The problem asks us to identify the number-pair that is different from the rest. We need to find a common pattern or relationship between the numbers in three of the pairs and then identify the pair that does not follow this pattern.
Correct Option: C) 12 : 15
This problem involves finding an unknown number given its relationship with another number, their HCF (Highest Common Factor), and LCM (Least Common Multiple). We will use the fundamental property that for any two positive integers, the product of the numbers is equal to the product of their HCF and LCM. We are given two conditions relating the HCF and LCM, which will allow us to find their values first.
Define variables:
Let the two numbers be \(N_1\) and \(N_2\).
Let HCF be \(H\).
Let LCM be \(L\).
We are given \(N_1 = 210\).
Formulate equations from given conditions:
Condition 1: The LCM of two numbers is 91 times their HCF.
\[ L = 91H \quad \text{(Equation 1)} \]
Condition 2: The sum of the HCF and LCM is 2760.
\[ H + L = 2760 \quad \text{(Equation 2)} \]
Solve for H and L:
Substitute Equation 1 into Equation 2:
\[ H + 91H = 2760 \]
\[ 92H = 2760 \]
\[ H = \frac{2760}{92} \]
\[ H = 30 \]
Now, substitute the value of \(H\) back into Equation 1 to find \(L\):
\[ L = 91 \times 30 \]
\[ L = 2730 \]
Apply the fundamental property of HCF and LCM:
For any two numbers \(N_1\) and \(N_2\), the product of the numbers is equal to the product of their HCF and LCM.
\[ N_1 \times N_2 = H \times L \]
We know \(N_1 = 210\), \(H = 30\), and \(L = 2730\). Substitute these values:
\[ 210 \times N_2 = 30 \times 2730 \]
Solve for the other number \(N_2\):
\[ N_2 = \frac{30 \times 2730}{210} \]
Simplify the expression:
\[ N_2 = \frac{30 \times 2730}{30 \times 7} \]
\[ N_2 = \frac{2730}{7} \]
\[ N_2 = 390 \]
Correct Option: C) 390
If \(\sqrt{3}\) sinθ − cosθ = 0 (θ is an acute angle), then the value of cos³θ − \(\sqrt{3}\) sin³θ will be:
\(\frac{\sqrt{3}}{4}\)
\(\frac{\sqrt{3}}{2}\)
-1
\( \frac{3}{8} \)
The problem requires us to find the value of an expression involving trigonometric functions, given an initial trigonometric equation. We will first solve the given equation to find the value of \(\theta\), or more precisely, the values of \(\sin\theta\) and \(\cos\theta\). Once these values are known, we can substitute them into the target expression to find its numerical value.
Solve the given equation:
We are given the equation:
\[ \sqrt{3} \sin\theta - \cos\theta = 0 \]Rearrange the equation to find the relationship between \(\sin\theta\) and \(\cos\theta\):
\[ \sqrt{3} \sin\theta = \cos\theta \]Divide both sides by \(\cos\theta\) (assuming \(\cos\theta \neq 0\)). If \(\cos\theta = 0\), then \(\sin\theta = \pm 1\), which would mean \(\sqrt{3}(\pm 1) - 0 = 0\), or \(\pm \sqrt{3} = 0\), which is not possible. So, \(\cos\theta \neq 0\).
\[ \sqrt{3} \frac{\sin\theta}{\cos\theta} = 1 \]\[ \sqrt{3} \tan\theta = 1 \]\[ \tan\theta = \frac{1}{\sqrt{3}} \]Since \(\theta\) is an acute angle, we know that \(\tan\theta = \frac{1}{\sqrt{3}}\) implies \(\theta = 30^\circ\).
Find the values of \(\sin\theta\) and \(\cos\theta\):
For \(\theta = 30^\circ\):
\[ \sin\theta = \sin(30^\circ) = \frac{1}{2} \]\[ \cos\theta = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]Substitute these values into the target expression:
The expression we need to evaluate is \(\cos^3\theta - \sqrt{3} \sin^3\theta\).
\[ \cos^3\theta - \sqrt{3} \sin^3\theta = \left(\frac{\sqrt{3}}{2}\right)^3 - \sqrt{3} \left(\frac{1}{2}\right)^3 \]Calculate the cubes:
\[ \left(\frac{\sqrt{3}}{2}\right)^3 = \frac{(\sqrt{3})^3}{2^3} = \frac{3\sqrt{3}}{8} \]\[ \left(\frac{1}{2}\right)^3 = \frac{1^3}{2^3} = \frac{1}{8} \]Substitute these back into the expression:
\[ \frac{3\sqrt{3}}{8} - \sqrt{3} \left(\frac{1}{8}\right) \]\[ = \frac{3\sqrt{3}}{8} - \frac{\sqrt{3}}{8} \]Combine the terms:
\[ = \frac{3\sqrt{3} - \sqrt{3}}{8} \]\[ = \frac{2\sqrt{3}}{8} \]\[ = \frac{\sqrt{3}}{4} \]A) The calculated value is \(\frac{\sqrt{3}}{4}\), which matches option A.
