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9901.

If (fog) (1)=3, g(1) = 2, g(1) = 1,then show that f(2)=3.

Answer»

SOLUTION :`(fog)(1)=3rArrf(G(1))g(1)=3rArrf(2)g(1)=3(thereforeg(1)=2)rArrf(2)1=3(thereforeg(1)=1)rArrf(2)=3`
Discussion
9902.

Find the distance covered by a body in a free fall within the time intervalt = a sec to t = b sec

Answer»


Answer :THEREFORE `DELTA v =g Delta t`, and HENCE , v(t) = GT , since `Delta t = t - 0 = t`
Discussion
9903.

Consider the inequality y ge ax^2 + bx + c, where a , b and c are all positive . Which of the following regions could be the solution set of the inequality ?

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ANSWER :A
Discussion
9904.

The point (3,-4) lies on both the circles x^2+y^2-2x+8y+13=0 and x^2+y^2-4x+6y+11=0 . Then the angle between the circles is

Answer»

`60^@`
`30^@`
`120^@`
`135^@`

ANSWER :D
Discussion
9905.

Differentiate.sqrtx.w.r.t. x^2

Answer»

SOLUTION :LET `y=sqrtx`and`z=x^2`
`dy/dx=1/(2sqrtx)`and `dz/dx=2X`
Now `dy/dz=(dy/dx)/(dz/dx)=(1/(2sqrtx))/(2x)=1/(4X^(3//2)`
Discussion
9906.

int_(0)^((pi)/(2)) ("4 sec x + 5 cosec x")/("sec x + cosec x") dx

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ANSWER :`(9pi)/(4)`
Discussion
9907.

The rangeof thefunction f(x)= (x^2 -x+1)/(x^2 +x+1) wherex in R , is

Answer»

`(-OO , 3]`
`(-oo,oo)`
`[3,oo)`
`[1/3,3]`

ANSWER :D
Discussion
9908.

int (x3sqrt(x^(2))+2 6sqrt(x))/(x(1+3sqrt(x))

Answer»


Answer :`(3)/(2)3sqrt(X^(2))+12tan^(-1)(6sqrt(x))+c`
Discussion
9909.

Integrate the following function : int(sintheta)/(1-4cos^(2)theta)d""theta

Answer»


ANSWER :`-(1)/(4)LOG|(1+2costheta)/(1-2costheta)|+C`
Discussion
9910.

For 0ltxlt(pi)/(6), all the values of tan^(2)3x cos^(2)x-4tan3xsin2x+16sin^(2)x lie in the interval

Answer»

`(0,(121)/(36))`
`(1,(121)/(9))`
`(-1,0)`
none of these

Solution :We have,
`tan^(2)3x cos^(2)x-4tan3xdsin2x+16sin^2)x`
`=((tan^(2)3x)/(tan^(2)x)-(8tan3x)/(tanx))sin^(2)x=((tan3x)/(tanx)-)^(2)sin^(2)x`
We KNOWN that
`1/3lt(tan3x)/(tanx)lt3`
`implies-11/3lt(tan3x)/(tanx)-4lt-1implies1lt((tan3x)/(tanx)-4)^(2)lt(121)/(9)`
Also, `0ltxlt(pi)/(9)implies0sinxlt1/2implies0ltsin^(2)xlt1/4`
`therefore0lt((tan3x)/(tanx)-4)^(2)sin^(2)xlt(121)/(36)`
Discussion
9911.

Let f: X to Yand g: Y to Z be two invertible functions. Then gof is also invertible with (gof)^(-1) = f^(-1)og^(-1).

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ANSWER :`(GOF)o(F^(-1)OG^(-1)) = I_(Z)`
Discussion
9912.

If the letters of the word PRISON are permuted in all possible ways and the words thus formed are arranged in dictionary order, find the rank of the word. PRISON

Answer»


ANSWER :438
Discussion
9913.

If a_(1), a_(2), a_(3), a_(4), a_(5) and a_(6) are six arithmetic means between 3 and 31, then a_(6) - a_(5) and a_(1) + a_(6) are respectively =

Answer»

5 and 34
4 and 35
4 and 34
4 and 36

Answer :C
Discussion
9914.

Compute the area of the figure bounded by the circle rho=3 sqrt2 a cos varphi and rho = 3a sin varphi

Answer»


ANSWER :`225 a^(2) (pi- "ARC " tan sqrt2- SQRT)`
Discussion
9915.

If A and B are two events such that P(A) = (1)/(2), P(B)= (1)/(3) and P(Acap B)= (1)/(4). Find : (i) P(A//B) (ii) P(B // A) (ii) P(A'//B) (iv)P(A'//B')

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ANSWER :`(5)/(8)`
Discussion
9916.

Write down the first three terms is the following expansions(3 + 5x)^(-7/3)

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Answer :`3^(-7/3), (-3^(-7/3) .35x)/(9) , (3^(-7/3) . 875 )/(81) x^2`
Discussion
9917.

The probability that a certain kind of component will survive a electrical test is 3/4. Find the probability that exactly 3 of the 5 components tested survive.

