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7551.

Let l_(1) and l_(2) be the two skew lines. If P, Q are two distinct points on l_(1) and R, S are two distinct points on l_(2), then prove that PR cannot be parallel to QS.

Answer»

Solution :LET equation of the line `l_(1)` be `vecr=veca+lamdavecb` and equation of the line `l_(2)` be `vecr=vecc+muvecd`,
where `veca-vecc, vecb and vecd` are non-coplanar.
Let the position vectors of points P and Q be `veca+lamda_(1)vecb and vecaa+lamda_(2)vecb`, respectively.
Let the position vectors of points R and S be `vecc+mu_(1)vecd and vecc+mu_(1)vecd`, respectively.
Then the lines PR and QS are parallel if and only if `vecc-veca+mu_(1)vecd-lamda_(1)vecb=k(vecc-veca+mu_(2)vecd-lamda_(2)vecb)`
i.e., `""(1-k)(vecc-veca)+(mu_(1)-kmu_(2))vecd - (lamda_(1)-klamda_(2))vecb=veco`
`therefore""1-k=0, mu_(1)-kmu_(2)=0, lamda_(1)-klamda_(2)=0`
i.e., `" "mu_(1)=mu_(2) and lamda_(1)=lamda_(2)` which is not POSSIBLE
Therefore, PR can not be parallel to QS.
Discussion
7552.

A person whose hobby is tossing a fair coin is to score one point for every tail and two points for every head. The person goes on tossing the coin, til his score reaches 100 or exceeds 100. Then the probability that his score attains exactly 100 is

Answer»

<P>`2/3-1/(3.2^(100)`
`2/3+1/(3.2^(100))`
`2/3-1/(2^(100))`
`2/3+1/(2^(100))`

SOLUTION : LET `E_(n)` denote the event that the score is `n`
`E_(n)=(E_(n-2)nnH)uu(E_(n-1)nnT)`
`P(E_(n))=1/2[P_(n-2)+P_(n-1)]`
`P_(n)+1/2P_(n-1)=P_(n-1)+1/2P_(n-2)`
`=P_(n-2)+1/2P_(n-3)=P_(2)+1/2P_(1)`
As, `P_(1)=P(T)=1/2impliesP_(2)=P((TnnT)uuH)=3/4`
`impliesP_(n)-2/3=-1/2[P_(n-1)-2/3]=(-1/2)^(2)[P_(n-2)-2/3]`
`=(-1/2)^(n-2)[P_(1)-2/3]=-1/2[1/2-2/3]`
`P_(100)=2/3+1/(3.2^(100))`
Discussion
7553.

Which of the following curves may represent the speed of the electron in a hydrogenatom as a functionof the principalquantumnumber n ?

Answer»





SOLUTION :K.E. `PROP Z^(2)rArrv prop z//n`
Discussion
7554.

If a , b , c are integers not all equal and w is a cube root of unity (w ne 1) then find the minimum value of |a + b w+ c w^(2)|

Answer»


ANSWER :A::B
Discussion
7555.

A professional baseball team wishes to average 45,500 season. Through the first 60 games of the season, the team has averaged 43,000 ticket purchases per game. Which of the following most closely approximates how many ticket purchases per game the team must average for the remainder of the season in order to hit its overall goal of an average of 45,500 ticket purchases per game for the season?

Answer»

46970
47880
48000
48220

Answer :A
Discussion
7556.

Find solution of ydx+xdy=xy(dy-dx)

Answer»
Discussion
7557.

Define a function phi:NtoN as follows phi(1)=1,phi(P^(n))=P^(n-1)(P-1) is prime and n epsilonN and phi(mn)=phi(m)phi(n) if m & n are relatively prime natural numbers. The number of natural numbers n such that phi(n) is odd is

Answer»

1
2
3
4

Answer :B
Discussion
7558.

Define a function phi:NtoN as follows phi(1)=1,phi(P^(n))=P^(n-1)(P-1) is prime and n epsilonN and phi(mn)=phi(m)phi(n) if m & n are relatively prime natural numbers. phi(8n+4) when n epsilonN is equal to

Answer»

`2phi(4n+2)`
`PHI(2n+1)`
`2PI(2n+1)`
`4PI(2n+1)`

ANSWER :C
Discussion
7559.

The plane passing the points (1,1,1),(1,-1,1) and (-1,3,-5) contains the point (K,1,2) then value of K = ..............

Answer»

`(-4)/(3)`
`(3)/(4)`
`(4)/(3)`
`(-3)/(4)`

ANSWER :C
Discussion
7560.

