We are given information about the relationship between the three angles of a triangle. We will use the property that the sum of angles in a triangle is 180° to set up equations and solve for the angles.
Let the three angles of the triangle be \(A\), \(B\), and \(C\).
According to the problem, two angles are in the ratio 1:2. Let these angles be \(A\) and \(B\). So, we can write:
\[ A : B = 1 : 2 \]This implies \(B = 2A\).
The sum of these two angles is equal to the third angle. So, \(A + B = C\).
Substitute \(B = 2A\) into the equation from step 3:
\[ A + 2A = C \] \[ 3A = C \]We know that the sum of all angles in a triangle is 180°:
\[ A + B + C = 180^\circ \]Substitute \(B = 2A\) and \(C = 3A\) into the sum equation:
\[ A + 2A + 3A = 180^\circ \]Combine the terms:
\[ 6A = 180^\circ \]Solve for \(A\):
\[ A = \frac{180^\circ}{6} \] \[ A = 30^\circ \]Now find the other angles:
\[ B = 2A = 2 \times 30^\circ = 60^\circ \] \[ C = 3A = 3 \times 30^\circ = 90^\circ \]The angles are 30°, 60°, and 90°. The smallest angle is \(A\).
A) 30° is the measure of the smallest angle, as calculated.
The question asks to identify the river in India where river dolphins are found. This requires knowledge of the distribution of river dolphins, specifically the Ganges River Dolphin (Platanista gangetica), which is native to the Indian subcontinent.
A) Ghaghara — The Ghaghara (also known as Sarayu in some stretches) is a major left-bank tributary of the Ganges River. It originates in the Himalayas and flows through Nepal and India before joining the Ganges. As part of the Ganges river system, it is known to be a habitat for the Ganges River Dolphin.
The question asks to identify the author of 'Natyashastra', a foundational text on performing arts in ancient India. This requires knowledge of prominent ancient Indian authors and their works.
Correct Option: C) Bharata Muni is widely credited as the author of 'Natyashastra', a seminal work on Indian performing arts.
A drum of water is \( \frac{3}{4} \) full. When 9 litres of water is drawn from it, it is \( \frac{1}{2} \)full. What is the capacity of the drum?
This problem involves fractions and basic algebra. We are given the initial and final fractional fullness of a drum and the amount of water drawn. We need to find the total capacity of the drum. We can set up an equation where the difference in the fractional fullness corresponds to the amount of water drawn.
The capacity of the drum is 36 litres.
B) 36 litres is the correct answer, as calculated above.
The question asks to identify the President of the Confederation of Indian Industry (CII) for the specific period of 2020-21. This requires knowledge of the past leadership of CII, a prominent business association in India.
Correct Option: A) Uday Kotak was the President of the Confederation of Indian Industry (CII) for the year 2020-21. He is the CEO and Managing Director of Kotak Mahindra Bank.
The question asks to identify the world's largest freshwater lake in terms of volume. This requires knowledge of the characteristics of major lakes around the world.
Lake Baikal, located in Siberia, Russia, is renowned for being the deepest lake in the world and containing the largest volume of freshwater. It holds approximately 22-23% of the world's fresh surface water.
D) Lake Baikal — Lake Baikal is the world's largest freshwater lake by volume, containing about 23,615 cubic kilometers of water. It is also the deepest lake in the world.
To pave a rectangular floor with square tiles using the least number of tiles, the side length of each square tile must be the largest possible common divisor of the floor's length and width. This is equivalent to finding the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of the given dimensions.
Identify the dimensions of the hall:
Understand the condition for the least number of tiles: For the least number of tiles, each tile must be as large as possible. Since the tiles are square and must perfectly fit the floor without any cutting, the side length of the square tile must be a common divisor of both the length and the width of the hall. To use the least number of tiles, this common divisor must be the greatest possible, i.e., the HCF of the length and width.
Find the HCF of 16 and 12:
The common factors are 1, 2, and 4. The highest common factor (HCF) is 4.