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ANSWER :`135/512`
Discussion
9918.

Find the number of distinct terms in the expansion(x+ y + z)^10+ (x + y - z)^10

Answer»
Discussion
9919.

If (1, - 2) is the focus and x + y -2 = 0 is the directrix of the ellipse 17x^2 - 2xy + 17x^2 - 32x + 76x + 86 - 0, then its eccentricityis

Answer»

`1/5`
`1/2`
`1/3`
`1/4`

ANSWER :C
Discussion
9920.

Find the order and degree of the differential equation, (d^(3)y)/(dx^(3))+2(d^(2)y)/(dx^(2))+ (dy)/(dx)= 0

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ANSWER :ORDER is 3 and DEGREEIS 1
Discussion
9921.

In a binomial distribution the sum and difference of mean and variance are 1.8 and 0.2 respectively. Find the parameters.

Answer»

<P>

ANSWER :`n=5,p=(1)/(5)`
Discussion
9922.

Determine, which

Answer»

SOLUTION :`(2,-5,0),(2,SQRT2,-3,sqrt2,-3) and (2,-1,3)` have same PROJECTION on x-axis.
Discussion
9923.

Consider the following population and year graph. Find the slope of the line AB and using it, find what will be the population in the year 2010?

Answer»


ANSWER :104.5 CRORES
Discussion
9924.

Evaluate: int(sin^(3)x/2)/(cosx/2sqrt(cos^(3)x+cos^(2)x+cosx))dx

Answer»

SOLUTION :`SEC^(-1)(SQRT(COSX+1/sqrt(cosx)))+C`
Discussion
9925.

If a_k is the coefficient of x^k in the expansion of (1 +x+x^2)^n for k = 0,1,2,……….,2n then a_1 +2a_2 + 3a_3 + ………+2n.a_(2n) =

Answer»

`-a_0`
`3^n`
`n.3^n`
`-n.3^n`

ANSWER :C
Discussion
9926.

Which of the following is incorrect :

Answer»

`(pi)^(1//pi)lt(e)^(1//e)`
`(10)^(1//10)gt(11)^(1//11)`
`(In2)^(1//(In2))lt(In3)^(1//(In3))`
`(tan^(-1)3)^(1//(tan^(-1))3)lt(tan^(-1)2)^(1//(tan^(-1)2))`

Solution :take`f(x) = x^(1//x)`
`log f(x) = (log^(x))/(x) "" rArr"" (f(x))/(f(x)) = (1-Inx)/(x^(2))`

`f.(x) = ((1-In x)/(x^(2))) . x^((1)/(X))`
For (x) is increasing for `0 lt xlt e`
And decreasing for `x gt e`
`pi^((1)/(pi)) lt e^((1)/(e))"" because pi gt e & f (x) darr`
`10^((1)/(10)) lt 11^((1)/(11) "" because 11 gt 10 gt e ,"" f(x) darr`
The function is increasing for `x lt e` , therefore`f(In 2) lt f (In 3) & f(tan^(-1) 2) lt f( tan^(-1)3)`
Discussion
9927.

A fixed parabola y^(2) = 4ax touches a variable parabola. Find the equation to the locus of the vertex of the variable parabola. Assume that the two parabolas are equal and the axis of the variable parabola remains parallel to the xaxis.

Answer»


ANSWER :`y^(2)=8AX`
Discussion
9928.

The plane 2x-3y+6z-11=0 makes an angle sin^(-1)(alpha) with X - axis. The value of alpha is equal to

Answer»

`(sqrt(3))/(2)`
`(sqrt(2))/(3)`
`(2)/(7)`
`(3)/(7)`

Answer :C
Discussion
9929.

int _(2) ^(3) (dx )/(sqrt(x -1) (5-x))

Answer»


ANSWER :`pi/6`
Discussion
9930.

Let f be a differentiable function on R and satisfying the integral equation xint_(0)^(x)f(t)dt-int_(0)^(x)(x-t)dt=e^(x-1) AA x in R thenf(1) equals to

Answer»


SOLUTION :N//A
Discussion
9931.

Find a unit vector in the direction of the vector (2, 3, 6).

Answer»


ANSWER :`((2)/(7),(3)/(7),(6)/(7))`
Discussion
9932.

If overset(a)underset(0)int(1)/(1+4x^(2))dx=(pi)/(8) then a=.....

Answer»


ANSWER :`:.a=(1)/(2)`
Discussion
9933.

(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+....

Answer»

`log_(2)E`
`log_(e )2`
`log_(e )3`
`log_(3) e`

Answer :B
Discussion
9934.

Write the principal value of sin^-1(frac(1)(2))

Answer»

Solution :`sin(-pi/6)=-sin(pi/6)=-1/2`and`-pi/6in [-pi/2,pi/2]`
`THEREFORE`The PRINCIPAL value of `sin^(-1)(-1/2)=-pi/6`
Discussion
9935.

If u=t^2and v=sint^2then dv/du=________.

