28278 + Interview Questions in Maths in Class 12 Page 211 InterviewSolution

10501.	A new sequence is obtained from the sequence of positive integers 1,2,…….., by deleting all the perfect squares. What is the 2015th term from the beginning of the new sequence ?
Answer» ANSWER :2060

Discussion

10502.	A black and red dice are rolled. a. Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. b. Find the conditional probability of obtainingthe sum 8, given that the red die resulted in a number less than 4.
Answer» ANSWER :(a) `(1)/(3)`, (B) `(1)/(9)`

Discussion

10503.	Solve the differential equation dy/dx=1+x+y^(2)+xy^(2) , when y=0, x=0
Answer» ANSWER :`1/(2SQRT2)LOG\|(x+ysqrt2)/(x-ysqrt2)\|=log\|x\|+c` is the SOLUTION

Discussion

10504.	If alpha and beta are the roots of the equation ax^(2)+bx+c=0then (alpha+beta)^(2) is …………..
Answer» `-4` `-1` `4` `-2` ANSWER :D

Discussion

10505.	If int(cos^(2)xsinx)/(sinx-cosx)dx=A log\|sinx-cosx\|+(1)/(8)(sin2x+cos2x)+C then A is equal to.
Answer» ANSWER :0.25

Discussion

10506.	The n ^(th) terms of the series 1 + (4)/(5) + (7)/(5 ^(2)) + (10)/(5 ^(3)) +………. is
Answer» ANSWER :`(3N -2)/( 5 ^(N -1))`

Discussion

10507.	Prove that the curves y^(2)=4x and x^(2)+y^(2)-6x+1=0 touch each other at the point (1, 2).
Answer» ANSWER :`m_(1)=m_(2)`

Discussion

10508.	If a, b and c are three positive real numbers such that a+bgec, then
Answer» `(a)/(1+a)+(b)/(1+b)ge(C)/(1+c)` `(a)/(1+a)+(b)/(1+b)LT(c)/(1+c)` `(a)/(1+a)+(b)/(1+b)GT(c)/(1+c)` None of these Answer :(a)

Discussion

10509.	Show that the function f defined as follows, is continuoius at x=2, but not differentiable at x=2. f(x)={{:(3x-2, 0 lt x le 1),(2x^(2)-x, 1 lt x le 2),(5x-4,x gt 2):}.
Answer» ANSWER :`X= 2`.

Discussion

10510.	The value of int_(0)^(1)(8log(1+x))/(1+x^(2))dx is :
Answer» `(pi)/(8) log2` `(pi)/(2)log2` `log2` `pi log2` Answer :4

Discussion

10511.	The value of int_(0)^(1) (x^( 2 alpha) - 1)/( log x) dx, if alpha = (2n-1)/( 2) is
Answer» `log n` `2 log n` `log 2N` `(1//2)log n` ANSWER :C

Discussion

10512.	A person has undertaken a construetion job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time it there is no strike and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time.
Answer» ANSWER :0.488

Discussion

10513.	Solve the following system of equation by using matrix method : x+y+z=6, y+3z-11=0" and "x+z=2y.
Answer» ANSWER :X = 1, y = 2 z = 3

Discussion

10514.	On the xy plane where O is the origin, given points, A(1,0), B(0,1) and C(1,1). Let P,Q, and R be moving points on the line OA, OB, OC respectively such that vec(OP)=45t(vec(OA)), vec(OQ)=60t(vec(OB)), vec(OR)=(1-t)(vec(OC)) with t gt 0. If the three points P,Q and R are collinear then the value of t is equal to
Answer» `1/106` `7/187` `1/100` none of these Solution :Again it is given that the POINT P,Q and R are collinear `rArr vec(PQ) = lambdavec(QR)` `rArr 15R(4vecj-3veci)=LAMBDA[(1-t)(HATI+hatj)-60thatj]` `=lambda[(1-t)hati+(1-6lthatj)]` `rArr (45t)/(t-1) = (4t)/(1-61t)` `rArr 3(1-61t)=4(t-1)` `rArr 3-183t=4t-4` `rArr 187t=7` `rArr t=7/187`

Discussion

10515.	If (2,-1,3) is the foot of the perpendicular drawn from the origin to the plane , then the equation of the plane is
Answer» `2x+y-3Z+6=0` `2x-y+3z - 14 = 0 ` `2x - y + 3z - 13 = 0 ` `2x+y+3z-10 = 0 ` ANSWER :B

Discussion

10516.	Discuss the continuity of the function f, where f is defined by f(x)={{:(2x," if "x lt 0),(0," if "0 le x le 1),(4x," if "x ge 1):}.
Answer» ANSWER :x=1 is the only POINT of DISCONTINUITY.

