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10651.

The value of a^x-b^y is (wherex=sqrt(log_ab)and y=sqrt(log_ba),agt0,bgt0 and a,bne1)

Answer»

1
2
0
-1

Answer :C
Discussion
10652.

Let vec(a),vec(b) and vec( c ) are three unit vectors such that vec(a)xx(vec(b)xx vec( c ))=(sqrt(3))/(2)(vec(b)+vec( c )). If the vectors vec(b) and vec( c ) are not parallel then the angle between vec(a) and vec(b) is ……….

Answer»

`(3pi)/(4)`
`(pi)/(2)`
`(2PI)/(3)`
`(5pi)/(6)`

Answer :D
Discussion
10653.

Which of the following matrice is invertible? [[1,0,0],[1,1,1],[2,-1,1]]

Answer»

SOLUTION :LET A=`[[1,0,0],[1,1,1],[2,-1,1]]`
`therefore absA= [[1,0,0],[1,1,1],[2,-1,1]]`
=`1[[1,1],[-1,1]]=1+1=2 ne 0`
`therefore` A is INVERTIBLE.
Discussion
10654.

Find the domain of the following following functions: (a) f(x)=(sin^(-1))/(x) (b)f(x)=sin^(-1)(|x-1|-2) (c ) f(x)=cos^(-1)(1+3x+2x^(2)) (d ) f(x)=(sin^(-1)(x-3))/(sqrt(9-x^(2))) (e ) f(x)="cos"^(-1)((6-3x)/(4))+"cosec"^(-1)((x-1)/(2)) (f) f(x)=sqrt("sec"^(-1)((2-|x|)/(4)))

Answer»


Solution :(a) `f(x)` is defined if `x in [-1,1] " and " x ne 0,`i.e.,
`x in [-1,0)cup (0,1]`
(b) `f(x)=sin^(-1)(|x-1|-2)`
SINCE the domain of `sin^(-1)x` is `[-1,1],f(x)` is defined if
`-1 le |x-1|-2 le 1`
or `1 le |x-1| le 3`
i.e., `-3 le x-1 le -1 " or " 1 lex -1 le 3`
i.e., `-2 le x le 0 " or " 2 le x le 4`
or domain `=[-2,0]cup [2,4]`
(c ) `-1 le 1+3x+2x^(2) le 1`
or ` 2x^(2)+3x+1 GE -1`
or` 2x^(2) +3x+2 ge 0 "(1)" `
and`2x^(2) +3x le 0 "(2)" `
From equation (2), `2x^(2) +3x le 0" or " 2x(x+(3)/(2)) le 0`
or `(-3)/(2) le x le 0 " or " x in [-(3)/(2),0]`
In equation (1), we get imaginary root for `2x^(2)+3x+2=0 " and " 2x^(2)+3x+2 ge 0` for all x. THEREFORE,
domain of function`=[-(3)/(2),0]`
(d) To define `f(x), 9-x^(2) gt 0 " or " -3 lt x lt 3 "(1) " `
`-1 le (x-3) le 1 " or " 2 le x le 4 "(2)" `
From equations (1) and (2), `2 le x lt 3," i.e., " x in [2,3).`
(e ) `f(x)="cos"^(-1)((6-3x)/(4))+"cosec"^(-1)((x-1)/(2))`
For ` "cos"^(-1)((6-3x)/(4)),-1 le (6-3x)/(4) le 1`
or `-4 le 6-3x le 4`
or `-10 le -3x le -2`
or `2//3 le x le 10//3 "(1)" `
For ` "cosec"^(-1)((x-1)/(2)),(x-1)/(2) le -1 " or "(x-1)/(2) ge 1`
i.e., `xle -1 " or " x ge 3 "(2)" `
From equation (1) and (2), ` x in [3,(10)/(3)].`
(f)`f(x)=sqrt("sec"^(-1)((2-|x|)/(4)))`
`sec^(-1)` function always TAKES positive values which are `[0,pi]-{pi/2}.`
Hence, the given function is defined if
`(2-|x|)/(4) le -1 " or " (2-|x|)/(4) ge 1`
i.e., `|x| ge 6 " or " |x| le -2 i.e.,x in (-oo,-6] cup [6,oo)`
Discussion
10655.

