Mathematics

CBSE Class 12

Practice to excel and get familiar with the paper pattern and the type of questions. Check you answers with answer keys provided.

Download the PDF Sample Papers Free for off line practice and view the Solutions online.

2.

Show that all the diagonal elements of a skew symmetric matrix are zero.

Let A [a_{ij}] be a skew symmetric matrix.

so,

a_{ij} =-a_{ji} for all i,j

⇒a_{ii} -a_{ii} for all values of i

⇒2a_{ii} =0

⇒a_{ii} =0 for all values of i

⇒a_{11} = a_{22} = a_{33} =..... a_{nn} =0

1789 Views

3.

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 = 4πr^{2}

3950 Views

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^{–1}) = (det A)^{k}.

3628 Views

8.

Let find a matrix D such that CD – AB = O.

Then 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

530 Views

9.

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 l^{2} + m^{2}+n^{2} = 1

Let the line make angle θ with the positive direction of the z-axis.

α = 90°, β = 60° γ = θ

Thus,

cos^{2} 90 + cos^{2}60 + cos^{2}θ =1

1682 Views

10.

Show that the function f(x) = 4x^{3} – 18x^{2} + 27x – 7 is always increasing on R.

The given function is f(x) =4x^{3} – 18x^{2} + 27x – 7

On differentiating both sides with respect to x, we get

f'(x) = 12x^{2}-36x +27

⇒ f'(x) = 3(4x^{2}-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.

1063 Views

Textbook Solutions | Additional Questions