This question involves analyzing a statement with inequality and equality relations between variables. We need to derive conclusions from the given statement and check which of the provided options logically follows.
Let's analyze each option based on the given statement: \(Q \geq W = E > R \leq T < Y\).
Only option A can be definitively concluded from the given statement.
Correct Option: A) Q > R < Y
To find the value of the given trigonometric expression, we will convert all trigonometric ratios into their sine and cosine forms and then simplify the expression using algebraic identities.
Write the given expression:
\[ \text{sec A} (1 - \text{cos A}) (\text{cosec A} + \text{cot A}) \]Convert all terms into sine and cosine forms:
\(\text{sec A} = \frac{1}{\text{cos A}}\)
\(\text{cosec A} = \frac{1}{\text{sin A}}\)
\(\text{cot A} = \frac{\text{cos A}}{\text{sin A}}\)
Substitute these into the expression:
\[ \frac{1}{\text{cos A}} (1 - \text{cos A}) \left( \frac{1}{\text{sin A}} + \frac{\text{cos A}}{\text{sin A}} \right) \]Simplify the terms inside the parentheses:
\[ \frac{1}{\text{cos A}} (1 - \text{cos A}) \left( \frac{1 + \text{cos A}}{\text{sin A}} \right) \]Combine the terms:
\[ \frac{(1 - \text{cos A})(1 + \text{cos A})}{\text{cos A} \cdot \text{sin A}} \]Use the algebraic identity \((a-b)(a+b) = a^2 - b^2\):
\[ \frac{1^2 - \text{cos}^2 \text{A}}{\text{cos A} \cdot \text{sin A}} = \frac{1 - \text{cos}^2 \text{A}}{\text{cos A} \cdot \text{sin A}} \]Apply the trigonometric identity \(\text{sin}^2 \text{A} + \text{cos}^2 \text{A} = 1\), which implies \(\text{sin}^2 \text{A} = 1 - \text{cos}^2 \text{A}\):
\[ \frac{\text{sin}^2 \text{A}}{\text{cos A} \cdot \text{sin A}} \]Cancel out one \(\text{sin A}\) term from the numerator and denominator:
\[ \frac{\text{sin A}}{\text{cos A}} \]Recognize that \(\frac{\text{sin A}}{\text{cos A}}\) is equal to \(\text{tan A}\):
\[ \text{tan A} \]Correct Option: C) tan A
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then evaluate each conclusion based on these diagrams. The statements are universal affirmative (All A are B), which implies a hierarchical relationship between the categories.
B) Only conclusion I follows
This question tests the understanding of image formation by a concave mirror, specifically when the object is placed between the pole (P) and the principal focus (F). We need to recall the ray tracing rules for concave mirrors or the mirror formula sign conventions to determine the nature, position, and orientation of the image.
Correct Option: D) virtual and erect
When an object is placed between the pole (P) and the principal focus (F) of a concave mirror, the reflected rays diverge and appear to come from a point behind the mirror. This results in a virtual, erect, and magnified image.
This problem can be solved by setting up a system of two linear equations based on the two given scenarios. We will define variables for the number of correct and incorrect answers and then solve for these variables. Since Sindhu attempted all questions, the total number of questions will be the sum of correct and incorrect answers.
Let \(C\) be the number of correct answers and \(W\) be the number of wrong answers.
Scenario 1: 3 marks for every correct answer and 1 mark deducted for every wrong answer. Sindhu scored 100 marks.
This can be written as the equation:
\[ 3C - 1W = 100 \quad \text{(Equation 1)} \]Scenario 2: 4 marks for every correct answer and 2 marks deducted for every incorrect answer. Sindhu would have scored 120 marks.
This can be written as the equation:
\[ 4C - 2W = 120 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
\[ \begin{align*} 3C - W &= 100 \ 4C - 2W &= 120 \end{align*} \]To solve this system, we can use the elimination method. Multiply Equation 1 by 2 to make the coefficient of \(W\) the same as in Equation 2:
\[ 2 \times (3C - W) = 2 \times 100 \]\[ 6C - 2W = 200 \quad \text{(Equation 3)} \]Subtract Equation 2 from Equation 3:
\[ \begin{align*} (6C - 2W) - (4C - 2W) &= 200 - 120 \ 6C - 2W - 4C + 2W &= 80 \ 2C &= 80 \ C &= \frac{80}{2} \ C &= 40 \end{align*} \]Substitute the value of \(C = 40\) into Equation 1 to find \(W\):
\[ 3(40) - W = 100 \]\[ 120 - W = 100 \]\[ W = 120 - 100 \]\[ W = 20 \]The number of correct answers is \(C = 40\) and the number of wrong answers is \(W = 20\).
