This is a logical reasoning problem involving the arrangement of boxes. We need to deduce the exact order of the six boxes based on the given clues and then identify the box immediately below U.
This means B is at position 1 (if we count from top to bottom).
Order: B
Order: B, C
Since B is at the top and C is immediately below it, V must be below C with two boxes in between B and V. This means V is at position 4.
Order: B, C, _, _, V
C is at position 2. If there are three boxes between P and C, and V is at position 4, P must be below V. This places P at position 6 (the bottom-most position).
Order: B, C, _, _, V, P
Since P is at the bottom (position 6), and there is only one box below U, U must be at position 5. This means P is below U.
Order: B, C, _, _, U, P
We have six boxes: A, B, C, P, U, V. The positions filled are B (1), C (2), V (4), U (5), P (6). The only remaining box is A, and the only remaining position is 3.
So the final order from top to bottom is:
From the final arrangement, the box immediately below U is P.
Correct Option: D) P
The problem requires us to modify a given set of numbers by adding 2 to their first digit. Then, for each new number formed, we need to check if the sum of its digits is divisible by three. The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.
Correct Option: D) Two
The question asks to identify a specific river in Rajasthan that was revived through community efforts supported by the NGO Tarun Bharat Sangh. This requires knowledge of environmental conservation efforts and specific geographical details related to Rajasthan.
D) Arvari River
The Arvari River, located in the Alwar district of Rajasthan, had dried up due to environmental degradation and lack of water conservation. Through the dedicated efforts of local villagers, guided and assisted by the NGO Tarun Bharat Sangh (led by Rajendra Singh, also known as the "Waterman of India"), the river was successfully revived. This project involved constructing traditional water harvesting structures like 'johads' (earthen check dams) and promoting sustainable water management practices, leading to the river flowing again after decades.
What is the quadratic equation whose roots are 2 and \( \frac{-1}{2} \)?
2x2 − 3x − 1 = 0
2x2 − 3x − 3 = 0
2x2 − 3x − 2 = 0
2x2 − 3x − 4 = 0
To find the quadratic equation given its roots, we can use the relationship between the roots and the coefficients of a quadratic equation. If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, then the equation can be written as \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\), or more generally as \(k(x^2 - (\alpha + \beta)x + \alpha\beta) = 0\) for some constant \(k\).
Identify the given roots:
Let the roots be \(\alpha = 2\) and \(\beta = \frac{-1}{2}\).
Calculate the sum of the roots:
\[ \alpha + \beta = 2 + \left(\frac{-1}{2}\right) = 2 - \frac{1}{2} = \frac{4-1}{2} = \frac{3}{2} \]
Calculate the product of the roots:
\[ \alpha\beta = 2 \times \left(\frac{-1}{2}\right) = -1 \]
Form the quadratic equation:
The general form of a quadratic equation with roots \(\alpha\) and \(\beta\) is:
\[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \]
Substitute the sum and product of the roots:
\[ x^2 - \left(\frac{3}{2}\right)x + (-1) = 0 \]
\[ x^2 - \frac{3}{2}x - 1 = 0 \]
Eliminate the fraction (if necessary):
To get integer coefficients, multiply the entire equation by the least common multiple of the denominators (which is 2 in this case):
\[ 2 \times \left(x^2 - \frac{3}{2}x - 1\right) = 2 \times 0 \]
\[ 2x^2 - 3x - 2 = 0 \]
This matches option C.
Correct Option: C) \(2x^2 - 3x - 2 = 0\)
In a triangle PQR, ∠P = 30° and the straight line ST is drawn parallel to QR, intersecting PQ at S and PR at T. Which of the following is always true?
A) \( \frac{PS}{PQ} \) = \( \frac{PT}{PR} \)
B) ST = \( \frac{QR}{2} \)
C) The triangle PST is similar to the triangle PQR.
Only (a) and (c)
Only (a) and (b)
All of (a), (b) and (c)
Only (a)
The problem describes a triangle PQR with a line ST parallel to QR, intersecting PQ and PR. We need to determine which of the given statements are always true based on the properties of similar triangles formed by a line parallel to one side of a triangle.
A) Only (a) and (c) are always true. Statement (a) is true because it represents the proportionality of sides in similar triangles. Statement (c) is true because the parallel line ST creates corresponding angles equal to those of the base, making \(\triangle PST\) similar to \(\triangle PQR\) by AAA similarity criterion.
