Find the value of the expression
[cosec(75°+θ) − sec(15°−θ) − tan(55°+θ) + cot(35°−θ)]
The problem requires evaluating an trigonometric expression involving cosec, sec, tan, and cot functions. The key to solving this is to use the complementary angle identities, which state that for any acute angle \( \theta \):
By applying these identities, we can convert terms into their complementary forms, which will allow for cancellation.
Consider the first pair of terms: \( \text{cosec}(75^\circ + \theta) - \text{sec}(15^\circ - \theta) \).
We know that \( \text{cosec}(A) = \text{sec}(90^\circ - A) \). Let's convert \( \text{cosec}(75^\circ + \theta) \) to its secant form:
\[ \text{cosec}(75^\circ + \theta) = \text{sec}(90^\circ - (75^\circ + \theta)) \]\[ = \text{sec}(90^\circ - 75^\circ - \theta) \]\[ = \text{sec}(15^\circ - \theta) \]Substitute this back into the first pair of terms:
\[ \text{cosec}(75^\circ + \theta) - \text{sec}(15^\circ - \theta) = \text{sec}(15^\circ - \theta) - \text{sec}(15^\circ - \theta) = 0 \]Now consider the second pair of terms: \( -\text{tan}(55^\circ + \theta) + \text{cot}(35^\circ - \theta) \).
We know that \( \text{tan}(A) = \text{cot}(90^\circ - A) \). Let's convert \( \text{tan}(55^\circ + \theta) \) to its cotangent form:
\[ \text{tan}(55^\circ + \theta) = \text{cot}(90^\circ - (55^\circ + \theta)) \]\[ = \text{cot}(90^\circ - 55^\circ - \theta) \]\[ = \text{cot}(35^\circ - \theta) \]Substitute this back into the second pair of terms:
\[ -\text{tan}(55^\circ + \theta) + \text{cot}(35^\circ - \theta) = -\text{cot}(35^\circ - \theta) + \text{cot}(35^\circ - \theta) = 0 \]Combine the results from step 2 and step 4 for the entire expression:
\[ [\text{cosec}(75^\circ + \theta) - \text{sec}(15^\circ - \theta)] - [\text{tan}(55^\circ + \theta) - \text{cot}(35^\circ - \theta)] \]\[ = 0 - 0 = 0 \]B) The value of the expression is \( 0 \) as shown by applying complementary angle identities to cancel out terms in pairs.
The question asks about a specific historical event where Indian footballer Baichung Bhutia refused to carry the Olympic torch. This requires knowledge of current affairs and significant events related to sports and international politics.
Correct Option: C) Baichung Bhutia, a renowned Indian footballer, refused to carry the Olympic torch for the 2008 Beijing Olympics in support of the Tibetan Independence Movement. This was a significant political statement made by an Indian sportsman on an international platform, drawing attention to the human rights situation in Tibet.
The question asks to identify which of the given options is NOT an international organization. An international organization is typically formed by treaties or agreements between two or more sovereign states, or is a body with members from multiple countries working towards common goals.
Correct Option: A) Defence Research and Development Organization. This is a national organization of India, not an international one.
To find the value of \(a+b\) when two quadratic equations have a common root, we can assume a common root, say \(\alpha\), and substitute it into both equations. Then, we can subtract one equation from the other to eliminate the \(\alpha^2\) term and solve for \(\alpha\). Finally, substitute the value of \(\alpha\) back into one of the original equations to find the relationship between \(a\) and \(b\).
Let the common root be \(\alpha\). Since \(\alpha\) is a root of both equations, it must satisfy both equations:
\[ \alpha^2 + a\alpha + b = 0 \quad \text{(1)} \] \[ \alpha^2 + b\alpha + a = 0 \quad \text{(2)} \]Subtract equation (2) from equation (1):
\[ (\alpha^2 + a\alpha + b) - (\alpha^2 + b\alpha + a) = 0 - 0 \] \[ \alpha^2 + a\alpha + b - \alpha^2 - b\alpha - a = 0 \] \[ (a - b)\alpha + (b - a) = 0 \]Factor out \((a - b)\) from the expression:
\[ (a - b)\alpha - (a - b) = 0 \] \[ (a - b)(\alpha - 1) = 0 \]Since it is given that \(a \neq b\), we know that \((a - b) \neq 0\). Therefore, for the product to be zero, we must have:
\[ \alpha - 1 = 0 \] \[ \alpha = 1 \]So, the common root is 1.