The question asks about the number of schedules in the Wildlife (Protection) Act, 1972, specifically as of 2021, before the 2022 amendment. This requires knowledge of the original structure of the Act or its structure prior to the recent amendment.
C) VI — As of 2021, the Wildlife (Protection) Act, 1972, had six schedules. These schedules categorized various species of wild animals and plants based on the degree of protection they required.
We need to find a two-digit number based on two given conditions. We can set up a system of linear equations using variables for the digits and then solve for them. Alternatively, we can test the given options against the conditions.
C) 24 is the correct number as it satisfies both conditions.
The problem asks us to find the fourth term in an analogy based on the relationship between the first two terms. We need to identify the pattern or rule that transforms "EFGH" into "VUTS" and then apply the same rule to "JKLM" to find the missing term.
Let's look at the alphabetical positions of the letters:
We can observe two types of relationships:
Method 1: Reverse Alphabetical Order and Position Sum
Notice that VUTS is the reverse of STUV. If we consider the letters from the end of the alphabet, V is the 5th letter from the end (Z, Y, X, W, V), U is the 6th, T is the 7th, and S is the 8th.
So, E (5th from start) corresponds to V (5th from end).
F (6th from start) corresponds to U (6th from end).
G (7th from start) corresponds to T (7th from end).
H (8th from start) corresponds to S (8th from end).
This means each letter in the first term is replaced by the letter that is at the same position from the end of the alphabet, and then the resulting sequence is reversed.
Alternatively, consider the sum of the alphabetical positions of corresponding letters (A=1, B=2, ... Z=26):
This indicates that each letter in the first term is replaced by its opposite letter in the alphabet (where A is opposite Z, B is opposite Y, etc.). Then, the order of these opposite letters is reversed to form the second term.
So, for EFGH, the opposite letters are VUTS. When reversed, it becomes STUV. However, the given second term is VUTS. This implies that the transformation is simply replacing each letter with its opposite letter, and then reversing the entire sequence of opposite letters.
Let's re-examine: E's opposite is V, F's opposite is U, G's opposite is T, H's opposite is S. So, EFGH becomes VUTS. The order is maintained for the opposite letters.
Method 2: Direct Mapping and Reversal
Let's consider the letters directly:
This is the reverse of the alphabetical order for each corresponding letter. E (5th) is mapped to V (22nd), F (6th) to U (21st), G (7th) to T (20th), H (8th) to S (19th). This means the 1st letter of EFGH corresponds to the 1st letter of VUTS, the 2nd to the 2nd, and so on. The relationship is that each letter is replaced by its corresponding letter from the opposite end of the alphabet, and the resulting sequence is then written in reverse order.
Let's try this: E (5th) -> V (22nd), F (6th) -> U (21st), G (7th) -> T (20th), H (8th) -> S (19th). So, EFGH maps to VUTS. This is a direct letter-to-letter mapping where each letter is replaced by its reverse alphabetical counterpart (A to Z, B to Y, etc.).
Now, let's find the opposite letters for JKLM:
So, JKLM transforms into QPON.
The derived term is QPON.
D) QPON is the correct answer, as it matches the pattern derived from the first pair.
In which of the following cities was the 45th G7 summit held in 2019?
The question asks for the location of the 45th G7 summit held in 2019. This is a factual question related to international current affairs and organizations. To answer correctly, one needs to recall the host city for that specific summit.
Correct Option: C) The 45th G7 summit was held in Biarritz, France, from August 24 to 26, 2019. France held the rotating presidency of the G7 in 2019.
The question asks about a specific Memorandum of Understanding (MoU) signed in January 2019 between India and another country for cooperation in the peaceful use of outer space. This requires knowledge of recent international agreements related to India's space program.
Correct Option: C) India and Finland
In January 2019, a Memorandum of Understanding (MoU) was signed between the Indian Space Research Organisation (ISRO) and the Finnish Ministry of Economic Affairs and Employment for cooperation in the peaceful uses of outer space. This agreement aims to foster collaboration in various areas of space science, technology, and applications.
The question asks for the definition of "Demographic Transition". We need to evaluate each option based on the widely accepted understanding of this concept in demography.
C) Demographic Transition is fundamentally a process describing the evolution of a society's population characteristics, including birth rates, death rates, and total population, over time. It is a model that explains how populations change from high mortality and fertility to low mortality and fertility as a country develops economically and socially.
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then check which conclusions logically follow from these diagrams.
Based on the evaluation, only Conclusion II definitely follows.
Correct Option: A) Only conclusion II follows.