This is a blood relations question. We need to deduce the relationship between Supriya and the woman by breaking down the statement given by Supriya into smaller, understandable parts, starting from the end of the statement.
Correct Option: A) Sister — The woman is the daughter of Supriya's parent (who is the only child of Supriya's grandmother). Therefore, the woman is Supriya's sister.
The question asks for the launch date of India's first satellite, Aryabhata. This is a factual question related to India's space history. Recalling key milestones in India's space program is essential for competitive exams.
A) April 19, 1975 is the correct date for the launch of India's first satellite, Aryabhata. This marked a significant milestone in India's space exploration journey.
The problem asks us to identify the number that is different from the rest, given four options. This implies that three of the numbers share a common property or pattern, while one does not. We need to analyze each number to find this underlying pattern.
From the analysis, options A, B, and C all follow the pattern \(n^2 + 1\) for some integer \(n\), whereas option D does not.
D) 59 — This number does not fit the pattern \(n^2 + 1\). While \(7^2 + 1 = 50\) and \(8^2 + 1 = 65\), 59 cannot be expressed in this form.
The question asks to identify the British Governor-General who introduced the "Doctrine of Lapse" policy in India. This policy was a significant annexation tool used by the British East India Company.
C) Lord Dalhousie devised and extensively implemented the policy of the Doctrine of Lapse. His tenure as Governor-General (1848-1856) is synonymous with this annexation policy, which contributed significantly to the expansion of British territories in India and was one of the causes of the Revolt of 1857.
The problem provides a ratio between two variables, \(x\) and \(y\), and asks us to find the ratio of two linear expressions involving \(x\) and \(y\). We can express one variable in terms of the other using the given ratio and then substitute this into the expression to simplify it.
Given the ratio \(x : y = 4 : 9\), we can write this as a fraction:
\[ \frac{x}{y} = \frac{4}{9} \]From this equation, we can express \(x\) in terms of \(y\):
\[ x = \frac{4}{9}y \]Now, we need to find the ratio of the two expressions: \(9x + 4y\) and \(18x + 3y\). Let's write this as a fraction:
\[ \frac{9x + 4y}{18x + 3y} \]Substitute the value of \(x = \frac{4}{9}y\) into the expression:
\[ \frac{9\left(\frac{4}{9}y\right) + 4y}{18\left(\frac{4}{9}y\right) + 3y} \]Simplify the terms in the numerator and denominator:
Numerator: \(9\left(\frac{4}{9}y\right) + 4y = 4y + 4y = 8y\)
Denominator: \[18\left(\frac{4}{9}y\right) + 3y = 2 \cdot 4y + 3y = 8y + 3y = 11y\]
Now, substitute these simplified terms back into the ratio:
\[ \frac{8y}{11y} \]Cancel out \(y\) from the numerator and denominator (assuming \(y \neq 0\), which must be true for the initial ratio to be defined):
\[ \frac{8}{11} \]Therefore, the ratio is \(8 : 11\).
D) 8 : 11 is the correct ratio obtained after substituting the relationship between \(x\) and \(y\) into the given expression and simplifying.
The question asks to identify the countries declared malaria-free by the WHO in May 2019. This requires knowledge of specific public health announcements made by the World Health Organization.
B) Algeria and Argentina were indeed declared malaria-free by the WHO in May 2019. Algeria achieved this status after a 27-year effort, with its last indigenous case reported in 2013. Argentina achieved this after 70 years of continuous effort, with its last indigenous case reported in 2010.
The question asks for the approximate total length of India's coastline, including its island territories. This is a factual question related to Indian geography. We need to recall the standard geographical data for India's coastline.
Correct Option: C) 7517 km is the approximate total length of India's coastline, including the Lakshadweep and Andaman & Nicobar Islands, which is a widely accepted geographical fact.
The question asks for the maximum strength of the Rajya Sabha, which is the Upper House of the Indian Parliament. This is a factual question related to the Indian Constitution and parliamentary structure.
Correct Option: D) 250
If x⁴ + \( \frac{1}{x^4} \) = 194, find x³ + \( \frac{1}{x^3} \)
To find the value of \(x^3 + \frac{1}{x^3}\), we first need to find the value of \(x + \frac{1}{x}\). We can derive \(x + \frac{1}{x}\) from the given equation \(x^4 + \frac{1}{x^4} = 194\) using algebraic identities.
