Trigonometric Functions

their sum is

To solve the problem, we need to find the value of given that where .

Let's break down the problem step by step:

1. From the equation , it is essential to understand the possible values for each component.
2. The maximum value of is 1, and similarly, and can also take a maximum value of 1. Knowing that the cosine function achieves a maximum value of 1 is crucial due to its range .
3. Substituting these maximum possible values to the original equation, we have:

which is less than . This contradiction indicates that achieving the maximum value for each component is not possible.

1. Therefore, an alternative combination must be assessed under the constraint given.
2. The provided equation suggests a special trigonometric identity or value. Consider particular angles for simplification.
3. Consider the possibility that . Evaluating , we calculate:

Note that could be evaluated with regards to specific trigonometric values. Nevertheless, to ensure solution consistency:

• For , compute and .
• Finally, summing these values gives:

This matches the given correct answer option: .

Hence, the answer is .

The correct answer is (D) : 4
Let



at x

If cos2A+cos2B+cos2C is minimum then A= B=C=60∘
So △ABC is equilateral
Now in-radias r=3
So in △MBD we have
Tan30∘=BDMD​=a/2r​=a6​
1/3​=a1​=a=63​
Perimeter of △ABC=183​
Area of △ABC=43​​a2=273​

The given equation is:

Simplify the Logarithms

Using the change of base formula:

Substitute these into the equation:

Simplify the Terms

Combine the terms:

Factorize:

Simplify further:

Relate Sine and Cosine

Exponentiate both sides:

Thus:

Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula:

Since , we take the positive root:

Find

Comparing with , we identify:

Thus:

Conclusion: The value of is 4. Therefore, the correct answer is .

Step 1: Analyzing the function for valid values of .

We need to check the conditions for the logarithmic and exponential parts:

• The argument of the logarithm must be greater than 0:
  
• The argument of the logarithm must be greater than 0:
  
• For the denominator to be valid, it must not be 0, so:
  and
• The denominator must not be 0, so:
  

Step 2: Combining the conditions.

From the above conditions, we get the domain of the function:

and

Thus, the domain of the function is .

Step 3: Conclusion.

The correct answer is option (4) .

We are tasked with analyzing the determinant of a given matrix and determining the maximum and minimum values of the resulting function . Let us proceed step by step:

1. The Given Matrix:
The matrix is:

2. Row Operations:
To simplify the determinant, perform the following row operations:

and :

3. Expanding the Determinant:
Expand the determinant about the first row:



Compute each minor determinant:



Substitute these values back into the expansion:



Simplify:



Using the Pythagorean identity :

4. Finding Maximum and Minimum Values:
The function depends on , which oscillates between and :





Thus:

5. Computing :
Using the values of and :



Compute each term:



Subtract:

Final Answer:
The value of is .

To evaluate the limit:

We begin by examining each component of the expression as .

1. . As , , so behaves like .
2. Consider the terms inside the square roots:
   • when : since , we have .
   • when : with and , we have .
3. Using first-order approximations for small ,
   • since .
   • .
4. The expression becomes:

1. Simplifying, we have:

Thus, the limit evaluates to:

The correct answer is .

Step 1: The problem asks us to find the number of seven-digit numbers that can be formed using the digits 1, 2, and 3, where the sum of the digits is 11.
Step 2: The equation we need to solve is where represent the number of times the digits 1, 2, and 3 are used, respectively. The constraint is that .
Step 3: This is a Diophantine equation, and we can find the number of non-negative integer solutions that satisfy both conditions using methods such as generating functions or combinatorics.
Step 4: After solving, the number of valid seven-digit numbers is found to be 161. Thus, the correct answer is (3).

The given expression involves inverse cotangents and some algebraic manipulation. By applying trigonometric identities and simplifying, we can show that the expression simplifies to .
Final Answer: .

Step 1: Understanding the Question:
We are asked to find the number of solutions to a trigonometric equation involving inverse tangent functions within a specific interval.

Step 2: Key Formula or Approach:
We use the formula for the sum of two inverse tangent functions: Here, and . The condition is .
This means . The given interval for x is , which satisfies the condition .

Step 3: Detailed Explanation:
Applying the formula to the given equation: We are given that this equals . Taking the tangent of both sides: Now, we solve for x: This is a quadratic equation in x. Using the quadratic formula : The two possible solutions are and .

Step 4: Checking the Solutions against the Interval:
The interval is .
. So the interval is approx .
Approximate values of the solutions: , .
.
This value lies inside the interval .
.
This value lies outside the interval .
Therefore, there is only one solution in the given interval.