The question asks to identify which country was NOT a co-host of the 2026 FIFA World Cup. This requires knowledge of the host nations for this specific event.
Correct Option: D) Brazil was not a co-host of the 2026 FIFA World Cup. The 2026 tournament will be jointly hosted by Canada, Mexico, and the USA. Brazil hosted the FIFA World Cup in 2014.
The question asks to identify the Ecomark logo of India. This requires knowledge of environmental labeling initiatives in India.
D) an earthen pot — The Ecomark logo in India is indeed an earthen pot. This symbol was chosen to represent the earth and its delicate balance, signifying products that are environmentally friendly.
The problem asks us to identify the odd one out from a series of letter-clusters. We need to find a pattern or rule that applies to four of the clusters and not to the fifth one. This usually involves analyzing the alphabetical positions of the letters within each cluster.
For AF, EJ, IN, and MR, the difference between the alphabetical positions of the second letter and the first letter is 5. For QU, this difference is 4.
Correct Option: A) QU is the odd one out because the difference between the alphabetical positions of its letters is 4, whereas for all other clusters, the difference is 5.
The question asks about the primary objective of the 'Sagarmala Project'. To answer this, one needs to recall the core purpose and scope of this major government initiative related to India's maritime sector.
Correct Option: D) The Sagarmala Project's primary objective is to reduce logistics costs for domestic cargo and EXIM (Export-Import) cargo by leveraging India's coastline and inland waterways. This is achieved through port modernization, new port development, port connectivity enhancement, and coastal community development.
The problem asks us to evaluate given courses of action based on a statement about increased air pollution due to metro construction. We need to determine which actions logically follow as reasonable and practical solutions to mitigate the problem without causing undue disruption.
The statement indicates that metro construction has increased air pollution by 30% in the short term. This is a problem that requires mitigation.
Based on this analysis, courses of action 2 and 3 are logical and practical responses to the problem stated.
A) Only 2 and 3 follow.
To find the value of \(k\), we will use the formula for the angle between two intersecting circles. The angle \(\theta\) between two circles with equations \(x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0\) and \(x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0\) is given by the formula:
\[ \cos \theta = \frac{2g_1g_2 + 2f_1f_2 - c_1 - c_2}{2\sqrt{g_1^2 + f_1^2 - c_1}\sqrt{g_2^2 + f_2^2 - c_2}} \]Given that the angle is 45°, we can substitute \(\cos 45° = \frac{1}{\sqrt{2}}\) into the formula and solve for \(k\).
Identify parameters for the first circle:
The equation of the first circle is \(x^2 + y^2 - 12x - 6y + 41 = 0\).
Comparing with \(x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0\), we get:
\(2g_1 = -12 \implies g_1 = -6\)
\(2f_1 = -6 \implies f_1 = -3\)
\(c_1 = 41\)
Identify parameters for the second circle:
The equation of the second circle is \(x^2 + y^2 + kx + 6y - 59 = 0\).
Comparing with \(x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0\), we get:
\(2g_2 = k \implies g_2 = \frac{k}{2}\)
\(2f_2 = 6 \implies f_2 = 3\)
\(c_2 = -59\)
Apply the angle formula:
The angle \(\theta\) between the circles is given as 45°, so \(\cos \theta = \cos 45° = \frac{1}{\sqrt{2}}\).
Substitute the values into the formula:
Numerator:
\[ 2g_1g_2 + 2f_1f_2 - c_1 - c_2 = 2(-6)\left(\frac{k}{2}\right) + 2(-3)(3) - 41 - (-59) \]\[ = -6k - 18 - 41 + 59 = -6k \]
Denominator terms:
\[ \sqrt{g_1^2 + f_1^2 - c_1} = \sqrt{(-6)^2 + (-3)^2 - 41} = \sqrt{36 + 9 - 41} = \sqrt{45 - 41} = \sqrt{4} = 2 \]\[ \sqrt{g_2^2 + f_2^2 - c_2} = \sqrt{\left(\frac{k}{2}\right)^2 + (3)^2 - (-59)} = \sqrt{\frac{k^2}{4} + 9 + 59} = \sqrt{\frac{k^2}{4} + 68} \]
Now, substitute these back into the formula:
\[ \frac{1}{\sqrt{2}} = \frac{-6k}{2(2)\sqrt{\frac{k^2}{4} + 68}} \]\[ \frac{1}{\sqrt{2}} = \frac{-6k}{4\sqrt{\frac{k^2}{4} + 68}} \]
\[ \frac{1}{\sqrt{2}} = \frac{-3k}{2\sqrt{\frac{k^2}{4} + 68}} \]
Solve for k:
Square both sides:
\[ \frac{1}{2} = \frac{9k^2}{k^2 + 272} \]
Cross-multiply:
\[ k^2 + 272 = 18k^2 \]\[ 17k^2 = 272 \]
\[ k^2 = \frac{272}{17} \]
\[ k^2 = 16 \]
\[ k = \pm 4 \]
B) ±4 is the correct value for k, as derived from the calculations.
