This question requires us to analyze the given statements and determine which conclusions logically follow. We need to evaluate each conclusion based on the information explicitly provided in the statements, without making any external assumptions.
Correct Option: B) Only conclusion B follows
To rearrange the jumbled sentences, we need to identify the logical flow of ideas. We should look for connections between sentences, such as cause and effect, problem and solution, or general statements followed by specific details. S1 and S6 are fixed, so we need to build a coherent narrative between them.
Thus, the correct order is 2, 1, 3, 4.
Correct Option: C) 2,1,3,4
This problem involves calculating the original price of an item given its depreciated value over a certain period and a constant annual depreciation rate. We will use the formula for compound depreciation to work backward and find the initial value.
The original price of the car was ₹7,50,000.
C) ₹7,50,000/- is the correct original price of the car after calculating backward from the depreciated value.
This is a syllogism problem that can be solved using Venn diagrams or by analyzing the relationships between the categories. We will draw Venn diagrams for each statement and then check the validity of the conclusions based on these diagrams.
Statement A: All papayas are fruits.
This means the set of 'Papayas' is entirely contained within the set of 'Fruits'.
Statement B: Some fruits are vegetables.
This indicates an intersection between the set of 'Fruits' and the set of 'Vegetables'. Some fruits are vegetables, but not all. Also, some vegetables are fruits, but not all.
Statement C: All vegetables are vegan.
This means the set of 'Vegetables' is entirely contained within the set of 'Vegan' items.
Combining the statements:
From these, we can infer that since some fruits are vegetables, and all vegetables are vegan, it logically follows that some fruits must also be vegan. Also, since all vegetables are vegan, and some fruits are vegetables, it means some fruits are vegan.
Evaluating the Conclusions:
Conclusion i: Some vegetables are papayas.
From statement A, all papayas are fruits. From statement B, some fruits are vegetables. However, there is no direct or indirect information to suggest that the specific subset of fruits that are papayas overlaps with the set of vegetables. It's possible that the fruits that are vegetables are not papayas. So, this conclusion does not necessarily follow.
Conclusion ii: Some vegans are fruits.
From statement B, some fruits are vegetables. From statement C, all vegetables are vegan. Since all vegetables are vegan, and some fruits are vegetables, it logically follows that those fruits which are vegetables must also be vegan. Therefore, some vegans are fruits. This conclusion follows.
Conclusion iii: Some vegans are papayas.
We know all papayas are fruits (Statement A). We also know all vegetables are vegan (Statement C). We know some fruits are vegetables (Statement B). However, there is no definite link established between papayas and vegetables, or papayas and vegans. The fruits that are vegetables might not include papayas. Therefore, we cannot conclude that some vegans are papayas. This conclusion does not necessarily follow.
Conclusion iv: Some fruits are papayas.
From statement A, "All papayas are fruits". If all papayas are fruits, it inherently means that some fruits are papayas (specifically, the fruits that are papayas). This is a direct implication of the statement. This conclusion follows.
Therefore, conclusions (ii) and (iv) logically follow.
C) Only (ii) and (iv) follow
A) Only (i) follows — Conclusion (i) "Some vegetables are papayas" does not necessarily follow. While papayas are fruits and some fruits are vegetables, there's no guarantee that the specific subset of fruits that are papayas overlaps with vegetables.
B) Only (iv) follows — Conclusion (iv) "Some fruits are papayas" follows, but conclusion (ii) "Some vegans are fruits" also follows. So, this option is incomplete.
D) Only (ii) and (iii) follow — Conclusion (ii) follows, but conclusion (iii) "Some vegans are papayas" does not necessarily follow. There is no direct link between papayas and vegetables/vegans established by the statements.
The question asks to identify the leaders of the Khilafat Movement. This requires knowledge of key figures in India's freedom struggle, particularly those associated with the Khilafat agitation.
Correct Option: A) Shaukat Ali and Muhammad Ali were the prominent leaders, often referred to as the 'Ali Brothers', who spearheaded the Khilafat Movement in India.
The question asks for the number of Himalayan peaks opened for trekking for domestic and foreign mountaineers as of August 2019. This is a factual question related to government policies concerning mountaineering in India.
Correct Option: A) 137. As of August 2019, the Indian government, through the Union Home Ministry, opened 137 Himalayan peaks for mountaineering and trekking for both domestic and foreign tourists. This was a significant policy change aimed at promoting adventure tourism.
