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14601.

Evaluate the following limits : Lt_(ntooo)(1^(3)+2^(3)+3^(3)+......+n^(3))/(n^(2)(n^(2)+1))

Answer»


ANSWER :`1/4`
Discussion
14602.

The vector equation of the plane through the point (1,1,-1) and passing through the line of intersection of the planes barr.(bari+3barj-bark) = 0 and barr.(barj+2bark) =0 is

Answer»

1)`barr.(bari+8barj+9bark) =0`
2)`barr.(bari+9barj+11bark) =6`
3)`barr.(bari-barj+13bark) = 0`
4)`barr.(bari+barj+bark) = 9`

Answer :A
Discussion
14603.

If r_(1)=8, r_(2)= 12r_(3)=24 then C=

Answer»

`(PI)/(4)`
`(pi)/(6)`
`(pi)/(3)`
`(pi)/(2)`

ANSWER :D
Discussion
14604.

If x + y = 2 pi // 3 and sin x / sin y = 2 , then the

Answer»

number of VALUES of `x in [0,4pi]` are 4
number of values of `in [0,4pi]` are 2
number of values of `y in [0,4pi]` are 4
number of values of `y in [0,4pi]` are 8

Answer :A::C
Discussion
14605.

The value of 'x' at which f(x) = cosx is stationary are given by

Answer»

`NPI,AAn in Z`
`(2n+1)(PI)/(2),AA in Z`
`(npi)/(4),AAn in Z`
`(npi)/(2),AAn in Z`

ANSWER :A
Discussion
14606.

For any sets A and B, prove that : (i) (A -B) cap B=phi (ii) A cup (B-A) =A cup B (iii) (A-B) cup (AcapB)=A (iv)(A cup B) -B = A-B (v) A- (A cap B) =A-B

Answer»

SOLUTION :(i) `(A-B) cap B =(A cap B') cap B = A cap(B' cap B) =A cap phi =phi`.
(ii) `A CUP (B-A )=A cup (B cap A') =(A cup B) cap (A cup A') =(A cup B) cap U=(A cup B)`.
(iii)` (A -B) cup (A cap B) =(A capB') cup (A cap B) =A cap (B'cup B)=A cap U=A`.
(IV) `A cup B) -B=(ACUP B) cap B'=(A cap B') =(A cap B') cup ( B cap B') =(A cap B') cup phi =ACAP B ' =A-B`
(v) ` A - (A cap B) =A cap (A cap B) '=A cap (A'cup B' ) =(A cap A') CUP (A cap B') =phi cup (A cap B') =(A cap B') =(A-B)`.
Discussion
14607.

sin^(-1)(sqrt(2)sin theta)+sin^(-1)(sqrt(cos 2 theta))=

Answer»

`(PI)/6`
`(pi)/4`
`(pi)/3`
`(pi)/2`

Answer :D
Discussion
14608.

A (3,2,0), B(5,3,2), C(-9,6,-3) are three points forming a triangle then the coordinates of the point in which the bisector of angleBAC meets BC is

Answer»


ANSWER :`((38)/(16),(57)/(16),(17)/(16))`
Discussion
14609.

If in two circles, arcs of the same length substend angles 60^(@) and 75^(@) at the centre, find the ratio of their radii.

Answer»


ANSWER :`5:4`
Discussion
14610.

Find the derivative of the function (8^(x))/(x ^(8))

Answer»


ANSWER :`(8 x ^(7) (x LOG 8-8))/(x ^(16))`
Discussion
14611.

If the d.c's of two lines are such that l+3m+n = 0, 3l^(2) + 5m^(2) - 4n^(2) = 0, then the angle between them is

Answer»

`pi//3`
`pi//2`
`pi//4`
`pi//6`

ANSWER :B
Discussion
14612.

In triangle ABC, if P, Q, R divides sides BC, AC, and AB, respectively, in the ratio k:1 (in order). If the ratio (( "area "Delta PQR)/("area " Delta ABC)) " is " 1/3 then k is equal to

Answer»

`1//3`
2
3
none of these

ANSWER :B
Discussion
14613.

Find 'x' if the angle between he lines with direction ratios (x,4,5) and (2,-1,2) is 90^(@)

Answer»


ANSWER :-3
Discussion
14614.

