If u, v and w (all positive) are the pth, qth and rth terms of a GP, then the determinant of the matrix \( \begin{bmatrix} \ln u & p & 1 \\ \ln v & q & 1 \\ \ln w & r & 1 \end{bmatrix} \) is
Let the coefficient of the middle term of the binomial expansion of \( (1 + x)^{2n} \) be \( \alpha \) and those of two middle terms of the binomial expansion of \( (1 + x)^{2n-1} \) be \( \beta \) and \( \gamma \). Which one of the following relations is correct?
Let \( A = \{x \in R : -1 \leq x \leq 1\} \), \( B = \{y \in R : -1 \leq y \leq 1\} \) and S be the subset of \( A \times B \), defined by \( S = \{(x, y) \in A \times B : x^{2} + y^{2} = 1\} \). Which one of the following is correct?
Let \( T_{r} \) be the \( r^{th} \) term of an AP for r = 1, 2, 3, .... If for some distinct positive integers m and n we have \( T_{m} = \frac{1}{n} \) and \( T_{n} = \frac{1}{m} \), then what is \( T_{mn} \) equal to?
Consider the following in respect of matrices A, B and C of same order:
1. \( (A + B + C)' = A' + B' + C' \)
2. \( (AB)' = A'B' \)
3. \( (ABC)' = C'B'A' \)
where A′ is the transpose of the matrix A.
Which of the above are correct?
The sum of the binary numbers \( (11011)_{2} \), \( (10110110)_{2} \) and \( (10011x0y)_{2} \) is the binary number \( (101101101)_{2} \). What are the values of x and y?
Let matrix B be the adjoint of a square matrix A, I be the identity matrix of same order as A. If k (≠ 0) is the determinant of the matrix A, then what is AB equal to?
