The rationalising factor of sqrt(5) + sqrt(3) is:

Question

The rationalising factor of sqrt(5) + sqrt(3) is:

Asked on March 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-26T07:22:11.009Z

1. The rationalising factor of $\sqrt{5} + \sqrt{3}$ is: Step 1: Identify the expression to be rationalized. The expression is $\sqrt{5} + \sqrt{3}$. Step 2: Determine the rationalizing factor. For an expression of the form $\sqrt{a} + \sqrt{b}$, its rationalizing factor is $\sqrt{a} - \sqrt{b}$. Therefore, the rationalizing factor of $\sqrt{5} + \sqrt{3}$ is $\sqrt{5} - \sqrt{3}$. The correct option is A. $$\boxed{\text{A) } \sqrt{5} - \sqrt{3}}$$ 2. In a cricket match two batsmen together scored 181 runs. The linear equation form for the statement is: Step 1: Define variables for the runs scored by each batsman. Let $x$ be the runs scored by the first batsman. Let $y$ be the runs scored by the second batsman. Step 2: Formulate the equation based on the given information. "Two batsmen together scored 181 runs" means the sum of their runs is 181. So, $x + y = 181$. The correct option is C. $$\boxed{\text{C) } x + y = 181}$$ 3. In a parallelogram, if the adjacent angles are in the ratio 2:3, then the measure of these adjacent angles are: Step 1: Recall the property of adjacent angles in a parallelogram. Adjacent angles in a parallelogram are supplementary, meaning their sum is $180^\circ$. Step 2: Set up an equation based on the given ratio. Let the adjacent angles be $2k$ and $3k$. $$2k + 3k = 180^\circ$$ Step 3: Solve for $k$. $$5k = 180^\circ$$ $$k = \frac{180^\circ}{5}$$ $$k = 36^\circ$$ Step 4: Calculate the measure of each angle. First angle: $2k = 2 \times 36^\circ = 72^\circ$ Second angle: $3k = 3 \times 36^\circ = 108^\circ$ The correct option is B. $$\boxed{\text{B) } 72^\circ \text{ and } 108^\circ}$$ 4. If the radius of a sphere is $2r$, then its volume will be: Step 1: Write down the formula for the volume of a sphere. The volume $V$ of a sphere with radius $R$ is given by: $$V = \frac{4}{3}\pi R^3$$ Step 2: Substitute the given radius into the formula. Given radius $R = 2r$. $$V = \frac{4}{3}\pi (2r)^3$$ Step 3: Simplify the expression. $$V = \frac{4}{3}\pi (2^3 r^3)$$ $$V = \frac{4}{3}\pi (8 r^3)$$ $$V = \frac{4 \times 8}{3}\pi r^3$$ $$V = \frac{32}{3}\pi r^3$$ The correct option is D. $$\boxed{\text{D) } \frac{32}{3}\pi r^3}$$

Accepted Answer · 2026-03-26T07:22:11.009Z

1. The rationalising factor of $\sqrt{5} + \sqrt{3}$ is: Step 1: Identify the expression to be rationalized. The expression is $\sqrt{5} + \sqrt{3}$. Step 2: Determine the rationalizing factor. For an expression of the form $\sqrt{a} + \sqrt{b}$, its rationalizing factor is $\sqrt{a} - \sqrt{b}$. Therefore, the rationalizing factor of $\sqrt{5} + \sqrt{3}$ is $\sqrt{5} - \sqrt{3}$. The correct option is A. $$\boxed{\text{A) } \sqrt{5} - \sqrt{3}}$$ 2. In a cricket match two batsmen together scored 181 runs. The linear equation form for the statement is: Step 1: Define variables for the runs scored by each batsman. Let $x$ be the runs scored by the first batsman. Let $y$ be the runs scored by the second batsman. Step 2: Formulate the equation based on the given information. "Two batsmen together scored 181 runs" means the sum of their runs is 181. So, $x + y = 181$. The correct option is C. $$\boxed{\text{C) } x + y = 181}$$ 3. In a parallelogram, if the adjacent angles are in the ratio 2:3, then the measure of these adjacent angles are: Step 1: Recall the property of adjacent angles in a parallelogram. Adjacent angles in a parallelogram are supplementary, meaning their sum is $180^\circ$. Step 2: Set up an equation based on the given ratio. Let the adjacent angles be $2k$ and $3k$. $$2k + 3k = 180^\circ$$ Step 3: Solve for $k$. $$5k = 180^\circ$$ $$k = \frac{180^\circ}{5}$$ $$k = 36^\circ$$ Step 4: Calculate the measure of each angle. First angle: $2k = 2 \times 36^\circ = 72^\circ$ Second angle: $3k = 3 \times 36^\circ = 108^\circ$ The correct option is B. $$\boxed{\text{B) } 72^\circ \text{ and } 108^\circ}$$ 4. If the radius of a sphere is $2r$, then its volume will be: Step 1: Write down the formula for the volume of a sphere. The volume $V$ of a sphere with radius $R$ is given by: $$V = \frac{4}{3}\pi R^3$$ Step 2: Substitute the given radius into the formula. Given radius $R = 2r$. $$V = \frac{4}{3}\pi (2r)^3$$ Step 3: Simplify the expression. $$V = \frac{4}{3}\pi (2^3 r^3)$$ $$V = \frac{4}{3}\pi (8 r^3)$$ $$V = \frac{4 \times 8}{3}\pi r^3$$ $$V = \frac{32}{3}\pi r^3$$ The correct option is D. $$\boxed{\text{D) } \frac{32}{3}\pi r^3}$$