If sinA=(3)/(5)"where "450^(@)ltAlt540^(@),then "cos"(A)/(2) is equal to

Answer»

`(1)/(sqrt(10))`
`-sqrt((3)/(10))`
`(sqrt(3))/(sqrt(10))`
None of the above

Solution :`sinA=(3)/(5),450^(@)ltAlt54^(@)`
`rArr225^(@)lt(A)/(2)lt270^(@)`
`COSA=(-4)/(5)(because A" LIES in Q 2")`

`therefore"cos"^(2)(A)/(2)=(1+cosA)/(2)=(1)/(10)`
`RARR"cos"(A)/(2)=(-1)/(sqrt(10))(because(A)/(2)" lies in Q 3")`
Discussion
10656.

A die is thrown 15 times. Getting a number greater than 5 is a success.Find the mean and variance of the number of successes ?

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SOLUTION :Here n=15, and `p=p(a numberlt5)=1/6` and `q=1-p=5/6` `thereforeMean=np=15/6=5/2` VARIANCE=`npq=15,1/6,5/6=25/12`
Discussion
10657.

Whichof the followingstatementsis false ?

Answer»

VECTORS `vec(P), vec(A)`and`vec(A) XX vec(B)`are linearlydependent
vectors `vec(P), vec(A)` and `vec(A) xx vec(B)`are linearlyindependent
`vec(P)`is orthogonal to `vec(B)`and has length `1//sqrt(2)`
Noneof these

Answer :B
Discussion
10658.

Find |vecb| if (veca + vecb).(veca - vecb) =8 and |veca| = 8|vecb|

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ANSWER :`SQRT(8/63)`
Discussion
10659.

d/dx (7x +5)^3

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`[21 (7X +5)^2]`
`[3 (7x +5)^2]`
`[7 (7x +5)^2]`
21

Answer :A
Discussion
10660.

The value sum_(r=0)^(7)tan^(2)""(pix)/(16) is equal to :

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28
35
21
None of these

Answer :B
Discussion
10661.

Statement-I : If P,Q,R,S are the ends of latus recta of the hyperbola x^(2)/(16)-y^(2)/9=1 then the area of rectangle of PQRS is 45 square units. Statement -II: The centre of the hyperbola ((x+y+1)^(2))/(3)-((x-y-3)^(2))/(6)=1 lies in second quadrant. Which of above statement is true

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only I
only II
both I and II
neither I nor II

Answer :A
Discussion
10662.

A,B,C are the centres of the three circles C_1,C_2,C_3such that C_1,C_2touch each other exernally and they both touch C_3 from inside then the radical centre of the circles is for triangle ABC is

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incentre
Excentre OPPOSITE to C
Excentre opposite to B
Excentre opposite to A

Answer :B
Discussion
10663.

If 9 digits (1 to 9) are arranged in the spaces of number 1263 _______6, what is the probability that the number is divisible by 9.

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ANSWER :`(1)`
Discussion
10664.

Find the equation of the parabola whose vertex and focus are on the positiveX-axis at a distance of a and a' from the origin respectively.

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ANSWER :`y^(2)=4(k-k)(x-k)`
Discussion
10665.

If p and q are chosen from the set (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), with replacement, the probability that the roots of the equation x^2 + 2px +q = 0 are real is

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0.84
0.16
0.62
0.38

Answer :A
Discussion
10666.

Fundamental theorem of definite integral : int_(0)^(pi/2)(cos2x)/((sinx+cosx)^(2))dx=...........

Answer»

`(PI)/(2)`
0
`-(pi)/(4)`
`(pi)/(4)`

ANSWER :B
Discussion
10667.

An equation a_(0) + a_(2)x^(2) + "……" + a_(99)x^(99) + x^(100) = 0 has roots .^(99)C_(0), .^(99)C_(1), C_(99)C_(2), "…..", .^(99)C_(99) The value of (.^(99)C_(0))^(2) + (.^(99)C_(1))^(2) + "….." + (.^(99)C_(99))^(2) is equal to

Answer»

`2a_(98) - a_(99)^(2)`
`a_(99)^(2) - a_(98)`
`a_(99)^(2) - 2a_(98)`
none of these

