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1651.

If two tangents drawn from the point (2, a) to the hyperbola are at right angles, then a equal to

Answer»


ANSWER :1.73
Discussion
1652.

Prove that the logarithmic function is increasing on (0, oo)

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ANSWER :`F'(X)GT 0`
Discussion
1653.

Column -1 : real valued function, Column -2: continuity of the function, Column - 3: differentiability of the function, Match the following Column(s) (Whre [.] denotes the greatest integer function and {.} fractional at function) Which of the following combination combination is correct?

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(II)(iii)(S)
(III)(I)(P)
(II)(ii)`(R)`
(III)(i)`(R)`

Solution :(I) `F(x)=||x-6|-|x-8||-|x^(2)-4|+3x-|x-7|^(3)` is continuous `AA x epsilon R` and not DIFFERENTIABLE at `x=-2, 2, 6, 7` & `8`
(II) `f(x)=(x^(2)-9)|x^(2)+11x+24|+sin|x-7|+cos|x-4|+(x-1)^(3//5)sin(x-1)` is continuous
`AA x epsilonR` and not differentiable at `x=-8` & `7`
(III) `f(x)={((x+1)^(3//5)-(3pi)/2, : xlt-1),((x-1/2)cos^(-1)(4x^(3)-3x), : -1le x le 1), ((x-1)^(5//3),:1ltxlt2):}` is discontinuous at `x=-1` & 1 not differentiable at `x=-1, -1/2` & 1
(IV) `f(x)=P{sinx}{cosx}+(sin^(3)pi{x})([x]), x epsilon [-1, 2PI]`
Let `g(x)=UNDERSET("cont. at" x=I) ubrace((sinpi{x})([x]))(sin^(2)pi{x})`
`g^(')(I^(+))=g^(')(I^(-))` so differentiable at `x=I` and for `{sinx}{cos}`
Doubtful poins for non differentiabililty are `x=0, (pi)/2, pi, (3pi)/2`
`:.{sinx}.{cosx}` is discontinuous at `x=0, (pi)/2, 2pi`
So not differentiable at `x=2npi, 2npi+(pi)/2`
Discussion
1654.

Column -1 : real valued function, Column -2: continuity of the function, Column - 3: differentiability of the function, Match the following Column(s) (Whre [.] denotes the greatest integer function and {.} fractional at function) Which of the following is correct?

Answer»

(I)(i)(S)
(IV)(i)(S)
(II)(ii)(S)
(II)(ii)(P)

Solution :(I) `f(X)=||x-6|-|x-8||-|x^(2)-4|+3x-|x-7|^(3)` is continuous `AA x epsilon R` and not differentiable at `x=-2, 2, 6, 7` & `8`
(II) `f(x)=(x^(2)-9)|x^(2)+11x+24|+sin|x-7|+cos|x-4|+(x-1)^(3//5)sin(x-1)` is continuous
`AA x epsilonR` and not differentiable at `x=-8` & `7`
(III) `f(x)={((x+1)^(3//5)-(3pi)/2, : xlt-1),((x-1/2)cos^(-1)(4x^(3)-3x), : -1le x le 1), ((x-1)^(5//3),:1ltxlt2):}` is DISCONTINUOUS at `x=-1` & 1 not differentiable at `x=-1, -1/2` & 1
(IV) `f(x)=P{sinx}{cosx}+(sin^(3)pi{x})([x]), x epsilon [-1, 2PI]`
Let `G(x)=underset("cont. at" x=I) ubrace((sinpi{x})([x]))(sin^(2)pi{x})`
`g^(')(I^(+))=g^(')(I^(-))` so differentiable at `x=I` and for `{sinx}{cos}`
Doubtful poins for non differentiabililty are `x=0, (pi)/2, pi, (3pi)/2`
`:.{sinx}.{cosx}` is discontinuous at `x=0, (pi)/2, 2pi`
So not differentiable at `x=2npi, 2npi+(pi)/2`
Discussion
1655.

int _(-1) ^(1) (x ^(4))/( 1 + e ^(x ^(7)))dx is

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`1/2`
0
`1/5`
NONE of these

ANSWER :C
Discussion
1656.

The general solution of differential equation : ydx-xdy=0 is :

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xy=c
`x=cy^2`
`y=cx^2`
x=cy

Answer :D
Discussion
1657.

Evaluate the following inegrals int sin mx sin nx dx

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Answer :`(1)/(2)[(SIN(m-n)x)/(m-n)-(sin(m+n)x)/(m+n)]+c`
Discussion
1658.

