To find the measure of the base angles in an isosceles triangle, we use two key properties: first, that the sum of angles in any triangle is 180°, and second, that the base angles of an isosceles triangle are equal.
A) 50° and 50° is correct because the sum of the angles is \(80^\circ + 50^\circ + 50^\circ = 180^\circ\), and the base angles are equal, satisfying the properties of an isosceles triangle.
The problem describes the relative positions of several persons standing in a straight row facing North. We need to deduce the complete arrangement and then count the total number of persons. We will build the arrangement step-by-step using the given clues.
Clue 1: "X is standing to the immediate left of W but immediate right of R."
This implies the sequence: R X W
Clue 2: "Z is standing to the immediate left of R."
Combining with the previous sequence: Z R X W
Clue 3: "There is only one person to the left of P."
This means P is the second person from the left end of the row. So, the arrangement starts with: _ P _ _ _ ...
Clue 4: "The person standing at the extreme right end is the only person to the right of Y."
This means Y is the second person from the right end of the row. So, the arrangement ends with: ... _ Y _
Clue 5: "Q is standing to the immediate left of Y."
Combining with the previous point, the end of the row is: ... Q Y _
Clue 6: "Only four persons are standing between P and Q."
Let's combine the known segments. We have Z R X W as a block. We also know P is the second from the left, and Q is the third from the right (since Y is second from right, and Q is immediate left of Y).
Let's place P. Since there is only one person to the left of P, the arrangement starts as: _ P
Now, let's try to fit Z R X W. If Z R X W is placed after P, then P must be the person to the left of Z. But P is the second person. So, the person to the left of P must be Z. This means Z is the first person.
So, the arrangement starts as: Z P R X W (This contradicts Z R X W, as P is between Z and R. Let's re-evaluate.)
Let's reconsider the position of P. "Only one person to the left of P" means P is the second person from the left. So, the arrangement starts like: \(\text{Person}_1\) P ...
Now consider the block Z R X W. If Z is the first person, then R is the second, X is the third, and W is the fourth. This means P cannot be the second person if Z R X W is the start of the row. So, P must be somewhere else.
Let's place Z R X W first. This is a fixed block. Z R X W.
Now, "There is only one person to the left of P". This means P is at position 2. So, the first person is to the left of P. Let's call the first person F.
F P _ _ _ _ _ _ _
We have Z R X W. If Z is the first person, then Z = F. So, the sequence starts Z P. But Z R X W means R is to the immediate right of Z. This is a contradiction. So, P is not immediately to the right of Z.
Let's restart the arrangement building carefully:
From (1) and (2): Z R X W (This is a block of 4 people)
From (3): There is only one person to the left of P. This means P is the 2nd person from the left end. Let the first person be \(\text{Person}_1\).
\(\text{Person}_1\) P _ _ _ _ _ _ _
From (4) and (5): Q is to the immediate left of Y, and Y is the second person from the right end. Let the last person be \(\text{Person}_N\).
_ _ _ _ _ _ Q Y \(\text{Person}_N\)
Now, let's combine these. We have the block Z R X W. Where can it fit?
Consider the statement "Only four persons are standing between P and Q."
Let's denote positions from left to right as 1, 2, 3, ...
Position of P = 2 (since one person is to its left).
Position of Q = Total persons - 2 (since Y is at Total persons - 1, and Q is to its left).
Let's try to build the sequence from left to right, placing P first:
\(\text{Person}_1\) P
Now, we have the block Z R X W. We know Z is to the left of R, R is to the left of X, X is to the left of W. This block must be placed somewhere. The only way for P to be the second person and for Z R X W to exist is if Z R X W comes after P, or P is part of Z R X W. P is not Z, R, X, or W based on the given relations.
So, the block Z R X W must be placed as a unit. Let's try to place it relative to P.
If Z R X W is to the right of P, then P is followed by Z R X W.
\(\text{Person}_1\) P Z R X W
Now, let's place Q and Y. We know there are 4 people between P and Q.
