Answer any two questions. [2x5=10] i) Evaluate using Logarithm Table. [3]4396.930.7532 ii) Solve the following equations. 2x3 + y = 18 , x + y4 = 12 iii) Find the equation of straight line which…

i) Evaluate using Logarithm Table: $\frac{\sqrt[3]{4396.93}}{0.7532}$

Step 1: Break down the expression

The expression is: $$\frac{\sqrt[3]{4396.93}}{0.7532}$$ We can rewrite it using logarithms as: $$\log \left(\frac{\sqrt[3]{4396.93}}{0.7532}\right)=\log (\sqrt[3]{4396.93})-\log (0.7532)$$

Step 2: Evaluate $\log (4396.93)$

Using a logarithm table (base 10):

• Find $\log (4396.93)$.
• From the table, $\log (4400)\approx 3.6435$.
• Adjust for the difference: $4396.93$ is $4400-3.07$.
• The correction factor for $-3.07$ is approximately $-0.0013$ (from logarithmic difference tables).
• Thus, $\log (4396.93)\approx 3.6435-0.0013=3.6422$.

Step 3: Evaluate $\log (\sqrt[3]{4396.93})$

$$\log (\sqrt[3]{4396.93})=\frac{1}{3}\log (4396.93)=\frac{1}{3}\times 3.6422=1.2141$$

Step 4: Evaluate $\log (0.7532)$

• $0.7532$ can be written as $7.532\times 10^{-1}$.
• $\log (7.532)\approx 0.8769$ (from the table).
• $\log (10^{-1})=-1$.
• Thus, $\log (0.7532)=0.8769-1=-0.1231$.

Step 5: Combine the logarithms

$$\log \left(\frac{\sqrt[3]{4396.93}}{0.7532}\right)=1.2141-(-0.1231)=1.2141+0.1231=1.3372$$

Step 6: Find the antilogarithm

• The antilogarithm of $1.3372$ is found by:
  • Characteristic: $1$ → $10^1=10$.
  • Mantissa: $0.3372$ → From the antilog table, $0.3372$ corresponds to $2.17$.
  • Thus, the result is $10\times 2.17=21.7$.

Final Answer: 21.7

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ii) Solve the following equations:

$$\frac{2x}{3}+y=18\text{(1)}$$ $$x+\frac{y}{4}=12\text{(2)}$$

Step 1: Simplify Equation (1)

Multiply both sides by 3 to eliminate the fraction: $$2x+3y=54\text{(1a)}$$

Step 2: Simplify Equation (2)

Multiply both sides by 4: $$4x+y=48\text{(2a)}$$

Step 3: Solve for $y$ in Equation (2a)

$$y=48-4x\text{(3)}$$

Step 4: Substitute $y$ into Equation (1a)

$$2x+3(48-4x)=54$$ $$2x+144-12x=54$$ $$-10x+144=54$$ $$-10x=54-144$$ $$-10x=-90$$ $$x=9$$

Step 5: Find $y$ using Equation (3)

$$y=48-4(9)=48-36=12$$

Final Answer: $x=9$, $y=12$.

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iii) Find the equation of a straight line passing through $(-2,5)$ with $x$-intercept three times the $y$-intercept.

Step 1: Define intercepts

Let the $y$-intercept be $a$. Then, the $x$-intercept is $3a$.

Step 2: Use intercept form of a line

The intercept form is: $$\frac{x}{3a}+\frac{y}{a}=1$$ Simplify: $$\frac{x}{3a}+\frac{y}{a}=1⟹\frac{x+3y}{3a}=1⟹x+3y=3a\text{(4)}$$

Step 3: Substitute the point $(-2,5)$ into Equation (4)

$$-2+3(5)=3a$$ $$-2+15=3a$$ $$13=3a⟹a=\frac{13}{3}$$

Step 4: Rewrite the equation

Substitute $a=\frac{13}{3}$ into Equation (4): $$x+3y=3\times \frac{13}{3}⟹x+3y=13$$

Step 5: Final equation

$$x+3y-13=0$$

Final Answer: The equation of the line is $x+3y-13=0$.