This is a linear arrangement puzzle. We need to deduce the seating arrangement of seven boys (B1 to B7) based on the given clues. Since they are facing South, "right" means the viewer's left and "left" means the viewer's right.
Clue 1: B2 sits at one of the extreme ends and is the neighbour of B4.
Since they are facing South, the extreme ends are positions 1 and 7. B2 can be at position 1 or 7.
Case 1: B2 is at position 1. Then B4 must be at position 2 (immediate right of B2, as they face South).
_B2_ _B4_ _ _ _ _ _
Case 2: B2 is at position 7. Then B4 must be at position 6 (immediate left of B2, as they face South).
_ _ _ _ _ _B4_ _B2_
Clue 2: B3 is to the immediate right of B4.
In Case 1: B4 is at position 2. B3 is to the immediate right of B4, so B3 is at position 3.
_B2_ _B4_ _B3_ _ _ _ _
In Case 2: B4 is at position 6. B3 is to the immediate right of B4, so B3 is at position 7. But B2 is already at position 7. This case is not possible.
Therefore, Case 1 is the correct arrangement so far:
_B2_ _B4_ _B3_ _ _ _ _
Clue 3: Only two boys sit between B1 and B5.
From the current arrangement, we have B2, B4, B3. The remaining positions are 4, 5, 6, 7. If B1 and B5 have two boys between them, they must be at positions 4 and 7, or 5 and 7, or 4 and 6. Let's explore.
Clue 4: B3 is not the neighbour of B5.
B3 is at position 3. This means B5 cannot be at position 2 (already B4) or position 4.
From Clue 3, if B1 and B5 have two boys between them, possible pairs are (B1, B5) at (4, 7) or (B5, B1) at (4, 7). If B5 is at 4, it's a neighbour of B3, which contradicts this clue. So B5 cannot be at position 4.
This means B5 must be at position 7. If B5 is at 7, and there are two boys between B1 and B5, then B1 must be at position 4.
Let's check: B1 at 4, B5 at 7. Two boys (positions 5 and 6) are between them. This fits.
So the arrangement becomes:
_B2_ _B4_ _B3_ _B1_ _ _ _B5_
Clue 5: B7 is not the neighbour of B1.
B1 is at position 4. Its neighbours are positions 3 (B3) and 5. So B7 cannot be at position 5.
The remaining boys are B6 and B7. The remaining positions are 5 and 6.
Since B7 cannot be at position 5, B7 must be at position 6.
This leaves position 5 for B6.
Final Arrangement:
_B2_ _B4_ _B3_ _B1_ _B6_ _B7_ _B5_
Let's verify all conditions:
Answer the question: Which of the following position of B6 is correct?
From the final arrangement, B6 is at position 5. B1 is at position 4. So B6 is to the immediate right of B1.
As per the final arrangement (B2 B4 B3 B1 B6 B7 B5), B1 is at position 4 and B6 is at position 5. Since they are facing South, position 5 is to the immediate right of position 4. Thus, B6 is to the immediate right of B1.
If X = \(\sqrt{1+\frac{\sqrt{3}}{2}}\) + \(\sqrt{1-\frac{\sqrt{3}}{2}}\) then the simplified value of X will be:
To simplify the given expression, we will first try to convert the terms inside the square roots into perfect squares. This can be achieved by recognizing that expressions of the form \(1 \pm \frac{\sqrt{3}}{2}\) can be related to \((a \pm b)^2\) or by multiplying the numerator and denominator by 2 to get \(\frac{2 \pm \sqrt{3}}{2}\) and then manipulating the numerator.
A) The simplified value of X is \(\sqrt{3}\).
The question asks to identify which of the given ISRO satellites is an experimental satellite. This requires knowledge of the primary purpose and classification of various ISRO missions.
Correct Option: A) YOUTHSAT was a joint Indo-Russian satellite mission primarily for stellar and atmospheric studies, making it an experimental and scientific research satellite.
The question asks to identify the primary mineral contained in bauxite. This requires knowledge of common ores and the metals they yield.
B) Aluminium — Bauxite is the chief ore of aluminium. It is a clay-like substance that is the world's main source of aluminium.
This question is based on circular seating arrangements. Seven friends P, Q, R, S, T, V, and W sit around a circular table facing the centre. We need to arrange them according to the given conditions and determine the position of T with respect to S. Since all the friends face the centre, left means clockwise and right means anticlockwise.
We need to find the position of T with respect to S.
Since everyone faces the centre of the circular table:
We can fix P at any position because rotating a circular arrangement does not change the relative positions of the friends.
R sits third to the left of P.
Starting from P, count three positions clockwise.
Therefore, R occupies the third clockwise position from P.
R sits immediately to the left of W.
Therefore, W must sit immediately anticlockwise from R.