The value of [(a-b)(b-c)(c-a)] is equal to

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0
1
2[ a b c ]
2

Solution :Now, `[(a-b)(b-c)(c-a)]`
`= (a-b).[(b-c) XX (c-a)]`
`= (a-b).[bxxc-bxxa-cxxc+cxxa]`
`= (a-b).[bxxc-bxxa+cxxa] ""[ :' c xx c = 0]`
`= a.[bxxc]- a.[bxxa] + a.[cxxa] - b.[bxxc]+b.[bxxa] - b.[cxxa]`
`= [abc] - 0 + 0 + 0 - [BCA]`
`= [abc] - [abc] = 0`
Discussion
1659.

Mean of a set of numbers is bar(x). If each number is increased by lambda, then the mean of new set is

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`BAR(X)`
`bar(x) + LAMBDA`
`lambda bar(x)`
`lambda`

ANSWER :B
Discussion
1660.

The solution of x - 1 = (x - [x]) (x - {x}), ( where [x] and {x} are integral and frectionalpart respectively of x ) is :

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X `in` R
` x in ` R ~ [ 1, 2).
x `in [ 1, 2)`
` x in `R ~ [ 1, 2] .

Answer :C
Discussion
1661.

An insect of ....................

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Solution :`v_(o)=v_(1)=omega A`
`=(sqrt(k/M))A`

`V_("rel")=sqrt(v_(0)^(2)+v_(1)^(2)+2v_(0)v_(1) COS 60^(@))`
`=sqrt(3) A sqrt(k/M)`
Discussion
1662.

The number of ways of dividing 100 scouts into 3 squads of 50, 30, 20 respectively is

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`(80!)/(60!50!40!)`
`(90!)/(50!40!30!)`
`(100!)/(50!30!20!)`
`(120!)/(70!50!20!)`

ANSWER :C
Discussion
1663.

Find by integration, the area of the region bounded by the lines 5x-2y=15,x+4=0 and the x-axis.

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ANSWER :`(35)/(2)`
Discussion
1664.

If the integral int(5tanx)/(tanx-2)dx=x+a log|sin x-2 cosx|+c then a is equal to

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`-1`
`-2`
1
2

Answer :4
Discussion
1665.

Let p: Maths is intersting and q : Maths is easy, then p rArr (~p vv q) is equivalent to

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It MATHS is EASY then it is INTERESTING
Either Maths is interesting or it easy
If Maths is interseting then it is easy
Maths is NEITHER interesting nor easy

Answer :C
Discussion
1666.

Evaluate : int _(-2)^(3) |1 - x^(2) | dx

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ANSWER :`(28)/(3)`
Discussion
1667.

Let a function f : XtoY is defined where X={0,1,2,3,….,9}, Y={0,1,2,…..,100} and f(5)=5, then the probability that the function of type f: xtoB where BsubeY is of bijective in nature is

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`(10!)/(sum_(r=1)^(101)r^(9)*^(100)C_(r-1))`
`("^(101)C_(9)*9!)/(sum_(r=1)^(101)r^(10)*^(100)C_(r))`
`("^(100)C_(9)*9!)/(sum_(r=1)^(101)r^(10)*^(101)C_(r))`
`("^(100)C_(9)*9!)/(sum_(r=1)^(101)r^(9)*^(100)C_(r-1))`

Solution :`(d)` Favourable cases for bijection is `"^(100)C_(9)xx9!`
For total cases
`'B'` is set of one ELEMENT, no. of function `-1`
`'B'` is set of two element, no. of function `-2^(9)`
`'B'` is set of three element, no. of function `-3^(9)`
.........
........
`'B'= Y`, No. of function `=101^(9)`
`:.` Total cases `=1+"^(100)C_(1)*2^(9)+^(100)C_(2)*3^(9)+....+^(100)C_(100)*101^(9)`
`=sum_(r=1)^(101)r^(9)*"^(100)C_(r-1)`
`:.` Required probability `=("^(100)C_(9)*9!)/(sum_(r=1)^(101)r^(9)*^(100)C_(r =1))`
Discussion
1668.

Find the domain and range of the following function : f: R rarr R , f(x) =1/(1-x^(2)), x ne pm1

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SOLUTION :N/A
Discussion
1669.

If f(x)= {((x^(2))/(a)-a",",x lt a),(0",",x=a),(a-(x^(2))/(a)",",x gt a):} then, ………

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`UNDERSET(x rarr a^(+))("LIM") F(x)= a`
`underset(x rarr a)("lim") f(x)= -a`
f is continuous at x=a
f is differentiable at x=a

Answer :C
Discussion
1670.

Which one of the following is aromatic ?

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SOLUTION : azulene exists as zwittor ION so both the RINGS are aromatic
Discussion
1671.

f : R rarr R , f(x) = cos x and g :R rarr R , g(x) = 3x^(2) then find the composite functions gof and fog.