The problem involves ratios and changes in quantities. We will use variables to represent the initial number of boys and girls based on their given ratio. Then, we will incorporate the change (admission of more boys) and the new ratio to form an equation. Solving this equation will give us the value of the variable, allowing us to find the initial numbers and their difference.
Initial Ratio: Let the initial number of boys be \(5x\) and the initial number of girls be \(7x\), where \(x\) is a common multiplier.
Change: Eight more boys were admitted. So, the new number of boys is \(5x + 8\). The number of girls remains \(7x\).
New Ratio: The new ratio of girls to boys is given as 1 : 1. This means the number of girls is equal to the new number of boys.
So, we can write the equation:
\[ 7x = 5x + 8 \]Solve for \(x\):
\[ 7x - 5x = 8 \]\[ 2x = 8 \]\[ x = \frac{8}{2} \]\[ x = 4 \]Calculate Initial Numbers:
Find the Difference: The difference between the number of boys and girls in the beginning was:
\[ \text{Difference} = \text{Number of girls} - \text{Number of boys} \]\[ \text{Difference} = 28 - 20 \]\[ \text{Difference} = 8 \]D) 08. The difference between the initial number of boys and girls is 8.
One root of the equation \( 2x^{2} - 8x - m = 0 \), is \( \frac{5}{2} \). The other root of the equation and the value of m are respectively:
\( \frac{5}{2} \) and -\( \frac{15}{2} \)
\( \frac{3}{2} \) and -\( \frac{15}{2} \)
-\( \frac{5}{2} \) and \( \frac{15}{2} \)
-\( \frac{3}{2} \) and \( \frac{15}{2} \)
We are given a quadratic equation and one of its roots. We need to find the other root and the value of the constant 'm'. We can use two properties of quadratic equations: first, if a value is a root, it must satisfy the equation; second, the sum and product of roots are related to the coefficients of the quadratic equation.
Substitute the given root into the equation:
The given equation is \( 2x^{2} - 8x - m = 0 \). One root is \( x = \frac{5}{2} \).
Substitute \( x = \frac{5}{2} \) into the equation:
\[ 2 \left( \frac{5}{2} \right)^{2} - 8 \left( \frac{5}{2} \right) - m = 0 \] \[ 2 \left( \frac{25}{4} \right) - 4 \times 5 - m = 0 \] \[ \frac{25}{2} - 20 - m = 0 \] \[ \frac{25 - 40}{2} - m = 0 \] \[ -\frac{15}{2} - m = 0 \] \[ m = -\frac{15}{2} \]Find the other root using the sum of roots property:
For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots (\( \alpha + \beta \)) is given by \( -\frac{b}{a} \).
In our equation \( 2x^{2} - 8x - m = 0 \), we have \( a=2 \), \( b=-8 \), and \( c=-m \).
Let the given root be \( \alpha = \frac{5}{2} \) and the other root be \( \beta \).
Sum of roots:
\[ \alpha + \beta = -\frac{b}{a} \] \[ \frac{5}{2} + \beta = -\frac{(-8)}{2} \] \[ \frac{5}{2} + \beta = \frac{8}{2} \] \[ \frac{5}{2} + \beta = 4 \] \[ \beta = 4 - \frac{5}{2} \] \[ \beta = \frac{8 - 5}{2} \] \[ \beta = \frac{3}{2} \]Verify the results:
The other root is \( \frac{3}{2} \) and the value of \( m \) is \( -\frac{15}{2} \).
B) \( \frac{3}{2} \) and -\( \frac{15}{2} \)
We need to find a number that satisfies two conditions: first, it leaves specific remainders when divided by 18, 21, and 24; second, it must be a multiple of 23. We will use the concept of LCM (Least Common Multiple) and the Chinese Remainder Theorem approach to find the general form of such a number, and then apply the divisibility condition by 23.
Analyze the remainder conditions:
The number, let's call it \(N\), leaves remainders 7, 10, and 13 when divided by 18, 21, and 24 respectively.
Since the difference between the divisor and the remainder is constant (11) in all cases, the number \(N\) can be expressed as \(\text{LCM}(18, 21, 24) \times k - 11\).
Calculate the LCM of 18, 21, and 24:
\[ \text{LCM}(18, 21, 24) = 2^3 \times 3^2 \times 7 = 8 \times 9 \times 7 = 72 \times 7 = 504 \]Thus, the number \(N\) can be written in the form \(504k - 11\), where \(k\) is a positive integer.
