This question tests basic knowledge about the behavior of a bar magnet when suspended freely. The Earth itself acts as a giant magnet, and a freely suspended magnet aligns itself with the Earth's magnetic field lines.
Correct Option: C) Geographical North- Geographical South
A convex lens forms a real and inverted image twice the size of the object. The magnification produced by the lens is equal to:
The problem asks for the magnification produced by a convex lens given that it forms a real and inverted image twice the size of the object. We need to recall the sign conventions for magnification in optics.
Understand Magnification: Magnification (\(m\)) is defined as the ratio of the height of the image (\(h_i\)) to the height of the object (\(h_o\)).
\[ m = \frac{h_i}{h_o} \]Interpret "twice the size of the object": This means the magnitude of the image height is twice the magnitude of the object height. So, \(|h_i| = 2 |h_o|\).
Interpret "real and inverted image": For an inverted image, by convention, the image height (\(h_i\)) is taken as negative if the object height (\(h_o\)) is taken as positive (which is standard for an erect object). Therefore, \(h_i\) will have an opposite sign to \(h_o\).
Combine the information: Since the image is inverted and twice the size, if we assume the object height \(h_o\) is positive, then the image height \(h_i\) must be negative and its magnitude must be twice that of \(h_o\). So, \(h_i = -2h_o\).
Calculate Magnification: Substitute this into the magnification formula:
\[ m = \frac{h_i}{h_o} = \frac{-2h_o}{h_o} = -2 \]C) -2 — The magnification is -2 because the image is inverted (negative sign) and twice the size of the object (magnitude 2).
The question asks to identify a correct feature of the federal system as enshrined in the Indian Constitution. We need to evaluate each option against the established characteristics of India's federal structure.
D) Dual government — The Indian Constitution establishes a dual polity, comprising the Union at the centre and the states at the periphery. Each is endowed with sovereign powers to be exercised in the field assigned to them by the Constitution. This is a core feature of a federal system.
The question asks to identify the court that upheld the constitutional validity of 27% OBC reservation in the All India Quota (AIQ) for medical and dental courses on January 20, 2022, in the case of Neil Aurelio Nunes and Ors vs. Union of India and Ors. This requires knowledge of significant judicial pronouncements related to reservations in India.
D) Supreme Court — The Supreme Court of India, on January 20, 2022, in the case of Neil Aurelio Nunes and Ors vs. Union of India and Ors, upheld the constitutional validity of 27% reservation for Other Backward Classes (OBCs) in the All India Quota (AIQ) for undergraduate and postgraduate medical and dental courses. This was a landmark interim order that allowed the counselling process for NEET-UG and NEET-PG to commence.
The question asks to identify the international organization with which the Government of India has signed the 115 million dollar Rejuvenating Watersheds for Agricultural Resilience through Innovative Development (REWARD) Programme. This requires knowledge of recent agreements and collaborations between the Indian government and international bodies.
Correct Option: A) The Government of India, along with the participating state governments of Karnataka and Odisha, signed a 115 million dollar loan agreement with the World Bank for the "Rejuvenating Watersheds for Agricultural Resilience through Innovative Development (REWARD) Programme". This program is designed to help national and state institutions adopt new approaches to watershed management to improve farmers' resilience to climate change, promote higher productivity, and ensure better livelihoods.
To find the number divisible by both 7 and 11, we can check each option for divisibility by 7 and 11. Alternatively, a number divisible by both 7 and 11 must also be divisible by their least common multiple (LCM). Since 7 and 11 are prime numbers, their LCM is their product, which is \(7 \times 11 = 77\). So, we need to find the number divisible by 77.
Since 1771 is divisible by both 7 and 11, it is the correct answer.
A) 1771 is divisible by both 7 and 11.
This question is based on a square-table seating arrangement. Six students, F, E, D, C, B, and A, sit around a square table facing the centre. Four students occupy the corners, while two occupy the centres of two sides. We need to arrange them according to the given conditions and identify who sits third to the left of B.
Since all students face the centre:
We can fix the position of one student because rotating the square table does not change the relative seating arrangement.
C and E are diagonally opposite each other.
Therefore, both occupy opposite corner positions.
For convenience, place C at the top-left corner and E at the bottom-right corner.
D and F are also diagonally opposite each other.
Since C and E already occupy one diagonal, D and F must occupy the other two corners.
