To find the sum of all interior angles of a polygon, we can use a standard formula that relates the number of sides of the polygon to the sum of its interior angles. A heptagon is a polygon with seven sides.
Therefore, the sum of all the interior angles of a heptagon is 900°.
Correct Option: B) 900°
This problem involves arranging six friends (P, Q, R, S, T, U) based on their heights using the given comparative statements. We will deduce their relative heights step-by-step to find the second tallest person.
"R is taller than only two other friends."
This implies that R is the 3rd tallest among the six friends. So, there are two friends taller than R and three friends shorter than R.
Arrangement so far: _ > _ > R > _ > _ > _
"T is shorter than only one friend."
This means T is the 2nd tallest among the six friends.
Arrangement so far: _ > T > R > _ > _ > _
"Q is shorter than R but taller than P."
From this, we know R > Q > P. Since R is the 3rd tallest, Q and P must be among the shorter friends.
Arrangement so far: _ > T > R > Q > P > _
"S is shorter than U."
This means U > S. We have two remaining friends, U and S, and one remaining position (the tallest) and one remaining position (the shortest).
From step 2, we know T is the 2nd tallest. This means the tallest person is still unknown. From step 3, Q and P are shorter than R. The only remaining position is the shortest. So, the order is: Tallest > T > R > Q > P > Shortest.
Since U > S, and U and S are the only friends left to place, U must be the tallest and S must be the shortest.
Final arrangement: U > T > R > Q > P > S
Identify the second tallest.
Based on the final arrangement, T is the second tallest.
A) T is the second tallest among all the friends, as derived from the step-by-step arrangement.
In this question, a group of numbers/symbols is coded using letters as per the table given below and the conditions which follow. The correct combination of codes following the conditions is your answer.
| Number/symbol | * | 2 | @ | 5 | ^ | + | 7 | # | % | 9 |
| Letter code | J | P | A | F | D | N | R | K | M | Z |
Conditions:
(i) If the first element is a symbol and the last, a number, the codes for these two (the first and the last elements) are to be interchanged.
(ii) If the first element is an odd number and the last, an even number, the first and last elements are to be coded as S.
Which of the following will be the code for 75#2?
The problem requires us to decode a given sequence of numbers/symbols using a provided table and then apply specific conditions to modify the coded sequence. We need to identify which condition applies to the given input "75#2" and then generate the final code.
B) SFKS is the correct code because the first element (7, an odd number) and the last element (2, an even number) satisfy Condition (ii), leading to both being coded as 'S'. The middle elements 5 and # are coded as F and K respectively from the table.
This problem requires us to substitute the given new meanings for the arithmetic operators into the expression and then evaluate the expression following the BODMAS/PEMDAS rule.
The original expression is: \(76 + 5 – 6 × 3 ÷ 4 = ?\)
Applying the substitutions, the new expression becomes:
\[ 76 - 5 \times 6 \div 3 + 4 = ? \]BODMAS/PEMDAS stands for Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
First, perform Multiplication and Division from left to right:
Perform multiplication:
\[ 76 - (5 \times 6) \div 3 + 4 \]\[ 76 - 30 \div 3 + 4 \]Perform division:
\[ 76 - (30 \div 3) + 4 \]\[ 76 - 10 + 4 \]Next, perform Addition and Subtraction from left to right:
Perform subtraction:
\[ (76 - 10) + 4 \]\[ 66 + 4 \]Perform addition:
\[ 66 + 4 = 70 \]D) 70 is the correct answer after applying the operator substitutions and following the order of operations.
This problem involves calculating the weight of a new person when an old person is replaced, leading to a change in the average weight of a group. We can solve this by understanding how the total weight of the group changes due to the replacement.
Let the initial number of persons be \(N = 6\).
Let the initial average weight be \(A_{\text{initial}}\) kg.
The total initial weight of the 6 persons is \(T_{\text{initial}} = N \times A_{\text{initial}} = 6 A_{\text{initial}}\) kg.
