HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

The value of objective function is maximum under linear constraints ______.

 Maximize: z = 3x + 5y  Subject to

x +4y ≤ 24                3x + y  ≤ 21 

x + y ≤ 9                     x ≥ 0 , y ≥0

The total cost function of a firm is C = x² + 75x + 1600 for output x. Find the  output for which the average cost ls minimum. Is C_A= C_m at this output?  

Find the joint equation of the pair of the line through the point (2, -1) and parallel to the lines represented by 2x² + 3xy - 9y² = 0.

Find the joint equation of the pair of the line through the point (2, -3) and parallel to the lines represented by x² + xy - y² = 0.

Show that the equation x² + 2xy + 2y² + 2x + 2y + 1 = 0 does not represent a pair of lines.

Show that the equation 2x² − xy − 3y² − 6x + 19y − 20 = 0 represents a pair of lines.

Show that the equation 2x² + xy - y² + x + 4y - 3 = 0 represents a pair of lines. Also, find the acute angle between them.

Find the separate equation of the line represented by the following equation:

(x - 2)² - 3(x - 2)(y + 1) + 2(y + 1)² = 0

Find the value of k, if the following equations represent a pair of line:

3x² + 10xy + 3y² + 16y + k = 0

Find the value of k, if the following equations represent a pair of line:

kxy + 10x + 6y + 4 = 0

Find the value of k, if the following equations represent a pair of line:

x² + 3xy + 2y² + x - y + k = 0

Find p and q, if the equation 2x² + 8xy + py² + qx + 2y - 15 = 0 represents a pair of parallel lines.

Equations of pairs of opposite sides of a parallelogram are x² - 7x + 6 = 0 and y² − 14y + 40 = 0. Find the joint equation of its diagonals.

Find the separate equation of the line represented by the following equation:

10(x + 1)² + (x + 1)(y - 2) - 3(y - 2)² = 0 

ΔOAB is formed by the lines x² - 4xy + y² = 0 and the line AB. The equation of line AB is 2x + 3y - 1 = 0. Find the equation of the median of the triangle drawn from O.

Find the coordinates of the points of intersection of the lines represented by x² − y² − 2x + 1 = 0

Find k, if the slopes of lines given by kx² + 5xy + y² = 0 differ by 1.

Find the length of the perpendicular (2, –3, 1) to the line `(x + 1)/(2) = (y - 3)/(3) = (z + 1)/(-1)`.

Find the co-ordinates of the foot of the perpendicular drawn from the point `2hati - hatj + 5hatk` to the line `barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk).` Also find the length of the perpendicular.