Consider a question and two statements:
Question:
Is \( 3x + 2y \) positive? Statement-I: \( x^{3} = -29.8 \) Statement-II: \( y^{3} = 3x \)
Which one of the following is correct in respect of the question and the statements?
A.
Statement-I alone is sufficient to answer the question
B.
Statement-II alone is sufficient to answer the question
C.
Both Statement-I and Statement-II are together sufficient to answer the question
D.
Both Statement-I and Statement-II are not sufficient to answer the question
C. Both Statement-I and Statement-II are together sufficient to answer the question
42.
Consider a question and two statements:
Question:
Does the equation \( ax^{2} + bx + c = 0 \) have real roots of opposite sign? Statement-I: The discriminant \( D > 0 \) Statement-II: \( c/a > 0 \)
Which one of the following is correct in respect of the question and the statements?
A.
Statement-I alone is sufficient to answer the question
B.
Statement-II alone is sufficient to answer the question
C.
Both Statement-I and Statement-II are together sufficient to answer the question
D.
Both Statement-I and Statement-II are not sufficient to answer the question
B. Statement-II alone is sufficient to answer the question
43.
Consider a question and two statements:
Question:
Is \( a^{2} + b^{2} + c^{2} - ab - bc - ca \) (a, b, c are distinct real numbers) always positive? Statement-I: \( a > b > c \) Statement-II: \( a + b + c = 0 \)
Which one of the following is correct in respect of the question and the statements?
A.
Statement-I alone is required to answer the question
B.
Statement-II alone is required to answer the question
C.
Both Statement-I and Statement-II are required to answer the question
D.
Neither Statement-I nor Statement-II is required to answer the question
D. Neither Statement-I nor Statement-II is required to answer the question
44.
Consider a question and two statements:
Question:
Is \( \frac{x^{6} + y^{6}}{x^{4} + y^{4}} \) always greater than \( \frac{x^{4} + y^{4}}{x^{2} + y^{2}} (x \neq y \neq 0) \)? Statement-I: \( x > y \) Statement-II: \( x^{2} + y^{2} > 2xy \)
Which one of the following is correct in respect of the question and the statements?
A.
Statement-I alone is required to answer the question
B.
Statement-II alone is required to answer the question
C.
Both Statement-I and Statement-II are required to answer the question
D.
Neither Statement-I nor Statement-II is required to answer the question
If \( \cos(x + y) = 0 \) and \( \sin(x - y) = \frac{1}{2} \), where \( x, y \in \left[0, \frac{\pi}{2}\right] \), then what is the value of \( \cot(2x - y) \)?
If \( \cos \theta + \sec \theta - 2 = 0 \), where \( 0 \leq \theta < \frac{\pi}{2} \), then what is the value of \( \cos^{4} \theta + \sec^{4} \theta - 2 \)?
