To arrange words in dictionary order, we compare them letter by letter from left to right. The word that has a letter appearing earlier in the alphabet at the first point of difference comes first.
Correct Step-by-step Comparison:
Correct Option: C) Possess, Process, Propose, Purpose
To find the area of a triangle when all three side lengths are given, we use Heron's formula. Heron's formula requires calculating the semi-perimeter first, and then applying the formula to find the area.
Identify the side lengths:
Let the sides of the triangle be \(a = 5 \text{ cm}\), \(b = 7 \text{ cm}\), and \(c = 11 \text{ cm}\).
Calculate the semi-perimeter (\(s\)):
The semi-perimeter is half the sum of the lengths of the sides of the triangle.
\[ s = \frac{a + b + c}{2} \]\[ s = \frac{5 + 7 + 11}{2} \]\[ s = \frac{23}{2} \]\[ s = 11.5 \text{ cm} \]Apply Heron's Formula:
Heron's formula for the area (\(A\)) of a triangle is:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Substitute the values into Heron's Formula:
\[ A = \sqrt{11.5(11.5 - 5)(11.5 - 7)(11.5 - 11)} \]\[ A = \sqrt{11.5(6.5)(4.5)(0.5)} \]Calculate the product inside the square root:
\[ A = \sqrt{11.5 \times 6.5 \times 4.5 \times 0.5} \]\[ A = \sqrt{335.4375} \]Calculate the square root:
\[ A \approx 18.31495 \text{ cm}^2 \]Recheck the calculation. Let's re-calculate the product: \(11.5 \times 6.5 \times 4.5 \times 0.5 = 11.5 \times 6.5 \times 2.25 = 74.75 \times 2.25 = 168.1875\).
So, the correct product is \(168.1875\).
\[ A = \sqrt{168.1875} \]\[ A \approx 12.96879 \text{ cm}^2 \]Round to two decimal places:
Rounding \(12.96879\) to two decimal places gives \(12.97 \text{ cm}^2\).
B) \(12.97 \text{ cm}^2\) is the calculated area of the triangle using Heron's formula, rounded to two decimal places.
The given series is an alternating sequence of squares and cubes of consecutive natural numbers. We need to identify this pattern and apply it to find the next term.
Correct Option: C) 36 — The series follows the pattern of squares and cubes of consecutive natural numbers: \(2^2=4\), \(3^3=27\), \(4^2=16\), \(5^3=125\). The next term should be \(6^2\), which is 36.
To find the number of sides of a regular polygon given its interior angle, we can use the formula for the interior angle of a regular polygon. Alternatively, we can first find the exterior angle, which is simpler, and then use its relationship with the number of sides.
Method 1: Using the interior angle formula
The formula for each interior angle of a regular polygon with \(n\) sides is given by:
\[ \text{Interior Angle} = \frac{(n-2) \times 180^{\circ}}{n} \]Given that each interior angle is \(165^{\circ}\), we can set up the equation:
\[ 165^{\circ} = \frac{(n-2) \times 180^{\circ}}{n} \]Multiply both sides by \(n\):
\[ 165n = (n-2) \times 180 \] \[ 165n = 180n - 360 \]Rearrange the terms to solve for \(n\):
\[ 360 = 180n - 165n \] \[ 360 = 15n \] \[ n = \frac{360}{15} \] \[ n = 24 \]Method 2: Using the exterior angle
The sum of an interior angle and its corresponding exterior angle of any polygon is \(180^{\circ}\). So, if the interior angle is \(165^{\circ}\), the exterior angle is:
\[ \text{Exterior Angle} = 180^{\circ} - \text{Interior Angle} \] \[ \text{Exterior Angle} = 180^{\circ} - 165^{\circ} \] \[ \text{Exterior Angle} = 15^{\circ} \]For any regular polygon, the sum of all exterior angles is \(360^{\circ}\). If there are \(n\) sides, and thus \(n\) exterior angles (all equal in a regular polygon), then:
\[ n \times \text{Exterior Angle} = 360^{\circ} \] \[ n \times 15^{\circ} = 360^{\circ} \] \[ n = \frac{360^{\circ}}{15^{\circ}} \] \[ n = 24 \]Both methods yield the same result.
A) 24 is the correct number of sides for a regular polygon with each interior angle measuring 165°.
To solve this letter series, we need to identify the pattern in which the letters are arranged. We will analyze the position of each letter in the English alphabet for both the first and second letters of each pair.
The pattern for the first letter is a decrease of 2 positions in the alphabet (13 - 2 = 11, 11 - 2 = 9, 9 - 2 = 7). Following this pattern, the next first letter should be \(7 - 2 = 5\), which corresponds to the letter E.