Discussion

10517.	If m_1, m_2, are slopes of the tangens to the hyperbola x^(2)//25-y^(2)//16=1 which pass through the point (6, 2) then
Answer» `m_(1)+m_(2)=24//11` `m_(1)+m_(2)=48//11` `m_(1)m_(2)=28//11` `m_(1)m_(2)=11//20` ANSWER :A

Discussion

10518.	Find values of k if area of triangle is 4 sq. units and vertices are ( i) (k,0),(4,0) ,( 0,2) "(ii) " ( -2,0),( 0,4),( 0,k )
Answer» ANSWER :`(i) 0.8 ,(II) 0.8 , `

Discussion

10519.	If in the triangleABC, B = 45^@, thena^4+ b^4 +c^4 =
Answer» `2a^2( b^2+ C^2)` `2c^2( a^2 +b^2)` `2b^2 (a^2 +c^2)` `2(a^2b^2+b^2c^2 +c^2 a^2)` ANSWER :C

Discussion

10520.	If two members of family (2+lambda)x+(1+2lambda)y-3(1+lambda) = 0 and line x+y=0 make an equilateral triangle, the the incentre of triangle so formed is
Answer» `((1)/(3), (1)/(3))` `((7)/(6), -(5)/(6))` `((5)/(6), (5)/(6))` `(-(3)/(2), -(3)/(2))` Solution :Given equation of family of lines is `(2x+y-3)+lambda(x+2y-3) =0, lambda in R.` The family of lines is concurrent at point of intersection of 2x+y-3 = 0 and x+2y-3=0 which is A(1,1). Two members `L_(1)" and "L_(2)` of this family and line x+y=0 form an equilateral TRIANGLE. Let foot of perpendicular from A(1, 1) on the line x+y=0 be M(h,K). `(h-1)/(1) = (k-1)/(1) = (-(1+1))/(2)` `therefore M-=(h,k)-=(0,0)` SINCE triangle is equilateral, AM is median, altitude and angle bisector. `AM= sqrt((1-0)^(2)+(1-0)^(2)) = sqrt(2)` Incentre coincides with centroid G such that AG `=(2)/(3)sqrt(2)`. `"Slope of " AM is 1 = "TAN" 45^(@)`. `therefore G-=(1-(2sqrt(2))/(3)"cos" 45^(@), 1-(2sqrt(2))/(3)"sin" 45^(@))` `-=((1)/(3),(1)/(3))`

Discussion

10521.	If f(x)=sin^(-1)[e^(x)]+sin^(-1)[e^(-x)] then which of the following hold(s) good ? [Note: where [k] denotes the larest integer than or equal to k.]
Answer» f(x) has isolated POINT remonable discontinuity at x=0 DOMAIN of f(x) is (-In 2, In 2) Range of f(x) CONTAINS exactly 2 elements. Rang of `f(x)={(pi)/(2)}` Solution :For domain of f(x), `e^(x)in(0,2)ande^(-x)in(0,2)` `:.x in(-ln2,ln2)` `"Now,"f(x)={:[(pi,,,x-0),((pi)/(2)",",,,x in(-ln2,ln2)-{0}):}`

Discussion

10522.	Find the differential equation of the family of (i) all non vertical lines in a plane (ii) all non horizontal lines in aplane
Answer» ANSWER :(i) 0 (II) 0

Discussion

10523.	Let a_(m)=underset("m times")(underbrace(111...1)), then which of the following is a prime number ?
Answer» `a_(4)` `a_(5)` `a_(91)` `a_(100)` Answer :B

Discussion

10524.	LetP(n)be thestatementn^2-n- 41is prime , thenwhich of thefollowingis nottrue ?
Answer» <P>`P(2)` `P(3)` `P(41)` NONEOF these Answer :C

Discussion

10525.	Find the area of the parallelogram whose diagonals are represented by -2hati+4hatj+4hatk and -4hati-2hatk
Answer» ANSWER :`6 SQRT(5)~~13.41` UNITS

