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.

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`

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`

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.

Concept: Methods of Integration - Integration by Parts

Find `intsqrtx/sqrt(a^3-x^3)dx`

Concept: Methods of Integration - Integration by Substitution

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Concept: Evaluation of Definite Integrals by Substitution

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

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

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.

Concept: Random Variables and Its Probability Distributions

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

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.

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.

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 `

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.

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.

Concept: Elementary Operation (Transformation) of a Matrix

Examine the maxima and minima of the function f(x) = 2x^{3} - 21x^{2} + 36x - 20 . Also, find the maximum and minimum values of f(x).

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.

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`

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

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).

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.

Concept: Variance of Binomial Distribution (P.M.F.)

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