Since Sindhu attempted all the questions, the total number of questions in the test is the sum of correct and wrong answers:
\[ \text{Total Questions} = C + W = 40 + 20 = 60 \]D) 60 is the correct answer. The total number of questions is the sum of correct and incorrect answers, which is \(40 + 20 = 60\).
Study the given graph and answer the question that follows.
The graph presents the total number of candidates who qualified for medical entrance examination from three reputed institutions over the six years.
In which year have the maximum number of candidates succeeded in all three institutes taken together?
To find the year with the maximum number of candidates who qualified from all three institutes, we need to sum the values for each year and compare them.
Correct Option: C) Year 6
The question asks about the end products of respiration in plants. Respiration is a biochemical process where organisms exchange gases with their environment and release energy from organic substances. We need to recall the balanced chemical equation for cellular respiration to identify its final products.
D) COâ‚‚, Hâ‚‚O and Energy
This option correctly lists carbon dioxide, water, and energy as the end products of aerobic respiration in plants. Plants respire continuously, day and night, to obtain energy for their life processes.
The question asks for a method to reduce the use of non-biodegradable products. Non-biodegradable products are materials that cannot be broken down by natural processes, leading to environmental pollution and waste accumulation. To reduce their use, we must choose alternatives that are either biodegradable or reusable.
Correct Option: B) By using eco-friendly packaging
We are given the sum of the present ages of A and B, and the ratio of their ages after 5 years. We need to find the present age of A. We can set up a system of linear equations based on the given information and solve for the unknown ages.
Let the present age of A be \(A\) years and the present age of B be \(B\) years.
According to the first statement, the sum of their present ages is 30 years:
\[ A + B = 30 \quad \text{(Equation 1)} \]After 5 years, A's age will be \(A + 5\) years and B's age will be \(B + 5\) years.
According to the second statement, the ratio of their ages after 5 years will be 3:2:
\[ \frac{A + 5}{B + 5} = \frac{3}{2} \]Cross-multiply to simplify the ratio equation:
\[ 2(A + 5) = 3(B + 5) \]\[ 2A + 10 = 3B + 15 \]\[ 2A - 3B = 15 - 10 \]\[ 2A - 3B = 5 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
Equation 1: \(A + B = 30\)
Equation 2: \(2A - 3B = 5\)
From Equation 1, express B in terms of A:
\[ B = 30 - A \]Substitute this expression for B into Equation 2:
\[ 2A - 3(30 - A) = 5 \]\[ 2A - 90 + 3A = 5 \]\[ 5A - 90 = 5 \]\[ 5A = 5 + 90 \]\[ 5A = 95 \]Solve for A:
\[ A = \frac{95}{5} \]\[ A = 19 \]So, the present age of A is 19 years.
To find B's age, substitute A = 19 into Equation 1:
\[ 19 + B = 30 \]\[ B = 30 - 19 \]\[ B = 11 \]Check the ratio after 5 years: A's age will be \(19 + 5 = 24\), B's age will be \(11 + 5 = 16\). The ratio is \[\frac{24}{16} = \frac{3}{2}\], which matches the given information.
D) 19 years
The 12th Hockey India Senior Male National Championship – 2022 was organised by:
The question asks about the organizer of the 12th Hockey India Senior Male National Championship in 2022. This is a factual question related to sports events and their venues/organizers. To answer this, one needs to recall or know the specific details of this particular championship.
Correct Option: C) Madhya Pradesh
The 12th Hockey India Senior Men's National Championship was organized by Hockey Madhya Pradesh and took place in Bhopal, Madhya Pradesh, from April 6 to April 17, 2022. Hockey Haryana won the championship by defeating Hockey Tamil Nadu in the final.