To find the mode of a given set of data, we need to identify the value that appears most frequently in the dataset. The mode is a measure of central tendency.
Correct Option: A) 13 is the mode because it appears 3 times, which is the highest frequency among all the given marks.
The question asks to identify a feature of the Indian Constitution that has been borrowed from the Constitution of the United Kingdom. This requires knowledge of the sources of various provisions in the Indian Constitution.
A) Single citizenship is a key feature of the Indian Constitution, meaning a person is a citizen of India and not of any particular state. This concept was adopted from the Constitution of the United Kingdom, which also follows a system of single citizenship.
To find the volume of a cube, we need to know the length of one of its edges. The volume of a cube is calculated by cubing the length of its edge.
Correct Option: D) The volume of the cube is \(729 \text{ m}^3\). This is obtained by cubing the edge length of 9 m.
The question asks for the maximum age for a subscriber to join the Atal Pension Yojana (APY), given the minimum age is 18 years. This requires knowledge of the eligibility criteria for the APY scheme as defined by the government.
D) 40 years — As per the guidelines of the Atal Pension Yojana (APY), any Indian citizen between the age of 18 and 40 years can subscribe to the scheme. The maximum age for entry into the scheme is 40 years.
The roots of the equation
\(\sqrt{3}\)x2 − 8x + 5\(\sqrt{3}\) = 0
are:
\(\sqrt{3},\ -\frac{5}{\sqrt{3}}\)
\(-\sqrt{3},\ -\frac{5}{\sqrt{3}}\)
\(-\sqrt{3},\ \frac{5}{\sqrt{3}}\)
\(\sqrt{3},\ \frac{5}{\sqrt{3}}\)
To find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\), we can use either factorization (splitting the middle term) or the quadratic formula. In this case, factorization is a suitable method.
Identify the coefficients: The given quadratic equation is \(\sqrt{3}x^2 - 8x + 5\sqrt{3} = 0\). Comparing this to \(ax^2 + bx + c = 0\), we have:
Calculate the product \(ac\):
\[ ac = (\sqrt{3})(5\sqrt{3}) = 5 \times (\sqrt{3} \times \sqrt{3}) = 5 \times 3 = 15 \]Find two numbers whose product is \(ac\) and sum is \(b\): We need two numbers that multiply to \(15\) and add up to \(-8\). These numbers are \(-3\) and \(-5\), because \((-3) \times (-5) = 15\) and \((-3) + (-5) = -8\).
Split the middle term: Rewrite the middle term \(-8x\) using the two numbers found:
\[ \sqrt{3}x^2 - 3x - 5x + 5\sqrt{3} = 0 \]Factor by grouping: Group the terms and factor out common factors:
\[ (\sqrt{3}x^2 - 3x) - (5x - 5\sqrt{3}) = 0 \]\[ \sqrt{3}x(x - \sqrt{3}) - 5(x - \sqrt{3}) = 0 \]Factor out the common binomial:
\[ (x - \sqrt{3})(\sqrt{3}x - 5) = 0 \]Set each factor to zero and solve for \(x\):
The roots are: \(\sqrt{3}\) and \(\frac{5}{\sqrt{3}}\).
Correct Option: D) \(\sqrt{3},\ \frac{5}{\sqrt{3}}\)
The question asks to identify the environmental movement that originated in Garhwal. This requires knowledge of the origins and locations associated with prominent environmental movements in India.
Correct Option: A) The Chipko Movement originated in the Garhwal Himalayas in the early 1970s. The movement's name comes from the word "chipko" which means "to hug" or "to cling to," referring to the villagers hugging trees to prevent them from being felled by contractors.
This is a syllogism problem. We need to analyze the given statements and determine which conclusions logically follow. We can use Venn diagrams or logical deductions to solve this.
D) Only conclusions II and III follow
The problem presents a number series, and we need to identify the pattern to find the missing term. We will analyze the relationship between consecutive numbers in the series to deduce the underlying rule.
The last operation before the question mark was \(12 + 3 = 15\). Following the pattern, the next operation should be multiplication by 3.
\[ ? = 15 \times 3 \]\[ ? = 45 \]If the missing term is 45, the next operation should be addition of 4.
\[ 45 + 4 = 49 \]This matches the given term 49.
The next operation should be multiplication by 4.
\[ 49 \times 4 = 196 \]This matches the given term 196.
The next operation should be addition of 5.
\[ 196 + 5 = 201 \]This matches the given term 201.
The pattern holds true for all terms.
B) 45 is the correct number that replaces the question mark, as it follows the established alternating pattern of addition and multiplication with increasing integers.
A, B and C invested Rs 10,000, Rs 12,000, and Rs 15,000, respectively, for 5, 6, and 9 months, respectively, in a business.
If A’s share in the profit is Rs 2,500, then the total profit is:
To solve this problem, we need to calculate the ratio of the investments made by A, B, and C, considering both the amount invested and the duration of the investment. The profit will be distributed in this ratio. Once we have the ratio and A's share of the profit, we can find the total profit.
D) 12,850 — The total profit is Rs 12,850, calculated based on the investment ratio and A's share.
The problem asks us to evaluate the expression \((\sin 32° \cos 58° + \cos 32° \sin 58°)\). This expression resembles a standard trigonometric identity. We will use the sum formula for sine, which is \(\sin(A+B) = \sin A \cos B + \cos A \sin B\).
Correct Option: C) 1