This problem can be solved using the principles of set theory, specifically by finding the number of students who are friends with at least one of L or A, and then subtracting this from the total number of students in the class.
Total number of students in the class: \[ \text{Total students} = 30 \]
Friends of L: L has 10 unique friends (friends only with L) and 5 friends common to A. So, the total number of friends L has is: \[ \text{Friends of L} = \text{Unique friends of L} + \text{Common friends} = 10 + 5 = 15 \]
Friends of A: A has a total of 17 friends in the class. \[ \text{Friends of A} = 17 \]
Friends common to L and A: It is given that 5 friends are common to A and L. \[ \text{Common friends (L \(\cap\) A)} = 5 \]
Number of students who are friends with at least L or A: Using the principle of inclusion-exclusion for two sets: \[ \text{Friends (L \(\cup\) A)} = \text{Friends of L} + \text{Friends of A} - \text{Common friends (L \(\cap\) A)} \] Substitute the values: \[ \text{Friends (L \(\cup\) A)} = 15 + 17 - 5 \] \[ \text{Friends (L \(\cup\) A)} = 32 - 5 \] \[ \text{Friends (L \(\cup\) A)} = 27 \] So, 27 students are friends with at least L or A.
Number of students who are friends with neither L nor A: This is the total number of students minus the number of students who are friends with at least L or A. \[ \text{Neither L nor A} = \text{Total students} - \text{Friends (L \(\cup\) A)} \] \[ \text{Neither L nor A} = 30 - 27 \] \[ \text{Neither L nor A} = 3 \]
Correct Option: B) 3
If a = (\( \frac{3}{7} \))b, then the value of \(\frac{8a-b}{2a+3b}\) is:
17
\( \frac{21}{17} \)
\( \frac{17}{2} \)
\( \frac{17}{27} \)
The problem provides a relationship between two variables, \(a\) and \(b\), and asks for the value of an algebraic expression involving \(a\) and \(b\). The most straightforward approach is to substitute the given relationship into the expression to eliminate one variable, or to find the ratio of \(a\) to \(b\) and then divide the numerator and denominator of the expression by \(b\) (or \(a\)).
We are given the equation:
\[ a = \left( \frac{3}{7} \right)b \]From the given equation, we can find the ratio \(\frac{a}{b}\):
\[ \frac{a}{b} = \frac{3}{7} \]The expression we need to evaluate is \(\frac{8a-b}{2a+3b}\). To use the ratio \(\frac{a}{b}\), we can divide both the numerator and the denominator by \(b\) (assuming \(b \neq 0\)).
\[ \frac{8a-b}{2a+3b} = \frac{\frac{8a-b}{b}}{\frac{2a+3b}{b}} \]This simplifies to:
\[ \frac{\frac{8a}{b} - \frac{b}{b}}{\frac{2a}{b} + \frac{3b}{b}} = \frac{8\left(\frac{a}{b}\right) - 1}{2\left(\frac{a}{b}\right) + 3} \]Now, substitute \[\frac{a}{b} = \frac{3}{7}\] into the simplified expression:
\[ \frac{8\left(\frac{3}{7}\right) - 1}{2\left(\frac{3}{7}\right) + 3} \]Calculate the terms in the numerator:
\[ 8 \times \frac{3}{7} - 1 = \frac{24}{7} - 1 = \frac{24}{7} - \frac{7}{7} = \frac{24-7}{7} = \frac{17}{7} \]Calculate the terms in the denominator:
\[ 2 \times \frac{3}{7} + 3 = \frac{6}{7} + 3 = \frac{6}{7} + \frac{21}{7} = \frac{6+21}{7} = \frac{27}{7} \]Finally, divide the result from step 6 by the result from step 7:
\[ \frac{\frac{17}{7}}{\frac{27}{7}} = \frac{17}{7} \times \frac{7}{27} = \frac{17}{27} \]Alternatively, one could substitute \(a = \frac{3}{7}b\) directly into the expression:
\[ \frac{8\left(\frac{3}{7}b\right) - b}{2\left(\frac{3}{7}b\right) + 3b} = \frac{\frac{24}{7}b - b}{\frac{6}{7}b + 3b} \]\[ = \frac{\left(\frac{24}{7} - 1\right)b}{\left(\frac{6}{7} + 3\right)b} = \frac{\left(\frac{24-7}{7}\right)b}{\left(\frac{6+21}{7}\right)b} \]\[ = \frac{\frac{17}{7}b}{\frac{27}{7}b} = \frac{17}{27} \]D) \(\frac{17}{27}\) is the correct value obtained after substituting the given relationship into the expression and simplifying.
The problem asks us to decode a word based on a given coding pattern. We need to identify the rule applied to transform "LOTUS" into "KPSVR" and then apply the same rule to "WATER".
The pattern observed is: \(-1, +1, -1, +1, -1\).
C) VBSFQ is the correct code for WATER, following the \(-1, +1, -1, +1, -1\) pattern.
The question asks to identify the first president of the Indian National Congress (INC). This requires knowledge of the history of the INC and its early leaders.
