A, B, C, D, E and F live on six different floors of the same building. The lowermost floor in the building is numbered 1, the floor above it is numbered 2, and so on till the topmost floor is numbered 6. B lives immediately below A and immediately above C. Only one person lives above E. Neither B nor C lives on the lowermost floor. A does not live on the topmost floor. Only two people live between A and D. Who lives immediately below C?
The problem asks us to identify the correct congruence statement between two triangles, \(\Delta ABC\) and \(\Delta DEF\), given that their corresponding sides are equal. We need to apply the SSS (Side-Side-Side) congruence theorem, which states that if three sides of one triangle are equal to three corresponding sides of another triangle, then the triangles are congruent.
Identify the given information about the sides of the two triangles:
Recall the SSS congruence theorem: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. When writing the congruence statement, the order of the vertices must match the corresponding equal sides.
Match the vertices based on the given side equalities:
Let's find a consistent mapping for the vertices:
There seems to be a mismatch in the direct mapping. Let's re-examine the given equalities carefully to establish the correct correspondence. The order of vertices in the congruence statement \(\Delta ABC \cong \Delta XYZ\) means that \(A \leftrightarrow X\), \(B \leftrightarrow Y\), and \(C \leftrightarrow Z\).
Let's try to establish the correspondence by picking one vertex from \(\Delta ABC\) and finding its corresponding vertex in \(\Delta DEF\). Consider vertex \(A\). The sides originating from \(A\) are \(AB\) and \(AC\).
The common vertex in \(DE\) and \(EF\) is \(E\). This suggests that \(A\) corresponds to \(E\).
Now consider vertex \(B\). The sides originating from \(B\) are \(AB\) and \(BC\).
The common vertex in \(DE\) and \(DF\) is \(D\). This suggests that \(B\) corresponds to \(D\).
Finally, consider vertex \(C\). The sides originating from \(C\) are \(AC\) and \(BC\).
The common vertex in \(EF\) and \(DF\) is \(F\). This suggests that \(C\) corresponds to \(F\).
So, the correspondence is \(A \leftrightarrow E\), \(B \leftrightarrow D\), \(C \leftrightarrow F\).
Therefore, \(\Delta ABC \cong \Delta EDF\).
A) \(\Delta ABC \cong \Delta EDF\)
This option correctly matches the vertices based on the given side equalities:
Since all three pairs of corresponding sides are equal, the triangles are congruent by SSS criterion, and the order of vertices is correctly matched.
B) \(\Delta ABC \cong \Delta FDE\)
Let's check the side correspondence:
C) \(\Delta ABC \cong \Delta DEF\)
Let's check the side correspondence:
Only one pair of sides matches, making this statement incorrect.
D) \(\Delta ABC \cong \Delta EFD\)
Let's check the side correspondence:
Only one pair of sides matches, making this statement incorrect.
\( \frac{5}{12} \) is the same as:
0.41\(\overline{6}\)
0.41\(\overline{7}\)
0.\(\overline{416}\)
0.4\(\overline{16}\)
To convert a fraction to a decimal, we perform division. For a repeating decimal, we identify the repeating block of digits. The notation \(\overline{X}\) indicates that the digit or block of digits \(X\) repeats infinitely.
Correct Option: A) \(0.41\overline{6}\) is the correct decimal representation of \(\frac{5}{12}\).
The question asks about the mechanism plants use for heat regulation. We need to identify the process that helps plants cool down, similar to sweating in animals.
Correct Option: C) Transpiration — This is the process where water vapor is released from plant leaves, and this evaporation has a cooling effect, helping the plant regulate its temperature. It is analogous to sweating in animals.
To solve this problem, we need to first write down the given word. Then, we apply the interchange rule to pairs of letters. Finally, we identify the ninth letter from the left in the new arrangement.
Correct Option: D) The ninth letter from the left in the rearranged word is 'D'.
The question asks to identify which of the given Himalayan peaks is located in India. This requires knowledge of the geographical locations of major Himalayan peaks.
Correct Option: C) Kamet is a prominent peak in the Garhwal Himalayas, situated in the Chamoli district of Uttarakhand, India.
To determine the number of completely filled shells for an element, we need to understand its electronic configuration based on its position in the periodic table. The period number indicates the total number of shells, and the group number (for p-block elements) helps determine the valence electrons, which in turn helps deduce the total number of electrons and thus the electronic configuration.
A) 3 — As determined from the electronic configuration, the first three shells (n=1, n=2, and n=3) are completely filled for an element in Group 17, Period 4.
This problem involves setting up and solving a system of linear equations based on the given ratios of ages at different points in time. We will use variables to represent the present ages and then form equations using the information about their ages two years ago and three years hence.
Let the present age of Subash be \(S\) years and the present age of Pranav be \(P\) years.
Two years ago:
Subash's age was \(S - 2\).
Pranav's age was \(P - 2\).
The ratio of their ages was 4 : 5. So, we can write the equation:
Cross-multiply to simplify the first equation:
\[ 5(S - 2) = 4(P - 2) \]\[ 5S - 10 = 4P - 8 \]\[ 5S - 4P = -8 + 10 \]\[ 5S - 4P = 2 \quad \text{(Equation 1)} \]Three years hence (from now):
Subash's age will be \(S + 3\).
Pranav's age will be \(P + 3\).
The ratio of their ages will be 5 : 6. So, we can write the equation:
Cross-multiply to simplify the second equation:
\[ 6(S + 3) = 5(P + 3) \]\[ 6S + 18 = 5P + 15 \]\[ 6S - 5P = 15 - 18 \]\[ 6S - 5P = -3 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
Equation 1: \(5S - 4P = 2\)
Equation 2: \(6S - 5P = -3\)
We need to find Pranav's present age, \(P\). We can eliminate \(S\).
Multiply Equation 1 by 6 and Equation 2 by 5:
\(6 \times (5S - 4P) = 6 \times 2 \implies 30S - 24P = 12 \quad \text{(Equation 3)}\)
\(5 \times (6S - 5P) = 5 \times (-3) \implies 30S - 25P = -15 \quad \text{(Equation 4)}\)
Subtract Equation 4 from Equation 3:
\[ (30S - 24P) - (30S - 25P) = 12 - (-15) \]\[ 30S - 24P - 30S + 25P = 12 + 15 \]\[ P = 27 \]Thus, Pranav's present age is 27 years.
Correct Option: D) 27 years
The problem requires us to first arrange the letters of the given word "FATIGUE" in alphabetical order. Then, we need to identify two specific letters: the fourth letter from the left and the second letter from the right in the rearranged word. Finally, we must count the number of letters in the English alphabet that fall between these two identified letters.
B) Twelve. As calculated, there are 12 letters between G and T in the English alphabetical series.
The question asks for the year in which the Asiatic Society was founded by Sir William Jones in Calcutta. This is a factual question related to Indian history, specifically the colonial period and the establishment of academic institutions.
Correct Option: B) 1784