3. Evaluate the following: cos1(t)dt C. dx b. J Jezx-1 In(x)dx V1-t2 a.

Q: c) dx E (sin cos) dt

A: We find ddx ∫xπ/2   sint2   cost3 dt

Q: 8. (t² + 3) cos(5t) dt

A:

Q: Given y= 5e3X + sin x, -is dx A) 5e3x + cosx B 5e3x – cosx 15e3x + cosx D 15е3х — сosx

A:

Q: f (t) = t(e-2t + cos at)

A:

Q: cos-1(2 x2)

A:

Q: solve D.E. by using homo. xsin-- ycos-) dx + xcos-dy = o

A: put y=vx dydx=v+xdvdx differential form  v+xdvdx=vxcosv-xsinvxcosv take x common and split the…

Q: dx/dt + 4x = cos(2t), x(0) = -1 Find x(t)

A:

Q: If Xy +y.1 find the value of dy dx 2 5- at the point Co,-1)

A:

Q: Solve the following equation for the given conditions y(0) = -1,y'(1) = 2 d²y dy + 4 dx :+ 4y = sin…

A: Given differential equation is d2ydx2+4dydx+4y=sin2x+cos2x      (1) with y(0)=-1, y'(1)=2 Auxiliary…

Q: (d^2 y)/〖dt〗^2 +16y=cos ⁡t

A: The given equation is; (d2 y)/(dt2) +16y=cos ⁡t

Q: Solve the given DE: dy ex – 2x cos y ey – x² sin y' - dx

A:

Q: Q3 Find dy/dx on the graph y3 sec/x2 + y2 = cos2x at point (1, 2).

A: The given function is, y3secx2+y2=cos2x

Q: Given the following DE (cos 2y – 3x?y?)dx + (cos 2y – 2x sin 2y – 2xy)dy = 0 -

A:

Q: Solve the following DE: [2x+ ycos(xy)] dx+ x cos(xy) dy = 0

A:

Q: Find dx/dt: x = 2[cos (t/2)]^2

A: x=2cost22

Q: dw "(1),y = sin (1),-: 2) w=+ x= cos at =3 Ans. -1 dt

A:

Q: Solve D.E using Exact from first order (x' +y' )dx+(2xy + cos y)dy

A: A differential equations M dx + N dy = 0 is exact if  My = Nx

Q: d²y Solve the following DE : dy + 2 dx + y = cos²x dx2

A:

Q: a) y = cos(e 2x) (e2x) -1

A: Differentiate wrt x

Q: B-The result of dy/dx for the following function is cos x y = 1+sin x

A: We have to find out derivative

Q: 14. Solve the equation d?x dx dt +2x sin 3 dr? dx given that at t= 0, x = 0 and %3D dt ment

A:

Q: 4. Find d2y/dx? for the following functions: a) y = sin x2 cos 2x b) y = cos (sec? 3x)

A: Given that  a)y=sinx2cos2x b)y=cos4sec23x find d2ydx2

Q: B-The result of dy/dx for the following function is COS X y 1+sin x

A:

Q: Q3 Find dy/dx on the graph y3 sec /x2 + y2 = cos2x at point (1, 2).

A: We have to find the derivative  of given expression at point (1, 2)

Q: Find dy/dt. y= cos ~ 10t + 11)

A:

Q: If y = x* cos' x-e* In( 7x +1) , then find dy at x=0. dx 3

A:

Q: Find dy : (a) y = cos(1 Ix + 5x) dx

A: Topic:- calculus

Q: find dx/dt x=(cos2(t/2))

A:

Q: 3x d. Evaluate dx COS cos (3t – 1) dt

A:

Q: L(cos at) = %3D s2 - a2 صواب İhi O

A: Given that  L(cosat)=s/(s^2-a^2) To find the condition is true or false

Q: T For y 10 cos ( 10x 4 - its amplitude is its period is its phase shift is

A:

Q: Q4: Find dy\dx if|cos'y +5y? = x?

A: The given equation iscos2y+5y2=x2to find dydx

Q: Evaluate J2 e* cosx dx 0.

A: To evaluate: The definite integral ∫0π2ex·cos x·dx. Concept used: Integration by parts:…

Q: A ) Find ſ cos 5t dt dy B) If y = x² +; , x= 2t -5 then find dt

A:

Q: Solve the following D.Es: d²x 1. di 2 dy +9x =0. when t-0, x-2 and dr dt 0. dt =2 cos? y dr 2.

A:

Q: T=V3 + cos 0 is given by:

A: Trrr

Q: (b) Calculate e cos(Tx) dx using integration by parts as many times as necessary.

A: Given: ∫excos(πx) dx The formula for integration by parts is given as…

Q: 5 Find *: (a) y = cos(11x + 2x) dy

A:

Q: dW Find at t=0, if dt .3, W = x*y+ zxy whe x = sin2t, y = cos ?

A:

Q: The general solution of the DE dx dy = cos?y is

A:

Q: b) ſ cos (Inx)dx=?

A:

Q: Q5) a) Use potential function to evaluate the following. e* Sin(y) dx + e*Cos(y)dy (o)

A: Q5) a)The given integral is, ∫(π2,0)(π2,π2)exsin(y) dx+ excos(y) dy

Q: Q2:B/ Find dy/dx f Via - "sin Vi dt y = + cos t dt

A:

Q: If x = 3 sin* t and y = 2 cos" t , then find dy at t = dx 4

A:

Q: (e) y' +2y : = cos(2x)

A:

Q: I. Find dy/dx in the following: 1. Y= sin In(x + 1)

A: To find the derivative of given function.

Q: If Xy +y'=1, find at the point Co,-1) 4P dx 2 5- the value of

A:

Q: d Cvaluate dx cos (r² + 1) dt

A:

Q: d. a) dt [8 sin 1 2t] = d. b) |7 cos s 1 5 dx d. 3u] = -1 c) 10 sec du

A:

Q: D. ( [ 3° cos"(5t) dt sin(z)

A:

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.