The pattern for the second letter is also a decrease of 2 positions in the alphabet (21 - 2 = 19, 19 - 2 = 17, 17 - 2 = 15). Following this pattern, the next second letter should be \(15 - 2 = 13\), which corresponds to the letter M.
The next pair in the series will be EM.
Correct Option: D) EM
The question asks to identify the incorrectly matched pair among the given options, where each pair consists of an Indian city and a river it is situated on. This requires knowledge of the geographical locations of major Indian cities relative to their rivers.
Correct Option: B) Jabalpur – Jhelum river
The problem can be solved by setting up an algebraic equation based on the given information. We are told that when 106 is subtracted from a number, the result is 47% of that number. We can use this to find the number and then calculate 11.5% of it.
Let the unknown number be \(x\).
According to the problem statement, when 106 is subtracted from the number, it reduces to its 47%. This can be written as an equation:
\[ x - 106 = 47\% \text{ of } x \]Convert the percentage to a decimal or fraction:
\[ 47\% = \frac{47}{100} \]Substitute this into the equation:
\[ x - 106 = \frac{47}{100}x \]To solve for \(x\), first, bring all terms involving \(x\) to one side of the equation:
\[ x - \frac{47}{100}x = 106 \]Combine the \(x\) terms:
\[ \left(1 - \frac{47}{100}\right)x = 106 \]\[ \left(\frac{100 - 47}{100}\right)x = 106 \]\[ \frac{53}{100}x = 106 \]Now, isolate \(x\) by multiplying both sides by \(\frac{100}{53}\):
\[ x = 106 \times \frac{100}{53} \]\[ x = 2 \times 100 \]\[ x = 200 \]The number is 200. The question asks for 11.5% of this number.
Calculate 11.5% of 200:
\[ 11.5\% \text{ of } 200 = \frac{11.5}{100} \times 200 \]\[ = 11.5 \times 2 \]\[ = 23 \]C) 23 — The number is 200, and 11.5% of 200 is 23.
The problem involves a square park surrounded by a path of uniform width. We are given the area of the path and the width of the path. We need to find the perimeter of the park. We can solve this by setting up an equation for the area of the path in terms of the park's side length, then solving for the side length, and finally calculating the perimeter.
Let the side length of the square park be \(x\) meters.
The area of the square park is \(A_{\text{park}} = x^2\) m\(^2\).
The path has a uniform width of 1.5 m all around the park. This means the outer square (park + path) will have a side length of \(x + 1.5 + 1.5 = x + 3\) meters.
The area of the square including the path is \(A_{\text{total}} = (x+3)^2\) m\(^2\).
The area of the path is the difference between the total area and the park's area:
\[ A_{\text{path}} = A_{\text{total}} - A_{\text{park}} \]\[ 225 = (x+3)^2 - x^2 \]Expand the equation:
\[ 225 = (x^2 + 6x + 9) - x^2 \]\[ 225 = 6x + 9 \]Solve for \(x\):
\[ 6x = 225 - 9 \]\[ 6x = 216 \]\[ x = \frac{216}{6} \]\[ x = 36 \text{ m} \]The side length of the park is 36 m.
The perimeter of the square park is \(P = 4 \times \text{side length}\):
\[ P = 4 \times 36 \]\[ P = 144 \text{ m} \]C) 144 m is the correct perimeter of the park.
The problem asks us to find the principal sum given the difference between compound interest (CI) and simple interest (SI) for 2 years at a specific annual rate. We can use the direct formula for the difference between CI and SI for 2 years or calculate CI and SI separately and then find the principal.
The formula for the difference between CI and SI for 2 years is given by:
\[ \text{CI} - \text{SI} = P \left( \frac{R}{100} \right)^2 \]Let \(P\) be the principal sum. We have:
\[ 16.32 = P \left( \frac{5}{100} \right)^2 \]To find \(P\), multiply both sides by 400:
\[ P = 16.32 \times 400 \]\[ P = 6528 \]The sum (principal) is ₹6,528.
B) 6,528 — The calculation shows that the principal sum is ₹6,528, which matches this option.
The question asks for the host city of the 2024 Olympic Games. This is a factual question related to current events and major international sporting events. Knowledge of the Olympic Games' host cities is essential for competitive exams.
Correct Option: A) Paris
The 2024 Summer Olympic Games are scheduled to be held in Paris, France. This will mark the third time Paris has hosted the Summer Olympics, having previously done so in 1900 and 1924.