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4351.

If z_(1) = 8 + 4i , z_(2) = 6 + 4i andz be a complex number such that Arg ((z - z_(1))/(z - z_(2))) = (pi)/(4) , then the locus of z is

Answer»

`(X - 7)^(2) + (y- 5)^(2) = 2`
`(x - 7)^(2) + (y+ 5)^(2) = 2`
`(x - 7)^(2) + (y - 5)^(2) = 4`
`(x + 7)^(2) + (y + 5)^(2) = 4`

Answer :A
Discussion
4352.

Find the chord of contact of(1,1 )with respect to the circlex^(2) + y^(2) =9

Answer»
Discussion
4353.

If alpha, beta, gamma are roots of x^(3) - px^(2) + qx - r = 0then (alpha + beta)^(-1) + (beta + gamma)^(-1) + (gamma + alpha)^(-1) =

Answer»

<P>`(q^(2) - 2pr)/(r^(2))`
`(p^(2) + q)/(PQ - r)`
`(p^(2) - 2Q)/(r^(2))`
none

Answer :2
Discussion
4354.

If the relation between x and y in order that the 20^(th) arithmetic mean between x and 2y is same as the 20^(th) arithmetic mean between 2x and y , (99 means being inserted in each case )is y = Kx, find K.

Answer»


ANSWER :4
Discussion
4355.

Let f(x)=lim_(nrarroo) (tan^(-1)(tanx))/(1+(log_(x)x)^(n)),x ne(2n+1)(pi)/(2) then

Answer»

`AA1ltxlt(pi)/(2),f(x)` is an identity function
`AA(pi)/(2)ltxltpi,` the graph of f(x) is a STRAIGHT LINE having y intercept of `-pi`
`AA(pi)/(2)ltxlte`, the graph of f(x) is a straight line having y intercept of `-pi`
`AAxgte, f(x)` is a constant function

Solution :`AA1 ltx LT(pi)/(2), tan^(-1)tanx=x`
`and 0 lt log_(e)lt log_(e).(pi)/(2)lt1`
`rArr""f(x)=x`
`AA(pi)/(2)ltx lte, tan^(-1)tanx=x-pi`
and `0lt log_(e)XLT1`
`THEREFORE""(log_(e)x)^(n)=0`
`rArr""f(x)=x-pi`
and for `x gt e, log_(e)xlt1, therefore (log_(e)x)^(n)rarroo`
`rArr""f(x)=0`
Discussion
4356.

Find P(A|B) ifP(B)=0.5 and P(A nnB)= 0.32

Answer»


ANSWER :0.64
Discussion
4357.

int (1)/(7 + 5 cos x )dx =

Answer»

`(1)/(sqrt(5)) tan^(-1)( (1)/(sqrt(3)) tan ""(X)/(2) ) + c `
`(1)/(sqrt(6)) tan^(-1) ((1)/(sqrt(6)) tan"" (x)/(2) ) + c `
`(1)/(sqrt(7)) tan^(-1)(tan""(x)/(2)) + c `
`(1)/(sqrt(4)) tan^(-1) (tan ""(x)/(2)) + c `

ANSWER :B
Discussion
4358.

Iff:R rarr R, S: R rarr R are defined by f(x) = 3x-4, g(x) = 5x-1 then, (fog^(-1))(2) =

Answer»

`11/5`
`-11/5`
`1/5`
`-1/5`

ANSWER :B
Discussion
4359.

Using properties of determinants, prove that |[1,1+p,1+p+q],[2,3+2p,4+3p+2q],[3,6+3p,10+6p+3q]| = 1

Answer»

SOLUTION :L.H.S. = `|[1,1+p,1+p+q],[0,1,2+p],[0,3,7+3p]|` by (`R_2toR_2-2R_1 and R_3toR_3-3R_1`)
=`|[1,1+p,1+p+q],[0,1,2+p],[0,0,1]|` by `R_3toR_3-3R_2`)
`= 1xx1xx1 = 1 =` R.H.S.
Discussion
4360.

Choose the correct . The general solution of the differential equation (y dx - x dy)/y =0 isa) xy =c b) x= cy^2c)y = cxd)y= cx^2

Answer»

XY = C
`X = cy^2`
y = CX
`y = cx^2`

ANSWER :C
Discussion
4361.

Find the following integrals int(4e^(3x)+1)dx

Answer»


ANSWER :`(4)/(3)e^(3X) + X + c`
Discussion
4362.