Discussion

10526.	If x^2 + 2x + n gt 10 for all real number x, then which of the following conditions is true ?
Answer» ` N LT 11` n = 10 n = 11 `n GT 11` Answer :D

Discussion

10527.	For events A and B P(bar(A cup B))= (1)/(6), P(barA)= (1)/(4) and P(A cap B) = (1)/(4) , Where (barA) is the complementary event of A then events A and B are
Answer» INDEPENDENT but not equiprobable Independent and are equiprobable Independent and MUTUALLY exclusive Equiprobable but not independent. Answer :A

Discussion

10528.	Solve system of linear equations, using matrix method in examples 7 to 14 5x+2y=4 7x+3y=5
Answer» ANSWER :`thereforex=2,y=-3`

Discussion

10529.	If the vectors bara = 3hati +2hatj +9hatk and hatb=hati+ mhatj + 3hatk are parallel then m is………… .
Answer» ANSWER :B::C

Discussion

10530.	If P= \|(x,1),(1,x)\| and Q= \|(x,1,1),(1,x,1),(1,1,x)\| " then " (dQ)/(dx)= _______
Answer» 1)3P 2)`-3P` 3)`1-3P` 4)`3P+1` ANSWER :A

Discussion

10531.	Let W denotethe words in the englishdicitionary define the relation R by R: {(x,y) in W xxWIthe words x and y have at least one letter in common }Then R is
Answer» REFLEXIVE, SYMMETRIC and not TRANSITIVE reflexive, symmetric and transitive reflexive, not symmetric and transitive not reflexive, symmetric and transitive ANSWER :A

Discussion

10532.	Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1). Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N.
Answer» Solution :Method I : `.^(N)C_(1) + 2.^(n)C_(2)+3.^(n)C_(3)+"….."+n.^(n)C_(n)` `=underset(R=1)overset(n)sumr.^(n)C_(r)` `= n underset(r=1)overset(n)sum.^(n-1)C_(r-1)` `= n(.^(n-1)C_(0) + .^(n-1)C_(1) + .^(n-1)C_(2) + .^(n-1)C_(3)+"…."+.^(n-1)C_(n-1))` `= n2^(n-1)` Method II : We have `(1+x)^(n) = .^(n)C_(0) + .^(n)C_(1)x+.^(n)C_(2)x^(2)+"...."+.^(n)C_(n)x^(n)` Differentiatingw.r.t x, we get `n(1+x)^(n-1)= .^(n)C_(1)+2 xx .^(n)C_(2)x + 3 xx .^(n)C_(3)x^(2) +"......" + n xx .^(n)C_(n)x^(n-1)` PUTTING `x = 1`, we get `n2^( n-1) = .^(n)C_(1)+2xx .^(n)C_(2) + "....." + n xx .^(n)C_(n)` USING `A.M. ge G.M.`, we get `(.^(n)C_(1)+2.^(n)C_(2)+3.^(n)C_(3)+".....+n.^(n)C_(n))/((n(n+1))/(2))` `[(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)]^((1)/((pi(n-1))/2))` `rArr (n2^(n-1))/((n(n+1))/(2))ge [(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)]^((2)/(n(n+1)))` `(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)le((2^(n))/(n+1))^((n(n+1))/(2))` or`(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)le((2^(n))/(n+1))^(.^(n+1)C_(2))`

Discussion

10533.	Two friends A and B have equal number of daughters of A and B . The probability that all the tickets go to daughters of A is 1/20.The number of daughters each of them have is
Answer» 4 5 6 3 Answer :D

Discussion

10534.	If the ellipse x^(2)+k^(2)y^(2)=k^(2)a^(2) is confocal with the hyperbola x^(2)-y^(2)=a^(2), then match the following lists and choose the correct code.
Answer» Solution :FOCUS of ellipse `-=(kae_(1),0)` and Focus of hyperbola `-=(ae_(2),0)` Now, `kae_(1)=ae_(2)` `rArr""sqrt(1-(a^(2))/(k^(2)a^(2)))=sqrt(1+(a^(2))/(a^(2)))` `rArr""k^(2)-1=2` `therefore""k= pmsqrt3` a. `(e_(2))/(e_(1))=k=SQRT3` ltBrgt b. Major axis of ellipse = 2ka and Transverse axis of hyperbola = 2A `therefore""(2ka)/(2a)=k=sqrt3` C. Since curves are confocal, they are orthogonal. `therefore""theta=(pi)/(2)` d. Latus rectum of ellipse and hyperbola are`(2a^(2))/(ak) and (2a^(2))/(a)`, respectively. `therefore""(2a//k)/(2a)=(1)/(k)=(1)/(sqrt3)`

