This is a syllogism problem. We need to evaluate the given conclusions based on the provided statements. We can use Venn diagrams or direct logical deduction to determine which conclusions follow.
Let's analyze each statement and conclusion:
Statements:
Analyzing the Conclusions:
Conclusion I: All grasses are flowers.
Conclusion II: All roots are flowers.
Conclusion III: Some flowers are fruits.
Based on the analysis, Conclusions II and III follow from the given statements.
C) Conclusions II and III follow
The question asks to identify the correct chemical reaction for the preparation of bleaching powder. We need to recall the common industrial methods for preparing bleaching powder and match them with the given options.
Correct Option: C) \(\text{Ca(OH)}_2 + \text{Cl}_2 \rightarrow \text{CaOCl}_2 + \text{H}_2\text{O}\)
This reaction correctly represents the industrial preparation of bleaching powder (calcium oxychloride) by passing chlorine gas over dry slaked lime (calcium hydroxide).
To solve this problem, we need to track Janhvi's movements step-by-step on a coordinate plane or by visualizing directions. We will calculate the net displacement in the East-West and North-South directions from her starting point (home).
A) 18 m, South. As calculated, her final position is 18 m south of her starting point.
How many languages are there in the 8th Schedule of the Indian Constitution as on June 2022?
The question asks about the number of languages recognized in the 8th Schedule of the Indian Constitution as of June 2022. This requires knowledge of the constitutional provisions regarding official languages and the amendments made to the 8th Schedule over time.
Correct Option: B) 22
As of June 2022, the 8th Schedule of the Indian Constitution recognizes 22 languages. These include Assamese, Bengali, Bodo, Dogri, Gujarati, Hindi, Kannada, Kashmiri, Konkani, Maithili, Malayalam, Manipuri, Marathi, Nepali, Odia, Punjabi, Sanskrit, Santhali, Sindhi, Tamil, Telugu, and Urdu.
The question asks to identify the correct statements regarding the principal foci of a lens. We need to recall the definitions of the first and second principal foci for a lens.
C) Both (a) and (c)
The problem involves ratios and differences. We can represent the number of chocolates in each box using a common multiplier (a variable) based on the given ratio. Then, we use the given difference to find the value of this multiplier, and finally calculate the number of chocolates in the larger box.
Represent the number of chocolates: Let the number of chocolates in the two boxes be \(7x\) and \(5x\), where \(x\) is a common multiplier. This maintains the given ratio of \(7:5\).
Set up the equation using the difference: We are given that the difference in the number of chocolates is 28. Since \(7x\) is greater than \(5x\), the difference is \(7x - 5x\).
\[ 7x - 5x = 28 \]
Solve for \(x\): Simplify the equation:
\[ 2x = 28 \]
Divide by 2 to find \(x\): \[ x = \frac{28}{2} \] \[ x = 14 \]
Calculate the number of chocolates in the larger box: The box with the larger number of chocolates corresponds to the ratio part 7, so it has \(7x\) chocolates.
Substitute the value of \(x\) into \(7x\): \[ \text{Number of chocolates in larger box} = 7 \times 14 \] \[ \text{Number of chocolates in larger box} = 98 \]
Correct Option: C) 98
The problem asks us to identify which of the given assumptions are implicit in the statement. We must consider the statement to be absolutely true and determine if the assumptions logically follow from it without adding external information.
C) Neither assumption I nor II is implicit
The question asks to identify the statement that is NOT necessarily correct when two parallel lines are intersected by a transversal. We need to recall the properties of angles formed when a transversal intersects parallel lines.
D) A pair of interior angles on the same side of the transversal have the same measure.
Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding / subtracting / multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(1, 8, 3)
(2, 21, 5)
If α and β are the roots of x² + x − 2 = 0, then the quadratic equation in x whose roots are \( \frac{1}{\alpha} + \frac{1}{\beta} \) and αβ is:
2x² + 5x − 2 = 0
2x² + 3x − 2 = 0
2x² − 5x + 2 = 0
2x² − 3x + 2 = 0
The problem requires us to find a new quadratic equation given the roots of an initial quadratic equation. We will first find the sum and product of the roots of the given equation. Then, we will use these values to determine the sum and product of the roots of the new quadratic equation. Finally, we will construct the new quadratic equation using the formula \(x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0\).
Analyze the given quadratic equation:
The given quadratic equation is \(x^2 + x - 2 = 0\).
For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \( \alpha + \beta = -b/a \) and the product of the roots \( \alpha \beta = c/a \).
Here, \(a = 1\), \(b = 1\), \(c = -2\).
Calculate the sum and product of roots for the given equation:
Sum of roots: \( \alpha + \beta = -\frac{1}{1} = -1 \).
Product of roots: \( \alpha \beta = \frac{-2}{1} = -2 \).
Identify the roots of the new quadratic equation:
The roots of the new quadratic equation are given as \[\frac{1}{\alpha} + \frac{1}{\beta}\] and \( \alpha \beta \).
Calculate the first root of the new equation:
Let the first root be \[r_1 = \frac{1}{\alpha} + \frac{1}{\beta}\].
We can simplify this expression:
\[ r_1 = \frac{\beta + \alpha}{\alpha \beta} \]
Substitute the values of \( \alpha + \beta \) and \( \alpha \beta \) from step 2:
\[ r_1 = \frac{-1}{-2} = \frac{1}{2} \]
Calculate the second root of the new equation:
Let the second root be \( r_2 = \alpha \beta \).
From step 2, we know \( \alpha \beta = -2 \).
So, \( r_2 = -2 \).
Calculate the sum of the roots for the new quadratic equation:
Sum of new roots \( S = r_1 + r_2 = \frac{1}{2} + (-2) \).
\[ S = \frac{1}{2} - 2 = \frac{1 - 4}{2} = \frac{-3}{2} \]
Calculate the product of the roots for the new quadratic equation:
Product of new roots \( P = r_1 \cdot r_2 = \left(\frac{1}{2}\right) \cdot (-2) \).
\[ P = -1 \]
Formulate the new quadratic equation:
The general form of a quadratic equation with roots \(r_1\) and \(r_2\) is \(x^2 - Sx + P = 0\).
Substitute the values of \(S\) and \(P\):
\[ x^2 - \left(\frac{-3}{2}\right)x + (-1) = 0 \]
\[ x^2 + \frac{3}{2}x - 1 = 0 \]
To eliminate the fraction, multiply the entire equation by 2:
\[ 2(x^2) + 2\left(\frac{3}{2}x\right) - 2(1) = 0 \]
\[ 2x^2 + 3x - 2 = 0 \]