Therefore, all four corner positions are occupied by C, D, E, and F.
A does not sit at a corner, so A must occupy the centre of a side.
A is an immediate neighbour of both E and D.
Therefore, A must sit at the centre of the side connecting the corners occupied by E and D.
This fixes the relative positions of E, A, and D.
B sits immediately to the right of F.
Since F faces the centre, its right direction is anticlockwise.
Also, no student sits between E and F or between C and D.
Combining these conditions places B at the centre of the side between C and F.
The remaining two sides have no students at their centres.
The clockwise order of the six students is:
C → B → F → E → A → D → C
The four corners are occupied by C, F, E, and D, while B and A occupy the centres of two sides.
All the given conditions are satisfied.
Since everyone faces the centre, move clockwise from B.
Therefore, A sits third to the left of B.
Correct Option: C) A
The clockwise seating arrangement is C, B, F, E, A, and D. Since all students face the centre, counting clockwise from B gives F first, E second, and A third. Therefore, the correct option is C) A.
The question asks to identify the river that marks the easternmost boundary of the Himalayas. This requires knowledge of the geographical extent of the Himalayan mountain range and the major rivers flowing in and around it.
Correct Option: C) The Brahmaputra River marks the easternmost boundary of the Himalayas. It forms a deep gorge after taking a hairpin bend around the Namcha Barwa peak, signifying the eastern limit of the range.
The value of \(\sqrt{729}\) + \(\sqrt{1681}\) + \(\sqrt{576}\) − \(\sqrt{1849}\) is:
To solve this problem, we need to find the square root of each number given in the expression and then perform the addition and subtraction operations in the correct order. Recognizing perfect squares or using prime factorization can help in finding the square roots efficiently.
We know that \(20^2 = 400\) and \(30^2 = 900\). The number ends in 9, so its square root must end in 3 or 7. Testing \(27^2\):
\[ 27 \ \times 27 \ --- \ 189 \ 540 \ --- \ 729 \]So, \(\sqrt{729} = 27\).We know that \(40^2 = 1600\) and \(50^2 = 2500\). The number ends in 1, so its square root must end in 1 or 9. Testing \(41^2\):
\[ 41 \ \times 41 \ --- \ 41 \ 1640 \ --- \ 1681 \]So, \(\sqrt{1681} = 41\).We know that \(20^2 = 400\) and \(30^2 = 900\). The number ends in 6, so its square root must end in 4 or 6. Testing \(24^2\):
\[ 24 \ \times 24 \ --- \ 96 \ 480 \ --- \ 576 \]So, \(\sqrt{576} = 24\).We know that \(40^2 = 1600\) and \(50^2 = 2500\). The number ends in 9, so its square root must end in 3 or 7. Testing \(43^2\):
\[ 43 \ \times 43 \ --- \ 129 \ 1720 \ --- \ 1849 \]So, \(\sqrt{1849} = 43\).The expression is \(\sqrt{729} + \sqrt{1681} + \sqrt{576} - \sqrt{1849}\).
Substituting the values:
\[ 27 + 41 + 24 - 43 \]Thus, the value of the expression is 49.
C) 49 is the correct answer, as calculated by finding each square root and then performing the arithmetic operations.
Find the positive value of x if 48 : x :: x : 75.
The problem involves finding a positive value of 'x' from a given proportion. A proportion states that two ratios are equal. We will convert the given proportion into an equation and solve for 'x'.
The given proportion is \(48 : x :: x : 75\). This can be written in fractional form as:
\[ \frac{48}{x} = \frac{x}{75} \]To solve for 'x', we cross-multiply the terms:
\[ 48 \times 75 = x \times x \] \[ 3600 = x^2 \]Now, we need to find the square root of 3600 to find the value of 'x':
\[ x = \sqrt{3600} \]To find the square root of 3600, we can recognize that \(3600 = 36 \times 100\). So,
\[ x = \sqrt{36 \times 100} \] \[ x = \sqrt{36} \times \sqrt{100} \] \[ x = 6 \times 10 \] \[ x = 60 \]The problem asks for the positive value of x. Since \(x = 60\) is positive, this is our answer.
B) 60 is the correct answer because when we solve the proportion \(48 : x :: x : 75\), we get \(x^2 = 3600\), which yields \(x = \pm 60\). Since the question asks for the positive value, \(x = 60\).