Show that all the diagonal elements of a skew symmetric matrix are zero.
Let A [aij] be a skew symmetric matrix.
aij =-aji for all i,j
⇒aii -aii for all values of i
⇒aii =0 for all values of i
⇒a11 = a22 = a33 =..... ann =0
The volume of a sphere is increasing at the rate of 3 cubic centimetres per second. Find the rate of increase of its surface area, when the radius is 2 cm.
Now, let S be the surface area of the sphere at any time t. then,
S = 4πr2
If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k.
Let find a matrix D such that CD – AB = O.
By equality of matrices we get,
2p +5r-3 = 0 ....(1)
3p +8r-43 = 0 ..(2)
2q +5s = 0 ......(3)
3q +8s-22 = 0 ..(4)
By solving (1) and (2) we get p = -191 and r = 77
Similarly, on solving (3) and (4) we get q = - 110 and s = 44
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of the z-axis.
suppose the direction cosines of the line be l,m,and n.
we know that l2 + m2+n2 = 1
Let the line make angle θ with the positive direction of the z-axis.
α = 90°, β = 60° γ = θ
cos2 90 + cos260 + cos2θ =1
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 is always increasing on R.
The given function is f(x) =4x3 – 18x2 + 27x – 7
On differentiating both sides with respect to x, we get
f'(x) = 12x2-36x +27
⇒ f'(x) = 3(4x2-12x+9)
⇒ f'(x) = 3(2x-3)2
Which is always positive for all x ε R.
Since, f'(x) ≥ 0 ∀ x ε R,
Therefore, f(x) is always increasing on R.