Solve the following.
\(\frac{\tan A}{1+\sec A}+\frac{1+\sec A}{\tan A}\) = ?
To solve this problem, we will combine the two fractions by finding a common denominator. Then, we will use trigonometric identities to simplify the numerator and denominator. Key identities to remember are \(\sec^2 A - \tan^2 A = 1\) and the definitions of \(\tan A\), \(\sec A\), and \(\text{cosec } A\) in terms of \(\sin A\) and \(\cos A\).
Given expression:
\[ \frac{\tan A}{1+\sec A} + \frac{1+\sec A}{\tan A} \]Find a common denominator, which is \((1+\sec A)(\tan A)\). Combine the fractions:
\[ \frac{(\tan A)(\tan A) + (1+\sec A)(1+\sec A)}{(1+\sec A)(\tan A)} \] \[ = \frac{\tan^2 A + (1+\sec A)^2}{(1+\sec A)(\tan A)} \]Expand the term \((1+\sec A)^2\):
\[ (1+\sec A)^2 = 1^2 + 2(1)(\sec A) + (\sec A)^2 = 1 + 2\sec A + \sec^2 A \]Substitute this back into the expression:
\[ = \frac{\tan^2 A + 1 + 2\sec A + \sec^2 A}{(1+\sec A)(\tan A)} \]Recall the trigonometric identity \(\sec^2 A - \tan^2 A = 1\), which can be rearranged to \(\tan^2 A + 1 = \sec^2 A\). Substitute this into the numerator:
\[ = \frac{(\tan^2 A + 1) + \sec^2 A + 2\sec A}{(1+\sec A)(\tan A)} \] \[ = \frac{\sec^2 A + \sec^2 A + 2\sec A}{(1+\sec A)(\tan A)} \] \[ = \frac{2\sec^2 A + 2\sec A}{(1+\sec A)(\tan A)} \]Factor out \(2\sec A\) from the numerator:
\[ = \frac{2\sec A (\sec A + 1)}{(1+\sec A)(\tan A)} \]Cancel out the common term \((1+\sec A)\) from the numerator and denominator:
\[ = \frac{2\sec A}{\tan A} \]Express \(\sec A\) and \(\tan A\) in terms of \(\sin A\) and \(\cos A\):
\[ \sec A = \frac{1}{\cos A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A} \]Substitute these into the expression:
\[ = \frac{2 \left(\frac{1}{\cos A}\right)}{\left(\frac{\sin A}{\cos A}\right)} \] \[ = 2 \cdot \frac{1}{\cos A} \cdot \frac{\cos A}{\sin A} \]Cancel out \(\cos A\):
\[ = 2 \cdot \frac{1}{\sin A} \]Recall that \(\frac{1}{\sin A} = \text{cosec } A\):
\[ = 2\text{ cosec } A \]Correct Option: C) 2 cosec A
This problem involves work and time. We can solve it by calculating the individual work rates of A and B. The work rate is the reciprocal of the time taken to complete the work. If A and B work together, their individual work rates add up to their combined work rate.
A and B together complete the work in 35 days. So, their combined work rate (amount of work done in one day) is:
\[ \text{Work rate of (A + B)} = \frac{1}{35} \text{ part of work per day} \]A alone completes the work in 60 days. So, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{60} \text{ part of work per day} \]The combined work rate of A and B is the sum of their individual work rates:
\[ \text{Work rate of (A + B)} = \text{Work rate of A} + \text{Work rate of B} \]\[ \frac{1}{35} = \frac{1}{60} + \text{Work rate of B} \]Subtract A's work rate from the combined work rate:
\[ \text{Work rate of B} = \frac{1}{35} - \frac{1}{60} \]The least common multiple (LCM) of 35 and 60 is 420.
\[ \text{Work rate of B} = \frac{12}{420} - \frac{7}{420} \]\[ \text{Work rate of B} = \frac{12 - 7}{420} \]\[ \text{Work rate of B} = \frac{5}{420} \]If B's work rate is \(\frac{1}{84}\) part of the work per day, then B alone will complete the work in the reciprocal of this rate:
\[ \text{Time taken by B} = \frac{1}{\text{Work rate of B}} = \frac{1}{\frac{1}{84}} = 84 \text{ days} \]B) 84 days is the correct answer. This is derived from calculating B's individual work rate and then finding the reciprocal to get the total time.
The question asks us to identify the relationship between the first two terms and apply the same relationship to find the fourth term. The relationship given is "Length : Meter". This implies that Meter is the standard unit of measurement for Length. We need to find the standard unit of measurement for Mass.
Correct Option: A) Kilogram
The problem asks us to identify numbers in a given sequence that are immediately preceded by a letter and immediately followed by a symbol. We need to scan the sequence and apply this condition to each number.
A) 4. There are 4 such numbers (4, 9, 3, 4) in the sequence that are immediately preceded by a letter and immediately followed by a symbol.
The question asks to identify the scheme launched in 2002 for ensuring drinking water supply to all villages. This requires knowledge of various government schemes and their launch years and objectives.
B) Swajaldhara Yojana was launched in 2002. It was a demand-driven and community-based scheme to provide safe drinking water in rural areas, empowering local communities to plan, implement, operate, and maintain their water supply schemes.
Find the value of x if \(\frac{x}{\sqrt{128}}=\frac{\sqrt{162}}{x}\)
To find the value of \(x\), we need to solve the given equation by cross-multiplication and simplifying the square roots. We will first simplify the terms under the square root to their simplest radical form.
First, simplify \(\sqrt{128}\):
\[ \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2} \]Next, simplify \(\sqrt{162}\):
\[ \sqrt{162} = \sqrt{81 \times 2} = \sqrt{81} \times \sqrt{2} = 9\sqrt{2} \]Correct Option: D) 12
This problem can be solved by using the algebraic identity for \((a-b)^2\). We are given the values of \((a-b)\) and \((a^2+b^2)\), and we need to find the value of \(ab\).
The relevant algebraic identity is:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]We can rearrange this identity to isolate the term \(2ab\):
\[ (a - b)^2 = (a^2 + b^2) - 2ab \]We are given:
Substitute these values into the rearranged identity:
\[ (5)^2 = (41) - 2ab \]Calculate the square of 5:
\[ 25 = 41 - 2ab \]Subtract 41 from both sides:
\[ 2ab = 41 - 25 \] \[ 2ab = 16 \]Divide by 2:
\[ ab = \frac{16}{2} \] \[ ab = 8 \]Correct Option: A) 8
The question asks to identify the first Indian to be appointed to the Viceroy's Executive Council. This is a factual question related to Indian history, specifically the administrative reforms during the British Raj.
Correct Option: C) Satyendra Prasanna Sinha
The question asks for the location of the INS Sardar Patel Naval Base. This requires knowledge of Indian Navy bases and their geographical locations.
Correct Option: C) Porbandar — INS Sardar Patel is a naval base of the Indian Navy located in Porbandar, Gujarat. It was commissioned to enhance the Navy's presence and operational capabilities on the western seaboard.
The question asks to identify the odd one out from a given list of words: Writer, Reader, Publisher, Printer. We need to analyze the role of each word in the context of creating and disseminating a written work.
D) Reader — A reader is the consumer of a written work, whereas a writer, publisher, and printer are all involved in the process of creating, producing, and distributing the written work.