Commerce (English Medium) Class 12 - CBSE Important Questions

Solve:

`2(y + 3) - xy  (dy)/(dx)` = 0, given that y(1) = – 2.

The order and degree of the differential equation `[1 + ((dy)/(dx))^2] = (d^2y)/(dx^2)` are ______.

Write the sum of the order and the degree of the following differential equation:

`d/(dx) (dy/dx)` = 5

Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.

`(dy)/(dx) + ycotx = 2/(1 + sinx)`

If m and n, respectively, are the order and the degree of the differential equation `d/(dx) [((dy)/(dx))]^4` = 0, then m + n = ______.

Find the general solution of the differential equation:

`(dy)/(dx) = (3e^(2x) + 3e^(4x))/(e^x + e^-x)`

The order and the degree of the differential equation `(1 + 3 dy/dx)^2 = 4 (d^3y)/(dx^3)` respectively are ______.

If `(a + bx)e^(y/x)` = x then prove that `x(d^2y)/(dx^2) = (a/(a + bx))^2`.

The degree of the differential equation `[1 + (dy/dx)^2]^3 = ((d^2y)/(dx^2))^2` is ______.

Write the number of vectors of unit length perpendicular to both the vectors `veca=2hati+hatj+2hatk and vecb=hatj+hatk`

Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector

`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.

If `veca=4hati-hatj+hatk` then find a unit vector parallel to the vector `veca+vecb`

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 \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .

Find the area of a parallelogram whose adjacent sides are represented by the vectors\[2 \hat{i} - 3 \hat{k} \text { and } 4 \hat{j} + 2 \hat{k} .\]

Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2

If `veca, vecb, vecc` are three vectors such that `veca.vecb = veca.vecc` and `veca xx vecb = veca xx vecc, veca ≠ 0`, then show that `vecb = vecc`.

If `|veca`| = 3, `|vecb|` = 5, `|vecc|` = 4 and `veca + vecb + vecc` = `vec0`, then find the value of `(veca.vecb + vecb.vecc + vecc.veca)`.

If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.

If `veca = 4hati + 6hatj` and `vecb = 3hatj + 4hatk`, then the vector form of the component of `veca` along `vecb` is ______.