To find the minimum value of the given trigonometric expression, we will use the fundamental trigonometric identity \( \sin^2\theta + \cos^2\theta = 1 \). We will rewrite the expression in terms of a single trigonometric function and then use the known range of that function.
The given expression is \( 9\sin^2\theta + 10\cos^2\theta \).
We know the identity \( \sin^2\theta + \cos^2\theta = 1 \). From this, we can write \( \sin^2\theta = 1 - \cos^2\theta \).
Substitute this into the given expression:
\[ 9(1 - \cos^2\theta) + 10\cos^2\theta \]Expand and simplify the expression:
\[ 9 - 9\cos^2\theta + 10\cos^2\theta \]\[ 9 + (10 - 9)\cos^2\theta \]\[ 9 + \cos^2\theta \]Now we need to find the minimum value of \( 9 + \cos^2\theta \).
We know that the range of \( \cos\theta \) is \( [-1, 1] \). Therefore, the range of \( \cos^2\theta \) is \( [0, 1] \) because squaring a number makes it non-negative.
To find the minimum value of \( 9 + \cos^2\theta \), we need to use the minimum value of \( \cos^2\theta \), which is \( 0 \).
Substitute the minimum value of \( \cos^2\theta \) into the expression:
\[ 9 + 0 = 9 \]Thus, the minimum value of \( 9\sin^2\theta + 10\cos^2\theta \) is \( 9 \).
D) 9 is the correct minimum value of the given expression, achieved when \( \cos^2\theta = 0 \) (i.e., \( \theta = \frac{\pi}{2} + n\pi \) for integer \( n \)).
The question asks for the Indus Valley Civilisation town whose name literally translates to 'mound of the dead'. This requires knowledge of the etymology of the names of major Indus Valley sites.
Correct Option: A) Mohenjo-daro is a prominent city of the Indus Valley Civilisation, located in present-day Sindh, Pakistan. Its name in Sindhi literally means 'Mound of the Dead'. This name was given due to the discovery of numerous skeletons during excavations, suggesting a catastrophic end or burial practices that led to the accumulation of remains.
To find the ratio A : C, we need to establish a relationship between A and C using the given equations. We can express B in terms of A from the first equation and then substitute this expression for B into the second equation to find a direct relationship between A and C.
From the first equation:
We are given \(4A = 5B\). To express B in terms of A, we can divide both sides by 5:
\[ B = \frac{4A}{5} \]From the second equation:
We are given \(6B = 7C\). To express C in terms of B, we can divide both sides by 7:
\[ C = \frac{6B}{7} \]Substitute B into the expression for C:
Now, substitute the value of B from Step 1 into the equation for C from Step 2:
\[ C = \frac{6}{7} \times \left(\frac{4A}{5}\right) \]\[ C = \frac{6 \times 4A}{7 \times 5} \]\[ C = \frac{24A}{35} \]Find the ratio A : C:
We have \(C = \frac{24A}{35}\). To find the ratio A : C, we can rearrange this equation:
\[ 35C = 24A \]\[ \frac{A}{C} = \frac{35}{24} \]Therefore, \(A : C = 35 : 24\).Correct Option: B) 35 : 24
This problem involves calculating the final displacement from a starting point after a series of movements in different directions. We can break down the movements into horizontal (East-West) and vertical (North-South) components and then find the net displacement.
Therefore, the park is 40 m to the east of the bangle stall.
C) 40 m to the east — Based on the calculations, the bangle stall is at a position 40 m west of the park. Therefore, the park is 40 m east of the bangle stall.