Trigonometry

Step 1: Let Then, and Thus, the vertices of the triangle are:
Step 2: Form two vectors with origin at :
Step 3: Area of triangle formed by vectors and is:
Step 4: Evaluate the determinant:
Step 5: Given:
Step 6: Final Answer (up to two decimal places):

Step 1: Compare the given equation with the standard form of an ellipse: Here,
Step 2: Eccentricity of an ellipse is given by:
Step 3: Substitute the values:
Step 4: Comparing with the given form , the denominator is: Final Answer (up to two decimal places):

Step 1: Let the vertices of the triangle be Given that the orthocentre is the origin .
Step 2: If the orthocentre is at the origin, then:




Step 3: Slope of : Hence slope of is: Equation of through : Since lies on :
Step 4: Slope of : Hence slope of is: Equation of through : Since lies on :
Step 5: Solve equations (1) and (2): Multiply by 3: Substitute in (1):
Step 6: Hence, the coordinates of the third vertex are:

Step 1: Simplify the given expression. Using the identity:
Step 2: Express in terms of cotangents. Divide numerator and denominator by :
Step 3: Take reciprocal to obtain .
Step 4: Interpret the result. The above relation implies: Hence, is the arithmetic mean of and . Therefore, are in arithmetic progression.
Hence, the correct answer is

Step 1: Use the range of . For real values, Given: This is the maximum possible sum. Hence, each term must be equal to .
Step 2: Find values of .
Step 3: Compute . But note that for to be defined, and equality at all three simultaneously implies , which makes the expression trivial and inconsistent with option patterns. Hence, the only consistent value satisfying the condition in general form is: Therefore, the correct answer is

Step 1: Write the function in piecewise form. Step 2: Check continuity at the critical point .

Left hand limit:
Right hand limit:
Value of the function at :
Step 3: Compare LHL, RHL and . Hence, is continuous at . Step 4: Check continuity elsewhere. Both expressions and are polynomials (or constants), which are continuous for all real . Step 5: Final conclusion.

Step 1: Examine option (A). In a vector space, This is a standard property of vector spaces. Hence, (A) is correct. Step 2: Examine option (B). This follows from the norm property: So, (B) is correct. Step 3: Examine option (C). Scalar multiplication of the zero vector always gives the zero vector: Hence, (C) is correct. Step 4: Examine option (D). The correct vector space property is: which is true independent of any scalar . However, option (D) incorrectly includes the condition “ ”, which is irrelevant and incorrect in this context. Step 5: Final conclusion. Thus, the statement which is not correct as written is option (D).

The formula relating average, sum, and count is: .
Given: .
Given: .
Total Sum .
Total Sum .