To find the curved surface area of a cylinder, we use the formula \(2 \pi r h\), where \(r\) is the radius and \(h\) is the height. We are given the radius and height, so we can directly substitute these values into the formula.
Correct Option: A) 960π cm² is the correct curved surface area calculated by substituting the given radius and height into the formula \(2 \pi r h\).
To solve this problem, we need to use the properties of quadratic equations. For a quadratic equation of the form \(ax^2 + bx + c = 0\), if one root is the reciprocal of the other, then the product of the roots is 1. We will use the formula for the product of roots, which is \(\frac{c}{a}\), to find the value of \(k\).
Identify the coefficients:
The given quadratic equation is \(4x^2 - 2x + k - 4 = 0\).
Comparing this with the standard form \(ax^2 + bx + c = 0\), we have:
Apply the condition for reciprocal roots:
If one root is the reciprocal of the other, let the roots be \(\alpha\) and \(\frac{1}{\alpha}\). The product of the roots is then:
\[ \alpha \times \frac{1}{\alpha} = 1 \]Use the product of roots formula:
For a quadratic equation \(ax^2 + bx + c = 0\), the product of the roots is given by \(\frac{c}{a}\).
So, we have:
\[ \frac{c}{a} = 1 \]Substitute the values of \(a\) and \(c\):
Substitute \(a = 4\) and \(c = k - 4\) into the equation:
\[ \frac{k - 4}{4} = 1 \]Solve for \(k\):
Multiply both sides by 4:
\[ k - 4 = 4 \]Add 4 to both sides:
\[ k = 4 + 4 \]\[ k = 8 \]Correct Option: D) 8
The value of \(k\) is 8, which satisfies the condition that one root of the given quadratic equation is the reciprocal of the other.
The problem involves arranging seven people in a straight row based on given conditions. We will use a step-by-step approach, starting with the most definitive conditions to fix positions, and then filling in the remaining people.
Correct Option: C) N is sitting at the extreme left end of the row.
The question asks to identify the world's largest basaltic impact crater lake found in India. This requires knowledge of prominent lakes in India and their geological origins.
Lonar Lake in Maharashtra is a unique lake formed by a meteorite impact during the Pleistocene Epoch. It is situated in a basaltic rock formation, making it a basaltic impact crater lake. It is considered the world's largest and only hyper-velocity impact crater in basaltic rock.
Correct Option: B) Lonar Lake
Select the term from among the given options that can replace the question mark (?) in the following series.
A3J5R, E6K6S, I9L7T, O12M8U, ?, A18O10W
The problem presents a series of alphanumeric terms. To find the missing term, we need to identify the pattern in the letters and numbers separately. Each term consists of a letter, followed by a number, then another letter, another number, and finally a letter. We will analyze the progression of each component.
Correct Option: B) U15N9V
Refer to the following number-symbol series and answer the question that follows.
1 * 2 9 8 4 ! ^ 3 5 4 # 9 ! $ 3 ^ 1 3 & 9 6 8 $ * $ 9 3
How many such symbols are there that are immediately preceded by a number and immediately followed by a number?
The problem asks us to identify symbols in a given series that are immediately preceded by a number and immediately followed by a number. We need to scan the series from left to right and apply this condition to each symbol.
1 * 2 9 8 4 ! ^ 3 5 4 # 9 ! \( 3 ^ 1 3 & 9 6 8 \) * \( 9 3*: Preceded by 1 (number), followed by 2 (number). Matches. (1 * 2)!: Preceded by 4 (number), followed by ^ (symbol). Does not match.^: Preceded by ! (symbol), followed by 3 (number). Does not match.#: Preceded by 4 (number), followed by 9 (number). Matches. (4 # 9)!: Preceded by 9 (number), followed by \) (symbol). Does not match.\(: Preceded by ! (symbol), followed by 3 (number). Does not match.^: Preceded by 3 (number), followed by 1 (number). Matches. (3 ^ 1)&: Preceded by 3 (number), followed by 9 (number). Matches. (3 & 9)\): Preceded by 8 (number), followed by * (symbol). Does not match.*: Preceded by \( (symbol), followed by \) (symbol). Does not match.$: Preceded by * (symbol), followed by 9 (number). Does not match.*, #, ^, &.Therefore, there are 4 such symbols.
B) 4 — As identified in the step-by-step analysis, there are 4 symbols (*, #, ^, &) that fit the given criteria.
Refer to the following letter, symbol series and answer the question.
D & C N ^ E G ^ Y T % @ G R I & * H K & Z A P * ^
How many consonants are there that are immediately preceded by a consonant and immediately followed by symbol?
The problem asks us to identify consonants that fit a specific pattern in a given series: a consonant immediately preceded by another consonant and immediately followed by a symbol. We need to scan the series from left to right, applying this condition to each element.
D) Three
The question asks about the product formed when UV radiation acts on oxygen molecules at higher levels of the atmosphere. This process is fundamental to the formation of the ozone layer.
A) Ozone is the correct answer because UV radiation initiates the process of ozone formation from oxygen molecules in the stratosphere.
8 : 4 :: 3.2 : x and 3 : 6 :: 6 : y. What is the ratio of x to y?
The problem involves finding the ratio of two variables, \(x\) and \(y\), which are determined from two separate proportion statements. We need to solve each proportion to find the values of \(x\) and \(y\) individually, and then calculate their ratio.
The first proportion is given as \(8 : 4 :: 3.2 : x\).
This can be written as a fraction equality:
\[ \frac{8}{4} = \frac{3.2}{x} \]Simplify the left side:
\[ 2 = \frac{3.2}{x} \]Multiply both sides by \(x\):
\[ 2x = 3.2 \]Divide by 2 to find \(x\):
\[ x = \frac{3.2}{2} \]\[ x = 1.6 \]The second proportion is given as \(3 : 6 :: 6 : y\).
This can be written as a fraction equality:
\[ \frac{3}{6} = \frac{6}{y} \]Simplify the left side:
\[ \frac{1}{2} = \frac{6}{y} \]Cross-multiply:
\[ 1 \times y = 2 \times 6 \]\[ y = 12 \]Now we need to find the ratio \(x : y\).
\[ x : y = 1.6 : 12 \]To simplify this ratio, we can first remove the decimal by multiplying both sides by 10:
\[ 1.6 \times 10 : 12 \times 10 \]\[ 16 : 120 \]Now, find the greatest common divisor (GCD) of 16 and 120. Both are divisible by 8.
Divide both sides by 8:
\[ \frac{16}{8} : \frac{120}{8} \]\[ 2 : 15 \]The ratio of \(x\) to \(y\) is \(2 : 15\).
C) 2 : 15 — This matches our calculated ratio of \(x\) to \(y\).
What is the mode of the following data?
43, 41, 46, 48, 43, 50, 54, 41, 48, 54, 46, 43, 48, 50, 46, 48, 50, 54
To find the mode of a given set of data, we need to identify the value that appears most frequently in the dataset. This involves counting the occurrences of each unique number.
\(43, 41, 46, 48, 43, 50, 54, 41, 48, 54, 46, 43, 48, 50, 46, 48, 50, 54\)
Comparing the frequencies:
The number \(48\) appears \(4\) times, which is more than any other number in the dataset.
Since \(48\) has the highest frequency, it is the mode of the data.
B) 48 is the correct option because it appears 4 times, which is the highest frequency among all the numbers in the given data set.