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Important Questions for HSC Science (General) 12th Board Exam - Maharashtra State Board - Mathematics and Statistics

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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.

Appears in 3 question papers
Chapter: [0.11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  `dy/dx `at t = `pi/4`

Appears in 3 question papers
Chapter: [0.13] Differentiation
Concept: Derivatives of Functions in Parametric Forms

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`

 

Appears in 3 question papers
Chapter: [0.13] Differentiation
Concept: Derivative > Derivative of Inverse Function

Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

                      = 0,                   if f (x) is an odd function.

Appears in 3 question papers
Chapter: [0.15] Integration
Concept: Methods of Integration - Integration by Parts

Find `intsqrtx/sqrt(a^3-x^3)dx`

Appears in 3 question papers
Chapter: [0.15] Integration
Concept: Methods of Integration - Integration by Substitution

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Appears in 3 question papers
Chapter: [0.15] Integration
Concept: Evaluation of Definite Integrals by Substitution

If y = P eax + Q ebx, show that

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

Appears in 3 question papers
Chapter: [0.17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.

Appears in 3 question papers
Chapter: [0.19] Probability Distribution
Concept: Random Variables and Its Probability Distributions

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

Appears in 2 question papers
Chapter: [0.01] Mathematical Logic
Concept: Mathematical Logic > Application of Logic to Switching Circuits, Switching Table.

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

Appears in 2 question papers
Chapter: [0.011000000000000001] Mathematical Logic
Concept: Mathematical Logic > Application of Logic to Switching Circuits, Switching Table.

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Appears in 2 question papers
Chapter: [0.012] Matrics
Concept: Elementary Operation (Transformation) of a Matrix

Prove that the volume of a parallelopiped with coterminal edges as  ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges  `bar i+barj, barj+bark `

Appears in 2 question papers
Chapter: [0.015] Vectors
Concept: Scalar Triple Product of Vectors

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Appears in 2 question papers
Chapter: [0.017] Linear Programming
Concept: Graphical Method of Solving Linear Programming Problems

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Appears in 2 question papers
Chapter: [0.02] Matrices
Concept: Elementary Operation (Transformation) of a Matrix

Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x). 

Appears in 2 question papers
Chapter: [0.022000000000000002] Applications of Derivatives
Concept: Maxima and Minima

Show that the height of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is `(2R)/sqrt3.`  Also find the maximum volume.

Appears in 2 question papers
Chapter: [0.022000000000000002] Applications of Derivatives
Concept: Maxima and Minima

Prove that `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)-a^2/2log|x+sqrt(x^2-a^2)|+c`

 

Appears in 2 question papers
Chapter: [0.023] Indefinite Integration
Concept: Methods of Integration - Integration by Parts

Order and degree of the differential equation `[1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2)` are respectively 

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

Appears in 2 question papers
Chapter: [0.026000000000000002] Differential Equations
Concept: Order and Degree of a Differential Equation

Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).

Appears in 2 question papers
Chapter: [0.027999999999999997] Binomial Distribution
Concept: Bernoulli Trials and Binomial Distribution

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

X = x 0 1 2 3 4
P(X = x) 0.08 0.15 0.45 0.27 0.05

Find the expected value and variance of X.

Appears in 2 question papers
Chapter: [0.027999999999999997] Binomial Distribution
Concept: Variance of Binomial Distribution (P.M.F.)
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