We start by evaluating . We know that . Now, substitute this value into the expression:
We know that , so the final answer is:
02
PYQ 2025
hard
mathematicsID: cbse-cla
The graph of a trigonometric function is as shown. Which of the following will represent the graph of its inverse?
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Official Solution
Correct Option: (2)
The graph of a trigonometric function and its inverse are symmetric about the line . - The given graph represents a trigonometric function like or , which is defined within the interval . - The graph of its inverse, such as , will be reflected across the line . Thus, the graph that corresponds to the inverse function is option .
03
PYQ 2025
hard
mathematicsID: cbse-cla
Solve for ,
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Official Solution
Correct Option: (2)
To solve the equation, we start by simplifying each term: - The first term is . We will consider the inverse tangent function and express in terms of an angle , where .
- The second term involves , which is a standard identity for , which equals . Thus, the equation becomes:
Simplifying further:
Now, take the tangent of both sides:
The solution gives .
04
PYQ 2025
medium
mathematicsID: cbse-cla
The graph shown below depicts:
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Official Solution
Correct Option: (3)
Let's analyze the features of the graph shown: - The graph has a range from and similar behavior mirrored across the X-axis (i.e., also in ), which corresponds to the principal values of the inverse cosecant function. - The curve is defined for and has vertical asymptotes at and , which is consistent with . - The graph is not periodic, which rules out trigonometric functions like or , which are periodic. Therefore, this graph corresponds to the inverse cosecant function: .
05
PYQ 2025
medium
mathematicsID: cbse-cla
Evaluate: .
Official Solution
Correct Option: (1)
We are asked to evaluate . The inverse sine function gives the angle such that and . Since is greater than , we need to adjust the angle so that it lies within the principal range . We can use the identity:
Thus, we have:
Therefore:
Thus, the value of is .
