The question asks to identify the world-famous scientist known for the theory of relativity. This is a direct knowledge-based question about prominent figures in physics.
Correct Option: A) Albert Einstein is widely recognized as the physicist who developed the theory of relativity, which includes both special relativity and general relativity. His work revolutionized our understanding of space, time, gravity, and the universe.
A boy does \( \frac{1}{4} \) th of the work done by a man in the same time. 15 men complete a particular work in 17 days. If 3 men are replaced by 3 boys, the work will be completed in:
This problem involves profit and loss percentages. We are given a relationship between the cost price of 'x' articles and the selling price of 'y' articles, along with the overall profit percentage. We need to find the ratio x : y.
Let \(\text{CP}\) be the cost price of one article and \(\text{SP}\) be the selling price of one article.
According to the problem statement, the cost price of \(x\) articles is equal to the selling price of \(y\) articles. This can be written as:
\[ x \times \text{CP} = y \times \text{SP} \]From this equation, we can find the ratio of \(\text{SP}\) to \(\text{CP}\):
\[ \frac{\text{SP}}{\text{CP}} = \frac{x}{y} \]We are given that the profit is 25%. The formula for profit percentage is:
\[ \text{Profit Percentage} = \left( \frac{\text{SP} - \text{CP}}{\text{CP}} \right) \times 100\% \]Substitute the given profit percentage into the formula:
\[ 25 = \left( \frac{\text{SP}}{\text{CP}} - 1 \right) \times 100 \]Divide by 100:
\[ \frac{25}{100} = \frac{\text{SP}}{\text{CP}} - 1 \]\[ 0.25 = \frac{\text{SP}}{\text{CP}} - 1 \]Add 1 to both sides:
\[ \frac{\text{SP}}{\text{CP}} = 1 + 0.25 \]\[ \frac{\text{SP}}{\text{CP}} = 1.25 \]Now, we have two expressions for \(\frac{\text{SP}}{\text{CP}}\). Equate them:
\[ \frac{x}{y} = 1.25 \]Convert 1.25 to a fraction:
\[ 1.25 = \frac{125}{100} = \frac{5}{4} \]Therefore,
\[ \frac{x}{y} = \frac{5}{4} \]The ratio \(x : y\) is \(5 : 4\).
Correct Option: C) 5 : 4
This problem involves the concept of division with remainders. We are given a number that leaves a specific remainder when divided by 45. We need to find the remainder when the same number is divided by 15. The key is to express the number in terms of the divisor and remainder, and then use the relationship between the divisors (45 is a multiple of 15).
Let the number be \(N\). According to the division algorithm, when \(N\) is divided by 45, it leaves a remainder of 28. This can be written as:
\[ N = 45q + 28 \]where \(q\) is the quotient and \(0 \le 28 < 45\).
We need to find the remainder when \(N\) is divided by 15. We can rewrite the expression for \(N\) by noting that 45 is a multiple of 15 (\(45 = 15 \times 3\)).
\[ N = (15 \times 3)q + 28 \]Now, we can group the terms to show divisibility by 15:
\[ N = 15(3q) + 28 \]The term \(15(3q)\) is clearly divisible by 15. So, the remainder when \(N\) is divided by 15 will depend on the remainder when 28 is divided by 15.
Divide 28 by 15:
\[ 28 = 15 \times 1 + 13 \]Here, the quotient is 1 and the remainder is 13.
Substitute this back into the expression for \(N\):
\[ N = 15(3q) + (15 \times 1 + 13) \] \[ N = 15(3q) + 15(1) + 13 \] \[ N = 15(3q + 1) + 13 \]From this form, we can see that when \(N\) is divided by 15, the quotient is \((3q + 1)\) and the remainder is 13.
A) 13 — As derived in the step-by-step explanation, the remainder when the number is divided by 15 is 13.
The problem states a relationship between the angles of a triangle. We will use the fundamental property that the sum of all angles in a triangle is \(180^\circ\) to find the measure of the third angle.
Let the three angles of the triangle be \(A\), \(B\), and \(C\).
According to the fundamental property of a triangle, the sum of its interior angles is \(180^\circ\). So, we have:
\[ A + B + C = 180^\circ \]The problem states that the sum of two angles of the triangle is equal to its third angle. Let's assume these two angles are \(A\) and \(B\), and the third angle is \(C\). So, we can write the given condition as:
\[ A + B = C \]Now, substitute the expression for \((A + B)\) from the second equation into the first equation:
\[ (A + B) + C = 180^\circ \]Substitute \(A + B = C\):
\[ C + C = 180^\circ \]Combine the terms involving \(C\):
\[ 2C = 180^\circ \]Solve for \(C\):
\[ C = \frac{180^\circ}{2} \] \[ C = 90^\circ \]Thus, the measure of the third angle is \(90^\circ\).
A) 90° is the correct answer. This angle indicates that the triangle is a right-angled triangle.
The problem involves calculating the total work done in terms of man-hours and then determining the number of days required to complete the same amount of work under new conditions. We will first calculate the total hours worked in the initial scenario and then use this total work to find the number of days needed in the second scenario, considering the daily working hours and lunch break.
Correct Option: A) 10 days is the correct answer. The total work is 75 man-hours. If the man works 7.5 hours per day, he will complete the work in 10 days.
This is a coding-decoding question where letters are assigned numerical values. We need to identify the pattern or rule used to assign numbers to 'J' and 'N' and then apply the same rule to find the numerical representation of 'JUNIOR'.
C) 1761318129 — This matches the calculated sequence of reverse alphabetical positions for each letter in JUNIOR.
If (a + b\(\sqrt{2}\))² = 19 + 6\(\sqrt{2}\), then a is equal to:
If the speeds of a train in 10 successive hours are a1, a2, a3, a4, a5, a6, a7, a8, a9, and a10, then the average speed of the train is:
harmonic mean of a1, a2, a3, a4, a5, a6, a7, a8, a9, and a10
geometric mean of a1, a2, a3, a4, a5, a6, a7, a8, a9, and a10
arithmetic mean of a1, a2, a3, a4, a5, a6, a7, a8, a9, and a10
median of a1, a2, a3, a4, a5, a6, a7, a8, a9, and a10
To find the final cost, we first calculate the cost of the extended warranty and add it to the laptop's price. Then, we calculate the GST on this total amount and add it to get the final cost.
Calculate the cost of the extended warranty:
The laptop costs ₹27,000. An extended warranty costs 15% extra.
\[ \text{Warranty Cost} = 15\% \text{ of } \text{Laptop Cost} \]\[ \text{Warranty Cost} = 0.15 \times 27,000 \]\[ \text{Warranty Cost} = \text{₹}4,050 \]Calculate the total cost before GST:
This is the sum of the laptop cost and the warranty cost.
\[ \text{Total Cost before GST} = \text{Laptop Cost} + \text{Warranty Cost} \]\[ \text{Total Cost before GST} = 27,000 + 4,050 \]\[ \text{Total Cost before GST} = \text{₹}31,050 \]Calculate the GST amount:
GST is 6% on the whole (total cost before GST).
\[ \text{GST Amount} = 6\% \text{ of } \text{Total Cost before GST} \]\[ \text{GST Amount} = 0.06 \times 31,050 \]\[ \text{GST Amount} = \text{₹}1,863 \]Calculate the final cost:
This is the total cost before GST plus the GST amount.
\[ \text{Final Cost} = \text{Total Cost before GST} + \text{GST Amount} \]\[ \text{Final Cost} = 31,050 + 1,863 \]\[ \text{Final Cost} = \text{₹}32,913 \]Correct Option: C) ₹32,913