Calculate the total surface area of a cone if its radius is \( \frac{r}{4} \) and slant Height is 4l.
2πr(l+r)
2πrl
πr\(\left(l+\frac{r}{16}\right)\)
πr(l+r)
To calculate the total surface area of a cone, we need to sum its curved surface area and the area of its circular base. We will use the given modified radius and slant height in the standard formulas.
The total surface area (\( \text{TSA} \)) of a cone is given by the sum of its curved surface area (\( \text{CSA} \)) and the area of its base (\( \text{A}_{\text{base}} \)).
\[ \text{TSA} = \text{CSA} + \text{A}_{\text{base}} \]The curved surface area of a cone is \( \pi \times \text{radius} \times \text{slant height} \).
\[ \text{CSA} = \pi r' l' \]The area of the circular base is \( \pi \times (\text{radius})^2 \).
\[ \text{A}_{\text{base}} = \pi (r')^2 \]So, the total surface area is:
\[ \text{TSA} = \pi r' l' + \pi (r')^2 = \pi r' (l' + r') \]Given, the radius of the cone is \( r' = \frac{r}{4} \).
Given, the slant height of the cone is \( l' = 4l \).
Substitute these values into the total surface area formula:
\[ \text{TSA} = \pi \left(\frac{r}{4}\right) \left(4l + \frac{r}{4}\right) \]Distribute \( \pi \frac{r}{4} \) inside the parenthesis:
\[ \text{TSA} = \left(\pi \frac{r}{4}\right) (4l) + \left(\pi \frac{r}{4}\right) \left(\frac{r}{4}\right) \]\[ \text{TSA} = \pi r l + \pi \frac{r^2}{16} \]We can factor out \( \pi r \) from both terms:
\[ \text{TSA} = \pi r \left(l + \frac{r}{16}\right) \]C) \( \pi r \left(l + \frac{r}{16}\right) \)
The question asks for the software responsible for translating a high-level source program into a form that a machine can understand. This involves identifying the role of different types of language processors in computer science.
Correct Option: C) A compiler is a software that translates a high-level source program into machine-understandable code (machine code or object code).
The problem asks us to identify the item that is different from the others in a given set of kitchen utensils. We need to categorize the items based on their primary function or type.
Correct Option: C) Pan
This problem involves simple interest. We are given the amounts after two different time periods. The key idea is that in simple interest, the interest earned each year is constant. By finding the difference in amounts and the difference in time periods, we can calculate the interest earned per year. Once the annual interest is known, we can find the principal amount and then the rate of interest.
Let the principal amount be \(P\) and the rate of simple interest be \(R\) per annum. The formula for the amount \(A\) after \(T\) years at simple interest is:
\[ A = P + \frac{P \times R \times T}{100} \]According to the problem statement:
The difference in the amounts is due to the interest earned in the difference of years:
\[ \text{Interest for } (7 - 5) \text{ years} = A_7 - A_5 \]\[ \text{Interest for } 2 \text{ years} = \text{₹}9,200 - \text{₹}7,500 = \text{₹}1,700 \]Now, calculate the interest for 1 year:
\[ \text{Interest for } 1 \text{ year} = \frac{\text{₹}1,700}{2} = \text{₹}850 \]Using the interest for 1 year, we can find the interest for 5 years:
\[ \text{Interest for } 5 \text{ years} = 5 \times \text{₹}850 = \text{₹}4,250 \]Now, calculate the principal amount \(P\). The amount after 5 years is the principal plus the interest for 5 years:
\[ P = A_5 - \text{Interest for } 5 \text{ years} \]\[ P = \text{₹}7,500 - \text{₹}4,250 = \text{₹}3,250 \]Finally, calculate the rate of interest \(R\). We know the principal \(P\), the interest for 1 year (\(I_1\)), and the time \(T=1\) year. The simple interest formula is \(I = \frac{P \times R \times T}{100}\).
\[ R = \frac{I_1 \times 100}{P \times T} \]\[ R = \frac{850 \times 100}{3250 \times 1} \]\[ R = \frac{85000}{3250} \]\[ R = \frac{8500}{325} \]\[ R = \frac{340}{13} \]\[ R \approx 26.1538\% \]Rounding to one decimal place, the approximate rate of interest is 26.2%.
D) 26.2% is the closest approximation to the calculated rate of interest of 26.1538%.
To find the largest number among five consecutive multiples of 2, we can represent these numbers algebraically. Since they are consecutive multiples of 2, they will differ by 2. We can set up an equation for their sum and solve for the first multiple, then determine the largest.
Let the first multiple of 2 be \(x\). Since the numbers are consecutive multiples of 2, they will be:
Here, \(x\) must be an even number.
The sum of these five numbers is given as 660. So, we can write the equation:
\[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 660 \]Combine like terms:
\[ 5x + (2 + 4 + 6 + 8) = 660 \] \[ 5x + 20 = 660 \]Subtract 20 from both sides of the equation:
\[ 5x = 660 - 20 \] \[ 5x = 640 \]Divide by 5 to find the value of \(x\):
\[ x = \frac{640}{5} \] \[ x = 128 \]So, the first number is 128. The five consecutive multiples of 2 are:
The largest number among these is 136.
Correct Option: D) 136
The question asks for the full form of 'IBEF' in the context of the 'India Lounge' and 'Make in India' programme. This requires knowledge of common abbreviations related to Indian trade and investment promotion.
D) India Brand Equity Foundation — This is the correct full form of IBEF. The India Brand Equity Foundation is a trust established by the Department of Commerce, Ministry of Commerce and Industry, Government of India. It is responsible for promoting and creating international awareness of 'Brand India' and disseminating knowledge about India to the global community, aligning perfectly with initiatives like the 'India Lounge' and 'Make in India'.
The question asks for the approximate percentage of India's population living in urban areas according to the 2011 Census. This is a factual recall question based on official census data.
Correct Option: B) 31.15% is the correct figure for the urban population percentage as per the 2011 Census of India.
The question asks about the type of object that forms shadows when placed in the path of light. Shadows are formed when an object blocks light from reaching a surface. We need to identify the property of an object that causes it to block light.
D) opaque — Opaque objects completely block the passage of light, thereby creating a region of darkness (a shadow) behind them where the light cannot reach.
Select the option that is related to the third term in the same way as the second term is related to the first term.
Pen : Writing :: EVM : ?
This is an analogy question where we need to find the relationship between the first pair of terms and apply the same relationship to the third term to find the fourth term.
Correct Option: C) Voting
An EVM (Electronic Voting Machine) is an instrument used for the process of voting. Just as a pen is used for writing, an EVM is used for voting. This establishes the same functional relationship between the pairs.
The problem asks us to find a positive number based on a given condition. We can translate the word problem into an algebraic equation and then solve for the unknown number. The condition states that when the number is added by 15, it is equal to 100 times its reciprocal.
Correct Option: C) 5