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SOLUTION :N/A
Discussion
1672.

Let u, v and w be such that |u|=1, |v|=2, |w|=3 If the projection v along u is equal to that of w along u and v, w are perpendicular to each other, then |u-v+w| is equal to

Answer»

2
`SQRT7`
`sqrt(14)`
14

Solution :SINCE, `|u| = 1, |v| = 2,|w| = 3`
The projectionof v ALONG `u = (v.u)/(|u|)` and theprojectionof w along `u = (w.u)/(|u|)`.
ACCORDINGTO givencondition.
`(v.u)/(|u|) = (w.u)/(|u|)`
`rArr v.u = w.u "…."(i)`
Since, v and w are perpendicularto each other.
`:. v. w = 0`
Now, `| u- v + w|^(2) = |u|^(2) +|v|^(2) +|w|^(2)`
`- 2u . v - 2v.w+2u.w`
`rArr |u- b- v + w|^(2) = 1 + 4+ 9 - 2 u . v + 0 + 2u . v` [from EQ. (i) ]
`rArr |u - v + w|^(2) = 1 + 4 + 9`
`:. |u- v + w| = sqrt(14)`
Discussion
1673.

The minimum length of the intercept between the coordinate axes made by a tangent of the ellipse (x^(2))/(64)+(y^(2))/(4)=1 is

Answer»

10
`(17)/(2)`
8
`(15)/(2)`

ANSWER :A::C
Discussion
1674.

If x+y=3 is tangent at (2,1) on hyperbola, intersects asymptotes at A and B such that AB=8sqrt(2). If centre of hyperbola is (-1,-1), then least possible value of sum of semi -transverset axis and semi -conjugate axis is psqrt(q) where HCF (p,q)=1 and sqrt(q) is irrational number. then, (p+q) is equal to

Answer»


SOLUTION :`(x-2)/(-1/(sqrt(2)))=(y-1)/(1/(sqrt(2)))=4sqrt(2)` or `-4sqrt(2)`
`:. A(-2,5)` & `B(6, -3)`
`:.` area of `/_\CAB=20`
`IMPLIES` sum `=2sqrt(20)=4sqrt(5)`
Discussion
1675.

The probability of choosing randomly a number c from the set {1,2,3,…,9} such that the quadratic equationx^(2) + 4x + c = 0has real roots is

Answer»

`1/9`
`2/9`
`3/9`
`4/9`

ANSWER :D
Discussion
1676.

Write down negations of It is raining and Mahanadi is flooded.

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SOLUTION :It is NEITHER RAIN nor MAHANADI is FLOODED.
Discussion
1677.

The mean of variable 1,2….. N whose corresponding frequencies are 1,2……. N is given by

Answer»

` (n+1)/(2) `
` ( 2N +1)/(6) `
` ( n(n+1))/(2)`
` (2n+1)/(3) `

ANSWER :D
Discussion
1678.

Find the logical quantifier of the following ; For every negative integer x,x^3 is also a negative integer.

Answer»

SOLUTION :For EVERY
Discussion
1679.

IF(x-a) /( x^2- 3x+2)takensall realvaluesforxin R, then

Answer»

`a=2`
`a LT2`
`1 lt a lt 2`
` 1 LE a le 2`

ANSWER :D
Discussion
1680.

If 0 < p

Answer»

E
p
q
0

Answer :C
Discussion
1681.

Form the differential equation representing thefamily of curves y = a sin ( x + b), where a, b are arbitrary constants.

Answer»


ANSWER :`(d^(2)y)/(DX^(2)) + y = 0`
Discussion
1682.

Find the probability of throwing at teast 3 sixes in 5 throws of a dia.

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Solution :In one throw of a die p(getting a 60=`1/6`
p(getting a NON 6 )=`5/6`
In 5 THROWS of a die
p(atleast 3 SIXES)=p(3 sixer)+p(4 sixer)+p(5 sixer)
= `"^5C_3(1/6)^3(5/6)^2+^5C_4(1/6)^4(5/6)+("^5C_5)(1/6)^5=250/6^5+25/6^5+1/6^5=276/6^5`
Discussion
1683.

Which of the following is the solution set of the equat where x in (0, 1), is equal to

Answer»

0
1
`-1`
NONE of these

Solution :`tan (sin^(-1) (cos (sin^(-1) x))) tan(cos^(-1) (sin (cos^(-1) x)))`
`= tan(sin^(-1) (cos (cos^(-1) SQRT(1 -x^(2)))))`
`tan {cos^(-1) (sin (sin^(-1) sqrt(1 -x^(2))))}`
`= tan (sin^(-1) sqrt(1 -x^(2)) tan (cos^(-1) sqrt(1 - x^(2)))`
`= tan (cos^(-1) x) tan (sin^(-1) x)`
`= tan(cos^(-1) x) tan (pi//2 - cos^(-1 x)`
`= tan (cos^(-1) x) cot (cos^(-1) x) = 1`
Discussion
1684.