Alternatively, using prime factorization:
The HCF is the product of the lowest powers of the common prime factors:
\[ \text{HCF}(16, 12) = 2^2 = 4 \]
Determine the length of each square tile: The HCF, which is 4, represents the largest possible side length for the square tiles that will perfectly fit the floor.
Correct Option: D) 4 meters
The length of each square tile must be 4 meters to ensure the least number of tiles are used while perfectly covering the floor.
Simplify
\(\frac{3}{4}\div\frac{5}{12}-\frac{1}{3}\times\frac{9}{5}\left(\frac{2}{3}\div\frac{1}{3}-1\right)\)
\(\frac{6}{7}\)
\(\frac{5}{6}\)
\(\frac{12}{5}\)
\(\frac{6}{5}\)
To simplify the given expression, we will follow the order of operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. We will evaluate the innermost bracket first, then perform division and multiplication from left to right, and finally addition and subtraction from left to right.
The expression inside the parenthesis is \[\left(\frac{2}{3}\div\frac{1}{3}-1\right)\].
First, perform the division:
\[ \frac{2}{3} \div \frac{1}{3} = \frac{2}{3} \times \frac{3}{1} = \frac{2 \times 3}{3 \times 1} = \frac{6}{3} = 2 \]Now, substitute this back into the parenthesis:
\[ 2 - 1 = 1 \]The original expression becomes:
\[ \frac{3}{4}\div\frac{5}{12}-\frac{1}{3}\times\frac{9}{5}(1) \]Which simplifies to:
\[ \frac{3}{4}\div\frac{5}{12}-\frac{1}{3}\times\frac{9}{5} \]The first term is \[\frac{3}{4}\div\frac{5}{12}\].
\[ \frac{3}{4} \div \frac{5}{12} = \frac{3}{4} \times \frac{12}{5} = \frac{3 \times 12}{4 \times 5} = \frac{36}{20} \]Simplify the fraction \(\frac{36}{20}\) by dividing both numerator and denominator by their greatest common divisor, which is 4:
\[ \frac{36 \div 4}{20 \div 4} = \frac{9}{5} \]The second term is \[\frac{1}{3}\times\frac{9}{5}\].
\[ \frac{1}{3} \times \frac{9}{5} = \frac{1 \times 9}{3 \times 5} = \frac{9}{15} \]Simplify the fraction \(\frac{9}{15}\) by dividing both numerator and denominator by their greatest common divisor, which is 3:
\[ \frac{9 \div 3}{15 \div 3} = \frac{3}{5} \]Now, substitute the simplified terms back into the expression:
\[ \frac{9}{5} - \frac{3}{5} \]Since the denominators are the same, subtract the numerators:
\[ \frac{9 - 3}{5} = \frac{6}{5} \]D) The simplified value of the expression is \(\frac{6}{5}\).
This problem can be solved using the principle of inclusion-exclusion, often visualized with Venn diagrams. We are given percentages of students who failed in one, two, or all three subjects. We need to find the percentage of students who failed in only Economics. We will use the formula for the number of elements in exactly one set.
The percentage of students who failed in only Economics can be found by subtracting the percentages of students who failed in Economics and other subjects, and then adding back the percentage of students who failed in all three subjects (because it was subtracted twice).
The formula for students failing in only Economics is:
\[ P(\text{only } E) = P(E) - P(E \cap G) - P(E \cap H) + P(E \cap G \cap H) \]B) 14% is the percentage of students who failed in only Economics, as calculated using the inclusion-exclusion principle.
This problem involves substituting given symbol transformations into a mathematical expression and then evaluating the new expression following the order of operations (BODMAS/PEMDAS).
The original expression is: \(40 \times 20 \div 28 - 4 + 2\)
Applying the transformations:
So, the new expression is: \(40 + 20 - 28 \div 4 \times 2\)
BODMAS/PEMDAS stands for Brackets/Parentheses, Orders/Exponents, Division, Multiplication, Addition, Subtraction. Operations are performed from left to right for Division/Multiplication and for Addition/Subtraction.
The expression is: \(40 + 20 - 28 \div 4 \times 2\)
D) 46 — After correctly applying the symbol transformations and following the order of operations (BODMAS/PEMDAS), the expression evaluates to 46.