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1451.

If the lines x - y - 1 = 0, 4x + 3y = k and 2x – 3y +1 = 0 are concurrent, then k is

Answer»

1
-1
25
5

Answer :C
Discussion
1452.

I : The locus of the midpoint of chords of the parabola y^(2)=4ax which subtends a right angle at the vetex is y^(2)=2a (x-4a) II : The locus of midpoint of chords of the parabola y^(2)=4ax which touch the circle x^(2)+y^(2)=a^(2) is (y^(2)-2ax)^(2)=a^(2)(y^(2)+4a^(2)) .

Answer»

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

ANSWER :C
Discussion
1453.

What can you say about the set, A,B,if A uu B = phi

Answer»

SOLUTION :If `A UU B =PHI IMPLIES then A =phi =B`
Discussion
1454.

Using differentials, find the approzimate value of each of the following up to 3 places of decimal :(82)^((1)/(4))

Answer»


ANSWER :`=3.0092`
Discussion
1455.

Compute theintegralsI = int_(0)^(1)(dx)/(1 + sqrt(x))

Answer»


ANSWER :2 - 2 In 2
Discussion
1456.

Two cards are drawn at random . Without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

Answer»

<P>

Solution :Let A be the event that the card DRAWN first is a black card and B be the event that the card drawn SECOND is a black card.
therefore P(B/A)=25/51
HENCE, the required probability
Discussion
1457.

Select the one which does not results in the formation of aromatic species .

Answer»




ANSWER :D
Discussion
1458.

If x+2y ge 10, 3x+4y le 24, x ge 0, y ge 0 then the minimum value of f=200x+500y is

Answer»

2300
2500
2700
3000

Answer :A
Discussion
1459.

Find the mean deviation about median for the following data

Answer»


ANSWER :`=10.34`
Discussion
1460.

Match the principal values ofcos^(-1) ( 8 x^(4) - 8 x^(2) + 1) given in column I withthe corresponding intervals of x givenin column II . For which it holds . {:(,"Column I",,"Column II"),(A,4 cos^(-1) x,p.,0 le x le 1/sqrt2),(B,4 cos^(-1)x - 2 pi,q.,1/sqrt2 le x le 1),(C,2pi - 4 cos^(-1)x,r.,-1le x le - 1/sqrt2),(D,4pi - 4 cos^(-1) x,s.,-1/sqrt2 le x le 0):}

Answer»


ANSWER :`A to q. B to s. C to p. D to r`
Discussion
1461.

If the latus rectum of a hyperbola subtend an angle of 60A^(@) at the other focus, then eccentricity of the hyperbola is

Answer»

2
`(SQRT(3)+1)/(2)`
`2sqrt(3)`
`sqrt(3)`

ANSWER :D
Discussion
1462.

Consider the lines L:(k+7)x-(k-1)y-(k-5)=0 where k is a parameter and the circle C:x^(2)+y^(2)+4x12y-60=0 Statement-1: Every member of L inttersects the circle 'C" at an angle of90^(@) Statement-2: Every member of L is tangent to the circle C.

Answer»

Statement-1 is TRUE, statement-2 is true, and statement-2 is CORRECT explanation for statement-1 .
Statement-1 is true, statement-2 is true, and statement-2 is NOT the correct explanation for statement-1.
Statement-1is true, statement-2 is false.
Statement-1 is false, statement-2 is true.

Solution :Center `(-2,-6)`. Substituting in L
`-2(k+7)+6(k-1)-4(k-5)=(-2k+6k-4k)-14-6+20`
Hence EVERY member of L passing through the centre of the circle`""IMPLIES""`cuts it at `90^(@)`.
Hence S-1 is true and S-2 is false.
Discussion
1463.

Solve the equation |{:(x+a,x,x),(x,x+a,x),(x,x,x+a):}|=0. (ane0)

Answer»


ANSWER :`THEREFOR x=-a/3`
Discussion
1464.

The modulus of sqrt2i-sqrt(-2i) is

Answer»

`-2`
`2`
`-2I`
`2i`

ANSWER :B
Discussion
1465.

