
Determine the common difference and find the general term formula \( T_n \). (5 Marks)
4.2
Determine which term in the sequence in **4.1** has a value of \( -147 \). (4 Marks)
4.3
A geometric sequence with alternating signs is given by: \( 1, -3, 9, -27, \ldots \). Determine its general term formula \( T_n \) and calculate the 10th term (\( T_{10} \)). (5 Marks)
4.4
A geometric pattern of matches forms adjacent hexagonal cells. Figure 1 (1 hexagon) has \( 6 \) matches, Figure 2 (2 hexagons) has \( 11 \) matches, and Figure 3 (3 hexagons) has \( 16 \) matches. Find the general formula for the matches, and solve for \(n\) if you have exactly \( 251 \) matchsticks. (6 Marks)
Question 5: Functions & Relationships [20 Marks]
5.1
Complete the output results in the quadratic relationship table for \( y = 2x^2 - 3 \):
(5 Marks)
| Input (\(x\)) | -2 | 0 | 1 | 3 |
|---|---|---|---|---|
| Output (\(y\)) | \(a\) | \(b\) | \(c\) | \(d\) |
5.2
Complete the table representing the rational relationship \( y = \frac{12}{x} + 2 \):
(5 Marks)
| Input (\(x\)) | -4 | 2 | 3 | 6 |
|---|---|---|---|---|
| Output (\(y\)) | \(e\) | \(f\) | \(g\) | \(h\) |
5.3
Determine the missing inputs for a spider diagram under the rule \( y = 3x - 5 \) if the corresponding outputs are \( \{ -11, -5, 10 \} \). (5 Marks)
5.4
Determine the linear relationship formula from the table: Input \(x \in \{-2, -1, 0, 1\}\) resulting in outputs \(y \in \{7, 4, 1, -2\}\). (5 Marks)
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Test 4
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120:00
June Exam - Test 4 [100 Marks]
Topics Covered: Whole Numbers & Finance, Integers & Rational Numbers, Exponents & Scientific Notation, Numeric & Geometric Patterns, and Functions & Relationships.
Question 1: Whole Numbers & Finance [20 Marks]
1.1
**Exchange Rates:** An investor converts \( \text{R}6\,000 \) to South African Rand into United States Dollars (USD) at an exchange rate of \( \$1\text{ USD} = \text{R}18,00 \). The bank deducts a \( 2\% \) transaction fee on the final ZAR amount before converting. Calculate the exact USD amount the investor receives. (5 Marks)
1.2
**Hire Purchase:** A refrigerator has an advertised cash price of \( \text{R}9\,500 \). Under a hire purchase agreement, you pay a \( 10\% \) deposit, and pay \( 24 \) monthly installments of \( \text{R}475 \) each. Calculate the simple interest rate per annum charged on the outstanding balance. (5 Marks)
1.3
**Complex Sharing Ratio:** Divide \( \text{R}10\,500 \) among three business partners A, B, and C such that A receives \( \frac{2}{3} \) of B's share, and C receives double of A's share. Calculate each partner's exact share. (5 Marks)
1.4
**Inverse Proportion:** When a swimming pool is filled using \( 5 \) identical taps running together, it takes \( 6\text{ hours} \). Calculate the time (in hours and minutes) it will take to fill the same pool if only \( 3 \) of those taps are used. (5 Marks)
Question 2: Integers & Rational Numbers [20 Marks]
2.1
Evaluate the following calculation using BODMAS: \( \frac{-3 \times (-4)^2 - \sqrt[3]{-64} \times (-5)}{\sqrt{25} - (-2^3)} \). (5 Marks)
2.2
Calculate the fractional expression, leaving your final answer as a simplified mixed fraction: \( \left(2 \frac{1}{2} - 1 \frac{3}{4}\right) \div \left(\frac{3}{8}\right)^2 \). (5 Marks)
2.3
**Multi-step Percentage:** A store offers a discount of \( 20\% \) on a jacket originally priced at \( \text{R}1\,500 \). During a clearance sale, the store offers an additional \( 10\% \) off the discounted price. Prove algebraically whether this consecutive discount yields the same final price as a single \( 30\% \) discount on the original price. (5 Marks)
2.4
A deep-sea diver ascends from a depth of \( -125\text{ m} \) relative to sea level. She ascends at a constant rate of \( 5\text{ m} \) every \( 2\text{ minutes} \) for \( 30\text{ minutes} \). Write down her new depth as a signed integer. (5 Marks)
Question 3: Exponents & Scientific Notation [20 Marks]
3.1
Simplify the exponential expression fully: \( \frac{2^{x+2} \cdot 4^{x-1}}{8^x} \). (5 Marks)
3.2
Simplify and write your final answer with positive exponents only: \( \frac{(3x^{-2}y^3)^{-2}}{9x^4y^{-4}} \). (5 Marks)
3.3
Solve the following exponential equation for \(x\): \( 3^{x+2} + 3^x = 90 \). (5 Marks)
3.4
Calculate the quotient fully, writing your final answer in standard scientific notation: \( \frac{1,8 \times 10^5}{3,0 \times 10^{-2}} \). (5 Marks)
Question 4: Numeric & Geometric Patterns [20 Marks]
4.1
Consider the non-linear pattern: \( 2, 5, 10, 17, \ldots \). Identify the general term rule \( T_n \) and determine the value of the \(30\)-th term. (5 Marks)
4.2
Consider the alternating geometric sequence: \( -3, 6, -12, 24, \ldots \). Determine the common ratio \( r \) and write down the general term formula \( T_n \). (5 Marks)