Answer»

`cos^2t`
`sint/t`
`SEC t^2`
`cost^2`

ANSWER :A
Discussion
9936.

A particle of mass 20 g is released with an initial velocity 5m//s along the curve from the point A, as shown in the figure. The pointA is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B, its angular momentum about O will be : (Take g=10m//s^2).

Answer»

`2kg-m^2//s`
`8kg-m^2//s`
`6kg-m^2//s`
`3kg-m^2//s`

SOLUTION :NA
Discussion
9937.

|x-1|/(x+2)lt(x ne -2)

Answer»


ANSWER :`(-INFTY,-2)CUP(-1/2,1)cup(1,infty)=(-infty,-2)cup(-1/2,infty)`
Discussion
9938.

The slope of the normal to the curve y = 2x 2 + 3 sin x at x = 0 is

Answer»

3
`(1)/(3)`
`-3`
`-(1)/(3)`

ANSWER :D
Discussion
9939.

The set (A cap B^(c ))^(c ) cup (B cap C)is equal to

Answer»

`A^(C ) CUP B cup C`
`A^(c ) cup B`
`A^(c ) cup C^(c )`
NONE of these

Answer :B
Discussion
9940.

The probability of India winning a test match against West-Indies is 1//2 .Assuming independence from match to match the probability that in 5match series India's 2nd win occurs at the third test is

Answer»

`(1)/(8)`
`(1)/(4)`
`(1)/(2)`
`(2)/(3)`

Answer :B
Discussion
9941.

bar(a)=(-3,1,0),bar(b)=(1,-1,-1) then |"Comp"_(bar(b))bar(a)| = …………

Answer»

`-(4)/(sqrt(3))`
`(4)/(sqrt(3))`
`-(4)/(sqrt(10))`
`(4)/(sqrt(10))`

ANSWER :B
Discussion
9942.

If a and b are two vectos such that a.b lt 0 and |a.b|=|axxb|, then the angle between a and b is

Answer»

`(3pi)/(4)`
`(2pi)/(3)`
`(pi)/(4)`
`(pi)/(3)`

Solution :Given,`|a *b| = | axxb|`
` RARR |a||b||cos THETA|=|a||b||SIN theta|`
[ where,`theta`is the angle between a and b]
`rArr |cos theta| = |sin theta| `
`rArrtheta = (pi)/(4) or (3pi)/(4) "" [ "as " 0 le theta le pi ]`
But`a*b lt 0 `
`therefore theta=(3pi)/(4) `
Discussion
9943.

Axis of the parabola x^2 - 3y - 6x + 6 = 0 is

Answer»

`X = -3`
`y = -1`
`x = 3`
`y=1`

ANSWER :C
Discussion
9944.

Intestinal enzyme which activate the pancreatic enzyme is :-

Answer»

Trypsin
Enterokinase
HCI
Bile Salt

Answer :A
Discussion
9945.

int e^(-(x)/(2))(sqrt(1 -sinx))/(1 + cosx )dx =

Answer»

`E^(-x//2) SEC ""(x)/(2) +c `
`-e^(-x//2) Sec ""(x)/(2) +c `
`e^(x//2) Sec ""(x)/(2) +c `
`-e^(x//2) Sec ""(x)/(2) +c `

ANSWER :B
Discussion
9946.

Investigate the following improper integrals for convergence : (a) int_(0)^(1)(e^(x)dx)/(sqrt(1-x^(3))), (b) int_(0)^(1)(x^(2)dx)/(root3((1-x^(2))^(5))), (c ) int_(0)^(1)sqrt((x)/(1-x^(4)))dx, (d) int_(0)^(1)(dx)/(1-x^(3)+x^(5)), (e ) int_(0)^(1)(dx)/(x-sinx), (f) int_(0)^(2)(1n(root4(x)+1))/(e^(tanx)-1)dx

Answer»


Answer :(a) It converges;
(B) DIVERGES;
(c ) converges;
(d) converges;
(e )diverges;
(F) converges.
Discussion
9947.

The slope of a straight line passing through A(-2, 3) is -4//3. The points on the line that are 10 unit away from A are

Answer»

only I is TRUE
only II is true
both I and II are true
NEITHER I nor II are true

ANSWER :C
Discussion
9948.

int_(-pi/4)^(pi/4)cos^2xdx

Answer»

SOLUTION :`int_(-pi/4)^(pi/4)cos^2xdx=int_(-pi/4)^(pi/4)[(1+cosx)/2]dx`
=`1/2int_(-pi/4)^(pi/4)dx+1/2int_(-pi/4)^(pi/4)COS2XDX`
=`1/2[x]_(-pi/4)^(pi/4)+1/4[sin2x]_(-pi/4)^(pi/4)`
=`1/2(pi/4+pi/4)+1/4(SIN(pi/4)+sin(pi/4))=pi/4+1/2`
Discussion
9949.

Evaluate the determinants |{:(x+4,x,x),(x,x+4,x),(x,x,x+4):}|

Answer»
Discussion
9950.

Evaluate the determinants |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|.

Answer»
Discussion