Differentiate sin^(2) xw.r.t. e^(cos x).

Answer»


ANSWER :`= -(2 COS X)/(E^(cos s))`
Discussion
7561.

Find the values of p for which both the roots of the equation 4x^2 - 20px + (25p^2 + 15p - 66) = 0 are less than 2.

Answer»


ANSWER :`(-INFTY, -1)`
Discussion
7562.

Write the range of the function y=sec^(-1)x.

Answer»


ANSWER :`[0,PI]-{pi/2}`
Discussion
7563.

The radius of the circle of least size that passes through (-2,1) and touches both axes is

Answer»

1
2
3
5

Answer :A
Discussion
7564.

If int (1)/(x (x^(5) - 1)(x^(5) + 1))dx = A log x^(5)B log (x^(5)- 1) " + C log "(x^(5) + 1) then (A, B , C) =

Answer»

`((-1)/(5), (1)/(10), (1)/(10)) `
`(- (5)/(4) , (1)/(8) ,- (1)/(8)) `
`(- (5)/(4) , -(1)/(8) ,(1)/(8)) `
`(1, (1)/(10) , - (1)/(10)) `

Answer :A
Discussion
7565.

Number of principal solution(s) of the equation sqrt(sin x) -1/(sqrt(sin x))= cos x, is :

Answer»

1
2
3
4

SOLUTION :`sqrt(SIN x) -1/(sqrt(sin x))= COS x`
`sin x + 1/(sin x) -2 = cos^(2)x = (1-sin^(2)x)`
Let `sin x = t`
`t + 1/t -2 = 1 - t^(2)`
`t^(2) - 2t + 1 = t(1-t)(1+t)`
`implies (1-t)[(1-t)-t(1+t)] = 0`
`implies (1-t)[1-2t-t^(2)] = 0`
`t = 1,t^(2) + 2t -1 = 0`
`sin x = 1, sin x = (-2+-sqrt(4+4))/(2)`
`sin x = 1, implies x = (pi)/2`
also `sin x = sqrt(2)-1 and cos x lt 0`
`implies` ONE solution in `(0,2 pi)`.
Discussion
7566.

Using the properties of determinants in Exercise 1 to 6, evaluate |{:(a+x,y,z),(x,a+y,z),(x,y,a+z):}|

Answer»


ANSWER :`thereoforeD=a^2(a+x+y+z)`
Discussion
7567.

For X ~ B (n = 10, p = 0.2), P (X=1) =

Answer»

`2XX(0.8)^9`
`2xx(0.8)^8`
`(0.8)^9`
`(0.8)^8`

ANSWER :A
Discussion
7568.

If0 ltx ltpithen( sin8x+ 7 sin6x+ 18 sin4x+12sin2x ) /(sin 7 x + 6 sin 5x+ 12 sin 3x )=

Answer»

2
`sin x `
`sin2x`
`2 COSX`

Answer :D
Discussion
7569.

IF two sides of a triangle are given by 3x^(2)-5xy+2y^(2) =0 and its orthocentre (2,1), then the equation of the third side of the triangle is

Answer»

5x-10y+1=0
10x+5y-1=0
5x-10y=21
10x+5y=21

Answer :D
Discussion
7570.

A studenthasto answer10 outof 13questionsin anexaminationchoosingat least5questionfromthe first6 questions. The numberofchoicesavailableto thestudentsis

Answer»

161
91
63
196

Answer :A
Discussion
7571.

Sigma_(r=0)^(oo) tan^(-1).(2)/(1+r^(2)+2r) is equal to

Answer»

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

SOLUTION :`f(theta)=UNDERSET(1)overset(theta^(2))(int)COS^(3)xcosecx SIN(theta^(2))dx`
`f(theta)=sin(theta^(2))underset(1)overset(theta^(2))(int)cos^(3)x cos ecx dx`
`f'(theta)=(2theta)cos(theta^(2))underset(1)overset(theta^(2))(int)cos^(3)xcos ecx dx+sin(theta^(2))(2theta)cos^(3)(theta^(2))cosec(theta^(2))`
`impliesf'(sqrt((pi)/(3)))=sqrt((pi)/(3))underset(1)overset(pi//3)(int)cos^(3)x cos ecx dx +(1)/(4)sqrt((pi)/(3))`
`implies(3)/(2sqrt(pi))f'(sqrt((pi)/(3)))=sqrt(3)/(2)underset(1)overset(pi//3)(int)cos^(3)xcos ecx dx + (sqrt(3))/(8)`
`implies(3)/(2sqrt(pi))f'(sqrt((pi)/(3)))=f (sqrt((pi)/(3)))+(sqrt(3))/(8)`
`implies (3)/(2sqrt(pi))f'(sqrt((pi)/(3)))-f(sqrt((pi)/(3)))=(sqrt(3))/(8)`
`impliesk=(sqrt(3))/(8)implies64k^(2)=3`
Discussion
7572.