Let PM be the perpendicualr from the point P(1, 2, 3) to x-y plane. If bar(OP) makes an angle theta with the positive dirction of z-axis and bar(OM) makes an angle phi with the positive direction of x axis, where O is the origin and theta and phi are acute angles, then

Answer»

`TAN theta=(SQRT(5))/(3)`
`sin theta sin phi=(2)/(sqrt(14))`
`tan phi =2`
`COS theta cos phi=(1)/(sqrt(14))`

Answer :A::B::C
Discussion
4363.

In which of the following replacement of Cl^(-) is most difficult ?

Answer»




ANSWER :D
Discussion
4364.

A man has 6 friends. The number of ways can he invite one or more of them to dinner is

Answer»

63
64
720
6

Answer :A
Discussion
4365.

Prove the following : sinA+sin3A+sin5A =sin3A(1+2cos2A)

Answer»

SOLUTION :L.H.S. = sinA+SIN3A+sin5A
= sin3A+sin5A+sinA
sin3A+2sin5A+A/2cos5A-A/2
= sin3A+2sin3Acos2A
sin3A(1+2cos2A)=R.H.S.
Discussion
4366.

Given that the two numbers appearing on throwing two dice are different. Find the probabitlity of the event the sum of numbers on the dice is 4.

Answer»


ANSWER :`1/15`
Discussion
4367.

If underset(n to oo)(lim)(e(1-1/n)^(n)-1)/(n^(alpha)), exists and is equal to l (l != 0), then the value of 12(l – alpha) is :

Answer»

4
3
6
7

Solution :Let `N = 1/X`
`l=UNDERSET(x to 0)(lim)(E(1-x)^(1//x)-1)/((1//x)^(alpha)) = underset(x to 0)(lim) (e.e^((ln(1-x))/(x))-1)/(x^(-alpha))`
`l=underset(x to 0)(lim)((ln(1-x)/(x)+1)/(x^(-alpha))) = underset(x to 0)(lim) ((-x/2-(x^2)/3......)/(x^(-alpha)))`
For limit to exist `alpha = -1`
`l = -1/2`.
Discussion
4368.

int (dx)/(cos(x+4)cos(x+2))=

Answer»

`(1)/(SIN2)LOG|cos(x+4)^2|+c`
`(1)/(2)log|(sec(x+2))/(sec (x+4))|+c`
`(1)/(sin2)log|(sec(x+4))/(sec(x+2))|+c`
`log|(sec(x+4))/(sec(x+2))+c|`

Answer :C
Discussion
4369.

If vec(x)+vec(y)+vec(z)=0 and |vec(x)|=|vec(y)|=|vec(z)|=2 If the angle between vec(y) and vec(z) and theta. Then cosec^(2)theta+cot^(2)theta= …………..

Answer»

`(4)/(3)`
`(5)/(3)`
`(1)/(3)`
1

Answer :B
Discussion
4370.

Solve the equation{:|( x+a,x,x),(x,x+a,x),(x,x,x+a) |:}=0,ane 0

Answer»


ANSWER :`X=(-a)/(3) `
Discussion
4371.

Approximatevalueof(31)^(1/5)is____

Answer»

`2.01`
`2.1`
`2.0125`
`1.9875`

ANSWER :D
Discussion
4372.

Integration by partial fraction : The value of int(x)/((x-2)(x-1))dx=....

Answer»

`"log"_(E)((x-2)^(2))/((x-1))+p`
`"log"_(e)((x-1)^(2))/((x-2))+p`
`(x-1)/(x-2)+p`
`2log_(e)((x-2)/(x-1))+p`

Answer :A
Discussion
4373.

int(1)/(x^(3)(1-x))dx=

Answer»

`log((X)/(1-x))-(2X+1)/(2x^(2))+c`
`log((1-x)/(x^(3)))+c`
`log((x^(3))/(1-x))+c`
`(x^(-2))/(-2)-log(1-x)+c`

Answer :A
Discussion
4374.

(cos 13^(@) - sin 13^(@))/(cos 13^(@) + sin 13^(@)) + (1)/(cot 148^(@)) =

Answer»

1
`-1`
0
`1//2`

ANSWER :C
Discussion
4375.

Let the point B be the reflection of the point A(2,3) with respect to the line 8x-6y-23=0 Let lceiling_A and lceiling_B respectively. Be circles of radil 2 and I which centres A and B respectively. Let T be a common tangent to the circles lceilingA and lceiling B such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing thorugh A and B, then the length of the line segment AC is .............