If alpha and beta are the coefficients of x^(8) and x^(-24) respectively, in the expansion of (x^(4) + 2 + (1)/( x^(4))) in powers of x, then (alpha)/(beta) is equal to :

Answer»

39
26
`(32)/(3)`
`(13)/(2)`

Answer :B
Discussion
1466.

If sinA=(2)/(sqrt(5))andcosB=(1)/(sqrt(10)) where A and B are acute angles , then what is A +B equal to ?

Answer»

`135^(@)`
`90^(@)`
`75^(@)`
`60^(@)`

SOLUTION :Given , `sinA=(2)/(sqrt(5))andcosB=(2)/(sqrt(10))`

`thereforecosA=(1)/(sqrt(5))sinB=(3)/(sqrt(10))`
`thereforesin(A+B)=sincosB+cosAsinB`
`=((2)/(sqrt(15)))((1)/(sqrt(10)))+((1)/(sqrt(5)))((3)/(sqrt(10)))`
`=(2)/(sqrt(50))+(3)/(sqrt(50))=(5)/(sqrt(50))=(5)/(sqrt(25xx2))=(5)/(5sqrt(2))=(1)/(sqrt(2))`
`sin(A+B)=(1)/(sqrt(2))rArrsin(90+45)=(1)/(sqrt(2))`
`(becausesin(90+45)=cos45^(@))`
`thereforeA=90^(@)andB=45^(@)`
`thereforeA+B=90^(@)+45^(@)=135^(@)`
Discussion
1467.

If a= .^(20)C_(0) + .^(20)C_(3) + .^(20)C_(6) + .^(20)C_(9) + "…..", b = .^(20)C_(1) + .^(20)C_(4) + .^(20)C_(7) + "……"' and c = .^(20)C_(2) + .^(20)C_(5) + .^(20)C_(8) + "…..", then Value of (a-b)^(2) + (b-c)^(2) + (c-a)^(2) is

Answer»

1
2
`2^(20)`
`2^(40)`

SOLUTION :`(a-b)^(2) + (b-c)^(2) + (c-a)^(2)`
`= 2(a+bomega+ comega^(2)) (a+bomega^(2) + comega)`
`= 2`
Discussion
1468.

Match the following

Answer»

<P>(I) (i) (P)
(I)(II) (P)
`(I) (i)(R )
(I) (III)(R )

ANSWER :d
Discussion
1469.

The tangents to the curve y = (x - 2)^(2) - 1 at its points of intersectio with the line x - y = 3, intersect at the point

Answer»

`((5)/(2), 1)`
`(-(5)/(2), -1)`
`((5)/(2), -1)`
`(-(5)/(2), 1)`

Solution :Given EQUATION of parabola is
`y = (x - 2)^(2) - 1`
`implies y = x^(2) - 4X + 3`
Now, let `(x_(1) y_(1))` be the print of intersection of TANGENTS of parabola (i) and line x - y = 3, then
Equation of CHORD of contact of point `(x_(1), y_(1))` w.r.t
parabola (i) is
`T = 0`
`implies (1)/(2) (y + y_(1) = xx_(1) - 2 (x + x_(1)) + 3`
`implies y + y_(2) = 2x (x_(1) - 2) - 4x_(1) + 6`
`implies 2x (x_(1) - 2) - y = 4 x_(1) + y_(1) - 6`, this equation represent the line x - y = 3 only, so on comparing, we get
`*(2(x_(1) - 2))/(1) = (-1)/(-1) = (4x_(1) + y_(1) - 6) `
`implies x_(1) = (5)/(2)` and `y_(1) = 1`
So, the REQUIRED point is `((5)/(2), - 1)`
Discussion
1470.

""^(2n)C_(n+1)+2. ""^(2n)C_(n) + ""^(2n) C_(n-1) =

Answer»

`""^((2n+2))C_(N+1)`
`""^((n+1))C_(n+1)`
`""^(n)C_(n)`
`""^((2n+1))C_(n)`

ANSWER :A
Discussion
1471.

Which one of the following viscera accommodates a largest blood volume at rest ?

Answer»

SPLEEN
Heart
Liver
Lungs

Answer :A
Discussion
1472.

If underset(r=0)overset(21)sum f((r)/(11)+2x)= constant AA x in R and f (x) is periodic, then period of f (x) is:

Answer»

1
`1//11`
2
1

Answer :C
Discussion
1473.