Match the following {:("Line", "Distance from origin"),(I. x-2y+1=0, (a) 7//sqrt(10)),(II. x+sqrt(3)y+2=0, (b)4//sqrt(5)), (III. 3x-y+7=0, (c) 1//sqrt(5)), (IV. 2x-y-4=0, (d)1), (, (e)7//10):}

Answer»

C, d, a, B
a, b, c, d
b, a, c, d
E, c, d, a

ANSWER :A
Discussion
7573.

Find the numerically greatest term (s) in the expansion of (3+7x)^(n)" when "x=(4)/(5),n=15

Answer»


ANSWER :`T_11 = ""^15C_10.3^5 . (28/5)^10`
Discussion
7574.

Elementary Transformation of a matrix: The following operation on a matrix are called elementary operations (transformations) 1.The interchange of any two rows (or columns) 2. The multiplication of the elements of any row (or column) by any nonzero number 3. The addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by any number Echelon Form of matrix : A matrix A is said to be in echelon form if (i) every row of A which has all its elements 0, occurs below row, which has a non-zero elements (ii) the first non-zero element in each non –zero row is 1. (iii) The number of zeros before the first non zero elements in a row is less than the number of such zeros in the next now.[ A row of a matrix is said to be a zero row if all its elements are zero]Note: Rank of a matrix does not change by application of any elementary operations For example [(1,1,3),(0,1,2),(0,0,0)],[(1,1,3,6),(0,1,2,2),(0,0,0,0)] are echelon forms The number of non-zero rows in the echelon form of a matrix is defined as its RANK. For example we can reduce the matrixA=[(1,2,3),(2,4,7),(3,6,10)] into echelon form using followingelementary row transformation. (i)R_2 to R_2 -2R_1 and R_3 to R_3 -3R_1 [(1,2,3),(0,0,1),(0,0,1)] (ii)R_2 to R_2 -2R_1 [(1,2,3),(0,0,1),(0,0,0)] This is the echelon form of matrix A Number of nonzero rows in the echelon form =2rArrRank of the matrix A is 2 Rank of the matrix [(1,1,1,-1),(1,2,4,4),(3,4,5,2)] is :

Answer»

1
2
3
4

Answer :C
Discussion
7575.

Elementary Transformation of a matrix: The following operation on a matrix are called elementary operations (transformations) 1.The interchange of any two rows (or columns) 2. The multiplication of the elements of any row (or column) by any nonzero number 3. The addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by any number Echelon Form of matrix : A matrix A is said to be in echelon form if (i) every row of A which has all its elements 0, occurs below row, which has a non-zero elements (ii) the first non-zero element in each non –zero row is 1. (iii) The number of zeros before the first non zero elements in a row is less than the number of such zeros in the next now.[ A row of a matrix is said to be a zero row if all its elements are zero]Note: Rank of a matrix does not change by application of any elementary operations For example [(1,1,3),(0,1,2),(0,0,0)],[(1,1,3,6),(0,1,2,2),(0,0,0,0)] are echelon forms The number of non-zero rows in the echelon form of a matrix is defined as its RANK. For example we can reduce the matrixA=[(1,2,3),(2,4,7),(3,6,10)] into echelon form using followingelementary row transformation. (i)R_2 to R_2 -2R_1 and R_3 to R_3 -3R_1 [(1,2,3),(0,0,1),(0,0,1)] (ii)R_2 to R_2 -2R_1 [(1,2,3),(0,0,1),(0,0,0)] This is the echelon form of matrix A Number of nonzero rows in the echelon form =2rArrRank of the matrix A is 2 The echelon formof the matrix [(1,3,4,3),(3,9,12,9),(1,3,4,1)]is :

Answer»

`[(1,3,4,3),(0,0,0,1),(0,0,0,0)]`
`[(1,2,4,3),(0,0,0,0),(0,0,0,-2)]`
`[(1,3,4,3),(0,0,0,-2),(0,0,0,0)]`
`[(1,3,4,-3/2),(0,0,0,1),(0,0,0,0)]`

ANSWER :C
Discussion
7576.

