HSC Commerce (Marathi Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

If A = [a_ij]_2×2, where a_ij = i – j, then A = ______

Find k, if A = `[(3, -2),(4, -2)]` and A² = kA – 2I, where I is identity matrix of order 2

If A = `[(2, 1),(0, 3),(1, -1)]` and B = `[(0, 3, 5),(1, -7, 2)]`, then verify (BA)^T = A^TB^T

Complete the following activity.

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`

R₁ ↔ R₂

`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`

R₂ – 4R₁, R₃ – 6R₁

`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`

`(-1)/5 "R"_2, (-1)/5 "R"_3`

`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`

R₃ – 2R₂

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`

By pre multiplying we get,

x + 2y + ( )z = ( )    .....(i)

y + 2z = 24    ......(ii)

–z = ( )      ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

If A = `[(2, 3),(a, 6)]` is a singular matrix, then a = ______.

Find x, y, z if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, -2),(1, 3)]}[(2),(1)] = [(x  + 1),(y - 1), (3z)]`

Find the inverse of the matrix A by using adjoint method.

where A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`

Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`

Find `("d"y)/("d"x)`, if y = [log(log(logx))]² 

Find `("d"^2y)/("d"x^2)`, if y = `"e"^((2x + 1))`

Find `("d"y)/("d"x)`, if y = `root(3)(((3x - 1))/((2x + 3)(5 - x)^2)`

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")`

If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`

If ax² + 2hxy + by² = 0, then prove that `(d^2y)/(dx^2)` = 0.

If the demand function is D = 50 – 3p – p². Find the elasticity of demand at p = 5 comment on the result

Choose the correct alternative:

Slope of the normal to the curve 2x² + 3y² = 5 at the point (1, 1) on it is 

Choose the correct alternative:

The function f(x) = x³ – 3x² + 3x – 100, x ∈ R is

The price P for the demand D is given as P = 183 + 120D − 3D², then the value of D for which price is increasing, is ______.

If the elasticity of demand η = 1, then demand is ______.