To find the net gain or loss, we need to calculate the total cost price and the total selling price of both items. Then, we compare these totals to determine if there's an overall profit or loss and express it as a percentage of the total cost price.
Given, \(\text{CP}_{\text{car}} = \text{₹}3,00,000\)
The car was sold at a profit of 10%.
\[ \text{Profit on car} = 10\% \text{ of } \text{₹}3,00,000 = \frac{10}{100} \times 3,00,000 = \text{₹}30,000 \]\[ \text{SP}_{\text{car}} = \text{CP}_{\text{car}} + \text{Profit on car} = 3,00,000 + 30,000 = \text{₹}3,30,000 \]Given, \(\text{CP}_{\text{bike}} = \text{₹}1,00,000\)
The bike was sold at a loss of 20%.
\[ \text{Loss on bike} = 20\% \text{ of } \text{₹}1,00,000 = \frac{20}{100} \times 1,00,000 = \text{₹}20,000 \]\[ \text{SP}_{\text{bike}} = \text{CP}_{\text{bike}} - \text{Loss on bike} = 1,00,000 - 20,000 = \text{₹}80,000 \]Since \(\text{Total SP} > \text{Total CP}\), there is a net gain (profit).
\[ \text{Net Gain} = \text{Total SP} - \text{Total CP} = 4,10,000 - 4,00,000 = \text{₹}10,000 \]The net result is a 2.5% gain.
B) 2.5% gain is the correct answer, as calculated from the total cost price and total selling price.
When two objects move towards each other, their relative speed is the sum of their individual speeds. This relative speed is used to calculate the time taken to cover the distance between them.
Since A and B are moving towards each other, their relative speed is the sum of their individual speeds.
\[ \text{Relative Speed} = S_A + S_B \]\[ \text{Relative Speed} = 10 \text{ km/h} + 14 \text{ km/h} \]\[ \text{Relative Speed} = 24 \text{ km/h} \]The time taken to meet is the total distance divided by their relative speed.
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]\[ \text{Time} = \frac{24 \text{ km}}{24 \text{ km/h}} \]\[ \text{Time} = 1 \text{ h} \]Correct Option: C) 1 h — The relative speed of the two persons moving towards each other is 24 km/h. To cover a distance of 24 km, they will take 1 hour.
To find the rate of interest for compound interest, we use the compound interest formula: \(A = P \left(1 + \frac{R}{100}\right)^T\). We are given the principal amount (P), the final amount (A), and the time period (T). We need to solve for the rate (R).
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
\[ 7986 = 6000 \left(1 + \frac{R}{100}\right)^3 \]
\[ \frac{7986}{6000} = \left(1 + \frac{R}{100}\right)^3 \]
Divide both numerator and denominator by their greatest common divisor. Both are divisible by 6.
\[ \frac{1331}{1000} = \left(1 + \frac{R}{100}\right)^3 \]
We know that \(1331 = 11^3\) and \(1000 = 10^3\).
\[ \left(\frac{11}{10}\right)^3 = \left(1 + \frac{R}{100}\right)^3 \]
\[ \frac{11}{10} = 1 + \frac{R}{100} \]
\[ \frac{11}{10} - 1 = \frac{R}{100} \]
\[ \frac{11 - 10}{10} = \frac{R}{100} \]
\[ \frac{1}{10} = \frac{R}{100} \]
\[ R = \frac{100}{10} \]
\[ R = 10 \]
The rate of interest is \(10\%\).
C) 10% is the correct rate of interest calculated using the compound interest formula.
If cos x - 3sin x = \(\sqrt{5}\) sin x, then the value of tan x is:
\(\frac{3-\sqrt{5}}{4}\)
\(\frac{3+\sqrt{5}}{4}\)
\(\frac{3-\sqrt{5}}{6}\)
\(3-\sqrt{5}\)
The problem asks us to find the value of \(\tan x\) given an equation involving \(\cos x\) and \(\sin x\). The strategy is to rearrange the given equation to isolate terms involving \(\sin x\) and \(\cos x\), and then divide by \(\cos x\) to express the equation in terms of \(\tan x\).
