
Let \( M \) be a \( 7 \times 7 \) matrix with entries in \( \mathbb{R} \) and having the characteristic polynomial \[ c_M(x) = (x - 1)^\alpha (x - 2)^\beta (x - 3)^2, \] where \( \alpha>\beta \). Let \( {rank}(M - I_7) = {rank}(M - 2I_7) = {rank}(M - 3I_7) = 5 \), where \( I_7 \) is the \( 7 \times 7 \) identity matrix.
If \( m_M(x) \) is the minimal polynomial of \( M \), then \( m_M(5) \) is equal to __________ (in integer).
In the given figure, graph of polynomial \(p(x)\) is shown. Number of zeroes of \(p(x)\) is
| Class | 0 – 15 | 15 – 30 | 30 – 45 | 45 – 60 | 60 – 75 | 75 – 90 |
|---|---|---|---|---|---|---|
| Frequency | 11 | 8 | 15 | 7 | 10 | 9 |
Leaves of the sensitive plant move very quickly in response to ‘touch’. How is this stimulus of touch communicated and explain how the movement takes place?
Read the following sources of loan carefully and choose the correct option related to formal sources of credit:
(i) Commercial Bank
(ii) Landlords
(iii) Government
(iv) Money Lende