4.3
A decreasing arithmetic sequence is given by: \( 45, 41, 37, 33, \ldots \). Determine the general term formula \( T_n \) and calculate which term in this sequence equals \( -115 \). (5 Marks)
4.4
A geometric pattern of matchsticks forms adjacent pentagons. Figure 1 (1 pentagon) has \( 5 \) matches, Figure 2 (2 pentagons) has \( 9 \) matches, and Figure 3 (3 pentagons) has \( 13 \) matches. Determine the general rule and solve for \( n \) if you have exactly \( 201 \) matchsticks. (5 Marks)
Question 5: Functions & Relationships [20 Marks]
5.1
Complete the quadratic relationship table outputs for \( y = -2x^2 + 3 \):
(5 Marks)
| Input (\(x\)) | -2 | 0 | 1 | 3 |
|---|---|---|---|---|
| Output (\(y\)) | \(a\) | \(b\) | \(c\) | \(d\) |
5.2
Complete the table representing the rational relationship \( y = \frac{24}{x + 1} \):
(5 Marks)
| Input (\(x\)) | -4 | 1 | 2 | 5 |
|---|---|---|---|---|
| Output (\(y\)) | \(e\) | \(f\) | \(g\) | \(h\) |
5.3
Complete the mapping diagram outputs where the inputs are fractional values \( \{ \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \} \) under the rule \( y = \frac{12}{x} + 2 \). (5 Marks)
5.4
Determine the linear relationship formula from the table: Input \(x \in \{-3, -1, 1, 3\}\) resulting in outputs \(y \in \{-10, -4, 2, 8\}\). (5 Marks)
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Test 5
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June Exam - Test 5 (Ultimate Master Challenge) [100 Marks]
Topics Covered: Whole Numbers & Finance, Integers & Rational Numbers, Exponents & Scientific Notation, Numeric & Geometric Patterns, and Functions & Relationships.
Question 1: Whole Numbers, Finance & Proportions [20 Marks]
1.1
**Hire Purchase:** A refrigerator has an advertised cash price of \( \text{R}8\,000 \). Under a hire purchase agreement, you pay a \( 10\% \) deposit, and simple interest is charged on the balance at a rate of \( 14\% \) per annum for \( 24 \text{ months} \). Calculate the total amount paid for the refrigerator overall and the exact monthly installment. (8 Marks)
1.2
An investment of \( \text{R}20\,000 \) is placed for \( 3 \) years. Compare the interest earned if accumulated at \( 10\% \) per annum simple interest versus \( 10\% \) per annum compounded annually. Calculate the difference in final interest. (8 Marks)
1.3
**Inverse Proportion:** It takes \(3\) workers exactly \(8\) days to build a brick boundary wall. Calculate how many workers would be required to build the same wall in \(6\) days. (4 Marks)
Question 2: Integers & Rational Numbers [20 Marks]
2.1
Evaluate the multi-stage BODMAS integer calculation: \( \frac{(-2)^4 \cdot (-3) - \sqrt{100 \div 4}}{(-3)^2 - 4} \). (8 Marks)
2.2
A research submarine is at a depth of \( -50\text{ m} \) relative to sea level. It dives an additional \( 120\text{ m} \), and then rises \( 80\text{ m} \). Calculate its final depth using signed integers. (4 Marks)
2.3
Find three consecutive integers whose sum is exactly \(123\). Set up an algebraic equation to represent this scenario and solve for the integers. (8 Marks)
Question 3: Exponents & Scientific Notation [20 Marks]
3.1
Simplify the base variable expression fully: \( \frac{10^{x+1} \cdot 2^{x-1}}{5^x \cdot 4^x} \). (7 Marks)
3.2
Solve the following exponential equation for \(x\): \( 4^{x-1} + 4^x = 80 \). (7 Marks)
3.3
Simplify fully and write your answer with positive exponents only: \( \frac{3^{2n-1} \cdot 9^{n+1}}{27^n} \). (6 Marks)
Question 4: Numeric & Geometric Patterns [20 Marks]
4.1
Consider the non-linear progression pattern: \( 3, 8, 15, 24, \ldots \)
Determine the formula \( T_n \) for the general term, and calculate the \(20\)-th term. (8 Marks)
Determine the formula \( T_n \) for the general term, and calculate the \(20\)-th term. (8 Marks)
4.2
A sequence of matchstick figures forms adjacent pentagonal houses. Figure 1 (1 house) has \( 6 \) matches, Figure 2 (2 houses) has \( 11 \) matches, and Figure 3 (3 houses) has \( 16 \). Find the matches needed for the \(40\)-th house. (6 Marks)
4.3
A geometric pattern with alternating signs is given by: \( -2, 4, -8, 16, \dots \). Write down the general term formula \( T_n \) and determine the value of the \(10\)-th term. (6 Marks)
Question 5: Functions & Relationships [20 Marks]
5.1
Complete the quadratic output relationship table for \( y = -3x^2 + 5 \):
(8 Marks)
| Input (\(x\)) | -1 | 0 | 2 | 3 |
|---|---|---|---|---|
| Output (\(y\)) | \(a\) | \(b\) | \(c\) | \(d\) |
5.2
Complete the table representing the fractional relationship: \( y = \frac{24}{x + 1} \):
(6 Marks)
| Input (\(x\)) | -4 | 1 | 2 | 5 |
|---|---|---|---|---|
| Output (\(y\)) | \(e\) | \(f\) | \(g\) | \(h\) |
5.3
Determine the linear relationship formula from the table: Input \(x \in \{-3, -1, 1, 3\}\) resulting in outputs \(y \in \{-10, -4, 2, 8\}\). (6 Marks)