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:

Problem: Evaluate $\frac{\sqrt[3]{4396.93}}{0.7532}$ using logarithm tables.

Step 1: Break down the expression

The expression can be rewritten using logarithm properties: $$\frac{\sqrt[3]{4396.93}}{0.7532}=\frac{(4396.93{)}^{1/3}}{0.7532}=(4396.93{)}^{1/3}\times (0.7532{)}^{-1}$$

Step 2: Apply logarithm properties

We use the property $\log (a^b)=b\log (a)$ and $\log (1/a)=-\log (a)$.

Let $X=\frac{\sqrt[3]{4396.93}}{0.7532}$. Then, $$\log X=\log \left((4396.93{)}^{1/3}\times (0.7532{)}^{-1}\right)=\frac{1}{3}\log (4396.93)-\log (0.7532)$$

Step 3: Find $\log (4396.93)$

From the logarithm table:

• $\log (4396)\approx 3.6432$ (since $4396=4.396\times 10^3$ and $\log (4.396)\approx 0.6432$).
• For the decimal part, we approximate $4396.93\approx 4397$. The difference between $\log (4396)$ and $\log (4397)$ is negligible for this calculation, so we use $\log (4396.93)\approx 3.6432$.

Step 4: Find $\log (0.7532)$

For $0.7532$, we use the property $\log (0.7532)=\log (7532)-\log (10000)=\log (7532)-4$. From the logarithm table:

• $\log (7532)\approx 3.8769$ (since $7532=7.532\times 10^3$ and $\log (7.532)\approx 0.8769$). Thus, $$\log (0.7532)=3.8769-4=-0.1231$$

Step 5: Calculate $\log X$

Substitute the values: $$\log X=\frac{1}{3}\times 3.6432-(-0.1231)=1.2144+0.1231=1.3375$$

Step 6: Find $X$ by antilogarithm

Now, find $X$ by taking the antilogarithm of $1.3375$:

• The characteristic is $1$, so $X$ lies between $10^1=10$ and $10^2=100$.
• The mantissa is $0.3375$. From the antilogarithm table, $0.3375$ corresponds to approximately $2.173$. Thus, $$X\approx 2.173\times 10^1=21.73$$

Verification using direct calculation:

For verification, compute directly: $$\sqrt[3]{4396.93}\approx 16.37\text{(using a calculator)}$$ $$\frac{16.37}{0.7532}\approx 21.73$$ The result matches, confirming our calculation.

Final Answer: 21.73

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ii) Solve the following equations:

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

Step 1: Rewrite the equations in standard form

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

Multiply equation (2) by 4 to eliminate the fraction: $$4x+y=48\text{(2a)}$$

Step 2: Solve for one variable

From equation (2a), solve for $y$: $$y=48-4x\text{(3)}$$

Step 3: Substitute into equation (1a)

Substitute $y$ from equation (3) into equation (1a): $$2x+3(48-4x)=54$$ $$2x+144-12x=54$$ $$-10x+144=54$$ $$-10x=54-144$$ $$-10x=-90$$ $$x=9$$

Step 4: Find $y$ using equation (3)

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

Verification:

Substitute $x=9$ and $y=12$ into the original equations:

1. $\frac{2(9)}{3}+12=6+12=18$ ✓
2. $9+\frac{12}{4}=9+3=12$ ✓

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

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iii) Find the equation of the straight line passing through the point $(-2,5)$ and having an x-intercept three times the y-intercept.

Step 1: Understand the intercepts

Let the y-intercept be $a$. Then, the x-intercept is $3a$ (since it is three times the y-intercept).

Step 2: Equation of the line in intercept form

The intercept form of a line is: $$\frac{x}{x\text{-intercept}}+\frac{y}{y\text{-intercept}}=1$$ Substituting the intercepts: $$\frac{x}{3a}+\frac{y}{a}=1$$

Step 3: Simplify the equation

Multiply through by $3a$ to eliminate denominators: $$x+3y=3a\text{(4)}$$

Step 4: Use the given point $(-2,5)$ to find $a$

Substitute $x=-2$ and $y=5$ into equation (4): $$-2+3(5)=3a$$ $$-2+15=3a$$ $$13=3a$$ $$a=\frac{13}{3}$$

Step 5: Substitute $a$ back into the equation

Now, substitute $a=\frac{13}{3}$ into equation (4): $$x+3y=3\times \frac{13}{3}$$ $$x+3y=13$$

Verification:

Check if the point $(-2,5)$ satisfies the equation: $$-2+3(5)=-2+15=13\text{✓}$$

Find intercepts:

• For y-intercept ($x=0$): $$0+3y=13⟹y=\frac{13}{3}$$
• For x-intercept ($y=0$): $$x+0=13⟹x=13$$ Check if $x$-intercept is three times the $y$-intercept: $$13=3\times \frac{13}{3}\text{✓}$$

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