Solution :`a_(0) + a_(1)x+a_(2)x+"...."+a_(99)x+x^(100) = 0` has root `.^(99)C_(0), .^(99)C_(1), .^(99)C_(2), "....", .^(99)C_(99)`.
or `a_(0) + a_(1)x+a_(2)x^(2) + "...."+a_(99)x^(99) + x^(100)`
`= (x-.^(99)C_(0))(x-.^(99)C_(1)) (x-.^(99)C_(2))"....."(x-.^(99)C_(99))`
Now, sum ofroot is
`.^(99)C_(0)+.^(99)C_(1)+.^(99)C_(2)+"...."+.^(99)C_(99)= - (a_(99))/("COEFFICIENT of" x^(100))`
or `a_(99) = - 2^(99)`
Also, sum of PRODUCTOF roots taken two at a time is
`(a_(99))/("coefficient of" x^(100))`
`:. underset(0 le i ltj le 99)(sumsum) .^(99)C_(i).^(99)C_(j) = ((underset(i=0)overset(99)sumunderset(j=0)overset(99)sum.^(99)C_(i).^(99)C_(j))- underset(i=0)overset(99)sum(.^(99)C_(i))^(2))/(2)`
`= ((underset(i=0)overset(99)sum.^(99)C_(i)2^(99))-underset(i=0)overset(99)sum(.^(99)C_(i))^(2))/(2)`
`= (2^(99)2^(99)-underset(i=0)overset(99)sum(.^(99)C_(i))^(2))/(2)`
`= (2^(198) - .^(198)C_(99))/(2)`
`(.^(99)C_(0))^(2)+ (.^(99)C_(1))^(2) + "......" + (.^(99)C_(99))^(2)`
`= (.^(99)C_(0) + .^(99)C_(1) + .^(99)C_(2) "......." + .^(99)C_(99))^(2) - 2 underset(0lei ltjle99)(sumsum).^(99)C_(i).^(99)C_(j)`
`=(-a_(99))^(2) - 2a_(98)`
`= a_(99)^(2) - 2a_(98)`
Discussion
10668.

Find the equation of circles determined by the following conditions. The centre at (1, 4) and passing through (-2, 1).

Answer»

Solution :`therefore` PC = `SQRT((1 + 2)^2 + (4 - 1)^2)`
= `sqrt(9 + 9) = sqrt(18)`
`therefore` Equation of the circle is
`(x - H)^2 + (y - k)^2 = a^2`
or `(x - 1)^2 + (y - 4)^2 = 18`
Discussion
10669.

A={1, 2, 3, ..., 9} ............

Answer»


SOLUTION :Favourable cases `=2.^(9)C_(3)`
Total arrangement `=9xx8xx7`
Required PROBABILITY `=(2xx .^(9)C_(3))/(9xx8xx7)=1/3`
Discussion
10670.

1/2. ""^nC_0 + ""^nC_1 + 2. ""^nC_2 + 2^2. ""^nC_3 + …….+ 2^(n-1) . ""^nC_n =

Answer»

`(3^(-N))/(2)`
`(4^(2n))/(2)`
`(4^n)/(2)`
`(3^n)/(2)`

Answer :D
Discussion
10671.

Find the value of (18^(3) + 7^(3) + 3*18*7*25)/((3^(6) + 6*243* + 15*81*4 + 20*27*8 + 15*9 *16*3*32 + 64))

Answer»


ANSWER :1
Discussion
10672.

Probability of an event that 10 persons A_(1), A_(2), ..., A_(10) with ages 70 years above expires in one year is (1)/(2). Then ......... is the probability that person A_(1) expire first in one year.

Answer»

`(1)/(10240)`
`(9217)/(10240)`
`(1023)/(24010)`
`(1023)/(10240)`

ANSWER :D
Discussion
10673.

P_(1):y^(2)=49x and P_(2):x^(2)=4ay are two parabolas. Equation of a tangent to the parabola P_(1) at a point where it intersects the parabola P_(2) is :

Answer»

`2x-y-4a=0`
`y=0`
`x-2y+4a-0`
`x-y-0`

ANSWER :C
Discussion
10674.

underset(x to oo)limsin((2)/(x))is equal to

Answer»

2
`(1)/(2)`
`OO`
0

Answer :A
Discussion
10675.

For two non-singular matrices A&B, show that adj (AB)=adj(AB)=(adjB)(adjA)

Answer»

Solution :We have (AB)(adj AB)=`|AB|I_(n)`
`|A|=|B| I_(n)`
`A^(-1)(AB)("adj"AB))=|A||B|A^(-1)`
`Rightarrow B"adj"(AB)=|B|adjA(THEREFORE A^(-1)(1)/(|A|)adjA)`
`Rightarrow B^(-1)B"adj"(AB)=|B|B^(-1)adjA`
`Rightarrow B^(-1)B "adj"(AB)|B|B^(-1)"adj"A`
`Rightarrow "adj"(AB)=("adj"B)("adjA")`
Discussion
10676.