Apply the divisibility condition by 23:
We are looking for the least multiple of 23 that satisfies the above condition. So, \(N\) must be a multiple of 23.
\[ N = 504k - 11 \]We need \(504k - 11\) to be divisible by 23.
Let's divide 504 by 23:
\[ 504 = 23 \times 21 + 21 \]So, \(504k - 11 \equiv 0 \pmod{23}\)
\[ (23 \times 21 + 21)k - 11 \equiv 0 \pmod{23} \]\[ 21k - 11 \equiv 0 \pmod{23} \]
We can write \(21k \equiv 11 \pmod{23}\). Since \(21 \equiv -2 \pmod{23}\), we have:
\[ -2k \equiv 11 \pmod{23} \]
To make the coefficient of \(k\) positive, we can add 23 to -2:
\[ 21k \equiv 11 \pmod{23} \]
Multiply both sides by -1 (or 22, since \(22 \equiv -1 \pmod{23}\)):
\[ -21k \equiv -11 \pmod{23} \]
\[ 2k \equiv 11 \pmod{23} \]
To find the inverse of 2 modulo 23, we can see that \(2 \times 12 = 24 \equiv 1 \pmod{23}\). So, multiply by 12:
\[ 12 \times 2k \equiv 12 \times 11 \pmod{23} \]
\[ 24k \equiv 132 \pmod{23} \]
\[ k \equiv 132 \pmod{23} \]
Now, divide 132 by 23:
\[ 132 = 23 \times 5 + 17 \]
So, \(k \equiv 17 \pmod{23}\).
The least positive integer value for \(k\) is 17.
Calculate the number \(N\):
Substitute \(k=17\) into the expression for \(N\):
\[ N = 504k - 11 = 504 \times 17 - 11 \]
\[ N = 8568 - 11 \]
\[ N = 8557 \]
Let's recheck the calculation for \(21k \equiv 11 \pmod{23}\).
If \(k=17\), then \(21 \times 17 = 357\).
\[ 357 \div 23 = 15 \text{ with remainder } 12 \]
This means \(k=17\) is not correct. Let's re-evaluate \(21k \equiv 11 \pmod{23}\).
We have \(21k \equiv 11 \pmod{23}\).
This is equivalent to \(-2k \equiv 11 \pmod{23}\).
Multiply by \(-1\) (or \(22\)): \(2k \equiv -11 \pmod{23}\).
Since \(-11 \equiv 12 \pmod{23}\), we have \(2k \equiv 12 \pmod{23}\).
Dividing by 2 (which is valid since 2 and 23 are coprime): \(k \equiv 6 \pmod{23}\).
The least positive integer value for \(k\) is 6.
Recalculate the number \(N\) with \(k=6\):
\[ N = 504k - 11 = 504 \times 6 - 11 \]
\[ N = 3024 - 11 \]
\[ N = 3013 \]
Verify the conditions for \(N = 3013\):
All conditions are satisfied.
B) 3013
If P = 2 + \(\sqrt{3}\), Q = 2 −\(\sqrt{3}\) then \( \frac{p}{q} \) = ?
\(\frac{7+4\sqrt{3}}{1}\)
4\(\sqrt{6}\) + 5
7 - 2\(\sqrt{6}\)
4\(\sqrt{3}\) - 5
To find the value of \( \frac{P}{Q} \), we need to substitute the given values of P and Q into the expression. Since the denominator involves a square root, we will rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.
\( P = 2 + \sqrt{3} \)
\( Q = 2 - \sqrt{3} \)
To rationalize the denominator, multiply the numerator and denominator by the conjugate of \( 2 - \sqrt{3} \), which is \( 2 + \sqrt{3} \).
\[ \frac{2 + \sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} \]For the numerator, we have \( (2 + \sqrt{3})(2 + \sqrt{3}) = (2 + \sqrt{3})^2 \). Using the identity \( (a+b)^2 = a^2 + 2ab + b^2 \):
\[ (2 + \sqrt{3})^2 = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]For the denominator, we have \( (2 - \sqrt{3})(2 + \sqrt{3}) \). Using the identity \( (a-b)(a+b) = a^2 - b^2 \):
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]Correct Option: A) \( \frac{7+4\sqrt{3}}{1} \)