Answer»


Solution :ACCRODING to given information the FIGURE is as following

From the figure, `AC=(2)/(sin THETA) .....(i)`
`becausesin theta =(1)/(CB)`
`(" from "Delta CPB) ......(ii)`
` and sin theta =(2)/(AC)=(2)/(CB+AB)("from "Delta CQA)......(iii)`
`because AB=AM+MB=2AM`
` =2(|(8xx2)-(6xx3)-23|)/(sqrt(64+36))=(2xx25)/(10)=500`
From Eqs. (ii) and (iii), we get
`sin theta=(1)/(CB)=(2)/(CB+AB)`
`rArr (1)/(CB)=(2)/(CB+5)`
`rArr CB+5=2GBrArr CB =5=(1)/(sin theta)`
From the EQ.(i),
`AC=(2)/(sin theta) =2xx5 =10,00`.
Discussion
4376.

Write two different vectors having same direction.

Answer»


ANSWER :An INFINITE NUMBER of POSSIBLE answer.
Discussion
4377.

The straight lines 2x+3y=5 and 6x-4y+k=0, k in R are the sides of [if the third line is not parallel any of these two lines]

Answer»

an EQUILATERAL triangle
RIGHT angled triangle
OBTUSE angled triangle
can not be the SIDES of a triangle

Answer :B
Discussion
4378.

findthe multiplerootsof12x^3 +40x^2 +39 x + 9=0

Answer»


ANSWER :`(-3)/(2),(-3)/(2),(-1)/(3)`
Discussion
4379.

Find the probability of getting atleast one head when 5 coins are tossed

Answer»


ANSWER :`31//32`
Discussion
4380.

Find the middle term(s) in the expansionof n in N ((1)/(2)x-3y)^(20)

Answer»


SOLUTION :N/A
Discussion
4381.

If alpha is a non-real root of x^(7) = 1 then alpha(1 + alpha) (1 + alpha^(2) + alpha^(4)) =

Answer»

2
-1
1
`-2`

ANSWER :B
Discussion
4382.

For a , b in R the maximum [(a-1)(b-1)+(1-sqrt(1-a^2))(1-sqrt(1-b^2))]

Answer»

`2+sqrt2`
`2-sqrt2`
`3+sqrt2`
`3+2sqrt2`

ANSWER :D
Discussion
4383.

Let T, be the r^(th) term of an A.P. whose first term is a and common difference is d If for some positive integers m, n, m != n, T_(m) = 1/n and T_(n) = 1/m, then a - d equals

Answer»

`1/m`
1
0
`1/m + 1/n`

ANSWER :C
Discussion
4384.

If x in (pi, 2pi)" then " int sqrt(1 - cos 2 x )dx =

Answer»

`sqrt(2) ` sin X + C
`- sqrt(2)` sin x + C
`sqrt(2) ` cos x + C
`-sqrt(2)` cos x + C

ANSWER :C
Discussion
4385.

IFA +B= 45 ^@, then( cot A -1)( cotB-1)isequalto

Answer»

1
`1/2`
`-1`
2

Answer :D
Discussion
4386.

Let A_1,A_2,A_3,...,A_n are n Points in a plane whose coordinates are (x_1,y_1),(x_2,y_2),....,(x_n,y_n) respectively. A_1A_2 is bisected at the point P_1,P_1A_3 is divided in the ratio 1:2 at P_2,P_2A_4 is divided in the ratio 1:3 at P_3,P_3A_5 is divided in the ratio 1:4 at P_4 and the so on until all n points are exhausted. find the coordinates of the final point so obtained.

Answer»

Solution :`P_1` is midpoint of `A_1A_2`.
`THEREFORE""P_1-=((x_1+x_2)/(2),(y_1+y_2)/(2))`
`P_2` DIVIDES `P_1A_3` in `1:2`.
`therefore""P_2-=((2((x_1+x_2)/2)+x_3)/(2+1),(2((y_1+y_2)/2)+y_3)/(2+1))`
`-=((x_1+x_2+x_3)/(3),(y_1+y_2+y_3)/(3))`
Now, `P_3` divides `P_2A_4` in ` 1:3`
`therefore""P_3-=((3.((x_1+x_2+x_3)/3)+x_4)/(3+1),(3.((y_1+y_2+y_3)/3)+y_4)/(3+1))`
`-=((x_1+x_2+x_3+x_4)/(4),(y_1+y_2+y_3+y_4)/(4))`
PROCEEDING in this manner, we get
`P_n-=((x_1+x_2+x_3+....x_n)/(n),(y_1+y_2+y_3+....y_n)/(n))`.
Discussion
4387.