The period of f(x) = cos (x /3) + sin (x/2) is

Answer»

`2PI`
`4PI`
`8pi`
`12pi`

ANSWER :4
Discussion
14615.

Sin^(-1)(sin3)+Cos^(-1)(cos7)-Tan^(-1)(tan5)=

Answer»

`PI-1`
`pi`
`3pi-1`
`2pi-1`

ANSWER :D
Discussion
14616.

If Tan = (1)/(2), Tan B = (1)/(3) then Tan (A+B) =

Answer»

-1
0
1
2

Answer :C
Discussion
14617.

Write the component statements of the following compound statements and check whether the compound statement is true or false. A line is straight and extends indefinitely in both directions.

Answer»
Discussion
14618.

If length of the sides of a triangle ABC are 4 cm, 5 cm and 6 cm, O is point inside the triangle ABC such that angle OBC = angle OCA = angle OAB = theta, then value of cot theta is .

Answer»

`(11sqrt7)/15`
`(4 SQRT7)/5`
`2/3 sqrt7`
`sqrt7`

ANSWER :a
Discussion
14619.

Let P(x) = ((1- cos 2 x + sin 2x)/(1+ cos2x + sin2x))^(2) + ((1+ cot x + cot^(2) x)/( 1+ tan x+ tan^(2) x) ), then the minimum value of P(x) equals

Answer»

1
2
4
16

Answer :B
Discussion
14620.

Differentiate each of the following from the first principle. (i) sqrt(sinx) , (ii) sqrt(cos x) (iii) sqrt(tanx) , (iv) sqrt(cosecx)

Answer»