Discussion

10535.	A certain coin is tossed with probability of showing head being 'p'. Let 'q' denotes the probability that when the coin is tossed four times the number of heads obtained is even. Then
Answer» there is no value of `p`, if `q=(1)/(4)` there is exactly ONE value of `p`, if `q=(3)/(4)` there are exactly three value of `p`, if `q=(3)/(5)` there are exactly four value of `p`, if `q=(4)/(5)` Solution :`(a,c)` We have `q=P(0H or 2H or 4H)` `:.q=p^(4)+^(4)C_(2)p^(2)(1-p)^(2)+(1-p)^(4)` `=8p^(4)-16p^(3)+12p^(2)-4p+1` `=((2p-1)^(4)+1)/(2)` Now put the values and check.

Discussion

10536.	Let f: (-oo, oo) rarr (-oo, oo) be acontinuous and differentiable function and let f'(.) denote the derivative of f(.).If f(0) = -2 and f'(x) le 3 for each x in [0, 2], then the largest possible value of f(2) is
Answer» 1 2 3 4 Answer :D

Discussion

10537.	Construct truth tables for the following and indicate which of these are tautologies [p rarr ( p vv q)] rarr [q rarr (p ^^ q)]
Answer» SOLUTION :

Discussion

10538.	The midpoint of chord formed by the polar of (-9,2) w.r.t x^(2)+y^(2)=100 is
Answer» `(4,-4/3)` `(-4,16/3)` `(-4,16/9)` `(4,16/3)` ANSWER :B

Discussion

10539.	The solution of (dy)/(dx) + (1)/(x) = (e^(y))/(x^(2)) is
Answer» `2X = (1+Cx^(2))E^(y)` `X = (1+Cx^(2))e^(y)` `2x^(2) = (1+Cx^(2))e^(-y)` `x^(2) = (1+Cx^(2))e^(-y)` ANSWER :A

Discussion

10540.	A family of curve S is given by S -= x^(2) +2xy +y^(2)-4x(1-lambda) - 4y (1+lambda) +4, then S = 0 represents
Answer» pair of straight LINE `AA lambda in R` straight line for exactly one value of `lambda` parabola `AA lambda in R -{0}` ellipse for three values of `lambda` SOLUTION :`S = 0 rArr x^(2) +2XY +y^(2) -4X -4Y +4 = 4lambda (y-x)` `rArr (x+y-2)^(2) = 4 lambda (y-x)` If `lambda=0`, it represents pair of straight line otherwise parabola.

Discussion

10541.	If 8x+5y le 40, 4x+3y ge 12, x ge 0, y ge 0 then the maximum value of f=3x+2y is
Answer» 5 12 24 25 Answer :C

Discussion

10542.	If 6 letters are placed at random in 6 addressed envelopes. Then the odds in favour of arranging them such that no letter goes into correct envelope is
Answer» `53 : 91` `91 : 53` `97 : 64` `64 : 97` ANSWER :A

Discussion

10543.	The transformed equation of x^(5) + 5x^(4) + 3x^(3) + x^(2) + x -1 = 0 , by eliminating second terms is
Answer» `y^(5) - 7Y^(3) + 12y^(2) - 7y = 0` `y^(3) - 2y + 1 = 0` `y^(4) - 4Y^(2) + 1 = 0` `y^(4) - 24y^(2) + 65y - 55 = 0` Answer :1

Discussion

10544.	If the volume of the tetrahedron with edges 2i + j -k, I + aj + k and I + 2j - k is one cubic unir then a =
Answer» 1 `-1` 2 `-2` ANSWER :C

Discussion

10545.	Two tangents are drawn from any point P on a given line L=0 to a given circle S=0, if these tangents touch the circle at point A and B, then locus of the circumcentre the DeltaPAB will be
Answer» A pair of straight lines A circle touching the given line `L=0` A circle touching the given circle `S=0` A line parallel to given line `L=0` Solution : The circumcirle of `DeltaPAB` PASSES through C with PC as its diameter. Therefore centre of this circle `(C_(1))` will be MID pt of PC. Let `L=ax+by+c=0` `P(x_(1).y_(1)) & C(p,q)` `C_(1)(H,k)` `h=(x_(1)+p)/(2)` `k=(y_(1)+q)/(2)` `p(x_(1),y_(1))` passes through `L=0` `impliesax_(1)+by_(1)+c=0` or `a(2h-p)+b(2k-q)+c=0` or `2ah+2bk-ap-bq+c=0` Locus of `(h,k)` is `ax+by=(ap+bq+c)/(2)` `implies` A line parallel to `L=0`