If the mean and standard deviation of 20 observations x_1,x_2…….x_20are 50 and 10 respectively , thenunderset( i=1) overset( 10) sum x_i^(2)is equal to

Answer»

` 2510`
`50200`
`52000`
`2600`

ANSWER :C
Discussion
10677.

The number of ways can five men sit around table so that all shall not have the same neighbours in any two arrangements is

Answer»

11
12
9
10

Answer :B
Discussion
10678.

lim_(xto oo)sqrt((x-sinx)/(x+cos^(2)x))=

Answer»

0
1
`OO`
`1/2`

ANSWER :B
Discussion
10679.

Let C_(1) and C_(2) be the graphs of the functions y=x^(2) and y=2x, respectively, where 0le x le 1." Let "C_(3) be the graph of a function y=f(x), where 0lexle1, f(0)=0. For a point P on C_(1), let the lines through P, parallel to the axes, meet C_(2) and C_(3) at Q and R, respectively (see figure). If for every position of P(on C_(1)), the areas of the shaded regions OPQ and ORP are equal, determine the function f(x).

Answer»

<P>

Solution :Let P be on `C_(1),y=x^(2) be (t,t^(2))`
`therefore"y co-ordinate of Q is also "t^(2)`
`"Now, Q on y =2x where "y=t^(2)`
`therefore""x=t^(2)//2`
`therefore""Q((t^(2))/(2),t^(2))`
For POINT R, x=t and it is on y=f(x)
`therefore""R(t,f(t))`
Given that,
Area OPQ = Area OPR
`rArr""int_(0)^(t^(2))(sqrt(y)-(y)/(2))dy=int_(0)^(t)(x^(2)-f(x))dx`
DIFFERENTIATING both sides w.r.t. t, we GET
`(sqrt(t^(2))-(t^(2))/(2))(2t)=t^(2)-f(t)`
`rArr""f(t)=t^(3)-t^(2)`
`rArr""f(x)=x^(3)-x^(2)`
Discussion
10680.

If I_(1)=int_(0)^(pi//4)x^(40)sin^(10)xdx, I_(2)=int_(0)^(pi//4)x sinxdx, then

Answer»

`I_(1) LT I_(2)`
`I_(2)=(1)/(SQRT2)(1-pi//4)`
`I_(1)gt I_(2)`
`I_(2)=(1)/(sqrt2)((pi)/(4)-2)`

ANSWER :A::B
Discussion
10681.

Find all integers a,b,c such that a^(2) = bc + 1, b^(2) = ca+1.

Answer»


ANSWER :`(1,1,0)(-1,-1,0), (-1,1,0),(-1,0,+1), (+1,0,-1),(0,1,-1),(0,-1,1)`
Discussion
10682.

Determine P(E|F). A die thrown three times, E: '4 appears on the third toss.F:'6 and 5 appears respectively on the two tosses'.

Answer»

SOLUTION :Here N (S) = `6^3` =216
E={(1,1,4),(1,2,4),(1,3,4),(1,4,4),(1,5,4),(1,6,4),(2,1,4),(2,2,4),(2,3,4),(2,4,4),(2,5,4),(2,6,4)
Discussion
10683.

Let a=a_(1)hati+a_(3)hatk, b=b_(1)hati+b_(3)hatk, c=c_(1)hati+c_(2)hatj+c_(3)hatk. If |c|=1 and (axxb)xxc=0, then |(a_(1),a_(2),a_(2)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))|is equal to

Answer»

0
1
`|a|^(2)|b|^(2)`
`|AXXB|^(2)`

Solution :GIVEN, `(axxb)xxc=0, " then " c = (axxb)/(|axxb|) `
`therefore (a xxb)*c = |axxb| `
`RARR |(a_(1),a_(2),a_(3)),(b_(1),b_(2), b_(3)),(c_(1),c_(2),c_(3))|=|axxb|^(2)`
Discussion
10684.

Find the distance of the point veca from the plane vecr*hatn=d measured parallel to the line vecr=vecb+vec(tc).

Answer»

Solution :A line parallel to `VECC` and passing through
`A(veca)` is `vecr=veca+lamdavecc`
A POINT for some `lamda` SATISFIES plane
`""(veca+lamdavecc)*HATN=d`
or `""lamda=(vecd-veca*hatn)/(vecc*hatn)`
`THEREFORE""` Distance `= (|vecd-hata*hatn|)/(|vecc*hatn|)|vecc|`.
Discussion
10685.