Solution :(i) Let y = `sqrt(SINX)`.
Let `deltay`be an increment in y, correspondingto an increment `deltax` in X.
Then, `y + deltay = sqrt(sin(x+deltax))`
`rArr deltay = sqrt(sin(x+deltax))-sqrt(sinx)`
`rArr (deltay)/(deltax) =(sqrt(sin(x+deltax))-sqrt(sinx))/(deltax)`
`rArr (dy)/(dx) = underset(deltararr 0)("lim") (deltay)/(deltax)`
`= underset(deltax rarr 0)("lim")(sqrt(sin(x+delta))-sqrt(sinx))/(deltax)`
`= underset(deltax rarr 0)("lim")({sinsqrt(x+deltax)-sqrt(sinx)})/(deltax)`
`= underset(deltax rarr 0)("lim")({sin(x+deltax)-sinx})/(deltax.{sqrt(sin(x+deltax))+ sqrt(sinx}})`
`= underset(deltaxrarr0)("lim")(2cos(x+(delta)/(2))sin((delta)/(2)))/({sqrt(sin(x+deltax)) +sqrtsinx}deltax)`.
`=underset(deltax rarr 0)("lim")cos(x+(deltax)/(2)). underset(deltax rarr 0)("lim") (sin((deltax)/2))/(((deltax)/(2)))`
`underset(deltax rarr 0)("lim")1/({sqrt(sin(x+deltax)) + sqrt(sinx)])`
`= {cosx.1.1/(2sqrt(sinx))} = (cosx)/(2sqrt(sinx))`.
Hence, `d/(dx) (sqrt(sinx)) = (cos x)/(2sqrt(sinx))`.
(ii) Let `y = sqrt(cos x)`.
Let `deltay` be an increment in y, correponding to an increment `deltax` in x.
Then, `y + deltay = sqrt(cos(c+deltax))`
`rArr deltay = sqrt(cos(x+deltax))-sqrt(cosx)`
`rArr (deltay)/(deltax) = ({sqrt(cos(x+deltax))-sqrt(cosx)})/(deltax)`
`rArr (dy)/(dx)=underset(deltax rarr 0)("lim") ({sqrt(cos(x+deltax))-sqrt(cosx)})/(deltax)`
`= underset(deltaxrarr0)("lim")({sqrt(cos(x+deltax))-sqrt(cosx)})/(deltax) xx ({sqrt(cos(x+deltax)) + sqrt(cosx)})/({sqrt(cos(x+deltax)) + sqrt(cosx)})`
`=underset(deltaxrarr0)("lim")({cos(x+deltax) - cos x})/(deltax.{sqrt(cos(x+deltax))+ sqrt(cosx)})`
`= underset(deltax rarr 0)("lim")(2SIN(x+(deltax)/(2))sin((deltax)/(2)))/(deltax.{sqrt(cos(x+deltax)) + sqrt(cosx)})`
`[ :' cos C - cos D = - 2sin ((C+D)/(2)) sin((C-D)/(2))]`
`=underset(deltax rarr 0)(-"lim")sin(x+(deltax)/(2)).underset(deltax rarr 0)("lim")(sin(deltax//2))/((deltax//2))`.
`underset(deltax rarr 0)("lim")(1)/({sqrt(cos(x+deltax))+sqrt(cosx)})`
`= (-sinx) xx 1 xx 1/(2sqrt(cosx))`
Hence, `(d)/(dx) (sqrt(cosx)) =(-sinx)/(2sqrt(cosx))`
(iii) Let `y = sqrt(TANX)`.
Let `deltay` be an increment in y. corresponding to an increment `deltax` in x.
Then, `y + deltax = sqrt(tan(x + deltax))`
`rArr deltay = sqrt(tan(x+deltax)) - sqrt(tanx)`
`rArr (deltay)/(deltax) = (sqrt(tan(x+deltax)) - sqrt(tanx))/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr 0)("lim") (deltay)/(deltax)`
`= underset(deltaxrarr0)("lim"){(sqrt(tan(x+deltax))- sqrt(tanx))/(deltax) xx (sqrt(tan(x+deltax))+sqrt(tanx))/(sqrt(tan(x+deltax))+sqrt(tanx))}`
`= underset(deltaxrarr0)("lim")(tan(x+deltax)-tanx)/(deltax[sqrt(tan(x+deltax))+sqrt(tanx)])`
`= underset(deltararr0)("lim")({(sin(x+deltax))/(cos(x+deltax))-(sinx)/(cosx)})/(deltax[sqrt(tan(x+deltax))+sqrt(tanx)])`
`= underset(deltaxrarr0)("lim")(sin(x+deltax)cosx - cos (x+deltax) sinx)/(cos(x+deltax) cosx. deltax. [sqrt(tan(x+deltax)) + sqrt(tanx)])`
`= 1/(cosx). underset(deltax rarr 0)("lim") (1)/(cos(x+deltax)) . underset(deltaxrarr0)("lim") (sindeltax)/(deltax).underset(deltararr0)("lim") (1)/((sqrt(tan(+deltax)) + sqrt(tanx)))`
`= ((1)/(cosx) . (1)/(cosx)1.(1)/(2sqrt(tanx)))= (sec^(2)x)/(2sqrt(tanx))`
Hence `d/(dx) (sqrt(tanx)) = (sec^(2)x)/(2sqrt(tanx))`.
Let `y = sqrt(cosecx)`.
Let `deltay` be an increment in y , correspondingto an increment `deltax` in x.
Then, `y +deltay = sqrt(cosec(x+deltax))`
`rArr deltay = sqrt(cosec(x+deltax)) - sqrt(cosecx)`
`rArr (dy)/(deltax) = (sqrt(cosec(x+deltax)) - sqrt(cosecx))/(deltax)`
`rArr (dy/(dx)= underset(deltax rarr 0)"lim" (dy)/(dx)`
`= underset(deltax rarr 0)("lim") (sqrt(cosec(x+deltax)) - sqrt(cosecx))/(deltax)`
`= underset(deltararr0)("lim")({(1)/(sqrt(sin(x+deltax))) -(1)/(sqrt(sinx))})/(deltax)`
`= underset(deltaxrarr0)("lim")({sqrt(sinx)-sqrt(sin(x+deltax))})/(deltax.sqrt(sin(x+deltax)).sqrt(sinx)) xx ({sqrt(sinx) +sqrt(sin(x+deltax))})/({sqrt(sinx) + sqrt(sin(x+deltax))})`
`= underset(deltararr0)("lim")({sinx-sin(x+deltax)})/({sqrt(sin(x+deltax)).sqrt(sin(x))}) xx (1)/(deltax.{sqrt(sinx) + sqrt(sin(x+deltax))})`
`= underset(deltararr0)("lim")(-2cosx(x+(deltax)/(2))sin((deltax)/(2)))/({sqrt(sin(x+deltax))}.sqrt(sinx).deltax.{sqrt(sinx)+sqrt(sin(x+deltax))})`
`= - underset(deltax)("lim")cos (x+(deltax)/(2)).underset(deltaxrarr0)("lim") (sin ((deltax)/(2)))/(((deltax)/(2)))`
`underset(deltararr0)("lim") (1)/(sqrt(sin(x+deltax)).sqrt(sinx)).underset(deltararr0)("lim")(1)/({sqrt(sinx)+sqrt(sin(x+deltax))})`
`= - cos xxx 1 xx (1)/(sqrt(sinx). sqrt(sinx)).(1)/((sqrt(sinx) +sqrt(sinx)))`
`= (-cosx)/(sinx).(1)/(2sqrt(sinx)) = - 1/2 sqrt(cosecx) cot x`.
Hence, `d/(dx) (sqrt(cosecx)) = - 1/2 sqrt(cosecx) cotx`.
Discussion
14621.