The clockwise arrangement becomes:
P → Vacant → W → R → Vacant → Vacant → Vacant
Exactly two friends sit between P and S.
Therefore, P and S must be separated by three seating positions in one direction.
There are two possible positions for S:
The third clockwise position is already occupied by R.
Therefore, S must occupy the third anticlockwise position from P, which is the fourth clockwise position.
The clockwise arrangement becomes:
P → Vacant → W → R → S → Vacant → Vacant
Q sits immediately to the left of V.
Therefore, Q must occupy the seat immediately clockwise from V.
The three remaining vacant seats are:
Among these vacant positions, only the fifth and sixth positions are adjacent.
Therefore:
The only remaining seat is assigned to T.
The final clockwise arrangement is:
P → T → W → R → S → V → Q → P
All seven friends have been placed according to the given conditions.
Starting from S, move towards the right, which is anticlockwise.
Therefore, T sits third to the right of S.
All the given conditions are satisfied.
Correct Option: B) Third to the right
The complete clockwise seating arrangement is P, T, W, R, S, V, and Q. Since all the friends face the centre, moving towards the right means moving anticlockwise. Counting from S gives R first, W second, and T third. Therefore, T is third to the right of S, making option B correct.
The question asks to identify the university whose Student Union Rashmi Samant became president of in February 2021. This is a factual recall question related to current affairs or general knowledge.
Correct Option: B) Rashmi Samant was elected as the President of the Oxford University Student Union in February 2021, making her the first Indian woman to hold this position. This was a significant news event at the time.
To find the least perfect square that has 3675 as a factor, we first need to find the prime factorization of 3675. Then, we need to ensure that all prime factors in the number we are looking for have even powers, which is a property of perfect squares. We will multiply 3675 by the smallest possible factors to make all prime powers even.
First, we find the prime factors of 3675:
\[ 3675 = 5 \times 735 \\ 735 = 5 \times 147 \\ 147 = 3 \times 49 \\ 49 = 7 \times 7 \]So, the prime factorization of 3675 is:
\[ 3675 = 3^1 \times 5^2 \times 7^2 \]For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of 3675, the prime factor 3 has an exponent of 1 (which is odd), while 5 and 7 have exponents of 2 (which are even).
To make the exponent of 3 even, we need to multiply \(3^1\) by \(3^1\). This means we need to multiply 3675 by 3.
Multiply 3675 by 3:
\[ \text{Least perfect square} = 3675 \times 3 \] \[ 3675 \times 3 = 11025 \]Let's check the prime factorization of 11025:
\[ 11025 = (3^1 \times 5^2 \times 7^2) \times 3^1 = 3^2 \times 5^2 \times 7^2 \]Since all exponents (2, 2, 2) are even, 11025 is a perfect square. It is also the least such number because we only multiplied by the minimum factor (3) required to make it a perfect square.
We can also verify that \(11025 = (3 \times 5 \times 7)^2 = (105)^2\).
Correct Option: A) 11025
The question asks to identify the second person to join the Individual Satyagraha Movement in 1940. This requires knowledge of the key figures involved in this specific movement and their sequence of participation.
Correct Option: B) Jawahar Lal Nehru was the second person to join the Individual Satyagraha Movement in 1940, following Vinoba Bhave.
We are given the ratios of incomes and expenditures of two individuals, A and B, and their savings. We need to find the difference between their incomes. We can set up algebraic equations based on the given ratios and the relationship that Income - Expenditure = Savings.
Let the incomes of A and B be \(3x\) and \(2x\) respectively, based on the ratio \(3:2\).
Let the expenditures of A and B be \(5y\) and \(3y\) respectively, based on the ratio \(5:3\).
We know that Savings = Income - Expenditure. Since each saves ₹2,000, we can set up two equations:
We need to solve these two linear equations for \(x\) and \(y\). We can use the elimination method. Multiply Equation 1 by 3 and Equation 2 by 5 to make the coefficients of \(y\) equal:
Subtract Equation 3 from Equation 4:
\[ (10x - 15y) - (9x - 15y) = 10000 - 6000 \]\[ 10x - 15y - 9x + 15y = 4000 \]\[ x = 4000 \]Now that we have the value of \(x\), we can find the incomes of A and B:
The difference between their incomes is:
\[ \text{Difference} = \text{Income of A} - \text{Income of B} \]\[ \text{Difference} = 12000 - 8000 = \text{₹}4,000 \]A) ₹4,000 is the correct difference between their incomes.
The question asks to identify the author of the famous science book 'A Brief History of Time'. This requires general knowledge about prominent scientists and their works.
Correct Option: D) Stephen Hawking is the author of 'A Brief History of Time'. This book, published in 1988, became a global bestseller and simplified complex cosmological concepts for the general public.