HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Solve the following system of equations by the method of reduction:

x + y + z = 6, y + 3z = 11, x + z = 2y.

Prove that the acute angle θ between the lines represented by the equation ax² + 2hxy+ by² = 0 is tanθ = `|(2sqrt(h^2 - ab))/(a + b)|` Hence find the condition that the lines are coincident.

If `y=cos^-1(2xsqrt(1-x^2))`, find dy/dx

Find `dy/dx if y=cos^-1(sqrt(x))`

find dy/dx if `y=tan^-1((6x)/(1-5x^2))`

If `y=sec^-1((sqrtx-1)/(x+sqrtx))+sin_1((x+sqrtx)/(sqrtx-1)), `

(A) x

(B) 1/x

(C) 1

(D) 0

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

Evaluate : `int x^2/((x^2+2)(2x^2+1))dx` 

Find: `I=intdx/(sinx+sin2x)`

If y = f(x) is a differentiable function of x such that inverse function x = f^–1 (y) exists, then prove that x is a differentiable function of y and `dx/dy=1/((dy/dx)) " where " dy/dx≠0`

Find p and q if the equation px² – 8xy + 3y² + 14x + 2y + q = 0 represents a pair of prependicular lines.

The probability mass function for X = number of major defects in a randomly selected
appliance of a certain type is 

Find the expected value and variance of X.

If y = f (x) is a differentiable function of x such that inverse function x = f ^–1(y) exists, then
prove that x is a differentiable function of y and 

`dx/dy=1/(dy/dx)`, Where `dy/dxne0`

Hence if `y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2`

then show that `dy/dx=1/sqrt(1-x^2)`, where  `|x|<1`

Evaluate: `∫8/((x+2)(x^2+4))dx` 

A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.

In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X).

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

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.