Let A = \(\begin{bmatrix}1 & 1 & -2\\2 & 1 & -3\\5 & 4 & -9\end{bmatrix}\)
By expanding along the first row, we have:
|A| = 1\(\begin{vmatrix}1 & -3\\4 & -9\end{vmatrix}\) - 1\(\begin{vmatrix}2 & -3\\5 & -9\end{vmatrix}\) - 2\(\begin{vmatrix}2 & 1\\5 & 4\end{vmatrix}\)
= 1(-9 + 12) – 1(-18 + 15) -2(8 – 5)
= 1(3) – 1 (-3) – 2(3)
= 3 + 3 – 6
= 6 – 6
= 0
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force
(iv) velocity (v) work done
Determine order and degree(if defined) of differential equation (ym)2 + (yn)3 + (y')4 + y5 =0
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres north-west (iii) 40°
(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2