Science (English Medium) Class 12 - CBSE Important Questions

Form the differential equation representing the family of curves `y² = m(a² - x²) by eliminating the arbitrary constants 'm' and 'a'. 

Solve the differential equation:  ` (dy)/(dx) = (x + y )/ (x - y )`

Form the differential equation representing the family of curves y = e²x (a + bx), where 'a' and 'b' are arbitrary constants.

Find the particular solution of the differential equation `x (dy)/(dx) - y = x^2.e^x`, given y(1) = 0.

Find the general solution of the differential equation `x (dy)/(dx) = y(logy - logx + 1)`.

Find the general solution of the differential equation:

`log((dy)/(dx)) = ax + by`.

Degree of the differential equation `sinx + cos(dy/dx)` = y² is ______.

Find the general solution of the differential equation:

(xy – x²) dy = y² dx

The sum of the order and the degree of the differential equation `d/dx[(dy/dx)^3]` is ______.

Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`

Show that the points A, B, C with position vectors `2hati- hatj + hatk`, `hati - 3hatj - 5hatk` and `3hati - 4hatj - 4hatk` respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, find the angle which `veca + vecb + vecc`make with `veca or vecb or vecc`

Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.

Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk`  and  `vecb = 3hati - 2hatj + 2hatk` . 

Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.

Let `veca` , `vecb` and `vecc` be three vectors such that `|veca| = 1,|vecb| = 2, |vecc| = 3.` If the projection of `vecb` along `veca` is equal to the projection of `vecc` along `veca`; and `vecb` , `vecc` are perpendicular to each other, then find `|3veca - 2vecb + 2vecc|`.

if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and  hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.

Projection of vector `2hati + 3hatj` on the vector `3hati - 2hatj` is ______.

A line l passes through point (– 1, 3, – 2) and is perpendicular to both the lines `x/1 = y/2 = z/3` and `(x + 2)/-3 = (y - 1)/2 = (z + 1)/5`. Find the vector equation of the line l. Hence, obtain its distance from the origin.