Let f(x) =e^(2x) =ae^(x)+1.Prove that f(x) cannotbe monotonicallydecreasingfor AA x in R for any value of 'a'

Answer»
Discussion
7577.

Elementary Transformation of a matrix: The following operation on a matrix are called elementary operations (transformations) 1.The interchange of any two rows (or columns) 2. The multiplication of the elements of any row (or column) by any nonzero number 3. The addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by any number Echelon Form of matrix : A matrix A is said to be in echelon form if (i) every row of A which has all its elements 0, occurs below row, which has a non-zero elements (ii) the first non-zero element in each non –zero row is 1. (iii) The number of zeros before the first non zero elements in a row is less than the number of such zeros in the next now.[ A row of a matrix is said to be a zero row if all its elements are zero]Note: Rank of a matrix does not change by application of any elementary operations For example [(1,1,3),(0,1,2),(0,0,0)],[(1,1,3,6),(0,1,2,2),(0,0,0,0)] are echelon forms The number of non-zero rows in the echelon form of a matrix is defined as its RANK. For example we can reduce the matrixA=[(1,2,3),(2,4,7),(3,6,10)] into echelon form using followingelementary row transformation. (i)R_2 to R_2 -2R_1 and R_3 to R_3 -3R_1 [(1,2,3),(0,0,1),(0,0,1)] (ii)R_2 to R_2 -2R_1 [(1,2,3),(0,0,1),(0,0,0)] This is the echelon form of matrix A Number of nonzero rows in the echelon form =2rArrRank of the matrix A is 2 Rank of the matrix [(1,1,1),(1,-1,-1),(3,1,1)] is :

Answer»

1
2
3
0

Answer :C
Discussion
7578.

For any four points P,Q,R,S |bar(PQ) xx bar(RS) - bra(QR) xx bar(PS)+ bar(RP) xx bar(QP)| is equal to 4 times the areof the triangle .

Answer»

PQR
QRS
PRS
PQS

Answer :B
Discussion
7579.

Statement-1: If z is a complex number satisfying (z-1)^(n) , n in N, then the locus of z is a straight line parallel to imaginary axis. Statement-2: The locus of a point equidistant from two given points is the perpendicular bisector of the line segment joining them.

Answer»

Statement-1 is True, Statement-2 is True: Statement-2 is a CORRECT EXP,anation for statement-1.
Statement-1 is true, statement -2 is true, Statement-2 is not a correct EXPLANATION for statement-1.
Statement-1 is True, statement-2 is false,
statement-1 is False, Statement-2 is true.

ANSWER :a
Discussion
7580.

Prove ""^(25)C_4+sum_(r=0)^4"^((29-r))C_3=^(30)C_4

Answer»


ANSWER :`""^(30)C_4`
Discussion
7581.

A towerstands atthecentreofa circularpark.Aand Bare twopointson theboundaryof theparksuchthatAB = (=a)subtendsan angleof 60^@at thefootof thetower,andthe angleof elevationof thetopof thetowerfromA or Bis 30^@ theheightof htetoweris

Answer»

`(a)/(SQRT(3))`
`a sqrt(3)`
`(2A )/( sqrt(3))`
`2a sqrt(3)`

ANSWER :A
Discussion
7582.

4^(th) terms of ((3x)/(5)-y)^(7)

Answer»


ANSWER :`""^(-7)C_(3) ((3)/(5))^(4) x^(4)y^(3)`
Discussion
7583.

int_(0)^(1//sqrt(2)) (sin^(-1))/((1-x^(2))^(3//2))dx

Answer»


ANSWER :`(pi)/(4)-(1)/(2)LOG 2`
Discussion
7584.

Which of the following is a solution of -7^2sqrt(3x-8+3)=-4x+7 for x.

Answer»

`-8`
`-0.18`
`0.18`
8

Answer :D
Discussion
7585.

If y= log ((1- x^(2))/(1+ x^(2))), then (dy)/(dx) is equal to

Answer»

`(4x^(3))/(1-x^(4))`
`(-4x)/(1-x^(4))`
`(1)/(4-x^(4))`
`(-4x^(3))/(1-x^(4))`

ANSWER :B
Discussion
7586.

If |{:(x,1,2),(1,0,3),(5,-1,4):}|=0 then find x

Answer»


ANSWER :-3
Discussion
7587.

If the LCM of a and 12 is 36, what are the possible values of a?

