Science (English Medium) Class 12 - CBSE Important Questions for Mathematics

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

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)`