Substitute \(\alpha = 1\) into either equation (1) or (2). Let's use equation (1):
\[ (1)^2 + a(1) + b = 0 \] \[ 1 + a + b = 0 \]Rearrange the equation to find the value of \(a + b\):
\[ a + b = -1 \]Correct Option: D) -1
The question asks to identify the individual who formulated the 'Peoples Plan' for India. This requires knowledge of key economic plans and their proponents in India's post-independence or pre-independence planning history.
Correct Option: B) MN Roy formulated the 'Peoples Plan' in 1945. This plan was based on Marxist principles and emphasized the nationalization of all production and distribution, aiming for a socialist transformation of the economy.
The problem states that the profit is divided in proportion to the capitals invested by the partners. We are given the total capital, the total profit, and the individual profit shares of two partners (A and B). We need to find the capital of the third partner (C).
The sum of profits received by A and B is:
\[ \text{Profit (A+B)} = \text{Profit (A)} + \text{Profit (B)} \]\[ \text{Profit (A+B)} = \text{₹}4,500 + \text{₹}5,500 = \text{₹}10,000 \]The total profit is given as ₹15,000. So, the profit received by C can be found by subtracting the combined profit of A and B from the total profit:
\[ \text{Profit (C)} = \text{Total Profit} - \text{Profit (A+B)} \]\[ \text{Profit (C)} = \text{₹}15,000 - \text{₹}10,000 = \text{₹}5,000 \]The profits are divided in proportion to their capitals. Let the capitals of A, B, and C be \(C_A\), \(C_B\), and \(C_C\) respectively, and their profits be \(P_A\), \(P_B\), and \(P_C\).
The ratio of profits is \(P_A : P_B : P_C = 4,500 : 5,500 : 5,000\).
We can simplify this ratio by dividing by 100:
\[ P_A : P_B : P_C = 45 : 55 : 50 \]Since the profit is divided in proportion to their capitals, the ratio of their capitals will be the same as the ratio of their profits:
\[ C_A : C_B : C_C = 45 : 55 : 50 \]The total capital is ₹33,000. Let the common ratio multiplier be \(x\). Then:
\[ C_A = 45x \]\[ C_B = 55x \]\[ C_C = 50x \]The total capital is the sum of individual capitals:
\[ C_A + C_B + C_C = 33,000 \]\[ 45x + 55x + 50x = 33,000 \]\[ 150x = 33,000 \]\[ x = \frac{33,000}{150} \]\[ x = 220 \]Now, find C's capital using the value of \(x\):
\[ C_C = 50x = 50 \times 220 \]\[ C_C = \text{₹}11,000 \]D) 11,000 — C's profit share is ₹5,000. Since profits are proportional to capital, and the total profit is ₹15,000 for a total capital of ₹33,000, C's capital is \[\frac{5000}{15000} \times 33000 = \frac{1}{3} \times 33000 = 11000\].
The question asks for the location in India where nuclear explosive devices were tested in 1998. This is a factual question related to India's nuclear program and history. Recalling the specific event and its location is key.
Correct Option: D) Pokharan is the correct answer. The Pokhran-II nuclear tests, also known as Operation Shakti, were a series of five nuclear bomb test explosions conducted by India at the Pokhran Test Range in May 1998. These tests made India the world's sixth nuclear power.
Solve the following.
\(\frac{0.8\times0.8\times0.8-0.7\times0.7\times0.7}{0.8\times0.8+0.56+0.7\times0.7}\) = ?
The problem involves simplifying an algebraic expression that resembles the difference of cubes formula. By recognizing the pattern \((a^3 - b^3) / (a^2 + ab + b^2)\), we can simplify the expression significantly.
C) 0.1 is the correct answer, obtained by simplifying the expression using the difference of cubes identity.
The question asks to identify the famous personality associated with the slogan "Do or Die". This requires knowledge of key figures in Indian history, particularly during the freedom struggle, and their iconic sayings or calls to action.
Correct Option: B) Mahatma Gandhi
Mahatma Gandhi gave the famous call to "Do or Die" (Karenge Ya Marenge) on August 8, 1942, at the Gowalia Tank Maidan in Bombay (now Mumbai) during the Quit India Movement. This was a powerful message urging Indians to make every effort to achieve complete independence or die in the attempt.
The question asks to identify a major factor responsible for deforestation. Deforestation refers to the clearing of forests for other land uses. We need to evaluate each option based on its impact on forest cover.
Correct Option: D) Agriculture is widely recognized as the leading cause of deforestation worldwide. Forests are cleared to make way for crop cultivation (e.g., soy, palm oil) and livestock grazing.