To find the area of the largest triangle that can be inscribed in a semi-circle, we need to maximize its base and height. The largest possible base for a triangle inscribed in a semi-circle will be the diameter of the semi-circle. The maximum height will be the radius of the semi-circle, achieved when the apex of the triangle is at the midpoint of the arc.
Correct Option: A) 36 cm²
The problem presents a series of three-letter codes based on the English alphabetical order. We need to identify the pattern in the progression of each letter's position within the alphabet to find the missing code.
Let's analyze the series: RJZ, PMW, NPT, LSQ, ?
We will examine the pattern for each position (first, second, and third letter) separately.
Combining these, the next term in the series is JVN.
B) JVN is the correct option because it follows the established patterns for all three letter positions: the first letter decreases by 2, the second letter increases by 3, and the third letter decreases by 3.
A sum of money, on compound interest, amounts to ₹578.40 in 2 years and to ₹614.55 in 3 years. The rate of interest per annum, in case of annual compounding, is:
This problem involves compound interest. When the interest is compounded annually, the amount after a certain number of years can be used to find the interest rate. The key idea is that the amount at the end of \(n\) years becomes the principal for the \((n+1)\)th year. Therefore, the interest earned in the third year is simply the difference between the amount at the end of 3 years and the amount at the end of 2 years. This interest is calculated on the amount at the end of 2 years.
Let the principal be \(P\) and the rate of interest be \(R\) per annum. The formula for the amount \(A\) after \(n\) years with compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]According to the problem, the amount after 2 years is ₹578.40. So,
\[ 578.40 = P \left(1 + \frac{R}{100}\right)^2 \quad \text{(Equation 1)} \]The amount after 3 years is ₹614.55. So,
\[ 614.55 = P \left(1 + \frac{R}{100}\right)^3 \quad \text{(Equation 2)} \]Divide Equation 2 by Equation 1:
\[ \frac{614.55}{578.40} = \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)^2} \]\[ \frac{614.55}{578.40} = 1 + \frac{R}{100} \]Calculate the value of the left side:
\[ \frac{614.55}{578.40} = 1.0625 \]Now, solve for \(R\):
\[ 1.0625 = 1 + \frac{R}{100} \]\[ 0.0625 = \frac{R}{100} \]\[ R = 0.0625 \times 100 \]\[ R = 6.25\% \]Convert the decimal percentage to a mixed fraction:
\[ 6.25\% = 6 \frac{25}{100}\% = 6 \frac{1}{4}\% \]B) The calculated rate of interest is \(6 \frac{1}{4}\)%, which matches option B.
Myopia, also known as nearsightedness, is a common vision defect where distant objects appear blurry. This occurs because the eye's lens focuses light in front of the retina instead of directly on it. To correct this, a lens that diverges light rays before they enter the eye is required. We need to identify which type of lens has this property.
Correct Option: C) a concave lens of suitable power. A concave lens diverges light rays before they enter the eye, effectively pushing the focal point back onto the retina, thus correcting the blurry vision for distant objects.
Anju ranked 18th from the bottom and 15th from the top in her class. How many students are there in her class?
This problem involves determining the total number of students in a class given a person's rank from both the top and the bottom. The key is to understand that the person herself is counted in both rankings. Therefore, to find the total number of students, we add the rank from the top and the rank from the bottom, and then subtract 1 (because the person is counted twice).
The formula to find the total number of students is:
\[ \text{Total students} = \text{Rank from top} + \text{Rank from bottom} - 1 \]Substitute the given values into the formula:
\[ \text{Total students} = 15 + 18 - 1 \] \[ \text{Total students} = 33 - 1 \] \[ \text{Total students} = 32 \]Thus, there are 32 students in the class.
A) 32 is the correct answer because adding Anju's rank from the top (15) and her rank from the bottom (18) gives 33. Since Anju herself is counted twice in this sum, we subtract 1 to get the actual total number of distinct students, which is 32.
The distance between the pole and focus of a spherical mirror, in terms of its radius of curvature R, is equal to:
The question asks for the relationship between the focal length (distance between the pole and focus) and the radius of curvature of a spherical mirror. This is a fundamental concept in optics.
A) \( \frac{R}{2} \) is the correct answer because the focal length (distance between the pole and focus) of a spherical mirror is half its radius of curvature.
The question asks about a specific event on June 26, 2024, where DRDO handed over a particular system to the Indian Navy. To answer this, one needs to recall recent news and developments related to DRDO and the Indian Navy.
Correct Option: D) Medium Range-Microwave Obscurant Chaff Rocket (MR-MOCR) was indeed handed over by DRDO to the Indian Navy on June 26, 2024. This system is crucial for enhancing the survivability of Indian Naval ships against advanced radar-guided missile threats.
The question asks to identify which solution can be distinguished by its smell. This requires understanding the properties of the substances listed and whether they emit a discernible odor when dissolved in water.
D) Dettol in water — Dettol is an antiseptic liquid with a characteristic strong, phenolic smell due to its active ingredient, chloroxylenol. When mixed with water, this smell is readily identifiable.
The problem asks us to find the values of X and Y such that the mathematical operation relating the first number to the second number in the pair X : 23 is the same as the operation relating 476 : Y. We need to identify this common pattern.
A) X = 322, Y = 34
As shown in the step-by-step analysis, if X = 322 and Y = 34, the relationship between the numbers in both pairs is consistent: the first number is 14 times the second number. For the first pair, \(322 = 14 \times 23\). For the second pair, \(476 = 14 \times 34\).
To find the least positive number that can be added to 2488 so that it is completely divisible by 3, 4, 5, and 6, we first need to find the Least Common Multiple (LCM) of these divisors. The LCM will be the smallest number that is divisible by all of them. Then, we will find the remainder when 2488 is divided by this LCM. The difference between the LCM and this remainder will be the least positive number to be added.
Find the LCM of 3, 4, 5, and 6:
To find the LCM, we take the highest power of all prime factors involved:
\[ \text{LCM}(3, 4, 5, 6) = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]So, the least number divisible by 3, 4, 5, and 6 is 60.
Divide 2488 by the LCM (60) and find the remainder:
\[ 2488 \div 60 \]We can perform long division:
\[ \begin{array}{r} 41 \\ 60 \overline{) 2488} \\ -240 \downarrow \\ \hline 88 \\ -60 \\ \hline 28 \end{array} \]The quotient is 41 and the remainder is 28.
Calculate the least positive number to be added:
To make 2488 completely divisible by 60, we need to add a number such that the sum is a multiple of 60. The current remainder is 28. The next multiple of 60 after \(2488 - 28 = 2460\) is \(2460 + 60 = 2520\).
The number to be added is the difference between the LCM and the remainder:
\[ \text{Number to be added} = \text{LCM} - \text{Remainder} = 60 - 28 = 32 \]If we add 32 to 2488, we get \(2488 + 32 = 2520\), which is completely divisible by 60 (since \(2520 = 60 \times 42\)).
Correct Option: C) 32