Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions

To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below:
 

What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.

Show that  \[\begin{vmatrix}y + z & x & y \\ z + x & z & x \\ x + y & y & z\end{vmatrix} = \left( x + y + z \right) \left( x - z \right)^2\]

x + y = 1
x + z = − 6
x − y − 2z = 3

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

If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A^-1Hence, solve the system of equations: 

x +y + z = 6

y + 3z = 11

and x -2y +z = 0

Find the inverse of the following matrix, using elementary transformations: 

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`

Find `int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `

Find `int_  sin ("x" - a)/(sin ("x" + a )) d"x"`

Find `int_  (log "x")^2 d"x"`

Find `int_  (sin2"x")/((sin^2 "x"+1)(sin^2"x"+3))d"x"`

Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration. 

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of types A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 4 hours available for assembling. The profit is ₹ 50 each for type A and ₹60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above  LPP and solve it graphically and find the maximum profit.

Find: `int_  (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`

Write the value of `|(a-b, b- c, c-a),(b-c, c-a, a-b),(c-a, a-b, b-c)|`

On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, everyone would have got ₹ 10 more. However, if there were 16 children more, everyone would have got ₹ 10 less. Using the matrix method, find the number of children and the amount distributed by Seema. What values are reflected by Seema’s decision?

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.

Find:
`int"dx"/sqrt(5-4"x" - 2"x"^2)`

Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.

Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`

Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`