What is the value of \(\frac{1-\tan^2A}{1+\tan^2A}\) ?
The problem asks us to simplify the given trigonometric expression \(\frac{1-\tan^2A}{1+\tan^2A}\). We will use fundamental trigonometric identities to transform the expression into one of the given options.
Recall the fundamental trigonometric identity relating tangent and secant:
\[ 1 + \tan^2A = \sec^2A \]Substitute this identity into the denominator of the given expression:
\[ \frac{1-\tan^2A}{1+\tan^2A} = \frac{1-\tan^2A}{\sec^2A} \]Now, express \(\tan A\) and \(\sec A\) in terms of \(\sin A\) and \(\cos A\):
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \sec A = \frac{1}{\cos A} \]Substitute these into the expression:
\[ \frac{1-\left(\frac{\sin A}{\cos A}\right)^2}{\left(\frac{1}{\cos A}\right)^2} = \frac{1-\frac{\sin^2A}{\cos^2A}}{\frac{1}{\cos^2A}} \]Simplify the numerator by finding a common denominator:
\[ \frac{\frac{\cos^2A - \sin^2A}{\cos^2A}}{\frac{1}{\cos^2A}} \]Multiply the numerator by the reciprocal of the denominator:
\[ \left(\frac{\cos^2A - \sin^2A}{\cos^2A}\right) \times \left(\frac{\cos^2A}{1}\right) \]Cancel out \(\cos^2A\):
\[ \cos^2A - \sin^2A \]Recall the double angle identity for cosine:
\[ \cos 2A = \cos^2A - \sin^2A \]Therefore, the given expression simplifies to \(\cos 2A\).
A) cos 2A is the correct answer because, by using trigonometric identities, the expression simplifies to \(\cos^2A - \sin^2A\), which is the formula for \(\cos 2A\).
To find the perimeter of the square, we first need to determine its area. The problem states that the area of the square is ten times the area of a given rectangle. We will calculate the area of the rectangle, then the area of the square, and finally the side length and perimeter of the square.
The dimensions of the rectangle are given as 5 cm \(\times\) 8 cm.
\[ \text{Area of rectangle} = \text{length} \times \text{width} \]\[ \text{Area of rectangle} = 8 \text{ cm} \times 5 \text{ cm} = 40 \text{ cm}^2 \]The problem states that the area of the square is ten times the area of the rectangle.
\[ \text{Area of square} = 10 \times \text{Area of rectangle} \]\[ \text{Area of square} = 10 \times 40 \text{ cm}^2 = 400 \text{ cm}^2 \]Let \(s\) be the side length of the square. The area of a square is given by \(s^2\).
\[ s^2 = 400 \text{ cm}^2 \]\[ s = \sqrt{400 \text{ cm}^2} \]\[ s = 20 \text{ cm} \]The perimeter of a square is given by \(4 \times s\).
\[ \text{Perimeter of square} = 4 \times s \]\[ \text{Perimeter of square} = 4 \times 20 \text{ cm} = 80 \text{ cm} \]Correct Option: B) 80 cm
To complete a task, A and B take 24 days, B and C take 12 days and A, B and C take 6 days. A and C will take:
This problem involves work and time. We can solve it by calculating the individual work rates of A, B, and C. The work rate is the reciprocal of the time taken to complete the task. Once individual rates are known, we can find the combined rate of A and C and then the time they take together.
