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

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)`

Find the distance between the point (−1, −5, −10) and the point of intersection of the line `(x-2)/3=(y+1)/4=(z-2)/12` and the plane x-y+z=5

If lines `(x−1)/2=(y+1)/3=(z−1)/4 and  (x−3)/1=(y−k)/2=z/1` intersect, then find the value of k and hence find the equation of the plane containing these lines.

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 equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Find the distance of the point (2, 12, 5) from the point of intersection of the line 

`vecr=2hati-4hat+2hatk+lambda(3hati+4hatj+2hatk) `

Find the equation of the plane through the line of intersection of `vecr*(2hati-3hatj + 4hatk) = 1`and `vecr*(veci - hatj) + 4 =0`and perpendicular to the plane `vecr*(2hati - hatj + hatk) + 8 = 0`. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.

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.

If the distance of the point (1, 1, 1) from the plane x – y + z + λ = 0 is `5/sqrt(3)`, find the value(s) of λ.

If the distance of the point (1, 1, 1) from the plane x – y + z + λ = 0 is `5/sqrt(3)`, find the value(s) of λ.

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

Find the coordinates of points on line `x/1 = (y - 1)/2 = (z + 1)/2` which are at a distance of `sqrt(11)` units from origin.

Find the coordinates of points on line `x/1 = (y - 1)/2 = (z + 1)/2` which are at a distance of `sqrt(11)` units from origin.

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.