Show that all the diagonal elements of a skew symmetric matrix are zero.
Let A = [aij] n × n be skew symmetric matrix
A is skew symmetric
∴ A = –A/
⇒ aij = –aji ∀ i, j
For diagonal elements i = j,
⇒ 2aii = 0
⇒ aii = 0
⇒ diagonal elements are zero.
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.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Answer the following as true or false.
\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Let f : N → N be defined by
State whether the function f is bijective. Justify your answer.
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,
)
Let f : X → Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f).
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
A balloon, which always remains spherical, has a variable diameter
\begin{align} \frac{3}{2}(2x+1)\end{align}
Find the rate of change of its volume with respect to x.
Determine order and degree(if defined) of differential equation yn + 2y' + siny = 0