One person weighing 72 kg is replaced by a new man. So, the total number of persons remains 6.
The average weight increases by 2 kg. So, the new average weight is \(A_{\text{new}} = A_{\text{initial}} + 2\) kg.
The new total weight of the 6 persons is \(T_{\text{new}} = N \times A_{\text{new}} = 6 (A_{\text{initial}} + 2)\) kg.
The change in total weight is due to the replacement. The new total weight can also be expressed as:
\[ T_{\text{new}} = T_{\text{initial}} - (\text{weight of old man}) + (\text{weight of new man}) \]Substitute the known values and expressions:
\[ 6 (A_{\text{initial}} + 2) = 6 A_{\text{initial}} - 72 + (\text{weight of new man}) \]Expand the left side:
\[ 6 A_{\text{initial}} + 12 = 6 A_{\text{initial}} - 72 + (\text{weight of new man}) \]Subtract \(6 A_{\text{initial}}\) from both sides:
\[ 12 = -72 + (\text{weight of new man}) \]Solve for the weight of the new man:
\[ \text{Weight of new man} = 12 + 72 \]\[ \text{Weight of new man} = 84 \text{ kg} \]B) 84 kg
We are given two conditions relating the ages of A and B: A's age is seven times B's age, and the sum of their ages is 48 years. We can set up a system of two linear equations with two variables (A's age and B's age) and solve for them. Alternatively, we can test the given options against the conditions.
Let A's age be \(A\) years.
Let B's age be \(B\) years.
Condition 1: A's age is seven times B's age.
\[ A = 7B \quad \text{(Equation 1)} \]Condition 2: The sum of their ages is 48 years.
\[ A + B = 48 \quad \text{(Equation 2)} \]Substitute Equation 1 into Equation 2:
\[ (7B) + B = 48 \] \[ 8B = 48 \]Divide by 8 to find B's age:
\[ B = \frac{48}{8} \] \[ B = 6 \text{ years} \]Now, substitute the value of \(B\) back into Equation 1 to find A's age:
\[ A = 7B \] \[ A = 7 \times 6 \] \[ A = 42 \text{ years} \]A's age = 42, B's age = 6.
Is A's age seven times B's age? \(42 = 7 \times 6\) (Yes, \(42 = 42\)).
Is the sum of their ages 48? \(42 + 6 = 48\) (Yes, \(48 = 48\)).
Both conditions are satisfied.
Correct Option: D) 42 and 6
To find the value of \(\alpha^2 + \beta^2\) for a quadratic equation, we first identify the coefficients of the quadratic equation. Then, we use Vieta's formulas to find the sum of the roots (\(\alpha + \beta\)) and the product of the roots (\(\alpha \beta\)). Finally, we use the algebraic identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\) to calculate the required value.
B) 85 is the correct value for \(\alpha^2 + \beta^2\).
The question asks to identify the first Indian woman sailor to qualify for the Olympics in April 2021. This is a factual question related to sports achievements and current affairs, requiring knowledge of specific historical milestones in Indian sports.
Correct Option: C) Nethra Kumanan became the first Indian woman sailor to qualify for the Olympics in April 2021. She achieved this feat at the Mussanah Open Championship in Oman, which served as an Asian Olympic qualifying event.
This problem involves analyzing a given inequality statement and determining which of the provided conclusions logically follows from it. We need to break down the statement into smaller parts and check each conclusion against these derived relationships.
The given statement is: \(I \ge R > U \le T > Q \ge O\).
Let's analyze each conclusion:
Only Conclusion 1 is true.
D) 1
The question asks why carbon monoxide (CO) is considered an air pollutant. This requires understanding the physiological effects of CO on the human body, particularly its interaction with blood components.
C) Because it reduces oxygen carrying capacity of blood. This is the primary and most critical reason why carbon monoxide is a dangerous air pollutant. It binds to hemoglobin, forming carboxyhemoglobin, which prevents oxygen from being transported to the body's tissues.