The question asks to identify the senior Indian cricketer who lost their Grade A status in the BCCI Central Contracts list for the 2021–22 season. This requires knowledge of the specific changes made to the central contracts during that period.
Correct Option: D) Cheteshwar Pujara was demoted from Grade A to Grade B in the BCCI Central Contracts for the 2021–22 season due to a decline in his performance.
To determine the final direction of the shop with respect to the starting point (office), we need to track Ajay's movements step-by-step, considering each turn and the distance covered. We can visualize this on a coordinate plane or by drawing a simple diagram.
A) South-east. The final position (8, -1) indicates that the shop is to the East (positive x-coordinate) and to the South (negative y-coordinate) of the office, placing it in the South-east direction.
The ratio of 93.5 : 35 is the same as:
To find the ratio of \(9^{3.5} : 3^5\), we need to express both terms with the same base. Since \(9\) can be written as \(3^2\), we can convert \(9^{3.5}\) to a power of \(3\). Then, we can simplify the ratio using the rules of exponents.
Express the first term with base 3:
We know that \(9 = 3^2\). So, \(9^{3.5}\) can be written as:
\[ 9^{3.5} = (3^2)^{3.5} \]Using the exponent rule \((a^m)^n = a^{m \times n}\):
\[ (3^2)^{3.5} = 3^{2 \times 3.5} = 3^7 \]Formulate the ratio:
Now the given ratio \(9^{3.5} : 3^5\) becomes:
\[ 3^7 : 3^5 \]Simplify the ratio:
A ratio \(a:b\) can be written as a fraction \(\frac{a}{b}\). So, we have:
\[ \frac{3^7}{3^5} \]Using the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\):
\[ \frac{3^7}{3^5} = 3^{7-5} = 3^2 \]This simplifies to:
\[ 3^2 = 9 \]So, the ratio is \(9:1\).
D) 9 : 1 is the correct ratio after simplifying \(9^{3.5} : 3^5\).
This problem involves setting up equations based on the given ratios and conditions about ages. We will use a variable to represent the common ratio factor for their current ages and then form an equation based on their ages three years later.
Let the present ages of Ram and Sita be \(5x\) and \(8x\) years, respectively, since their ages are in the ratio 5:8.
After three years, their ages will be:
According to the problem, the sum of their ages three years later will be 58 years. So, we can write the equation:
\[ (5x + 3) + (8x + 3) = 58 \]Now, solve the equation for \(x\):
\[ 5x + 3 + 8x + 3 = 58 \] \[ 13x + 6 = 58 \] \[ 13x = 58 - 6 \] \[ 13x = 52 \] \[ x = \frac{52}{13} \] \[ x = 4 \]The present age of Ram is \(5x\). Substitute the value of \(x\):
\[ \text{Ram's present age} = 5 \times 4 = 20 \text{ years} \]D) 20 years is the present age of Ram.
A) 22 years: This would be Ram's age if \(x\) were 4.4, or if there was a calculation error in the sum or subtraction. For instance, if the sum was 61 instead of 58, then \(13x = 55\), which doesn't give an integer \(x\).
B) 21 years: This would be Ram's age if \(x\) were 4.2. This is not the result obtained from the equation.
C) 25 years: This would be Ram's age if \(x\) were 5. This would happen if the sum of ages after 3 years was 65 instead of 58 (i.e., \(13x + 6 = 65 \implies 13x = 59\), which is not 65). Or if the ratio was different.
This problem involves setting up an algebraic equation based on the given age conditions. We need to find the number of years after which the man's age will be twice his daughter's age.
Let \(x\) be the number of years after which the man will be twice as old as his daughter.
According to the problem, after \(x\) years, the man's age will be twice his daughter's age. So, we can write the equation:
\[ 50 + x = 2 \times (22 + x) \]First, distribute the 2 on the right side:
\[ 50 + x = 44 + 2x \]Now, rearrange the terms to solve for \(x\). Subtract \(x\) from both sides:
\[ 50 = 44 + 2x - x \] \[ 50 = 44 + x \]Subtract 44 from both sides:
\[ 50 - 44 = x \] \[ x = 6 \]After 6 years:
Is the man's age twice his daughter's age? \(56 = 2 \times 28\). Yes, it is.
Correct Option: D) 6