Start with the given equation:
\[ \cos x - 3\sin x = \sqrt{5} \sin x \]Move all terms involving \(\sin x\) to one side of the equation:
\[ \cos x = \sqrt{5} \sin x + 3 \sin x \]Factor out \(\sin x\) from the terms on the right side:
\[ \cos x = (\sqrt{5} + 3) \sin x \]To find \(\tan x\), which is \(\frac{\sin x}{\cos x}\), divide both sides of the equation by \(\cos x\) (assuming \(\cos x \neq 0\)):
\[ 1 = (\sqrt{5} + 3) \frac{\sin x}{\cos x} \]\[ 1 = (\sqrt{5} + 3) \tan x \]Solve for \(\tan x\):
\[ \tan x = \frac{1}{\sqrt{5} + 3} \]Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \((3 - \sqrt{5})\):
\[ \tan x = \frac{1}{(3 + \sqrt{5})} \times \frac{(3 - \sqrt{5})}{(3 - \sqrt{5})} \]Apply the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to the denominator:
\[ \tan x = \frac{3 - \sqrt{5}}{3^2 - (\sqrt{5})^2} \]\[ \tan x = \frac{3 - \sqrt{5}}{9 - 5} \]\[ \tan x = \frac{3 - \sqrt{5}}{4} \]A) The calculated value of \(\tan x\) is \(\frac{3-\sqrt{5}}{4}\), which matches option A.
The question asks about the Vedic affiliation of the Mundaka Upanishad. To answer this, one needs to recall the traditional classification of Upanishads based on the Veda they are associated with.
Correct Option: A) The Mundaka Upanishad is one of the most prominent Upanishads and is traditionally associated with the Atharva Veda. It is known for containing the famous motto "Satyameva Jayate" (Truth alone triumphs), which is India's national motto.
To find the difference between the place values of two digits in a number, we first need to determine the place value of each specified digit. The place value of a digit depends on its position in the number. Once we have the individual place values, we subtract the smaller place value from the larger one.
The difference is 199600.
B) 199600 — This is the correct difference calculated by subtracting the place value of 4 from the place value of 2.
Solve the given equation.
\(\sqrt{544^2-256^2}\)= ?
The problem requires us to evaluate an expression involving a square root of a difference of squares. We can use the algebraic identity \((a^2 - b^2) = (a - b)(a + b)\) to simplify the expression before taking the square root. This will make the calculation much easier than squaring the large numbers directly.
Identify the expression:
\[ \sqrt{544^2 - 256^2} \]Recognize the form \(a^2 - b^2\), where \(a = 544\) and \(b = 256\). Apply the identity \((a^2 - b^2) = (a - b)(a + b)\):
\[ 544^2 - 256^2 = (544 - 256)(544 + 256) \]Calculate the values inside the parentheses:
\(544 - 256 = 288\)
\(544 + 256 = 800\)
Substitute these values back into the expression:
\[ \sqrt{288 \times 800} \]Simplify the product inside the square root. We can factor out perfect squares to make the calculation easier:
\[ \sqrt{288 \times 800} = \sqrt{(144 \times 2) \times (100 \times 8)} \] \[ = \sqrt{144 \times 100 \times 2 \times 8} \] \[ = \sqrt{144 \times 100 \times 16} \]Take the square root of each perfect square factor:
\[ \sqrt{144} = 12 \] \[ \sqrt{100} = 10 \] \[ \sqrt{16} = 4 \]Multiply these results:
\[ 12 \times 10 \times 4 = 120 \times 4 = 480 \]Thus, \(\sqrt{544^2 - 256^2} = 480\).
A) 480 is the correct answer as calculated by simplifying the expression using the difference of squares identity.
The problem provides a ratio between two variables, \(x\) and \(y\). We need to find the ratio of two expressions involving \(x\) and \(y\). The most straightforward approach is to express one variable in terms of the other using the given ratio and then substitute it into the expression. Alternatively, we can divide the expression by one of the variables (e.g., \(y\)) to simplify it into a form that only depends on the ratio \(\frac{x}{y}\).
