HSC Commerce (Marathi Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions

Following table shows the all India infant mortality rates (per '000) for years 1980 to 2010:

Fit the trend line to the above data by the method of least squares.

Evaluate.

`int (5x^2 - 6x + 3) / (2x -3) dx`

Find `dy / dx` if, x = `e^(3t), y = e^sqrt t` 

Solve the following.

If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`

Find `dy/dx` if, x = e^3t, y = `e^sqrtt`

Solve the following differential equation:

`"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0`

Find the equation of the line of regression of Y on X for the following data:

n = 8, `sum(x_i - barx).(y_i - bary) = 120, barx = 20, bary = 36, sigma_x = 2, sigma_y = 3`

Solve the following problem :

Five jobs must pass through a lathe and a surface grinder, in that order. The processing times in hours are shown below. Determine the optimal sequence of the jobs. Also find the idle time of each machine.

Solve the following problem :

Find the sequence that minimizes the total elapsed time to complete the following jobs. Each job is processed in order AB.

Determine the sequence for the jobs so as to minimize the processing time. Find the total elapsed time and the idle times for both the machines.

A toy manufacturing company produces five types of toys. Each toy has to go through three machines A, B, C in the order ABC. The time required in hours for each process is given in the following table.

Solve the problem for minimizing the total elapsed time.

Choose the correct alternative:

If x = 2am, y = 2am², where m be the parameter, then `("d"y)/("d"x)` = ? 

If x = `"a"("t" - 1/"t")`, y = `"a"("t" + 1/"t")`, where t be the parameter, then `("d"y)/("d"x)` = ?

State whether the following statement is True or False:

If x = 2at, y = 2a, where t is parameter, then `("d"y)/("d"x) = 1/"t"`

State whether the following statement is True or False:

If x = 5m, y = m, where m is parameter, then `("d"y)/("d"x) = 1/5`

If x = `(4"t")/(1 + "t"^2)`, y = `3((1 - "t"^2)/(1 + "t"^2))`, then show that `("d"y)/("d"x) = (-9x)/(4y)` 

Find `("d"y)/("d"x)`, if x = e^m, y = `"e"^(sqrt("m"))`

Solution: Given, x = e^m and y = `"e"^(sqrt("m"))`

Now, y = `"e"^(sqrt("m"))`

Diff.w.r.to m,

`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`

∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))`    .....(i)

Now, x = e^m

Diff.w.r.to m,

`("d"x)/"dm" = square`    .....(ii)

Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`

∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`

∴  `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`

State whether the following statement is True or False:   

A homogeneous differential equation is solved by substituting y = vx and integrating it

Choose the correct alternative:

The slope of the line of regression of y on x is called the ______

Choose the correct alternative:

If the lines of regression of Y on X is y = `x/4` and X on Y is x = `y/9 + 1` then the value of r is

Choose the correct alternative:

u = `(x - 20)/5` and v = `(y - 30)/4`, then b_xy =