We know the identity \( (a+b)^2 = a^2 + b^2 + 2ab \). Let \(a = x^2\) and \(b = \frac{1}{x^2}\). Then,
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + \frac{1}{x^4} + 2 \cdot x^2 \cdot \frac{1}{x^2} \]\[ \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + \frac{1}{x^4} + 2 \]Given \(x^4 + \frac{1}{x^4} = 194\), substitute this value:\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 194 + 2 \]\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 196 \]Taking the square root of both sides:\[ x^2 + \frac{1}{x^2} = \sqrt{196} \]\[ x^2 + \frac{1}{x^2} = 14 \] (Since \(x\) is usually considered real and positive in such problems, we take the positive root).Now, we use the same identity again for \(a=x\) and \(b=\frac{1}{x}\):
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \cdot x \cdot \frac{1}{x} \]\[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \]Substitute the value of \(x^2 + \frac{1}{x^2} = 14\):\[ \left(x + \frac{1}{x}\right)^2 = 14 + 2 \]\[ \left(x + \frac{1}{x}\right)^2 = 16 \]Taking the square root of both sides:\[ x + \frac{1}{x} = \sqrt{16} \]\[ x + \frac{1}{x} = 4 \]We use the identity \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \). Let \(a=x\) and \(b=\frac{1}{x}\). Then,
\[ \left(x + \frac{1}{x}\right)^3 = x^3 + \frac{1}{x^3} + 3 \cdot x \cdot \frac{1}{x} \left(x + \frac{1}{x}\right) \]\[ \left(x + \frac{1}{x}\right)^3 = x^3 + \frac{1}{x^3} + 3 \left(x + \frac{1}{x}\right) \]We need to find \(x^3 + \frac{1}{x^3}\), so rearrange the formula:\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3 \left(x + \frac{1}{x}\right) \]Substitute the value \(x + \frac{1}{x} = 4\):\[ x^3 + \frac{1}{x^3} = (4)^3 - 3(4) \]\[ x^3 + \frac{1}{x^3} = 64 - 12 \]\[ x^3 + \frac{1}{x^3} = 52 \]B) The calculated value of \(x^3 + \frac{1}{x^3}\) is 52.
To find the number of smaller cubes that can be cut from a larger cuboid, we need to calculate the volume of the cuboid and the volume of one small cube. Then, divide the volume of the cuboid by the volume of one cube. Alternatively, we can divide the dimensions of the cuboid by the side of the cube along each axis and multiply the results.
The dimensions of the cuboid are given as length \(L = 27 \text{ cm}\), breadth \(B = 18 \text{ cm}\), and height \(H = 21 \text{ cm}\).
\[ \text{Volume of cuboid} = L \times B \times H \]\[ \text{Volume of cuboid} = 27 \text{ cm} \times 18 \text{ cm} \times 21 \text{ cm} \]\[ \text{Volume of cuboid} = 54432 \text{ cm}^3 \]The side of the cube is given as \(a = 3 \text{ cm}\).
\[ \text{Volume of cube} = a^3 \]\[ \text{Volume of cube} = (3 \text{ cm})^3 \]\[ \text{Volume of cube} = 27 \text{ cm}^3 \]The number of cubes that can be cut from the cuboid is the ratio of the volume of the cuboid to the volume of one cube.
\[ \text{Number of cubes} = \frac{\text{Volume of cuboid}}{\text{Volume of cube}} \]\[ \text{Number of cubes} = \frac{54432 \text{ cm}^3}{27 \text{ cm}^3} \]\[ \text{Number of cubes} = 378 \]Alternatively, we can find how many cubes fit along each dimension:Number of cubes along length \(L\):
\[ N_L = \frac{27 \text{ cm}}{3 \text{ cm}} = 9 \]Number of cubes along breadth \(B\):
\[ N_B = \frac{18 \text{ cm}}{3 \text{ cm}} = 6 \]Number of cubes along height \(H\):
\[ N_H = \frac{21 \text{ cm}}{3 \text{ cm}} = 7 \]Total number of cubes:\[ \text{Total cubes} = N_L \times N_B \times N_H \]\[ \text{Total cubes} = 9 \times 6 \times 7 \]\[ \text{Total cubes} = 54 \times 7 \]\[ \text{Total cubes} = 378 \]B) 378