Answer»


ANSWER :9, 18, 36
Discussion
7588.

int_(-2)^(2) |[x]| dx is equal to

Answer»

1
2
3
4

Answer :D
Discussion
7589.

x=2 + t^(3), y= 2t^(2). If ((d^(2)y)/(dx^(2)))/(((dy)/(dx))^(n)) is constant then n= ………

Answer»

4
1
0
3

Answer :A
Discussion
7590.

Let f(sin x)lt 0and f(sin x) lt 0for all x in

Answer»

`(pi/4,pi/2)`
`(0,pi/4)`
`(0,pi/2)`
`(pi/2,pi/2)`

Solution : We have G(X)=f(sin x)+ f(COSX)
`rArrg(x)=f'(sin x) cos^2 x+f' (cos x) sin ^2X -f(sin x)sin x-f(cos x) cos x`
`rArrg'(x) gt 0 " for all " x in (0,pi//2)`
Also `g(pi/4)=1/(sqrt(2))f((1)/(sqrt(2)))-(1/sqrt(2))f(1/sqrt(2))=0`
`thereforeg(x) lt ` 0 for all `x in (0,pi/4)` and `g(x) gt 0 ` for all `x in (pi/4,Pi/2)`
`rArrg(x)` is decreasing on `(0,pi/4)` and increasing on `(pi/,pi/2)`
Discussion
7591.

Integrate the function in Exercise. x log 2x

Answer»


ANSWER :`(X^(2))/(2).log2x-(x^(2))/(4)+C`
Discussion
7592.

Every function is invertible.

Answer»


SOLUTION :N/A
Discussion
7593.

f(x) = max { sin x, cos x,1/2}the areaof theregionboundedby thecurvey=f(x), X-axisY-axisandx=2piis……..Sq. units

Answer»

`((5PI)/(12)+3)`
`((5pi)/(12)+sqrt(2))`
`((5pi)/(12) + sqrt(3))`
`((5pi)/(12) + sqrt(2) + sqrt(3))`

ANSWER :D
Discussion
7594.

If y= e^(x^(3)) find (dy)/(dx)

Answer»


ANSWER :`E^(x^(3)).3x^(2)`
Discussion
7595.

A black and a red die are rolled.Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5 .

Answer»


ANSWER :(a) `(1)/(3)`, (B) `(1)/(9)`
Discussion
7596.

Prove that :int_(0)^(pi//2) x . cot x dx =(pi)/(2)log 2

Answer»
Discussion
7597.

If A=(cos12^(@)-cos36^(@))(sin96^(@)+sin24^(@))and B=(sin60^(@)-sin12^(@))(cos48^(@)-cos72^(@)),then what is(A)/(B) equal to ?

Answer»

`-1`
0
1
0

Solution :Given`A=(COS12^(@)-COS36^(@))(sin96^(@)+sin24^(@))`
`B=(sin60^(@)-sin12^(@))(cos48^(@)-cos72^(@))`
`(A)/(B)=([2sin24^(@)sin12^(@)][2sin60^(@)cos36^(@)])/([2cos36^(@)sin24^(@)][2sin60^(@)sin12^(@)])`
`rArr(A)/(B)=1`
Discussion
7598.

Reflection of the line x+y+1=0 in the line lx+my+n=0 is

Answer»

`(x+y+1)(L+m)-2(l^(2)+m^(2))(lx+my+n)=0`
`(x+y+1)(l^(2)+m^(2))-2(l+m)(lx+my+n)=0`
`(l+m+1)(x+y)-2(lx+my)(l^(2)+m^(2))=0`
NONE of these

ANSWER :B
Discussion
7599.

Three points with position vectors a, b and c will be collinear, if there exist scalars x, y and z such that (where x + y + z = 0)

Answer»

X a + y b = z c
XA + yb + zc = 0
xa + yb + zc = 0
xa + yb = c

Solution :Let A, B and C be the POINTS with POSITION vectors a, b and c respectively. These points will be collinear, if
`AB = LAMBDA AC`
`implies b-a=lambda(c-a)`
`implies (lambda-1)a+b+(-lambda)c=0`
`implies xa+yb+zc=0`
where, `x = lambda - 1, y = 1, z = - lambda`
`implies xa + yb + zc = 0` such that
x + y + z = 0
Discussion
7600.

Prove that the function given by f(x)=x^(3)-3x^(2)+3x-100 is increasing in R.

Answer»


ANSWER :`F'(X) GE 0`
Discussion