If the sequence is \(\text{Person}_1\) P Z R X W, then the people between P and Q would be Z, R, X, W. This means Q must be immediately after W. So, the sequence becomes:
\(\text{Person}_1\) P Z R X W Q
Now, we add Y and the last person. Q is to the immediate left of Y, and Y is second from the right end.
So, the sequence is: \(\text{Person}_1\) P Z R X W Q Y \(\text{Person}_N\)
Let's verify all conditions with this sequence:
All conditions are satisfied.
Now, count the total number of persons:
\(\text{Person}_1\) (1) + P (1) + Z (1) + R (1) + X (1) + W (1) + Q (1) + Y (1) + \(\text{Person}_N\) (1) = 9 persons.
D) Nine
The question asks to identify the Indian cueist who won his eighth title at the 19th Asian 100 UP Billiards Championship in 2022. This is a factual question related to sports achievements.
Correct Option: C) Pankaj Advani won his eighth title at the 19th Asian 100 UP Billiards Championship 2022. He defeated Dhruv Sitwala in the final match held in Doha, Qatar.
The question asks to identify the state where the Palitana temples, a significant Jain pilgrimage site, are located. This requires knowledge of important religious sites in India and their geographical locations.
Correct Option: D) The Palitana temples are a major Jain pilgrimage site located on Shatrunjaya hill in the Bhavnagar district of Gujarat, India. It is considered one of the holiest places for Jains.
This question asks about the fundamental function of an electric generator. To answer this, recall the basic principle of how an electric generator operates and what forms of energy it deals with.
Correct Option: A) mechanical energy into electrical energy
To evaluate \(999^3\), we can use the algebraic identity for the cube of a binomial, specifically \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\). This method simplifies the calculation by avoiding direct multiplication of large numbers.
So, \(999^3 = (1000 - 1)^3\).
Correct Option: A) 997002999
A cylindrical rod has an outer curved surface area of 2800 cm². If the length of the rod is 100 cm, then the outer radius (in cm) of the rod, correct to two places of decimal, is:
Take π = \( \frac{22}{7} \)
The problem asks us to find the outer radius of a cylindrical rod given its outer curved surface area and length. We will use the formula for the curved surface area of a cylinder and substitute the given values to solve for the radius.
The curved surface area (also known as lateral surface area) of a cylinder is given by:
\[ A = 2 \pi r h \]where \(r\) is the radius and \(h\) is the height (or length) of the cylinder.
Multiply both sides by 7 and divide by 4400:
\[ r = \frac{2800 \times 7}{4400} \]Rounding to two decimal places, we get:
\[ r \approx 4.45 \text{ cm} \]D) 4.45 cm is the calculated outer radius of the rod, correct to two decimal places.
Which of the following terms will replace the question mark (?) in the given series to make it logically complete?
BKS36, GPX48, LUC60, QZH72,?
To solve this series, we need to identify the pattern in the letters and the numbers separately. We will analyze the progression of the first letter, second letter, third letter, and the numerical part of each term.
Analyze the first letter:
Analyze the second letter:
Analyze the third letter:
Analyze the numerical part:
Combine the results:
The next term in the series will have the first letter V, the second letter E, the third letter M, and the number 84. So, the term is VEM84.
Correct Option: D) VEM84
Which of the following is an INCORRECT relation between V, I and R?
V=\(\frac{R}{I}\)
I=\(\frac{V}{R}\)
R=\(\frac{V}{I}\)
V=IR
The question asks to identify the incorrect relationship among voltage (V), current (I), and resistance (R). This requires knowledge of Ohm's Law and its various forms.
A) V=\(\frac{R}{I}\)
This relation is incorrect. According to Ohm's Law, \(\text{V} = \text{IR}\) or \(\text{V} = \frac{\text{R}}{\text{I}}\) is not a valid form. The correct form for V is \(\text{V} = \text{IR}\).
The problem asks us to identify how many even numbers in the given series are immediately preceded by an odd number and immediately followed by an odd number. We need to scan the series and look for the pattern: Odd Number - Even Number - Odd Number.
A) 4 is the correct answer because we identified 4 instances where an even number is immediately preceded by an odd number and immediately followed by an odd number in the given series.