If y=e^(sin^(-1)x)+e^(cos^(-1)x),0ltxlt1, then

Answer»

`(DY)/(dx)=0`
`(dy)/(dx)=(PI)/(2)`
does not exist
none of these

Answer :A
Discussion
10686.

For all complex numbers z_(1) , z_(1) satisfying |z_(1)| = 12 and |z_(2) - 3 - 4i| = 5 , the minimum value of |z_(1) - z_(2)| is

Answer»

0
2
7
17

Answer :B
Discussion
10687.

(i) If f_(n)(x)=(n+1)/((n-1)!)x^(n), then find the value of int_(0)^(1)[sum_(n=1)^(oo)f_(n)(x)]dx. (ii) Evaluate int_(0)^(2)lim_(nrarroo)int_(r=0)^(n)(x^(r+1))/(r!)5^(r)dx.

Answer»


ANSWER :E
Discussion
10688.

Let" *" be a binary operation on N given by a"*"b=LCM of a and b. Find20"*"16.

Answer»


ANSWER :80
Discussion
10689.

Compute the area of the loop of the curve y^(2) = x^(2) [(1 + x)/(1 –x)].

Answer»


ANSWER :`2-(pi//2)` SQ. UNITS
Discussion
10690.

If one root is greater then 3 and other root is less than 1 'a' belongs to

Answer»

`PHI`
`(-1,2)CUP(2,3)`
`(-INFTY,2)`
`(2,infty)`

ANSWER :A
Discussion
10691.

Find the unit vector in the direction of the vector vecr_1-vecr_2, where vecr_1 = hati+2hatj+hatk and vecr_2 = 3hati+hatj-5hatk.

Answer»

Solution :`vecr_1-vecr_2 = hati+2hatj+hatk-3hati-HATJ+5hatk`
=`-2hati+hatj+6hatk`
`|vecr_1-vecr_2| = SQRT(4+1+36) = sqrt(41)`
Unit vector ALONG `vecr_1-vecr_2`
= `(-2hati+hatj+6hatk)/sqrt(41) = -2/sqrt(41) hati + 1/sqrt(41) hatj + 6/sqrt(41) hatk`.
Discussion
10692.

If log_(e )((1+x)/(1-x))=a_(0)+a_(1)x+a_(2)x^(2)+…oo then a_(1), a_(3), a_(5) are in

Answer»

A.P.
G.P.
H.P.
A.G.P.

ANSWER :C
Discussion
10693.

For m=n and m, n in N then the value of int_(0)^(pi) cos m x cos n x dx =

Answer»

0
`PI/2`
`pi`
1

Answer :B
Discussion
10694.

Let alpha chord of a circle be that chord of the circle which subtends an angle alpha at the center. If x+y=1 is a chord of x^(2)+y^(2)=1, then alpha is equal to

Answer»

`PI //4`
`pi//2`
`pi//6`
`x+y=1` is not a chord

Solution :
Discussion
10695.

The two lines x=ay+b, z=cy+d and x=a'y+b', z=c'y+d will be perpendicular ,if and only if

Answer»

aa'+bb'+cc'=0
(a+a')(b+b')+(C+c')=0
aa'+cc'+1=0
aa'+bb'+cc'+1=0

ANSWER :C
Discussion
10696.

If hat(a), hat(b) and hat(c) are unit vector inclined to each other at an angle of pi/3 then the absolute value of [hat(a)+hat(b)hat(b)+hat(c)hat(c) + hat(a)] equals

Answer»

`SQRT2`
`SQRT3`
`3sqrt2`
`2SQRT3`

ANSWER :A
Discussion
10697.

Thereare 12boysto beseatedon2benches, 6on eachlench. Twoofthemdesiretositon thebenchand twootherson theother. Thenumberof waysin whichtheboyscan beseatedon thebenches is

Answer»

`300xx8!`
`900xx8!`
`600xx8!`
`90xx8!`

ANSWER :B
Discussion
10698.

Find the angle between the curves y^(2)=8x and 4x^(2)+y^(2)=32.

Answer»


ANSWER :3
Discussion
10699.

A dry mixture consists of 3 cups of flour for every 2 cups of sugar. How much sugar is in 4 cups of the mixture?

Answer»


ANSWER :`(8)/(5)`
Discussion
10700.

Find the second order derivatives of the function tan^(-1) x.

Answer»


ANSWER :`-(2X)/((1+x^2)^2)`
Discussion