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Solve the following:
Find the area enclosed between the circle x2 + y2 = 1 and the line x + y = 1, lying in the first quadrant.
Concept: Area Bounded by the Curve, Axis and Line
Find the area of the region lying between the parabolas 4y2 = 9x and 3x2 = 16y
Concept: Area Between Two Curves
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
Prove that:
`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`
Concept: Differential Equations
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Concept: Differential Equations
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Concept: Formation of Differential Equations
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
For the differential equation, find the particular solution (x – y2x) dx – (y + x2y) dy = 0 when x = 2, y = 0
Concept: Differential Equations
A fair coin is tossed 8 times. Find the probability that it shows heads at least once
Concept: Bernoulli Trials and Binomial Distribution
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.)
Given X ~ B(n, p) if p = 0.6 and E(X) = 6, find n and Var(X).
Concept: Mean of Binomial Distribution (P.M.F.)
If the mean and variance of a binomial distribution are 18 and 12 respectively, then n = ______.
Concept: Binomial Distribution
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.)
If X ~ B (n, p) and E(X) = 6 and Var (X) = 4.2, then find n and p.
Concept: Variance of Binomial Distribution (P.M.F.)
Prove the theorem of parallel axes about moment of inertia
Concept: Theorems of Perpendicular and Parallel Axes
Write the necessary equation for maximum safety, speed and state the significance of each term involved in it.
Concept: Banking of Roads
Define moment of inertia. State its SI unit and dimensions.
Concept: Angular Momentum or Moment of Linear Momentum
The moment of inertia of a circular loop of radius R, at a distance of R/2 around a rotating axis parallel to horizontal diameter of the loop is ______
Concept: Moment of Inertia as an Analogous Quantity for Mass
A flywheel is revolving with a constant angular velocity. A chip of its rim breaks and flies away. What will be the effect on its angular velocity?
Concept: Angular Momentum or Moment of Linear Momentum