I : The coefficient of x^24 in (1+3x+6x^2 + 10x^3 +....oo)^(2//3) is 25. II : The coefficient ofx^7 in (1+2x+3x^2 + 4x^2 + …..oo)^(-3) is 3

Answer»

only I is true
only II is true
both I and II are true
neither I nor II true

Answer :A
Discussion
1474.

The radius of the circle passing throgh the foci of the ellipse (x^2)/16 + (y^2)/9 = 1 and haivng centre (0,3) is

Answer»

4
3
`SQRT(12)`
`7/2`

ANSWER :A
Discussion
1475.

Consider the letters of the word MATHEMATICS. Set of repeating letters = {M, A, T}, set of non repeating letters = {H, E, I, C, S} :

Answer»

<P>

ANSWER :`A to Q; B to R; C to P; D to T`
Discussion
1476.

Integrate the functions 1/(sqrt(ax-x^(2)))

Answer»
Discussion
1477.

Given I_(m) = int_(1)^(e ) (log x)^(m) dx. If (I_m)/( K) + (l_(m-2) )/(L) =e then values of K and L are

Answer»

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

ANSWER :A
Discussion
1478.

Let A be the sum of the digits of the number (4444)^(4444)and B be the sum of the digits of the number A. Find the sum of the digits of the number B.

Answer»


ANSWER :7
Discussion
1479.

Show that y=log(1+x)-(2x)/(2+x), x gt -1, is an increasing function of x throughout its domain.

Answer»


ANSWER :`(0, OO)`
Discussion
1480.

If a, b and c are the three vectors mutually perpendicular to each other to form a right handed system and |a|=1, |b|=3 and |c|=5, then [a-2b b-3c c-4a] is equal to

Answer»

0
`-24`
3600
`-215`

SOLUTION :GIVEN that, `|a|=1, |b|=3` and `|c|=5`
`:. [(a-2b,b-3c,c-4a)]`
`=(a-2b).{(b-3c)XX(c-4a)}`
`=(a-2b). {bxxc-4bxx+12 xx}`
`=(a-2b).(a+4c+12b)`
`=a.a-24b.b=1-24xx9`
`=1-216=-215`
Discussion
1481.

If veca = (2,1), vecb = (-1,0), find 3veca + 2vecb.

Answer»

SOLUTION :`3veca+2vecb` = 3(2,1) + 2(-1,0)
(6-2, 3+0) = (4, 3)
Discussion
1482.

Ifx=sqrt(a^(sin^(-1)t)),y=sqrt(a^(cos^(-1)t)),a gt 0and -1 lt t lt 1. show that (dy)/(dx)=-(y)/(x),

Answer»


ANSWER :`-(y)/(X)`
Discussion
1483.

If y = (1+x)(1+x^(2))(1+x^(4)), then (dy)/(dx) at x = 1 is

Answer»

20
28
1
0

Answer :B
Discussion
1484.

int_(0)^(5)sqrt(25-x^(2))dx=.............

Answer»

`(25pi)/(4)`
`(PI)/(4)`
`(25)/(4)`
`25pi`

ANSWER :A
Discussion
1485.

Find the principal argument arg z, when z = (-2)/(1 + isqrt(3))

Answer»


ANSWER :`(2PI)/(3)`
Discussion
1486.

If 1, -2, 3 are the roots of ax^(3) + bx^(2) + cx + d = 0 then the roots of ax^(3) + 3bx^(2) + 9cx + 27d = 0are

Answer»

`2,4,6`
`3,4,5`
`1/2 ,1,3/2`
`-1,0,1`

ANSWER :A
Discussion
1487.

Find the locus of the third vertex of a right angled triangle , the ends of whose hypotenuse are (4,0) and (0,4)

Answer»


ANSWER :`X^(2)+y^(2)-4x-4y=0`
Discussion
1488.

Show that the circles x^(2)+y^(2)-2x=0 and x^(2)+y^(2)+6x-6y+2=0 touch each other. Find the coordinates of the point of contact. Is the point of contact external of internal?

Answer»


ANSWER :`(-1/5,3/5)` TOUCHES EXTERNALLY
Discussion
1489.

