PUC Science 2nd PUC Class 12 - Karnataka Board PUC Important Questions

If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.

Evaluate :

`int_e^(e^2) dx/(xlogx)`

Evaluate the following integral:

Evaluate:

`∫ (1)/(sin^2 x cos^2 x) dx`

Evaluate the following integral:

Evaluate each of the following integral:

`int "dx"/(("x" - 8)("x" + 7))`=

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Find the differential equation representing the curve y = cx + c².

Write the integrating factor of the following differential equation:

(1+y²) dx−(tan⁻¹ y−x) dy=0

Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`

Solve the following differential equation:
\[\text{ cosec }x \log y \frac{dy}{dx} + x^2 y^2 = 0\]

For the following differential equation, find a particular solution satisfying the given condition:

\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]

If `veca and vecb` are two vectors such that `|veca+vecb|=|veca|,` then prove that vector `2veca+vecb` is perpendicular to vector `vecb`

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

The two adjacent sides of a parallelogram are `2hati-4hatj-5hatk and 2 hati+2hatj+3hatj` . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

If `veca and vecb` are perpendicular vectors, `|veca+vecb| = 13 and |veca| = 5` ,find the value of `|vecb|.`