ISC (Arts) Class 12 - CISCE Question Bank Solutions for Mathematics

If y = P e^ax + Q e^bx, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Find the following integrals:

`int (x^3 - x^2 + x - 1)/(x - 1) dx`

Evaluate `int(x^3+5x^2 + 4x + 1)/x^2  dx`

Find the differential equation of the family of concentric circles `x^2 + y^2 = a^2`

Evaluate `int tan^(-1) sqrtx dx`

The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.

The Cartesian equation of the line is 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, –5, 0) and parallel to the given line.

Find the image of a point having the position vector: `3hati - 2hatj + hat k` in the plane `vec r.(3hati - hat j + 4hatk) = 2`

Find the value of k if M  = `[(1,2),(2,3)]` and `M^2 - km - I_2 = 0`

Show that the rectangle of the maximum perimeter which can be inscribed in the circle of radius 10 cm is a square of side `10sqrt2` cm.

Solve the differential equation:

`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1

Solve the differential equation:

(1 + y²) dx = (tan⁻¹ y − x) dy

For the following differential equation, find a particular solution satisfying the given condition:

\[x\left( x^2 - 1 \right)\frac{dy}{dx} = 1, y = 0\text{ when }x = 2\]

The cost function of a product is given by C(x) =`x^3/3 - 45x^2 -  900x + 36` where x is the number of units produced. How many units should be produced to minimise the marginal cost?

The marginal cost function of x units of a product is given by 2MC= 3x² -10x +3x² The cost of producing one unit is Rs. 7. Find the cost function and average cost function. 

Find the matrix X for which:
`[(5, 4),(1,1)]` X=`[(1,-2),(1,3)]`

Find the Cartesian equation of the plane, passing through the line of intersection of the planes `vecr. (2hati + 3hatj - 4hatk) + 5 = 0`and `vecr. (hati - 5hatj + 7hatk) + 2 = 0`  intersecting the y-axis at (0, 3).

The total cost function for x units is given by C(x) = `sqrt(6x + 5) + 2500`. Show that the marginal cost decreases as the output x increases.

If the total cost function is given by `C = x + 2x^3 - 7/2x^2`, find the Marginal Average Cost function (MAC).

Solve the differential equation:

`(xdy - ydx)  ysin(y/x) = (ydx + xdy)  xcos(y/x)`.

Find the particular solution satisfying the condition that y = π when x = 1.