If z is a complex number such that |z| = 2 , find the maximum and minimum value of |z - 2 + 3i|

Answer»


ANSWER :A::B::C
Discussion
1490.

If alpha , beta , gamma are the roots of x^(3) + qx + r = 0 then the equation whose roots (beta)/(gamma) + (gamma)/(beta) , (gamma)/(alpha) + (alpha)/(beta), (alpha)/(beta) + (beta)/(alpha)is

Answer»

`x^(3) - q^(2) - x^(2) - 2qr^(2) x - R^(4) = 0 `
`rx^(3) + q(1 - r) x^(2) + (1 - r)^(3) = 0`
`x^(3) - 2q x^(2) + q^(2) x + r^(2) = 0 `
`r^(2) x^(3) + 3r^(2) x^(2) + (3r^(2) + q^(3)) x + 2q^(3) + r^(2)= 0`

Answer :4
Discussion
1491.

If a, b, x, y in R, omega ne 1, is a cube root of unity and (a + b omega)^(7) = x + y omega, then (b + a omega)^(7) equals

Answer»

`y + X OMEGA`
`-y - x omega`
`x + y omega`
`-x-y omega`

ANSWER :A
Discussion
1492.

Differentiate.sinx.w.r.t.cotx

Answer»

Solution :Lety=sin XAND `z = COT X.
dy/dx=cos x `and `dz/dx=-cosec^2x`
`dy/dz=(dy/dx)/(dz/dx)=(COSX)/-(cosec^2x)=-cosxsin^2x`
Discussion
1493.

Find the equation of the circum circle of the triangle formed by the linesx- y -2=0, 2x - 3y + 4=0, 3x - y + 6 =0

Answer»


Answer :` x^(2)+ y^(2) -24X +16Y -52 =0 `
Discussion
1494.

A circle passes through origin and has its centre on y = x. If it cuts x^2+y^2-4x-6y+10=0orthogonally then the equation of the circle is

Answer»

`x^2+y^2-x-y=0`
`x^2+y^2-6x-4y=0`
`x^2+y^2-2x-2y=0`
`x^2+y^2+2x+2y=0`

ANSWER :C
Discussion
1495.

Let (hatp xx vecq) xx (hatp.vecq)vecq =(x^(2)+y^(2))vecq + (14-4x-6y)vecp Where hatp and hatq are two non-collinear vectors vecp is unit vector and x,y are scalars. Then the value of (x+y) is

Answer»

4
5
6
7

Solution :`(vecp xx vecq) xx HATP+(hatp.vecc)vecq=(X^(2)+y^(2))vecq+(14-4x-6y)hatp`
`rArr (hatp.hatp)vecq+(1-4x-6y)hatp`
Since `hatp` and `vecq` are non-zero non-collinear.
We can COMPARE coefficients of vectors `hatp` and `hatq`.
`therefore 1+hatp.vecq=x^(2)+y^(2)`.............(i)
And `hatp.vecq=4x+6y-14`............(II)
From (i) and (ii), we get
`rArr x^(2) + y^(2)-4x-6y+13=0`
`rArr (x-2)^(2)+(y-3)^(2)=0`
x=2 and y=3
Discussion
1496.

Find the value of sin^(-1)("sin"(3pi)/5).

Answer»


ANSWER :`(2PI)/5`
Discussion
1497.

By what percent did the number of school with Pupil/Teacher ratio less than 16 increase in January 1999 over January 1998?

Answer»

`-2%`
`0%`
`2%`
`10%`

ANSWER :D
Discussion
1498.

If f(y)=f(x^(2)+2) and f'(3)=5, then (dy)/(dx) at x=1 is :

Answer»

10
15
25
5

Answer :A
Discussion
1499.

The point of concurrence of all conjugate lines of the line 5x + 7y - 78 = 0 with respect to the circle x^(2) + y^(2) + 6x + 8y - 96 = 0is

Answer»

(-2, 3)
(3, -2)
(3,2 )
(2,3)

ANSWER :D
Discussion
1500.

If OT is the semi- minor axisof an ellipse A and B are its foci andangle ATB is right angle , then the eccentricity of that ellipse is

Answer»

1
`1/sqrt3`
`1/sqrt2`
`1/2`

ANSWER :C
Discussion