The question asks to identify the name of a one-stop education portal designed by the Ministry of Human Resource Development (now Ministry of Education) that caters to a wide range of learners, from elementary students to research scholars and teachers.
B) SAKSHAT — SAKSHAT is the one-stop education portal launched by the Ministry of Human Resource Development (MHRD), Government of India. It was designed to provide educational content and resources to students, teachers, and lifelong learners, covering a wide spectrum from elementary education to research.
To find the mean of the first ten even natural numbers, we first need to identify these numbers. Then, we will sum them up and divide by the count of the numbers (which is 10).
Even natural numbers are positive integers divisible by 2. The first ten even natural numbers are:
\[ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \]Sum \(S = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20\)
This is an arithmetic progression with \(n=10\) terms, first term \(a_1 = 2\), and last term \(a_{10} = 20\). The sum can be calculated using the formula:
\[ S = \frac{n}{2} (a_1 + a_n) \]\[ S = \frac{10}{2} (2 + 20) \]\[ S = 5 \times 22 \]\[ S = 110 \]The mean is the sum of the numbers divided by the count of the numbers.
\[ \text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} \]\[ \text{Mean} = \frac{110}{10} \]\[ \text{Mean} = 11 \]Correct Option: C) 11
The mean of the first ten even natural numbers is 11.
The question asks to identify the writ that allows a court to order the presentation of an arrested person or to set them free if the arrest is unlawful. This directly relates to the protection of personal liberty against unlawful detention, which is the primary function of one specific writ.
Correct Option: B) Habeas corpus
The question asks to identify the Nobel Prize winner known for founding a grassroots movement to combat deforestation. This requires knowledge of prominent Nobel laureates and their contributions, particularly in environmental activism.
C) Wangari Maathai was a Kenyan social, environmental, and political activist who founded the Green Belt Movement, an environmental non-governmental organization focused on tree planting, environmental conservation, and women's rights. She was awarded the Nobel Peace Prize in 2004 for her contribution to sustainable development, democracy, and peace.
To find the circumference of a semicircular compound, we need to calculate the length of its curved part (half the circumference of a full circle) and add the length of its diameter (the straight part).
The circumference of a full circle is given by the formula \(C = 2\pi r\). For a semicircle, the curved part is half of this.
\[ \text{Curved part} = \frac{1}{2} \times 2\pi r = \pi r \]Substitute the given radius \(r = 35\) m and use the approximation \(\pi = \frac{22}{7}\):
\[ \text{Curved part} = \frac{22}{7} \times 35 = 22 \times 5 = 110 \text{ m} \]The diameter \(d\) of a circle is twice its radius \(r\).
\[ d = 2r = 2 \times 35 = 70 \text{ m} \]The total circumference of the semicircular compound is the sum of its curved part and its straight part (diameter).
\[ \text{Total Circumference} = \text{Curved part} + \text{Diameter} \] \[ \text{Total Circumference} = 110 \text{ m} + 70 \text{ m} = 180 \text{ m} \]Correct Option: B) 180 m
The question asks to identify the river whose section was declared as National Waterway-2 in 1988. This requires knowledge of India's National Waterways and their respective rivers and declaration dates.
Based on this information, the Brahmaputra River is the correct answer.
C) Brahmaputra — The Brahmaputra River, specifically the stretch from Sadiya to Dhubri (891 km), was declared as National Waterway-2 (NW-2) in 1988. This makes it one of the most important waterways in the North-Eastern region of India.
The question asks to identify the person who, along with Motilal Nehru, founded the Swaraj Party to advocate for participation in council politics. This requires knowledge of the early 20th-century Indian nationalist movement and the key figures involved in the formation of the Swaraj Party.
A) CR Das was the co-founder of the Swaraj Party along with Motilal Nehru. He was a prominent lawyer and nationalist leader who believed that nationalists should enter the legislative councils to oppose the government from within.
We need to calculate the son's current age. We are given the average age of the husband and wife 20 years ago, and the current average age of the family (husband, wife, and son). We will use the concept of average age to find the sum of ages at different points in time and then deduce the son's age.
Correct Option: B) 16 years
To find the coefficient of \(x\) in the expansion of \((x+3)^3\), we can use the binomial theorem or simply expand the expression directly using the formula for \((a+b)^3\).
D) 27 — As derived from the binomial expansion, the term with \(x\) is \(27x\), making \(27\) the coefficient of \(x\).
The ratio of the area of a circle and that of an equilateral triangle, where the diameter of the circle is equal to the sides of the equilateral triangle, is:
π : \(\sqrt{2}\)
π : \(\sqrt{3}\)
π : 1
\(\sqrt{3}\) : π
To find the ratio of the area of a circle to that of an equilateral triangle, we need to express both areas in terms of a common variable. The problem states that the diameter of the circle is equal to the side of the equilateral triangle. We will use this relationship to find the ratio.
Define variables:
Let the diameter of the circle be \(d\).
Let the side of the equilateral triangle be \(a\).
According to the problem statement, \(d = a\).
Calculate the area of the circle:
The radius of the circle, \(r\), is half of its diameter. So, \(r = \frac{d}{2}\).
The area of a circle is given by the formula \(A_{\text{circle}} = \pi r^2\).
Substitute \(r = \frac{d}{2}\) into the formula:
\[ A_{\text{circle}} = \pi \left(\frac{d}{2}\right)^2 = \pi \frac{d^2}{4} \]Calculate the area of the equilateral triangle:
The area of an equilateral triangle with side \(a\) is given by the formula \(A_{\text{triangle}} = \frac{\sqrt{3}}{4} a^2\).
Since \(a = d\), substitute \(d\) for \(a\):
\[ A_{\text{triangle}} = \frac{\sqrt{3}}{4} d^2 \]Find the ratio of the areas:
The ratio of the area of the circle to that of the equilateral triangle is:
\[ \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\pi \frac{d^2}{4}}{\frac{\sqrt{3}}{4} d^2} \]Simplify the ratio:
Cancel out the common terms \(\frac{d^2}{4}\) from the numerator and the denominator:
\[ \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\pi}{\sqrt{3}} \]Express the ratio:
The ratio is \(\pi : \sqrt{3}\).
Correct Option: B) The ratio of the area of the circle to that of the equilateral triangle is \(\pi : \sqrt{3}\), which matches this option.