If |z| =3 , the area of the triangle whose sides are z , omega zand z + omega z (where omega is a complex cube root of unity) is

Answer»

`(9 sqrt3)/(4)`
`(8 sqrt3)/(2)`
`(5)/(2)`
`(8 sqrt3)/(3)`

ANSWER :A
Discussion
1685.

Let a kind of bacteria grow in such a way that at time t sec, there are t^(3//2) bacteria. Find the rate of growth at time t=4 hours.

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ANSWER :180 bacteria/sec
Discussion
1686.

Consider the proposition : "If the pressure increases, then the volume decreases". The negation of this propositions is

Answer»

If the PRESSURE does not INCREASES the VOLUME does not DECREASE
If the volume increases, the pressure decrease
Pressure increases and volume does not decreases
If the volume decrease, then the pressure increases

Answer :C
Discussion
1687.

If |(z - 4bari)/(z-2)| = 2 then the locus of z is

Answer»

a CIRCLE with centre 0
a REAL circle
an imaginary circle
line

ANSWER :B
Discussion
1688.

" if " int (sin 2x- cos 2x) dx=(1)/(sqrt(2)) sin (2x-k)+c " then " k=?

Answer»

`-(5PI)/(4)`
`(PI)/(4)`
`-(pi)/(4)`

ANSWER :A
Discussion
1689.

I : I xx (a xx i) + j xx (a xx j + k xx (a xx k) = 2a II : I xx [(a xx b) xx i) + j xx [(a xx b) xx j) + k xx [(a xx b) xx k) = 0

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only I is ture
Only II is ture
both I and II are true
Neither I nor II are true

Answer :A
Discussion
1690.

The locus of the point (x, y) which is equidistant from the points (a + b, b - a) and (a - b, a + b) is

Answer»

AX = by
ax + by = 0
bx + AY = 0
bx - ay = 0

Answer :D
Discussion
1691.

Let veca=hati-hatj+hatk, vecb=2hati+hatj+hatk and vecc=hati+hatj-2hatk, then the value of [(veca, vecb, vecc)] is equal to

Answer»


ANSWER :-7
Discussion
1692.

Find the coefficient of x^7 in (1+ x^2)^4 (1 + x)^7

Answer»


ANSWER :323
Discussion
1693.

Evaluate: int(sin^(-1)x^(3))/(sqrt(1-x)^(2))dx

Answer»

Solution :We have `int((LNX)^N)/(X)x=int(lnx)^(n)1/xdx=int(lnx)^(n)d(lnx)=(lnx)^(n+1)/(n+1)+C`
Discussion
1694.

If int_(0)^(pi//2) sin^(4) x cos^(2)x dx = (pi)/(32) then int_(0)^(pi//2) cos^(4) x sin^(2) x dx=

Answer»

0
1
`pi/32`
`pi/16`

ANSWER :C
Discussion
1695.

Find the value of int_(-(pi)/(2))^(pi/2)|sinx|dx.

Answer»


ANSWER :2
Discussion
1696.

int_(2/sqrt3)^2(dx)/(x(sqrtx^2-1)

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SOLUTION :`int_(2/sqrt3)^2(DX)/(X(sqrtx^2-1)dx)=[sec^(-1)x]_0^1`
`sec^(-2)-sec^(-1)(2/sqrt3)`
`cos^(-1)-cos^(-1)((sqrt3)/2)=pi/6`
Discussion
1697.

A paramagnetic material has 10^(28) atoms//m^(3). Its magnetic susceptibility at temperature 350K is 2.8xx10^(-4). Its susceptibility at 300K is :

Answer»

`3.267xx10^(-4)`
`3.672xx10^(-4)`
`3.726xx10^(-4)`
`2.672xx10^(-4)`

SOLUTION :NA
Discussion
1698.

(1+omega)(1+omega^(2))(1+omega^(5))..........2n factors =

Answer»

0
1
`-1`
2

Answer :B
Discussion
1699.

int1/(sqrtx+xsqrtx)dx=

Answer»

`tan^(-1)sqrtx+C`
`2LOG(sqrtx+1)+C`
`2tan^(-1)sqrtx+C`
`1/2tan^(-1)sqrtx+C`

ANSWER :C
Discussion
1700.

if a sinphi-bcosphi show that, acosphi+bsinphi=pmsqrt(a^2+b^2-c^2)

Answer»


ANSWER : `acosphi+bsinphi=pmsqrt(a^2+b^2-c^2)`
Discussion