A and B take 24 days to complete the task. So, their combined work rate is:
\[ (\text{A} + \text{B})' = \frac{1}{24} \text{ part/day} \]B and C take 12 days to complete the task. So, their combined work rate is:
\[ (\text{B} + \text{C})' = \frac{1}{12} \text{ part/day} \]A, B, and C take 6 days to complete the task. So, their combined work rate is:
\[ (\text{A} + \text{B} + \text{C})' = \frac{1}{6} \text{ part/day} \]We know \((\text{A} + \text{B} + \text{C})'\) and \((\text{A} + \text{B})'\). Subtracting the latter from the former gives C's work rate:
\[ \text{C}' = (\text{A} + \text{B} + \text{C})' - (\text{A} + \text{B})' = \frac{1}{6} - \frac{1}{24} \]\[ \text{C}' = \frac{4 - 1}{24} = \frac{3}{24} = \frac{1}{8} \text{ part/day} \]We know \((\text{A} + \text{B} + \text{C})'\) and \((\text{B} + \text{C})'\). Subtracting the latter from the former gives A's work rate:
\[ \text{A}' = (\text{A} + \text{B} + \text{C})' - (\text{B} + \text{C})' = \frac{1}{6} - \frac{1}{12} \]\[ \text{A}' = \frac{2 - 1}{12} = \frac{1}{12} \text{ part/day} \]Add their individual work rates:
\[ (\text{A} + \text{C})' = \text{A}' + \text{C}' = \frac{1}{12} + \frac{1}{8} \]\[ (\text{A} + \text{C})' = \frac{2 + 3}{24} = \frac{5}{24} \text{ part/day} \]The time taken is the reciprocal of their combined work rate:
\[ \text{Time}(\text{A} + \text{C}) = \frac{1}{(\text{A} + \text{C})'} = \frac{1}{\frac{5}{24}} = \frac{24}{5} \text{ days} \]A) \( \frac{24}{5} \) days. This matches our calculated result.
The problem requires us to decipher a given coding pattern by comparing the original phrase with its coded version. Once the pattern is identified, we apply it to the new word 'DRAGON' to find its code.
Let's look at the first letters of each word and their corresponding coded letters:
This is also inconsistent. Let's consider a different type of substitution.
Let's re-examine 'TREE' → 'ASEE'.
This looks like a substitution cipher where each letter is mapped to another specific letter. Let's try to find common letters.
Let's list the mappings we've found:
Now, let's apply these mappings to the word 'DRAGON'.
So, 'DRAGON' codes to 'CSQJKM'.
Correct Option: D) CSQJKM
To find the number of prime numbers less than 50, we need to list all natural numbers from 1 to 49 and then identify which of them are prime. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The prime numbers less than 50 are:
\(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\)
Counting these numbers, we find there are 15 prime numbers.
Correct Option: C) 15
The problem requires us to solve a linear equation involving percentages to find the value of the unknown variable 'x'. We need to calculate the percentage values, perform the addition, and then isolate 'x'.
To find 20% of 160, we can write 20% as a fraction \(\frac{20}{100}\) or a decimal 0.20.
\[ 20\% \text{ of } 160 = \frac{20}{100} \times 160 \]\[ = \frac{1}{5} \times 160 \]\[ = 32 \]To find 10% of 50, we can write 10% as a fraction \(\frac{10}{100}\) or a decimal 0.10.
\[ 10\% \text{ of } 50 = \frac{10}{100} \times 50 \]\[ = \frac{1}{10} \times 50 \]\[ = 5 \]The given equation is: \(20\% \text{ of } 160 + 10\% \text{ of } 50 = x - 1\)
Substitute the calculated values:
\[ 32 + 5 = x - 1 \]First, add the numbers on the left side:
\[ 37 = x - 1 \]\[ \]To isolate \(x\), add 1 to both sides of the equation:\[ 37 + 1 = x \]\[ x = 38 \]C) 38 is the correct value of x.
The question asks to identify the state with which the folk dance 'Bardo Chham' is associated. This requires knowledge of Indian folk dances and their respective regions.
Correct Option: C) Arunachal Pradesh
The question asks to identify the Governor-General under whom the policy of 'paramountcy' was initiated. This requires knowledge of the British East India Company's administrative policies and the timelines of various Governor-Generals in India.
B) Lord Hastings – He initiated the new policy of 'paramountcy' during his tenure as Governor-General (1813-1823). Under this policy, the East India Company claimed that its authority was paramount or supreme, and therefore, its power was greater than that of Indian states.
The question asks for the city where India's first bio-refinery plant was inaugurated. This is a factual question related to developments in renewable energy and biofuels in India. Recalling key events and locations related to such inaugurations is essential.
Correct Option: A) The first bio-refinery plant in India, which converts agricultural residue into ethanol, was inaugurated in Pune, Maharashtra. This plant was established by Praj Industries.
This is a statement and conclusion type of question. We need to analyze the given statement carefully and determine which of the conclusions logically follow from the information provided in the statement, without introducing any outside knowledge or assumptions.
B) Neither I nor II conclusion follows