Arts (English Medium) Class 12 - CBSE Important Questions

Find the area bounded by the circle x² + y² = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.

Find the particular solution of the differential equation

(1 – y²) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.

Find the general solution of the following differential equation : 

`(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0`

Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.

If y = P e^ax + Q e^bx, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Find the general solution of the following differential equation:

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

Read the following passage:

Based on the above, answer the following questions:

1. Show that (x² – y²) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
2. Solve the above equation to find its general solution. (2)

Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1

Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel

Assertion (A): If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin² α + sin² β + sin² γ = 2.

Reason (R): The sum of squares of the direction cosines of a line is 1.

Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines. 

`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " and "vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk)`

If the Cartesian equations of a line are ` (3-x)/5=(y+4)/7=(2z-6)/4` , write the vector equation for the line.

Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines `(x - 8)/3 = (y + 19)/(-16) = (z - 10)/7` and `(x - 15)/3 = (y - 29)/8 = (z - 5)/(-5)`

If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.

Read the following passage and answer the questions given below.

Based on the above information, answer the following questions:

1. Find the shortest distance between the given lines.
2. Find the point at which the motorcycles may collide.

Equation of a line passing through (1, 1, 1) and parallel to z-axis is ______.

If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.

The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is ______.

Find the distance between the lines:

`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;

`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.