To find the value of C, we first need to determine the value of B. B is defined as the sum of the first three even natural numbers. Once B is found, we can use the definition of the third proportional to calculate C, given 29 and B.
The first even natural number is 2.
The second even natural number is 4.
The third even natural number is 6.
B is the sum of these numbers:
\[ B = 2 + 4 + 6 \]\[ B = 12 \]If C is the third proportional of two numbers, say 'a' and 'b', it means that a, b, and C are in continued proportion. This can be written as:
\[ \frac{a}{b} = \frac{b}{C} \]Here, \(a = 29\) and \(b = B = 12\). We need to find C.
\[ \frac{29}{12} = \frac{12}{C} \]Cross-multiply to solve for C:
\[ 29 \times C = 12 \times 12 \]\[ 29C = 144 \]\[ C = \frac{144}{29} \]D) 4.97 is the correct value of C when rounded to two decimal places.
The question asks to identify the river that flows through Rajasthan from the given options. This requires knowledge of the geographical distribution of major Indian rivers.
Correct Option: C) Luni is a significant river of western Rajasthan. It originates in the Pushkar valley of the Aravalli Range, near Ajmer, and flows southwest through the Thar Desert before losing its water in the marshy lands of Rann of Kutch in Gujarat.
The question asks to identify the winner of the 2024 US Open Women's Singles title. This requires knowledge of recent major tennis tournament results.
Aryna Sabalenka defeated Jessica Pegula in the final to win the 2024 US Open Women's Singles title. This was her second Grand Slam singles title.
Correct Option: C) Aryna Sabalenka
To find the number divisible by both 6 and 31, we need to check two conditions for each option:
Alternatively, since 6 and 31 are coprime (their greatest common divisor is 1), a number divisible by both 6 and 31 must be divisible by their product, \(6 \times 31 = 186\).
Let's check each option:
Divisibility by 6:
Since 3720 is divisible by both 2 and 3, it is divisible by 6.
Divisibility by 31:
\[ \frac{3720}{31} = 120 \]Since 3720 is divisible by 31, it is divisible by both 6 and 31.
Divisibility by 6:
Thus, 2132 is not divisible by 6.
Divisibility by 6:
Thus, 4916 is not divisible by 6.
Divisibility by 6:
Thus, 2858 is not divisible by 6.
A) 3720 is divisible by both 6 and 31, as shown in the step-by-step analysis.
The question asks for the primary basis of seat allocation in the Rajya Sabha (Council of States) among different States and Union Territories in India. This is a fundamental aspect of the Indian parliamentary system, specifically related to the composition of the Upper House.
Correct Option: B) Population of each state or union territory. The allocation of seats in the Rajya Sabha is primarily based on the population of each state or union territory, as laid down in the Fourth Schedule of the Indian Constitution.
To determine the nature of the roots of a quadratic equation, we use the discriminant formula. For a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the discriminant is given by \( \Delta = b^2 - 4ac \). The nature of the roots depends on the value of the discriminant:
Correct Option: D) no real roots
This problem involves substituting given symbol meanings into an equation and then solving it using the BODMAS/PEMDAS rule. First, replace the operators in the given equation according to the provided rules. Then, perform the mathematical operations in the correct order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.
The original equation is: \(55 โ 8 รท 3 + 15 ร 5 = ?\)
Applying the new meanings:
So, the new equation becomes: \(55 ร 8 + 3 โ 15 รท 5 = ?\)
The order of operations is Division, Multiplication, Addition, then Subtraction.
First, perform Division:
\[ 15 รท 5 = 3 \]The equation is now: \(55 ร 8 + 3 โ 3 = ?\)
Next, perform Multiplication:
\[ 55 ร 8 = 440 \]The equation is now: \(440 + 3 โ 3 = ?\)
Next, perform Addition:
\[ 440 + 3 = 443 \]The equation is now: \(443 โ 3 = ?\)
Finally, perform Subtraction:
\[ 443 โ 3 = 440 \]So, the value is \(440\).
A) 440 is the correct answer, as calculated by correctly substituting the operators and following the BODMAS rule.
This problem involves calculating the work done by pipes filling a cistern. We need to determine the individual filling rates of Pipe A and Pipe B, then calculate how much of the cistern is filled by Pipe A working for the entire duration and by Pipe B working for a partial duration. By setting up an equation for the total work done, we can find the time Pipe B was active.
Pipe A fills the cistern in 84 minutes. So, its rate is:
\[ \text{Rate of A} = \frac{1}{84} \text{ cistern/minute} \]Pipe B fills the cistern in 36 minutes. So, its rate is:
\[ \text{Rate of B} = \frac{1}{36} \text{ cistern/minute} \]The cistern is completely filled in a total of 42 minutes.
Let \(t\) be the time (in minutes) for which Pipe B is open. Since Pipe B is turned off after some time, Pipe B works for \(t\) minutes.
Pipe A works for the entire duration, which is 42 minutes.
Fraction filled by Pipe A in 42 minutes:
\[ \text{Work by A} = \text{Rate of A} \times \text{Time A works} = \frac{1}{84} \times 42 = \frac{42}{84} = \frac{1}{2} \]Fraction filled by Pipe B in \(t\) minutes:
\[ \text{Work by B} = \text{Rate of B} \times \text{Time B works} = \frac{1}{36} \times t = \frac{t}{36} \]The total work done (filling the entire cistern) is 1.
\[ \text{Work by A} + \text{Work by B} = 1 \]\[ \frac{1}{2} + \frac{t}{36} = 1 \]Subtract \(\frac{1}{2}\) from both sides:
\[ \frac{t}{36} = 1 - \frac{1}{2} \]\[ \frac{t}{36} = \frac{1}{2} \]Multiply both sides by 36:
\[ t = \frac{1}{2} \times 36 \]\[ t = 18 \text{ minutes} \]So, Pipe B should be turned off after 18 minutes.
A) 18 minutes โ This is the correct time after which Pipe B should be turned off for the cistern to be filled in 42 minutes.
The question asks for the sex ratio in India according to the 2011 Census. This is a factual recall question based on demographic data. The sex ratio is defined as the number of females per 1000 males.
Correct Option: A) 943 females per 1000 males
The 2011 Census of India officially recorded the sex ratio as 943 females for every 1000 males. This is a key demographic indicator often tested in competitive exams.