Discussion

10546.	Prove the inequalities : (a) 202^(303) gt 303^(202) (b) 202! Lt 100^(2000
Answer» Answer :(a) `101^2` (B) `1 XX 100 xx 199 xx 200 LT 100^4`

Discussion

10547.	Let f:RRrarrRR be a continuous function defined defined by f(x)=(1)/(e^(x)+2e^(-x)) Statement - I : f(c)=(1)/(3)"for some "c inRR Statement - II 0ltf(x)le(1)/(2sqrt(2))" for all "x in RR
Answer» Statement -I is True , Statement -II is True , Statement -II is a CORRECT explanation for statement -I Statement -I is True , Statement -II is True , Statement -II is not a correct explanation for Statement -I Statement -I is True , Statement -II is FALSE . Statement - I is false , Statement -II is True . ANSWER :A

Discussion

10548.	The centre of the ellipse (x+y-2)^(2)/9+(x-y)^(2)/16=1 is
Answer» ANSWER :(1,1) Length of LR=`(9 sqrt(2))/(4)` DIRECTRIX `x-y+-(16)/(sqrt(7))=0`

Discussion

10549.	The value I= int_(-1)^(1) ( cos^(-1) x + (x^(7) -3x^(5) + 7x^(3) -x) /( cos ^(2) x ) ) dx is
Answer» `PI//2` `0` `2PI` `pi` ANSWER :D

Discussion

10550.	4 moles of H_(2) at 500 K is mixed with 2 moles of He at 400 K. The mixture attains a temperature T and volume V. Now the mixture is compressed adiabatically to a volume V' and temperature T'. If (T')/T=(V/(V'))^(n), find the value of 26 n.
Answer» ANSWER :`23.0`

Discussion

Explore topic-wise InterviewSolutions in .

A new sequence is obtained from the sequence of positive integers 1,2,…….., by deleting all the perfect squares. What is the 2015th term from the beginning of the new sequence ?

A black and red dice are rolled. a. Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. b. Find the conditional probability of obtainingthe sum 8, given that the red die resulted in a number less than 4.

Solve the differential equation dy/dx=1+x+y^(2)+xy^(2) , when y=0, x=0

If alpha and beta are the roots of the equation ax^(2)+bx+c=0then (alpha+beta)^(2) is …………..

If int(cos^(2)xsinx)/(sinx-cosx)dx=A log|sinx-cosx|+(1)/(8)(sin2x+cos2x)+C then A is equal to.

The n ^(th) terms of the series 1 + (4)/(5) + (7)/(5 ^(2)) + (10)/(5 ^(3)) +………. is

Prove that the curves y^(2)=4x and x^(2)+y^(2)-6x+1=0 touch each other at the point (1, 2).

If a, b and c are three positive real numbers such that a+bgec, then

Show that the function f defined as follows, is continuoius at x=2, but not differentiable at x=2. f(x)={{:(3x-2, 0 lt x le 1),(2x^(2)-x, 1 lt x le 2),(5x-4,x gt 2):}.

The value of int_(0)^(1)(8log(1+x))/(1+x^(2))dx is :

The value of int_(0)^(1) (x^( 2 alpha) - 1)/( log x) dx, if alpha = (2n-1)/( 2) is

Solve the following system of equation by using matrix method : x+y+z=6, y+3z-11=0" and "x+z=2y.

If (2,-1,3) is the foot of the perpendicular drawn from the origin to the plane , then the equation of the plane is

Discuss the continuity of the function f, where f is defined by f(x)={{:(2x," if "x lt 0),(0," if "0 le x le 1),(4x," if "x ge 1):}.

If m_1, m_2, are slopes of the tangens to the hyperbola x^(2)//25-y^(2)//16=1 which pass through the point (6, 2) then

Find values of k if area of triangle is 4 sq. units and vertices are ( i) (k,0),(4,0) ,( 0,2) "(ii) " ( -2,0),( 0,4),( 0,k )

If in the triangleABC, B = 45^@, thena^4+ b^4 +c^4 =

If two members of family (2+lambda)x+(1+2lambda)y-3(1+lambda) = 0 and line x+y=0 make an equilateral triangle, the the incentre of triangle so formed is

If f(x)=sin^(-1)[e^(x)]+sin^(-1)[e^(-x)] then which of the following hold(s) good ? [Note: where [k] denotes the larest integer than or equal to k.]