Understand the given ratio: We are given that \(x : y = 2 : 3\). This can be written as a fraction:
\[ \frac{x}{y} = \frac{2}{3} \]Express one variable in terms of the other: From the above, we can write \(x\) in terms of \(y\): Multiply both sides by \(y\):
\[ x = \frac{2}{3}y \]Substitute into the expression: We need to find the ratio \((4x + 3y) : (2x + 5y)\). Let's substitute \(x = \frac{2}{3}y\) into both parts of the ratio:
First part:
\[ 4x + 3y = 4\left(\frac{2}{3}y\right) + 3y = \frac{8}{3}y + 3y \]To add these terms, find a common denominator:
\[ \frac{8}{3}y + \frac{9}{3}y = \frac{8y + 9y}{3} = \frac{17y}{3} \]Second part:
\[ 2x + 5y = 2\left(\frac{2}{3}y\right) + 5y = \frac{4}{3}y + 5y \]To add these terms, find a common denominator:
\[ \frac{4}{3}y + \frac{15}{3}y = \frac{4y + 15y}{3} = \frac{19y}{3} \]Form the new ratio: Now, form the ratio of the two simplified expressions:
\[ (4x + 3y) : (2x + 5y) = \frac{17y}{3} : \frac{19y}{3} \]This can be written as a fraction:
\[ \frac{\frac{17y}{3}}{\frac{19y}{3}} \]Cancel out the common terms \(y\) and \(3\):
\[ \frac{17}{19} \]So, the ratio is \(17 : 19\).
Given \(x : y = 2 : 3\), which means \[\frac{x}{y} = \frac{2}{3}\].
We need to find the ratio \((4x + 3y) : (2x + 5y)\). Divide both the numerator and the denominator of the ratio by \(y\):
\[ \frac{4x + 3y}{2x + 5y} = \frac{\frac{4x}{y} + \frac{3y}{y}}{\frac{2x}{y} + \frac{5y}{y}} = \frac{4\left(\frac{x}{y}\right) + 3}{2\left(\frac{x}{y}\right) + 5} \]Substitute the value of \[\frac{x}{y} = \frac{2}{3}\] into the expression:
\[ \frac{4\left(\frac{2}{3}\right) + 3}{2\left(\frac{2}{3}\right) + 5} = \frac{\frac{8}{3} + 3}{\frac{4}{3} + 5} \]Simplify the numerator and the denominator:
Numerator: \[\frac{8}{3} + 3 = \frac{8}{3} + \frac{9}{3} = \frac{17}{3}\]
Denominator: \[\frac{4}{3} + 5 = \frac{4}{3} + \frac{15}{3} = \frac{19}{3}\]
Form the final ratio:
\[ \frac{\frac{17}{3}}{\frac{19}{3}} = \frac{17}{19} \]Thus, the ratio is \(17 : 19\).
B) \(17 : 19\) is the correct ratio obtained after substituting the given ratio into the expression and simplifying.
The problem presents a number series: 5, 13, 29, ?. We need to identify the pattern or rule governing this series to find the missing number.
The differences are 8 and 16. It appears that each difference is double the previous difference (\(16 = 2 \times 8\)).
Following this pattern, the next difference should be \(2 \times 16 = 32\).
Add this new difference to the last known term in the series:
\[ \text{Missing Term} = 29 + 32 = 61 \]Another common pattern for such series is \(2n + k\). Let's check if the series follows the rule \(2 \times \text{previous term} + k\).
This pattern also holds true. So, for the next term:
\[ \text{Missing Term} = 2 \times 29 + 3 = 58 + 3 = 61 \]A) 61 — Both identified patterns (doubling the difference and \(2n+3\)) consistently lead to 61 as the next number in the series.
To find the strongest boy, we need to establish a hierarchy of strength based on the given statements. We will represent "stronger than" with a greater than sign (>) and "weaker than" with a less than sign (<).
Now let's combine all the inequalities:
Let's build the complete chain:
Putting it all together, the strongest relationships are:
\[ \text{Sujeet} > \text{Akhil} = \text{Zeeshan} > \text{Rohan} > \text{Mathew} > \text{Paul} \]and also
\[ \text{Sujeet} > \text{Akhil} = \text{Zeeshan} > \text{Rohan} > \text{Subhash} > \text{Paul} \]From this combined hierarchy, Sujeet is at the top.
Correct Option: C) Sujeet