The value of
(sinθ+cosecθ)2+(cosθ+secθ)2 is:
7 - cot2θ + tan2θ
5 + cot2θ + tan2θ
7 + cot2θ + tan2θ
5 - cot2θ + tan2θ
To find the value of the given expression, we will expand each squared term using the identity \((a+b)^2 = a^2 + b^2 + 2ab\). Then, we will use fundamental trigonometric identities such as \(\sin^2\theta + \cos^2\theta = 1\), \(\text{cosec}^2\theta = 1 + \cot^2\theta\), and \(\sec^2\theta = 1 + \tan^2\theta\), along with reciprocal identities like \(\sin\theta \cdot \text{cosec}\theta = 1\) and \(\cos\theta \cdot \sec\theta = 1\).
Expand the first term \((\sin\theta + \text{cosec}\theta)^2\):
\[ (\sin\theta + \text{cosec}\theta)^2 = \sin^2\theta + \text{cosec}^2\theta + 2\sin\theta \text{cosec}\theta \]Simplify the product term using \(\sin\theta \cdot \text{cosec}\theta = 1\):
\[ \sin^2\theta + \text{cosec}^2\theta + 2(1) = \sin^2\theta + \text{cosec}^2\theta + 2 \]Expand the second term \((\cos\theta + \sec\theta)^2\):
\[ (\cos\theta + \sec\theta)^2 = \cos^2\theta + \sec^2\theta + 2\cos\theta \sec\theta \]Simplify the product term using \(\cos\theta \cdot \sec\theta = 1\):
\[ \cos^2\theta + \sec^2\theta + 2(1) = \cos^2\theta + \sec^2\theta + 2 \]Add the expanded forms of both terms:
\[ (\sin^2\theta + \text{cosec}^2\theta + 2) + (\cos^2\theta + \sec^2\theta + 2) \]\[ = \sin^2\theta + \cos^2\theta + \text{cosec}^2\theta + \sec^2\theta + 4 \]Apply the identity \(\sin^2\theta + \cos^2\theta = 1\):
\[ 1 + \text{cosec}^2\theta + \sec^2\theta + 4 \]\[ = 5 + \text{cosec}^2\theta + \sec^2\theta \]Apply the identities \(\text{cosec}^2\theta = 1 + \cot^2\theta\) and \(\sec^2\theta = 1 + \tan^2\theta\):
\[ 5 + (1 + \cot^2\theta) + (1 + \tan^2\theta) \]\[ = 5 + 1 + \cot^2\theta + 1 + \tan^2\theta \]\[ = 7 + \cot^2\theta + \tan^2\theta \]Thus, the value of the expression is \(7 + \cot^2\theta + \tan^2\theta\).
C) \(7 + \cot^2\theta + \tan^2\theta\) is the correct result obtained after simplifying the given expression using algebraic expansion and fundamental trigonometric identities.
The question asks to identify the group in the periodic table that primarily consists of metals. To answer this, we need to recall the general classification of elements in the periodic table based on their groups.
A) Group 2 — Group 2 elements are known as alkaline earth metals, and all elements in this group (Beryllium, Magnesium, Calcium, Strontium, Barium, Radium) are metals.
The question asks to identify the Prime Minister of India during the Eighth Five-Year Plan. To answer this, one needs to know the time period of the Eighth Five-Year Plan and the Prime Ministers who served during that specific period.
Correct Option: A) P. V. Narasimha Rao was the Prime Minister of India from 1991 to 1996, which covers the majority of the Eighth Five-Year Plan period (1992-1997). He is credited with initiating significant economic reforms during this time.
This is a logical reasoning problem involving the arrangement of five boxes. We need to deduce the exact order of the boxes from the given clues and identify the box at the bottommost position. We will use a step-by-step method to build the arrangement.
Let's combine the clues. We have 5 positions for the boxes. Let's denote them from top (1) to bottom (5).
From Clues 3 and 4, neither A nor Y can be at position 1. This means the box at position 1 must be either B, X, or Z.
Consider the B-X-Z block. Since X is between B and Z, and there are 5 boxes, this block takes up 3 consecutive positions.
Case 1: B-X-Z arrangement (from top to bottom)
Case 2: Z-X-B arrangement (from top to bottom)
From Clue 5, A is below Y. So, Y must be above A.
From Clue 2, Z and A are not adjacent.
Let's try to place Y and A.
Since A is not topmost (Clue 3) and Y is not topmost (Clue 4), the topmost box must be B or Z (from the B-X-Z / Z-X-B block).
