HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

Show that four points A, B, C and D whose position vectors are 

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

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

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

Find: `int(x+3)sqrt(3-4x-x^2dx)`

Find: `I=intdx/(sinx+sin2x)`

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours  `("Given"  sqrt(2) = 1.414)`

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.

Write the following compound statement symbolically.

Nagpur is in Maharashtra and Chennai is in Tamil Nadu. 

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(p ∧ ~ q) → (~ p ∧ ~ q)

Find k, if the sum of the slopes of the lines represented by x² + kxy – 3y² = 0 is twice their product.

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line `(x + 3)/(5) = (y - 1)/(2) = (z + 4)/(3)`.

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 derivative of the inverse of function y = 2x³ – 6x and calculate its value at x = −2

If y = cos(m cos^–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. Find the velocity when 𝑡 = 2 sec

Find `intsqrtx/sqrt(a^3-x^3)dx`

Choose the correct alternative:

`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`