PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

Find the value of p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l₁.

Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Integrate the functions:

`(1+ log x)^2/x`

Find x, y, a and b if

`[[2x-3y,a-b,3],[1,x+4y,3a+4b]]`=`[[1,-2,3],[1,6,29]]`

Find the values of a, b, c and d from the following equations:`[[2a + b,a-2b],[5c-d,4c + 3d ]]`= `[[4,- 3],[11,24]]`

Find x, y and z so that A = B, where`A= [[x-2,3,2x],[18z,y+2,6x]],``b=[[y,z,6],[6y,x,2y]]`

`If [[x,3x- y],[2x+z,3y -w ]]=[[3,2],[4,7]]` find x,y,z,w

`If [[x-y,z],[2x-y,w]]=[[-1,4],[0,5]]`Find X,Y,Z,W.

`If [[x + 3 , z + 4 ,     2y-7 ],[4x + 6,a-1,0 ],[b-3,3b,z + 2c ]]= [[0,6,3y-2],[2x,-3,2c-2],[2b + 4,-21,0]]`Obtain the values of a, b, c, x, y and z.

`If [[2x +1    5x],[0     y^2 +1]]``= [[x+3   10],[0      26 ]]`, find the value of (x + y).

`If [[xy          4],[z+6     x+y ]]``=[[8     w],[0     6]]`, then find the values of X,Y,Z and W . 

Given an example of

a row matrix which is also a column matrix,

For what values of x and y are the following matrices equal?

`A=[[2x+1   2y],[0              y^2 - 5y]]``B=[[x + 3      y^2 +2],[0        -6]]`

Find the values of x and y if

`[[X + 10,Y^2 + 2Y],[0, -4]]`=`[[3x +4,3],[0,y^2-5y]]`

For what values of a and b if A = B, where

`A = [[a + 4        3b],[8        -6]]   B = [[2a +2              b^2+2],[8                    b^2  - 5b]]`

Disclaimer: There is a misprint in the question, b² − 5b should be written instead of b² − 56.

If  \[\begin{bmatrix}x + 3 & 4 \\ y - 4 & x + y\end{bmatrix} = \begin{bmatrix}5 & 4 \\ 3 & 9\end{bmatrix}\] , find x and y

Find the value of x from the following: `[[2x - y          5],[ 3                         y ]]` = `[[6            5 ],[3         - 2\]]`

Find the value of x, if  \[\begin{bmatrix}3x + y & - y \\ 2y - x & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ - 5 & 3\end{bmatrix}\]