Find the differential equation of the family of (i) all non vertical lines in a plane (ii) all non horizontal lines in aplane

Let a_(m)=underset("m times")(underbrace(111...1)), then which of the following is a prime number ?

LetP(n)be thestatementn^2-n- 41is prime , thenwhich of thefollowingis nottrue ?

Find the area of the parallelogram whose diagonals are represented by -2hati+4hatj+4hatk and -4hati-2hatk

If x^2 + 2x + n gt 10 for all real number x, then which of the following conditions is true ?

For events A and B P(bar(A cup B))= (1)/(6), P(barA)= (1)/(4) and P(A cap B) = (1)/(4) , Where (barA) is the complementary event of A then events A and B are

Solve system of linear equations, using matrix method in examples 7 to 14 5x+2y=4 7x+3y=5

If the vectors bara = 3hati +2hatj +9hatk and hatb=hati+ mhatj + 3hatk are parallel then m is………… .

If P= |(x,1),(1,x)| and Q= |(x,1,1),(1,x,1),(1,1,x)| " then " (dQ)/(dx)= _______

Let W denotethe words in the englishdicitionary define the relation R by R: {(x,y) in W xxWIthe words x and y have at least one letter in common }Then R is

Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1). Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N.

Two friends A and B have equal number of daughters of A and B . The probability that all the tickets go to daughters of A is 1/20.The number of daughters each of them have is

If the ellipse x^(2)+k^(2)y^(2)=k^(2)a^(2) is confocal with the hyperbola x^(2)-y^(2)=a^(2), then match the following lists and choose the correct code.

A certain coin is tossed with probability of showing head being 'p'. Let 'q' denotes the probability that when the coin is tossed four times the number of heads obtained is even. Then

Let f: (-oo, oo) rarr (-oo, oo) be acontinuous and differentiable function and let f'(.) denote the derivative of f(.).If f(0) = -2 and f'(x) le 3 for each x in [0, 2], then the largest possible value of f(2) is

Construct truth tables for the following and indicate which of these are tautologies [p rarr ( p vv q)] rarr [q rarr (p ^^ q)]

The midpoint of chord formed by the polar of (-9,2) w.r.t x^(2)+y^(2)=100 is

The solution of (dy)/(dx) + (1)/(x) = (e^(y))/(x^(2)) is

A family of curve S is given by S -= x^(2) +2xy +y^(2)-4x(1-lambda) - 4y (1+lambda) +4, then S = 0 represents

If 8x+5y le 40, 4x+3y ge 12, x ge 0, y ge 0 then the maximum value of f=3x+2y is

If 6 letters are placed at random in 6 addressed envelopes. Then the odds in favour of arranging them such that no letter goes into correct envelope is

The transformed equation of x^(5) + 5x^(4) + 3x^(3) + x^(2) + x -1 = 0 , by eliminating second terms is

If the volume of the tetrahedron with edges 2i + j -k, I + aj + k and I + 2j - k is one cubic unir then a =

Two tangents are drawn from any point P on a given line L=0 to a given circle S=0, if these tangents touch the circle at point A and B, then locus of the circumcentre the DeltaPAB will be

Prove the inequalities : (a) 202^(303) gt 303^(202) (b) 202! Lt 100^(2000

Let f:RRrarrRR be a continuous function defined defined by f(x)=(1)/(e^(x)+2e^(-x)) Statement - I : f(c)=(1)/(3)"for some "c inRR Statement - II 0ltf(x)le(1)/(2sqrt(2))" for all "x in RR

The centre of the ellipse (x+y-2)^(2)/9+(x-y)^(2)/16=1 is

The value I= int_(-1)^(1) ( cos^(-1) x + (x^(7) -3x^(5) + 7x^(3) -x) /( cos ^(2) x ) ) dx is

4 moles of H_(2) at 500 K is mixed with 2 moles of He at 400 K. The mixture attains a temperature T and volume V. Now the mixture is compressed adiabatically to a volume V' and temperature T'. If (T')/T=(V/(V'))^(n), find the value of 26 n.