If `bar("a"), bar("b")` and `bar("c")` are position vectors of the points A, B, C respectively and `5bar("a") - 3bar("b") - 2bar("c") = bar(0)`, then find the ratio in which the point C divides the line segement BA

Concept: Section formula

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

Find the derivative of the inverse of function y = 2x^{3} – 6x and calculate its value at x = −2

Concept: Derivatives of Inverse Functions

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

A car is moving in such a way that the distance it covers, is given by the equation s = 4t^{2} + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

Concept: Derivatives as a Rate Measure

Find the values of x, for which the function f(x) = x^{3} + 12x^{2} + 36đť‘Ą + 6 is monotonically decreasing

Concept: Increasing and Decreasing Functions

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

Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`

Concept: Methods of Evaluation and Properties of Definite Integral

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

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

Concept: Formation of Differential Equations

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.)

The probability that certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

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

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

Concept: Basic Concepts of Trigonometric Functions

Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0

Concept: Trigonometric Functions > General Solution of Trigonometric Equation of the Type

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

Concept: Trigonometric Functions > Trigonometric equations

Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).

Concept: Pair of Straight Lines > Pair of Lines Passing Through Origin - Combined Equation

Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by 5x^{2} +2xy- 3y^{2} = 0.

Concept: Pair of Straight Lines > Pair of Lines Passing Through Origin - Homogenous Equation

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