An equilateral triangle is inscribed in the parabola y^(2)=4ax whose vertex is at the vertex of the parabola .Find the length of its side.

Answer»


ANSWER :`8sqrt(3A)` UNITS
Discussion
14622.

A =(cos 9^(@) - sin 9^(@))/(cos 9^(@) + sin 9^(@)) , B = (cos 21^(@) + sin 21^(@))/(cos21^(@) -sin21^(@)) and C= tan 20^(@) + tan40^(@) + sqrt(3)tan20^(@) tan40^(@) then descending order is:

Answer»

A,B,C
B,C,A
C,A,B
C,B,A

Answer :B
Discussion
14623.

If the middle term of (1/x +x in x )^(10) is equal to7 . 7/8 , then the valueof x is .

Answer»

`2N pi +pi/6 , N in Z`
`NPI +pi/6 , n in Z`
`npi + (-1)^(n). Pi/6 , n in Z`
`npi +(-1)^(n). Pi/3 , n in Z `

Answer :C
Discussion
14624.

Find the equation of pair of lines intersecting at (2, -1) and Perpendicular to pair6x^(2)-13xy-5y^(2)=0

Answer»


ANSWER :`6X^(2)-13xy-5y^(2)-37x+16y+45=0`
Discussion
14625.

Let p be '' It is cold '' and q be ''It is raining '' Give simple verbal sentence which describe the following statements.~(p^^q)

Answer»

SOLUTION :~`(p^^q)`:It is not the CASE it is COLD and it is RAINING
Discussion
14626.

Let p be '' It is cold '' and q be ''It is raining '' Give simple verbal sentence which describe the following statements.prArr(~q)

Answer»

SOLUTION :`prArr`(~Q): If it is COLD, then it is not RAINING.
Discussion
14627.

If sin^(2)theta gt cos^(2) theta then

Answer»

`NPI+(pi)/2LT THETA lt N pi +(pi)/4,, n in Z`
`n pi +(3pi)/2 lt theta lt n pi +(5PI)/4, n in Z`
`n pi +(pi)/4 lt theta lt n pi +(3pi)/4, n in Z`
`n pi +(3pi)/4lt theta lt n pi +(5pi)/4, n in Z`

Answer :C
Discussion
14628.

Find the area of the region bounded by the linesy=|x-1| and y=3-|x|

Answer»


ANSWER :4 SQ. UNITS
Discussion
14629.

If x,y,z are in A.P, then ( sin x - sin z)/( cos z - cos x) is equal to

Answer»

`TAN y `
`COT y `
`SIN y `
`COS y`

ANSWER :D
Discussion
14630.

Calculate the three -yearly moving averagesand display these and the originalfigures on the same graph.{:("Year ", 2001, 2002, 2003, 2004, 2005, 2006, 2007),("Values "," "20," " 40," " 50," "70," " 80," " 100," " 130):}

Answer»


ANSWER :TREND VALUESARE : 36.7, 53.3, 66.7, 83.3 and 103.3
Discussion
14631.

Simiplify :(-(1)/(2)- (sqrt3)/(2)i)^(2)

Answer»


ANSWER :`-(1)/(2) + (SQRT3)/(2)i`
Discussion
14632.