In the following, [x] denotes thegreatest integer less than or equal to x. {:(,"Column I",,"Column II"),(A.,x|x|,p.,"continuous in(-1, 1)"),(B.,sqrt|x|,q.,"differentiable in (-1, 1) "),(C.,x+[x],r.,"strictly increasing (-1, 1)"),(D.,|x-1|+|x+1|" in(-1,1)",s.,"not differentiable atleast at one point in(-1, 1)"):}

Answer»


Solution :A. x|x| is continuous,DIFFERENTIABLE and strictly increasing in (-1, 1).
B. `sqrt|x|` is continuous in (-1, 1) and not differentiable at x = 0.
C. x+[x] is strictly increasing in (-1, 1) and DISCONTINUOUS atx = 0
`rArr` not differentiable at x = 0.
D. `|x-1|+|x+1|=2" in "(-1, 1)`
`rArr ` The FUNCTION is continuous and differentiable in (-1, 1) .
Discussion
4388.

If P(A) = 2/3, P(B) = 4/9 and P(AcapB)=cap, then P(A'capB') is greater than or equal to

Answer»

`37/45`
`1/5`
`4/5`
`41/45`

ANSWER :A::B::C
Discussion
4389.

Evaluate the following integrals. int(dx)/(e^(x)+e^(2x))

Answer»


ANSWER :`log((E^(x)+1)/(e^(x)))-1/e^(x)+c`
Discussion
4390.

Let alpha=sqrt(19-8sqrt3)+sqrt(7+4sqrt3)and beta=sqrt(83-18sqrt2)-sqrt(6-4sqrt2),then log_(2)((alpha)/(beta)) lies in the interval

Answer»

`(-2,-1)`
`((-1)/(2),0)`
`(0,1)`
`(-1,(-1)/(2))`

Solution :`ALPHA=(4-sqrt3)+(2+sqrt3)=6AND beta=(9-sqrt2)-(2-sqrt2)=7.`
`thereforelog_(2)((alpha)/(beta))=log_(2)((6)/(7))`
`As,(1)/(sqrt2)lt6/7lt1implies(-1)/(2)ltlog_(2)((6)/(7))LT0`
Discussion
4391.

There are fifteen players for a cricket match In how many ways the 11 players can be selected including a particular player?

Answer»


ANSWER :1001
Discussion
4392.

Write the equation of the plane passing through x-axis and y-axis.

Answer»

SOLUTION :Any plane perpendicular to Z-axis has EQUATION z = 4 As it passes through `(1,-2,4) we have 4 = k `therefore` The REQUIRED equation is z = 4.
Discussion
4393.

(1/(2!)+1/(4!)+1/(6!)+ ....oo)/(1+1/(3!)+1/(5!)+....oo)

Answer»

`(E+1)/(e-1)`
`(e)/(e-1)`
`1/(e-1)`
`(e-1)/(e+1)`

ANSWER :D
Discussion
4394.

Find the area under the given curves a given line :y=x^4,x=1,x=5 and x-axis

Answer»

SOLUTION :AREA `=overset5underset1intydx=overset5underset1int x^4dx=[x^5/5]_1^5`
`1/5[5^5-1^5]=624.8`sq.units
Discussion
4395.

Evaluate underset(0)overset(pi)int sinxdx

Answer»


ANSWER :2
Discussion
4396.

The radical axis of x^(2) +y^(2) -2ax =0and x^(2) +y^(2)-2by =0is common tangent to the circles if

Answer»

` a GT B`
` a LT b`
` a= b `
` a=1 , b=2`

ANSWER :C
Discussion
4397.

A firm has the following total cost and demand function: C(x)=x^3/3-7x^2+11x+50,x=100-pFind the profit function.

Answer»

<P>

ANSWER :`p(X)=x^3.3+6x^2-11x-50`
Discussion
4398.

IF [x] denotesthegreatestinteger le x then[ 2/3] +[2/3 +1/99] +[2/3 +2/(99)]+ ………. [2/3 +(98 )/(99)] =

Answer»

99
98
66
65

Answer :C
Discussion
4399.

Find the combined equation of the lines through the origin : (1) each making an ange of 45^(@) with the line 3x + y = 2. (2) each making an angle of pi//6 with the line 3x + y - 6 = 0 . (3) which form an equilateral triangle with the line3x + 4y = 8.

Answer»

Solution :`2X^(2) + 3xy - 2y^(2) = 0`
(2) `13X^(2) + 12xy - 3y^(2) = 0`
(3) `39x^(2) - 96xy + 11y^(2) = 0`.
Discussion
4400.

Let A and B be two finite sets having m and n elements respectively such that m gt n. A mapping is selected at random from the set of all mappings from A to B. The probability that the mapping selected is an injective mapping is

Answer»

`(N!)/((n-m)!m^(n))`
`(n!)/((n-m)!n^(m))`
`(m!)/((n-m)!n^(m)`
none

Answer :D
Discussion