Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

The distance of the point P (a, b, c) from the x-axis is 

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio

If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are

The angle between the two diagonals of a cube is

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos² α + cos² β + cos²γ + cos² δ is equal to

Find the unit vector in the direction of vector \[\overrightarrow{PQ} ,\]

Show that the points \[A \left( 2 \hat{i} - \hat{j} + \hat{k} \right), B \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right), C \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)\] are the vertices of a right angled triangle.

Find the value of x for which \[x \left( \hat{i} + \hat{j} + \hat{k} \right)\] is a unit vector.

If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k} \text{ and }\vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a unit vector parallel to \[2 \vec{a} - \vec{b} + 3 \vec{c .}\] 

If \[\overrightarrow{AO} + \overrightarrow{OB} = \overrightarrow{BO} + \overrightarrow{OC} ,\] prove that A, B, C are collinear points.

Show that the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear.

If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find \[\frac{dy}{dx}\] When  \[\theta = \frac{\pi}{3}\] .

If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0 .\]

Find : \[\int\frac{2 \cos x}{\left( 1 - \sin x \right) \left( 1 + \sin^2 x \right)}dx\] .

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of golds while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, formulate LPP to maximize profit.

Prove that : \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) = \frac{\pi}{4} + \frac{1}{2} \cos^{- 1} x^2 ;  1 < x < 1\].

Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .

If \[\vec{a}  \times  \vec{b}  =  \vec{c}  \times  \vec{d}   \text { and }   \vec{a}  \times  \vec{c}  =  \vec{b}  \times  \vec{d}\] , show that \[\vec{a}  -  \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .