This question asks us to identify implicit assumptions in a given statement. An implicit assumption is something that is taken for granted or presupposed by the statement, even if not explicitly stated. We need to evaluate each assumption to see if the statement logically requires it to be true.
Correct Option: A) Neither assumption I nor II is implicit.
The problem involves trigonometric identities. We are given an equation relating sine and cosine functions of angles involving \(\theta\). The key to solving this is to use the complementary angle identity, which states that \(\cos x = \sin(90^\circ - x)\) or \(\sin x = \cos(90^\circ - x)\). By converting one of the trigonometric functions to the other, we can equate the angles and solve for \(\theta\).
We are given the equation:
\[ \sin 3\theta = \cos(\theta - 6^\circ) \]We know that \(\cos x = \sin(90^\circ - x)\). We can apply this identity to the right side of the equation:
\[ \cos(\theta - 6^\circ) = \sin(90^\circ - (\theta - 6^\circ)) \]\[ \cos(\theta - 6^\circ) = \sin(90^\circ - \theta + 6^\circ) \]\[ \cos(\theta - 6^\circ) = \sin(96^\circ - \theta) \]Now, substitute this back into the original equation:
\[ \sin 3\theta = \sin(96^\circ - \theta) \]For the sine of two angles to be equal (assuming both angles are acute, which they will be in this context), the angles themselves must be equal:
\[ 3\theta = 96^\circ - \theta \]Add \(\theta\) to both sides of the equation:
\[ 3\theta + \theta = 96^\circ \]\[ 4\theta = 96^\circ \]Divide by 4:
\[ \theta = \frac{96^\circ}{4} \]\[ \theta = 24^\circ \]If \(\theta = 24^\circ\):
\[ \sin(3\theta) = \sin(3 \times 24^\circ) = \sin(72^\circ) \]\[ \cos(\theta - 6^\circ) = \cos(24^\circ - 6^\circ) = \cos(18^\circ) \]We know that \(\sin(72^\circ) = \cos(90^\circ - 72^\circ) = \cos(18^\circ)\). Since \(\sin(72^\circ) = \cos(18^\circ)\), our value of \(\theta = 24^\circ\) is correct.
Correct Option: A) 24°
The question asks for the establishment year of the first Indian Cricket Club, the Calcutta Cricket Club. This is a factual question requiring knowledge of historical dates related to sports in India.
Correct Option: A) 1792
The Calcutta Cricket Club, the oldest cricket club in India and one of the oldest in the world, was established in 1792. It played a significant role in promoting cricket in India during the British Raj.
x% of y is y% of:
The problem asks us to find an unknown value such that "x% of y" is equal to "y% of that unknown value". We can translate the percentage expressions into mathematical equations and then solve for the unknown.
A) x is the correct answer because when we set up the equation based on the problem statement, we find that the unknown value is equal to \(x\).
The question asks to identify the first authentic work of prose in Hindi by Devaki Nandan Khatri. This requires knowledge of prominent Hindi literary works and their authors.
Correct Option: B) Chandrakanta — Devaki Nandan Khatri's Chandrakanta (1888) is a landmark work in Hindi literature. It is a fantasy novel that gained immense popularity and is often credited with being one of the first authentic and widely read prose works in modern Hindi, contributing significantly to the language's development and readership. Its intricate plot, magical elements, and engaging narrative style made it a sensation.
The question asks for the founding location of the International Criminal Police Commission (ICPC), the predecessor to INTERPOL, in 1923. This is a factual question requiring knowledge of the history of international organizations.
Correct Option: D) Vienna
The International Criminal Police Commission (ICPC), which later became INTERPOL, was indeed founded in Vienna, Austria, in 1923 during an International Police Congress.
If x + \( \frac{1}{x} \) = 6, then value of x² + \( \frac{1}{x^2} \) is:
This problem involves algebraic manipulation, specifically squaring a binomial expression. We are given an equation relating \(x\) and \(1/x\), and we need to find the value of an expression involving \(x^2\) and \(1/x^2\). The key is to square the given equation on both sides.
The given equation is:
\[ x + \frac{1}{x} = 6 \]To find \(x^2 + \frac{1}{x^2}\), we can square both sides of the given equation. Recall the algebraic identity \((a+b)^2 = a^2 + b^2 + 2ab\). Here, let \(a=x\) and \(b=\frac{1}{x}\).
\[ \left(x + \frac{1}{x}\right)^2 = (6)^2 \]Expand the left side using the identity:
\[ x^2 + \left(\frac{1}{x}\right)^2 + 2 \cdot x \cdot \frac{1}{x} = 36 \]Simplify the expression. Notice that \(x \cdot \frac{1}{x} = 1\).
\[ x^2 + \frac{1}{x^2} + 2 \cdot 1 = 36 \]Further simplify:
\[ x^2 + \frac{1}{x^2} + 2 = 36 \]Subtract 2 from both sides to isolate \(x^2 + \frac{1}{x^2}\):
\[ x^2 + \frac{1}{x^2} = 36 - 2 \]Calculate the final value:
\[ x^2 + \frac{1}{x^2} = 34 \]Correct Option: D) 34
The problem involves decoding a word based on a given set of coding rules. We need to identify the pattern or relationship between the original letters and their coded forms. Once the pattern is established, we apply it to the word "JOY" to find its coded form.
Let's find the position of each letter in the English alphabet (A=1, B=2, ..., Z=26).
The consistent pattern is that each letter is shifted forward by 4 positions in the alphabet (a +4 shift).
J becomes N, O becomes S, and Y becomes C.
Therefore, JOY is coded as NSC.
C) NSC is the correct code for JOY, following the +4 shift pattern for each letter.
To find the percentage gain or loss, we first need to determine the profit or loss amount. Given the ratio of cost price (CP) to selling price (SP), we can assume proportional values for CP and SP. Then, we calculate the profit or loss and express it as a percentage of the cost price.
Understand the given ratio:
The ratio of the cost price (CP) to the selling price (SP) is given as \(4 : 5\).
This means:
\[ \frac{\text{CP}}{\text{SP}} = \frac{4}{5} \]Assume values for CP and SP:
Let the cost price (CP) be \(4x\) and the selling price (SP) be \(5x\), where \(x\) is a common multiplier. For simplicity, we can assume \(x=1\), so:
Determine if there is a gain or loss:
Since the selling price (\(5\)) is greater than the cost price (\(4\)), there is a gain (profit).
Calculate the gain amount:
\(\text{Gain} = \text{SP} - \text{CP}\)
Calculate the percentage gain:
The percentage gain is calculated using the formula:
D) 25% gain. The selling price is higher than the cost price, indicating a gain. The calculated percentage gain is 25%.
To determine which pair of numbers is co-prime, we need to find the greatest common divisor (GCD) for each pair. Two numbers are co-prime (or relatively prime) if their GCD is 1. We will list the factors for each number in a pair and identify their common factors.
A) 6 and 35 — As determined in the step-by-step analysis, the greatest common divisor of 6 and 35 is 1. This means they share no common factors other than 1, making them co-prime.