Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Write the direction cosines of the vector \[\hat{i} + 2 \hat{j} + 3 \hat{k}\].

Write the direction cosines of the vectors \[- 2 \hat{i} + \hat{j} - 5 \hat{k}\].

The vector cos α cos β \[\hat{i}\] + cos α sin β \[\hat{j}\] + sin α \[\hat{k}\] is a

A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10,000. If the profit is Rs 60 per unit for the product A and Rs 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.

If l_i, m_i, n_i, i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^T = I, where \[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]

If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − A^T is a skew-symmetric matrix.

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).

A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of ₹ 12 and ₹ 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?

The solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x} + \frac{\phi\left( \frac{y}{x} \right)}{\phi'\left( \frac{y}{x} \right)}\] is

\[\frac{dy}{dx} = \frac{\sin x + x \cos x}{y\left( 2 \log y + 1 \right)}\]

x (e^2y − 1) dy + (x² − 1) e^y dx = 0

\[\frac{dy}{dx} + 1 = e^{x + y}\]

\[\frac{dy}{dx} = \left( x + y \right)^2\]

cos (x + y) dy = dx

\[\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}\]

\[\frac{dy}{dx} = \frac{y\left( x - y \right)}{x\left( x + y \right)}\]

(x + y − 1) dy = (x + y) dx

\[\frac{dy}{dx} - y \cot x = cosec\ x\]

\[\frac{dy}{dx} - y \tan x = - 2 \sin x\]

\[\frac{dy}{dx} - y \tan x = e^x \sec x\]