If ΔABC and ΔPQR are similar and \( \frac{BC}{QR} \) = \( \frac{1}{3} \), find \(\frac{\text{ar}(\triangle PQR)}{\text{ar}(\triangle BCA)}\)
This problem involves the properties of similar triangles. When two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. We are given the ratio of corresponding sides and need to find the ratio of their areas.
Correct Option: A) 9 is the correct answer because the ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Since \[\frac{BC}{QR} = \frac{1}{3}\], then \[\frac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle PQR)} = \left(\frac{1}{3}\right)^2 = \frac{1}{9}\]. Therefore, \[\frac{\text{ar}(\triangle PQR)}{\text{ar}(\triangle BCA)} = 9\].
The question asks to identify which city is NOT part of the 'Golden Triangle' tourist circuit in India. This requires knowledge of the cities that constitute this well-known tourist route.
Correct Option: A) Lucknow is the capital city of Uttar Pradesh and is known for its historical and cultural heritage, but it is not a part of the Golden Triangle tourist circuit. The Golden Triangle specifically refers to the circuit connecting Delhi, Agra, and Jaipur.
The problem involves a coding-decoding pattern where letters are assigned numerical values. We need to identify the rule used to convert letters into numbers and then apply that rule to find the value of "BASIC".
D) 85
This question tests general knowledge about famous autobiographies by Indian sports personalities. The task is to identify the author of the book "The Test of My Life".
Correct Option: C) Yuvraj Singh is the author of the autobiography "The Test of My Life". The book chronicles his journey through cricket, his diagnosis and battle with cancer, and his eventual return to the sport.
The length of each side of an equilateral triangle is 2\(\sqrt[4]{3}\) cm. What is the area of this triangle?
To find the area of an equilateral triangle, we use the standard formula for the area of an equilateral triangle, which is given by \( \frac{\sqrt{3}}{4} \times \text{side}^2 \). We are given the length of each side, so we will substitute this value into the formula and simplify.
Identify the given information:
The length of each side of the equilateral triangle, let's call it \(a\), is \(2\sqrt[4]{3}\) cm.
Recall the formula for the area of an equilateral triangle:
The area \(A\) of an equilateral triangle with side length \(a\) is given by:
Substitute the given side length into the formula:
Substitute \(a = 2\sqrt[4]{3}\) into the area formula:
Simplify the expression:
First, calculate \( (2\sqrt[4]{3})^2 \):
Substitute this back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \times (4\sqrt{3}) \]Perform the final multiplication:
\[ A = \frac{\sqrt{3} \times 4\sqrt{3}}{4} \]\[ A = \frac{4 \times (\sqrt{3})^2}{4} \]\[ A = \frac{4 \times 3}{4} \]\[ A = 3 \text{ cm}^2 \]B) 3 cm² is the correct area of the equilateral triangle.
The problem asks us to identify a common characteristic among the given numbers (29, 43, 59, 53, 89) and then select an option that shares this characteristic. We need to examine properties like prime numbers, divisibility, or specific number patterns.
D) 79
The question asks to identify the president of the 1916 Lucknow Session of the Indian National Congress, which is notable for the reunification of the Moderate and Extremist factions.
B) Ambika Charan Majumdar presided over the historic 1916 Lucknow Session of the Indian National Congress, where the Moderates and Extremists reunited, and the Lucknow Pact was signed.
The problem asks us to decode a word based on a given coding pattern. We need to identify the rule by which "PASSPORT" is transformed into "RCUUNMPR" and then apply the same rule to "DOMINANT".
A) FQOKLYLR is the correct option as it matches the derived coded word.
To solve this problem, we need to add several decimal numbers. The key is to align the decimal points vertically before performing the addition. This ensures that digits of the same place value are added together correctly.
Write down the numbers one below the other, ensuring that their decimal points are perfectly aligned. If a number does not explicitly have a decimal point, it is understood to be at the end of the number (e.g., \(1254\) is \(1254.000\)).
\[ \begin{array}{r} 1254.000 \\ 125.400 \\ 12.540 \\ + \quad 1.254 \\ \hline \end{array} \]Add the digits in each column, starting from the rightmost column (thousandths place) and moving left, carrying over to the next column when the sum exceeds 9.
Place the decimal point in the result directly below the aligned decimal points in the numbers being added.
The sum obtained is \(1393.194\).
B) \(1393.194\) is the correct sum obtained by aligning the decimal points and adding the numbers.
To find the number that does not belong to the series, we need to identify the pattern or rule governing the given sequence of numbers. Once the rule is established, we can check each number to see if it conforms to the pattern.
D) 26 — The established pattern for the series is \(n_{i+1} = 2n_i - 1\). Following this pattern, the term after 13 should be \(2 \times 13 - 1 = 25\). However, the given term is 26. Therefore, 26 does not belong to the series.