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Verify Rolle's theorem for the following functions: f(x)= sin x + cos x - 1, x in [0, (pi)/(2)]

Find the order and degree, if defined, of each of the following differential equations: (i) (dy)/(dx) - cos x = 0 (ii) xy (d^(2)y)/(dx^(2)) + x((dy)/(dx))^(2) - y (dy)/(dx) = 0 (iii) y''' + y^(2) + e^(y)'= 0

Find the numerically greatest terms in the expansion of (3x-4y)^(14) when x = 8,y = 3

Provethatsin ^(-1) x + cos^(-1)x=pi/2 , x in [-1,1]

int(x^(2)+1)/(x^(4)-x^(2)+1)dx=...+c

Let l_1=int_0^ooln(x+1/x)(dx)/(1+x^2) and l_2=int_0^(pi//2)(theta/(sintheta))^2 d theta, then which of the following is/are correct ?

If a,b,c are three non-coplanar vectors and if r is any vector then r =

~[(~p)^^q] is logically equivalent to

A monkey is climbing with uniform acceleration of a=4m//s^2. Other side of rope is connected with pulley. All pulley are frictionless.Find mass of the monkey (in kg) for which pulley P_1 (of mass 10 kg) remain in rest.

Consider the following parametric equation of a curve : x(theta)=|cos 4 theta| cos theta y(theta)|cos 4 theta|sin theta for 0 le theta le 2 pi Which of the following graphs represents the curve ?

Which one of the following is not true always?

An insurance company insured 2000 scooters and 3000 motorcycles. The probability of an accident involving a scooter is 0.01 and that of a motorcycle is 0.02. An insured vehicle met with an accident. Find the probability that the accidented vehicle was a motorcycle.

27passengersare totravel bya doublebuswhichcan accommodate12in theupperdeckand 15in thelowerdeck . Thenumberof waysthatthesepassengersare distributedif10 refuseto sitin theupperdeckand 10refuseto sitinthe lowerdeckis

y = sin^(-1)((2x)/(1 + x^2))

E_1, E_2 are events of a sample sapce such that P(E_1) =1/4 , P(E_2|E_1)=1//2, P(E_1|E_2)=1//4. Then P(E_1 |E_2)+P(E_1 | bar(E_2))=

Find the value of k for which the function f(x) f(x)={(kx+1,xlex),(cosx,xgtpi):}is continuous at x = pi

Let n be the largest integer that is the product of exactly 3 distinct prime numbers x, y and 10x + y where x and y are digits. What is the sum of digits of n ?

The total revenue received from the sale of x units of a product is given by R(x) = 20 x - 0.5x^(2). Find Marginal revenue

Two particles of of a medium disturbed by the wave propagation are at x_(1) = 0 and x_(2) = 1 cm. The wave is propagating in positive x-direction. The displacement of the particles is given by the equation : y_(1) = (2sin3pit)cm and y_(2) = 2sin(3pit- pi//8)cm (t is in second

Evaluate inte^(x)(1+x)/((2+x)^(2))dx

Evaluateint{2x+1)(x^2+x+1)^10dx

Choose the correct answer. The number of arbitrary constains in the general solution of a differential equation of fourth order is

15 persons are sitting in a row. In how many ways we can select three of them if adjacent persons are not selected?

Eighteen guest have to be sated. Half on each side of long table. Four particlular guest desire to sit on one particular side and three others on the other side. Determine the number of ways in which seating arrangements can be made.

On Z, define R as follows: a,b in Z, aRb if 3|(a^(2) - b^(2)), then R is an equivalence relation on Z. Equivalence class containing 1, that is, [1] is given by

If A, B are symmetric matrices of same order, then AB-BA is a

A box contains 3 green and 5 white balls. Three balls are drawn from it one after the other with replacement. Find probability distribution of the number of green balls.

Determine the order of smallness of the quantity beta with respect of the infinitesimal alpha. (a) beta=cos alpha-cos 2 alpha, (b) beta=tan alpha-sin alpha

Find the area enclosed between the curves y=4x^(2) and y=x^(2)+3

If the equation 3x + 3y + 5 = 0 is written in﻿ the form x cos alpha + y sin alpha = p, then the﻿ value of sin alpha + cos alpha is

Integrate the function (5x+3)/(sqrt(x^(2)+4x+10))

Find the number of ways of selecting pair of black squares in chessboard such that they have exactly one common corner

a,b are non-zero vectors, c is given non-zero scalar such that a is perpendicular to b. Then the vector x satisfying the equaitons a. x = c and a xx x = b is

Consider the following statements: (1)Mode can be computed from histrogram (b) Median is not independent of change of scale©Which of these is/are correct

The objective function of a linear programming problem is ..........

Compute P(A|B), if P(B) = 0.5 andP(A cap B)=0.32

Four persons A, B, C, D are to speak at a function along with 6 others. If they all speak in random order, the probability that A speaks before B, B before C, C before D is

The unit vector perpendicular to the vectors hati-hatj and hati +hatjforming a right handedsystem is

If 2.4^(2k+1) + 3^(3k+1) = 11t and 2.4^(2k+3) + 3^(3k+4) = 11 (pt + 3^(q)), where k, t in Z^(+), then (p, q) =

y dx + x log ((y)/(x))dy - 2xdy = 0

One microscope slide is placed on top of another with their left edges in contact and a human hair under the right edge of the upper slide. As a result, a wedge of air exists between the slides. An interference pattern results when monochromatic light is incident on the wedge.

Theangle"bet"^(n)twoline( x+3)/(3)= (y-1)/(5) = (z +3)/(4) and( x+1)/(1)=(4-y)/(-1) =(z-5)/(2)is……..

Differentiate w.r.t x the function (5x)^(3 cos 2x)

What is the slope of the line containing the points (-2,7) and (3, -3) ?

If f (x) = {{: ((X -|X|)/(X),"when x" lt o),(5x^(2) + a, "when" 0 le x le 1 ),(b((x^(2) - 1)/(x^(2) - 3x + 2)), "when" 1 lt x lt 3 ),(-14, "when x"ge 3):} is a continuous function on R, then (a, b) =

If . ( tan theta + (sin. (theta)/(2) +cos. (theta)/(2)))/(1+2i sin. (theta)/(2)) is purely imaginary. Then thenumber of values of theta in [0,2pi] is

If f(x)=cos^(-1)(sgn((2[x])/(3x-[x]))), where sgn ( ) denotes the signum function and [.] dentoes the greatest integer functions, then which of the following is/are correct?

Relation " similar" in triangles in a plane is :

Two litre glass flask contains some mercury. k is found that at all temperatures the volume of the air inside the flask remains the same. The volume of mercury inside the flask is (a_(g) = 9 xx 10^(-6) ^(@)C^(-1) gamma_(Hg) = 1.8 xx 10^(-4)^(@)C^(-1))

If the equation 3x + 3y + 5 = 0 is written in the form x cos alpha + y sin alpha = p, then the value of sin alpha + cos alpha is