To find the quadratic equation given the sum and product of its roots, we use the standard form of a quadratic equation: \(x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0\). We will substitute the given values into this formula and then simplify to match one of the options.
Recall the standard form of a quadratic equation: If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, the equation can be written as:
\[ x^2 - (\alpha + \beta)x + (\alpha \beta) = 0 \]Identify the given sum and product of roots: Given, sum of the roots \( (\alpha + \beta) = \frac{3}{5} \). Given, product of the roots \( (\alpha \beta) = \frac{2}{5} \).
Substitute the values into the standard form: Substitute the sum and product of roots into the equation from Step 1:
\[ x^2 - \left(\frac{3}{5}\right)x + \left(\frac{2}{5}\right) = 0 \]Eliminate the denominators to get integer coefficients: To remove the fractions, multiply the entire equation by the least common multiple of the denominators, which is 5:
\[ 5 \left(x^2 - \frac{3}{5}x + \frac{2}{5}\right) = 5 \times 0 \] \[ 5x^2 - 5 \left(\frac{3}{5}\right)x + 5 \left(\frac{2}{5}\right) = 0 \] \[ 5x^2 - 3x + 2 = 0 \]Compare with the given options: The derived equation \(5x^2 - 3x + 2 = 0\) matches option C.
Correct Option: C) 5x² − 3x + 2 = 0
All 48 students of a class are standing in a row facing the north. Yuvika is 7th from the extreme right end of the row. Malini is 22nd from the extreme left end. How many students are standing between Malini and Yuvika?
To find the number of students between Malini and Yuvika, we first need to determine their positions from a common reference point (either left or right end). We are given the total number of students and the positions of Yuvika from the right and Malini from the left. We can convert Yuvika's position from the right to her position from the left, or vice versa, and then calculate the number of students between them.
If Yuvika is 7th from the right, it means there are 6 students to her right. Her position from the left can be calculated as:
\[ \text{Yuvika's position from left} = \text{Total students} - \text{Position from right} + 1 \]\[ \text{Yuvika's position from left} = 48 - 7 + 1 = 42 \]Now we have both positions from the left end:
Since Yuvika is further to the right (position 42) than Malini (position 22), the number of students between them can be found by subtracting Malini's position from Yuvika's position and then subtracting 1 (because we are looking for students *between* them, not including them).
\[ \text{Students between Malini and Yuvika} = \text{Yuvika's position from left} - \text{Malini's position from left} - 1 \]\[ \text{Students between Malini and Yuvika} = 42 - 22 - 1 \]\[ \text{Students between Malini and Yuvika} = 20 - 1 = 19 \]Therefore, there are 19 students standing between Malini and Yuvika.
Correct Option: D) 19
The question asks to identify the plant used by Gregor Mendel for his groundbreaking genetics experiments. This is a direct factual recall question from biology.
A) Garden peas. Gregor Mendel conducted his famous experiments on inheritance using the garden pea plant (Pisum sativum). He studied various traits like seed shape, seed color, pod shape, pod color, flower color, flower position, and stem height, which were easily distinguishable and allowed him to formulate the basic principles of heredity.
This is a direction sense problem. We need to work backward from the final known direction to determine the initial direction. We will trace Sohan's path and turns, then reverse the turns to find the starting orientation.
Let's verify this:
Therefore, he started cycling from his home in the North direction.
Correct Option: D) North
This problem requires us to substitute the given new meanings for mathematical operators into the expression and then evaluate the expression following the order of operations (BODMAS/PEMDAS).
The original equation is: \(18 + (18 − 2) × 9 + 10 = ?\)
Applying the substitutions:
The new equation becomes:
\[ 18 - (18 \times 2) \div 9 - 10 = ? \]First, solve the operation inside the parentheses:
\[ 18 \times 2 = 36 \]B) 4 is the correct answer after applying the given operator substitutions and following the order of operations.
This question asks about the specific plant structures where waste products are stored. We need to recall the functions of the given options in plant physiology.
Correct Option: C) Old xylem. Old xylem, especially heartwood, is a common site for the deposition of metabolic waste products in plants. These wastes can include resins, gums, tannins, and crystals, which are stored in the non-functional xylem cells.
This question asks about the term (duration) of a Panchayat Body in India. This is a factual question related to the Indian Constitution and local self-governance.
Correct Option: C) 5 years is the correct term for a Panchayat Body in India, as stipulated by the Constitution.
The question asks for the launch year of the 'Kanya Shiksha Pravesh Utsav' initiative by the Ministry of Women and Child Development, Government of India. This requires knowledge of recent government schemes and their launch dates.
D) 2022 — The 'Kanya Shiksha Pravesh Utsav' was indeed launched in March 2022 by the Ministry of Women and Child Development to enroll out-of-school girls in the age group of 11-14 years.
The question asks to identify a major difficulty faced during barter exchange, especially in large economies. Barter is a system of exchange where goods or services are directly exchanged for other goods or services without using a medium of exchange like money. We need to consider the inherent problems of such a system.
Correct Option: B) Lack of a common measure of value was a significant difficulty faced during barter exchange in large economies. Without a standard unit to express the worth of different goods and services, determining fair exchange rates becomes extremely complex and inefficient, hindering trade on a large scale.