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

Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]

The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution

The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting

Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?

What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?

Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is

Which of the following differential equations has y = C₁ e^x + C₂ e^−x as the general solution?

The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]

The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.

Classify the following measures as scalars and vectors:
(i) 15 kg
(ii) 20 kg weight
(iii) 45°
(iv) 10 meters south-east
(v) 50 m/sec²

Classify the following as scalars and vector quantities:
(i) Time period
(ii) Distance
(iii) displacement
(iv) Force
(v) Work
(vi) Velocity
(vii) Acceleration

Answer the following as true or false:
\[\vec{a}\] and \[\vec{a}\]  are collinear.

Answer the following as true or false:
Two collinear vectors are always equal in magnitude.

Answer the following as true or false:
Zero vector is unique.

Answer the following as true or false:
Two vectors having same magnitude are collinear.

Answer the following as true or false:
Two collinear vectors having the same magnitude are equal.

If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors having the same initial point. What are the vectors represented by \[\vec{a}\] + \[\vec{b}\]  and \[\vec{a}\] − \[\vec{b}\].

If \[\vec{a}\] is a vector and m is a scalar such that m \[\vec{a}\] = \[\vec{0}\], then what are the alternatives for m and \[\vec{a}\] ?

Five forces \[\overrightarrow{AB,}   \overrightarrow { AC,} \overrightarrow{ AD,}\overrightarrow{AE}\] and \[\overrightarrow{AF}\] act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 \[\overrightarrow{AO,}\] where O is the centre of hexagon.

If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that \[\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}\]