This is a syllogism problem. We need to evaluate the given conclusions based on the provided statements. We can use Venn diagrams or logical deduction to determine which conclusions logically follow.
Analyze Statement 1: All biscuits are millets.
This means that the set of 'biscuits' is entirely contained within the set of 'millets'. If we draw Venn diagrams, the circle for 'biscuits' would be inside the circle for 'millets'.
Analyze Statement 2: All millets are grains.
This means that the set of 'millets' is entirely contained within the set of 'grains'. In Venn diagrams, the circle for 'millets' would be inside the circle for 'grains'.
Combine the statements:
From step 1, we have: Biscuits \(\subset\) Millets.
From step 2, we have: Millets \(\subset\) Grains.
Combining these, we get: Biscuits \(\subset\) Millets \(\subset\) Grains. This implies that the set of 'biscuits' is also entirely contained within the set of 'grains'.
Evaluate Conclusion 1: All biscuits are grains.
Based on our combined understanding from step 3 (Biscuits \(\subset\) Grains), this conclusion logically follows. If all biscuits are millets, and all millets are grains, then all biscuits must necessarily be grains.
Evaluate Conclusion 2: All grains are millets.
From Statement 2, we know that all millets are grains (Millets \(\subset\) Grains). However, this does not mean that all grains are millets. There could be grains that are not millets (e.g., rice, wheat, which are grains but not necessarily millets). The statement only establishes a one-way relationship. Therefore, this conclusion does not logically follow.
Correct Option: A) Only conclusion 1 follows
This question tests general knowledge about famous autobiographies of Indian cricketers. To answer correctly, one needs to recall the title of Sachin Tendulkar's autobiography.
Correct Option: C) Sachin Tendulkar
The question asks to identify the river on which the Almatti Dam is constructed. This is a factual question related to Indian geography and major river projects. Knowledge of important dams and their associated rivers is crucial for competitive exams.
Correct Option: D) Krishna
\( \frac{2}{3} \), \( \frac{4}{6} \), \( \frac{6}{9} \) are:
Improper fractions
Equivalent fractions
Irreducible fractions
Like fractions
To determine the relationship between the given fractions, we need to simplify each fraction to its lowest terms and compare them. We will then evaluate each option based on the definitions of different types of fractions.
The first fraction is \( \frac{2}{3} \). The greatest common divisor (GCD) of 2 and 3 is 1. So, this fraction is already in its simplest form.
The second fraction is \( \frac{4}{6} \). We can divide both the numerator and the denominator by their GCD, which is 2.
\[ \frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \]The third fraction is \( \frac{6}{9} \). We can divide both the numerator and the denominator by their GCD, which is 3.
\[ \frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \]After simplification, all three fractions are equal to \( \frac{2}{3} \).
Correct Option: B) Equivalent fractions
All the given fractions \( \frac{2}{3} \), \( \frac{4}{6} \), and \( \frac{6}{9} \) simplify to the same value, \( \frac{2}{3} \). Therefore, they are equivalent fractions.
The question asks to identify the founder of the religion Din-i-Ilahi. This requires knowledge of prominent Mughal emperors and their religious policies or initiatives.
Correct Option: A) Akbar is widely known for his policy of religious tolerance and his attempt to create a new syncretic faith called Din-i-Ilahi (meaning "Religion of God"). He introduced this in 1582, drawing elements from various religions like Islam, Hinduism, Zoroastrianism, and Christianity, aiming to establish a common spiritual path for his diverse subjects.
To find the area of a parallelogram, we use the formula: Area = base × height. We are given the base and the height of the parallelogram, so we can directly substitute these values into the formula to calculate the area.
Therefore, the area of the metal piece is 108 sq.cm.
Correct Option: B) 108 sq.cm
To solve this problem, we need to follow the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
Thus, the value of the given expression is \(-20\).
C) -20 is the correct answer, as derived from following the order of operations.
To find the year that will have the same calendar as a given year, we need to calculate the number of odd days (remainder when the total number of days is divided by 7) accumulated from the given year until the cumulative odd days become a multiple of 7 (i.e., 0 odd days). A normal year has 1 odd day (365 days \(= 52 \times 7 + 1\)), and a leap year has 2 odd days (366 days \(= 52 \times 7 + 2\)). The calendar repeats when the total number of odd days accumulated from the given year becomes 0 (or a multiple of 7).
Correct Option: C) 2011
To find the average speed for a journey where the distance traveled in each direction is the same but speeds are different, we can use the formula for average speed, which is total distance divided by total time. Alternatively, for two equal distances covered at different speeds, a specific formula for average speed can be used.
Define variables:
Let the distance between point A and point B be \(D\) km.
Speed from A to B (\(v_1\)) = 40 km/hr.
Speed from B to A (\(v_2\)) = 60 km/hr.
Calculate time taken for each leg of the journey:
Time taken to travel from A to B (\(t_1\)) is given by:
Time taken to travel from B to A (\(t_2\)) is given by:
\[ t_2 = \frac{\text{Distance}}{\text{Speed}_2} = \frac{D}{60} \text{ hours} \]Calculate total distance and total time:
Total distance traveled = Distance from A to B + Distance from B to A = \(D + D = 2D\) km.
Total time taken = \[t_1 + t_2 = \frac{D}{40} + \frac{D}{60}\] hours.
To add the fractions, find a common denominator, which is 120:
\[ \text{Total Time} = \frac{3D}{120} + \frac{2D}{120} = \frac{5D}{120} = \frac{D}{24} \text{ hours} \]Calculate average speed:
Average speed is defined as total distance divided by total time:
Alternative Method (Harmonic Mean):
When the distance traveled is the same in both directions (or for two equal segments), the average speed can be calculated using the formula:
Substitute the given values:
\[ \text{Average Speed} = \frac{2 \times 40 \times 60}{40 + 60} = \frac{2 \times 2400}{100} = \frac{4800}{100} = 48 \text{ km/hr} \]Correct Option: C) 48 km/hr
To determine if a pair of numbers are co-primes, we need to find their greatest common divisor (GCD). Two numbers are co-primes (or relatively prime) if their GCD is 1. We will check each option by finding the GCD of the given pairs.
A) 59, 97 — As determined in Step 2, the GCD of 59 and 97 is 1, making them co-primes.