If the arrangement is B-X-Z:
Now we need to place Y and A. A must be below Y. Z and A cannot be adjacent. So A cannot be at position 4. This means A must be at position 5. If A is at 5, then Y must be at 4 (since Y is above A). But this would make Z and Y adjacent. Let's check: Y-A. If A is at 5, Y is at 4. This means Z is at 3 and Y is at 4, so Z and Y are adjacent. This doesn't violate any rules yet. Let's check Z and A not adjacent. If A is at 5, Z is at 3. They are not adjacent. This arrangement is possible: B, X, Z, Y, A.
Let's verify all conditions for B, X, Z, Y, A:
This arrangement satisfies all conditions.
Let's consider the other possibility for the block: Z-X-B.
Now we need to place Y and A. A must be below Y. Z and A cannot be adjacent. Since Z is at position 1, A cannot be at position 2. This means A must be at position 4 or 5. If A is at 4, then Y must be at 2 or 3. But X is at 2 and B is at 3. So Y cannot be at 2 or 3. This means A cannot be at 4. So A must be at 5. If A is at 5, then Y must be at 4 (since Y is above A). This gives Z, X, B, Y, A.
Let's verify all conditions for Z, X, B, Y, A:
Both arrangements (B, X, Z, Y, A) and (Z, X, B, Y, A) satisfy all conditions.
However, the question asks for the box at the bottommost position. In both valid arrangements, the box at the bottommost position is A.
Therefore, A is at the bottommost position.
B) A is the box kept at the bottommost position.
This is a logical reasoning problem involving arrangement. We need to deduce the living arrangement of five people (R, S, T, U, V) on five different floors (1 to 5) based on the given clues. We will use a table to keep track of the floor assignments and eliminate possibilities.
D) S lives immediately below R (R is on Floor 4, S is on Floor 3).
The problem requires us to transform a given number based on specific rules for its even and odd digits, then identify digits that appear more than once in the new number, and finally sum those unique repeating digits.
A) 12 is the correct sum of the digits that appear more than once in the new number.
Refer to the following letter, number, symbol series and answer the question.
(Left) S 7 & D M # Z 6 % 3 T 2 * H G = C ? 4 Y B (Right)
How many such symbols are there which are immediately preceded by a number and also immediately followed by a letter?
The problem asks us to identify symbols in a given series that are immediately preceded by a number and immediately followed by a letter. We need to scan the series from left to right and apply this condition to each symbol.
Therefore, there are two such symbols that satisfy the given condition.
A) Two is the correct answer because we identified two symbols ('&' and '*') that are immediately preceded by a number and immediately followed by a letter.
The question asks to identify the ex-officio Chairman of the Rajya Sabha. This is a direct question about the constitutional roles of high-ranking officials in the Indian parliamentary system.
Correct Option: C) Vice-President of India — As per Article 64 of the Indian Constitution, the Vice-President of India is the ex-officio Chairman of the Rajya Sabha. This means that the person holding the office of Vice-President automatically performs the duties of the Chairman of the Rajya Sabha.
The question asks about the application of tungsten based on its high resistivity. We need to understand what high resistivity implies in electrical applications and then evaluate each option.
Re-evaluation based on common knowledge and typical exam patterns: Tungsten's high resistivity means it resists current flow, leading to heat generation. This property, combined with its high melting point, makes it excellent for filaments in incandescent bulbs (to generate light and heat) and sometimes in heating elements. For transmission lines, materials with LOW resistivity are essential to minimize energy loss. Therefore, the statement that tungsten is used in transmission lines due to high resistivity is fundamentally incorrect.
Given the provided answer 'B', there might be a misunderstanding or a flawed question/answer key. However, if forced to choose based on the premise that the provided answer is correct, one would have to assume an unusual context. But in standard physics, high resistivity is detrimental for transmission lines.
Let's assume there's a typo in the question or the provided answer key, and the question implicitly asks where high resistivity is a beneficial property. Then, electric bulbs would be the correct answer. But if we must adhere to the provided correct option 'B', then the premise of the question (high resistivity being a reason for use in transmission lines) is incorrect.
Since I must provide an explanation for the given correct answer 'B', I will explain why 'B' is incorrect based on physics principles, but acknowledge that the question's premise leading to 'B' is flawed.
Correct Option: B) Transmission lines — This option is actually incorrect based on the property mentioned. Tungsten has high resistivity, which means it resists the flow of electric current. For transmission lines, materials with very low resistivity (like copper or aluminum) are used to minimize energy loss during power transmission. High resistivity would lead to significant energy loss as heat (\(P = I^2 R\)) over long distances, making tungsten unsuitable for transmission lines. Therefore, the premise that tungsten is used in transmission lines due to its high resistivity is false.
This is a direction-based reasoning problem. We need to determine the direction Shyam is facing based on the position of his shadow during the evening. The key information is the time of day (evening) and the relative position of Shyam's shadow to Ram.
A) South. Based on the step-by-step analysis, if Ram is facing North, Shyam must be facing South to be talking face-to-face with Ram.