Correct Option: D) Womesh Chandra Banerjee was the first president of the Indian National Congress, presiding over its first session in Bombay in 1885.
A vendor bought toffees at 10 for a rupee. How many for a rupee must he sell to gain 25%?
To solve this problem, we need to first determine the cost price (CP) of one toffee. Then, we calculate the selling price (SP) of one toffee required to achieve a 25% gain. Finally, we find out how many toffees must be sold for a rupee at this calculated selling price.
The vendor bought 10 toffees for 1 rupee.
\[ \text{Cost of 1 toffee} = \frac{\text{Total Cost}}{\text{Number of Toffees}} = \frac{1 \text{ Rupee}}{10 \text{ Toffees}} = \text{Rs. } 0.10 \text{ per toffee} \]To gain 25%, the selling price must be 125% of the cost price.
\[ \text{SP} = \text{CP} \times \left(1 + \frac{\text{Gain Percentage}}{100}\right) \]\[ \text{SP} = 0.10 \times \left(1 + \frac{25}{100}\right) \]\[ \text{SP} = 0.10 \times (1 + 0.25) \]\[ \text{SP} = 0.10 \times 1.25 \]\[ \text{SP} = \text{Rs. } 0.125 \text{ per toffee} \]If one toffee is sold for Rs. 0.125, then the number of toffees sold for 1 rupee will be:
\[ \text{Number of toffees} = \frac{1 \text{ Rupee}}{\text{SP of 1 toffee}} \]\[ \text{Number of toffees} = \frac{1}{0.125} \]\[ \text{Number of toffees} = 8 \]Thus, the vendor must sell 8 toffees for a rupee to gain 25%.
Correct Option: B) 8
Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the ships are 45° and 60° respectively. If the lighthouse is 81 m high, then the distance between two ships is:
\(\frac{1+\sqrt{3}}{81\sqrt{3}}\) m
\(\frac{1+\sqrt{3}}{\sqrt{3}}\) m
\(\frac{81(1+\sqrt{3})}{\sqrt{3}}\) m
\(\frac{81}{\sqrt{3}}\) m
The problem involves finding the distance between two ships on opposite sides of a lighthouse, given the height of the lighthouse and the angles of elevation from each ship. This can be solved using trigonometry, specifically the tangent function, which relates the angle of elevation to the height of the lighthouse and the horizontal distance from the ship to the lighthouse.
C) \(\frac{81(1+\sqrt{3})}{\sqrt{3}}\) m. This matches our calculated distance between the two ships.
This question is based on coding and decoding using reverse alphabetical positions. We need to identify how CHARGER is coded as 129 and apply the same rule to HINGES. The coding rule involves assigning each letter its position when the English alphabet is counted in reverse order, from Z to A, and then adding these values.
In normal alphabetical order, A = 1, B = 2, C = 3, and so on up to Z = 26.
In reverse alphabetical order, the values are:
The reverse alphabetical position of a letter is calculated using:
\(\text{Reverse position}=27-\text{Normal position}\)
The word CHARGER contains seven letters: C, H, A, R, G, E, and R.
Find their reverse alphabetical positions:
Add these values:
\(24+19+26+9+20+22+9=129\)
Therefore, CHARGER is coded as 129, matching the given example.
The code is obtained by adding the reverse alphabetical positions of all the letters in the word.
\(\text{Code}=\text{Sum of reverse alphabetical positions}\)
The word HINGES contains six letters: H, I, N, G, E, and S.
Find their reverse alphabetical positions:
Add the reverse alphabetical positions:
\(19+18+13+20+22+8\)
\(=37+13+20+22+8\)
\(=50+20+22+8\)
\(=100\)
Therefore, HINGES is coded as 100.
The same reverse alphabetical position rule produces the given code and the required answer. Therefore, option B is correct.
Correct Option: B) 100
The coding rule involves adding the reverse alphabetical positions of the letters. For HINGES, the values are H = 19, I = 18, N = 13, G = 20, E = 22, and S = 8. Their sum is \(19+18+13+20+22+8=100\). Therefore, option B is correct.
The question asks to identify the fourth Veda, given the other three. This requires knowledge of the four principal Vedas in Hinduism.
Correct Option: A) Atharvaveda is the fourth Veda, completing the set of the four principal Vedas.
The question asks to identify the individual appointed as UNICEF's global Goodwill Ambassador in 2018. This requires knowledge of prominent appointments made by international organizations.
Correct Option: A) Millie Bobby Brown was appointed as UNICEF's global Goodwill Ambassador in November 2018, becoming the youngest person to hold this title at the age of 14. She advocates for children's rights and issues such as education, violence, poverty, and bullying.
The question asks to identify the term for workers who are highly qualified, skilled, and primarily engage in mental work. This requires understanding the common classifications of labor based on the nature of their work.
A) White collar workers are individuals who perform professional, managerial, or administrative work. Their jobs typically require higher education, specialized skills, and involve intellectual or mental tasks rather than physical labor. Examples include doctors, lawyers, engineers, and teachers.