If the letters of the word ASSASSINATION are arranged at random. Find the Probability that (a) Four S's come consecutively in the word (b) Two I's and two N's come together (c ) All A's are not coming togerther (d) No two A's are coming together.

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ANSWER :(i) `(2)/(143)`
Discussion
14633.

Find the perpendicular distance from the point (-3,4) to the straight 5x-12y=2.

Answer»


ANSWER :5
Discussion
14634.

Three cards are drawn at random from a pack of 52 cards. The probability of drawing cards which has a king, a queen and a jack is .....

Answer»

`(64)/(5525)`
`(16)/(5525)`
`(128)/(5525)`
none of these

Answer :B
Discussion
14635.

Find the value of(ii) tan(855^(@))

Answer»


ANSWER :-1
Discussion
14636.

If x = sin alpha + sin beta, y = cos alpha + cos betathen tan alpha + tan beta =

Answer»

`(8XY)/(2 (y ^(2) - X ^(2)) + (x ^(2) + y ^(2)) (x ^(2) + y ^(2)-2 ))`
`(4XY)/(( y^(2) -x ^(2)) + ( x ^(2)+ y ^(2)) (x ^(2) + y ^(2) - 2))`
`(8xy)/((x ^(2) + y ^(2)) (x ^(2) + y ^(2) -2))`
`4xy`

ANSWER :A
Discussion
14637.

Write the negation of the following statement q :5+4=8

Answer»

SOLUTION :`~q:5+4!= 8,
Discussion
14638.

sin 70^(@)cos10^(@)- cos70^(@) sin 10^(@) =_________

Answer»


ANSWER :`(SQRT3)/(2)`
Discussion
14639.

While shuffling a pack of 52 playing cars, 2 are accidentally dropped. Find the probability that the missing cards to be of different colours

Answer»

`(29)/(52)`
`(1)/(2)`
`(26)/(51)`
`(27)/(51)`

ANSWER :C
Discussion
14640.

bar(a), bar(b), bar(c) are three vectors of which every pair is non-collinear. If the vectors bar(a)+bar(b), bar(b)+bar(c) are collinear with bar(c), bar(a) respectively, then bar(a)+bar(b)+bar(c)=

Answer»

`BAR(a)`
`bar(B)`
`bar(C)`
`bar(0)`

ANSWER :D
Discussion
14641.

If f(x)=logx,x=2,deltax=0.02 then df=

Answer»

`LOG(1.01)`
0.01
`-log(1.01)`
`-0.01`

ANSWER :B
Discussion
14642.

1^(2) + 3^(2) + 5^(2) + …. upto nterms =

Answer»

`( n (2n-1) (2n+1))/( 6)`
`(n (2n-1) (2n+1) )/(3)`
`(n (n+1) (2n+1) )/( 12)`
`(n (n+1) (2n-1) )/(6)`

Answer :B
Discussion
14643.

The coefficient of variation of two distribution are 60 and 70. The standard deviations are 21 and 16 respectively, then their mean is

Answer»


ANSWER :22.85
Discussion
14644.

If x+iy = (a+ib)/(a-ib), prove that x^(2) + y^(2)=1.

Answer»


ANSWER :1
Discussion
14645.

The triangle formed by the pair of lines x^(2)+16x-y^(2)=0 and x-2y+13=0 is

Answer»

EQUILATERAL
ISOSCELES
RIGHT ANGLED
scalane

Answer :A
Discussion
14646.

Period of 2 sin^(3) x - 3 cos^(3) x is

Answer»

`(pi)/(2)`
`pi`
`2PI`
`4PI`

Answer :C
Discussion
14647.

Find the multiplicative inverse of 2-3i.

Answer»


ANSWER :`2/13+ 3/13i`
Discussion
14648.

Find the distance of P(3, -2, 4) from the origin.

Answer»


ANSWER :`SQRT(29)`
Discussion
14649.

Find the roots of the equations. Q. 5x^(2)-x+4=0.

Answer»


ANSWER :`(1pmsqrt(79)i)/(10)`
Discussion
14650.

For x in (0,pi) the equation sinx+2sin2x-sin3x=3 has

Answer»

INFINITELY MANY solutions
three solutions
one solution
no solution

Answer :D
Discussion