28278 + Interview Questions in Maths in Class 12

1.	[veca, veca+vecb, veca+vecb+vecc] is :
Answer» `[veca vecb VEC C]` 0 `2[veca, vecb, vecc]` `[veca, vecb, vecc]^(2)` SOLUTION :N/A

Discussion

2.	State which of the following are positive ?cos 315^@
Answer» Solution :`COS 315^@` is +ve as `315^@` LIES in 4TH quadrant and cos is +ve there.

Discussion

3.	According to the table above , for what value of x does g(f(x))=-1 ?
Answer» 2 3 4 8 Answer :A

Discussion

4.	Let the normals are drawn (alpha, beta ) to the hyperbola xy = 1 " and " (x_(i) , y_(i)) , l = 1,2,3,4be the feel of the co-normal points .If the algebraic sum of the perpendicular distances drawn from (x_(i) , y_(i)), l = 1, 2, 3, 4 onto a variable line vanishes, then the varable line passes through the point (alpha/gamma , beta/gamma ), then the value of gamma be __________ .
Answer» Statement -1 is TRUE,Statement -2 is True , Statement - 2 is a CORRECT EXPLANATION for Statement - 3 Statement -1 is True, Statement - 2 is True , Statement -2 is NOT a correct explanation for Statement - 3 Statement -1 True, Statement - 2 is False. Statement -1 ,is False, Statement - 2 is True. SOLUTION :N/A

Discussion

5.	(a)Maximise z = 4x + y subject to constraints : x + y le 50 3x + y le90 x ge0 y ge 0 by graphical method. (b) Find the value of K, if f(x) = {{:(Kx + 1, "if"x le pi),(cos x, "if " x gt pi):} is continuous at x = pi.
Answer» ANSWER :MAXIMUM of Z = 120 at (30, 0)

Discussion

6.	If veca, vecb and vecc are the position vectors of the vertices A,B and C. respectively , of triangleABC. Prove that the perpendicualar distance of the vertex A from the base BC of the triangle ABC is (\|vecaxxvecb+vecbxxvecc+veccxxveca\|)/(\|vecc-vecb\|)
Answer» SOLUTION :`\|VEC(BC)xxvec(BA)\|=\|vecaxxvecb+vecb+vecbxxvecc+veccxxveca\|` `\|vec(BC)\|\|vec(BA)\|sin B=\|vecaxxvecbxxvecbxxvecc+veccxxveca\|` `\|vecc-vecb\| (ABsinB)=\|vecaxxvecb+vecbxxvecc+veccxxveca\|` Therefore, the length of PERPENDICUALR from A on BC is `AL=ABsinB=(\|vecaxxvecb+vecbxxvecc+veccxxveca\|)/(\|vecb-vecc\|)`

Discussion

7.	Ifthetais the angle between any two vectors veca and vecb, then \|veca.vecb\|=\|veca xx vecb\| when theta is equal to
Answer» 0 `(PI)/(4)` `(pi)/(2)` `pi` SOLUTION :N/A

Discussion

8.	int_(0)^(pi//4) [sqrt(tan x)+sqrt(cot x)]dx
Answer» `PI/SQRT(2)` `pi/2` `(3PI)/(sqrt(2))` `pi` ANSWER :A

Discussion

9.	Show that the line lx+my+n=0 is a normal to the circles S=0 iff gl+mf=mn.
Answer» ANSWER :`HARR LG+ MF = N `

Discussion

10.	Let S = {(x,y) in N xx N, x^(2) -y^(2) = 10,21,954}, then
Answer» `S = PHI` S contains exactly ONE element S is finite and contains at LEAST two elements S is an INFINITE set Answer :A

Discussion

11.	If f(x)=\|{:(2cos^(2)x,sin2x,-sinx),(sin2x,2sin^(2)x,cosx),(sinx,-cosx,0):}\|" then "int_(0)^(pi//2)[f(x)+f'(x)]dx=
Answer» 0 1 `PI//2` `pi` ANSWER :D

Discussion

12.	Using binomial theorem, prove that 8^(n)-7n always leaves remainder 1 when divided by 49.
Answer» Solution :For any TWO numbers a and b if we can FIND the numbers q and r such that `a=bq+r`, then we SAY that b divides a with q as QUOTIENT and r as remaindder. Thus, in order to show that `8^(n)-7n` leaves remainer 1 when divided by 49, we prove that `8^(n)-7n=49k+1`, where k is some natural NUMBER. Now, `8^(n)-7` can be written as `(1+7)^(n)-7n` Now, we can write `(1+7)^(n)+{1+^(n)C_(1)7^(1)+.^(n)C_(2)7^(2)+.^(n)C_(3)7^(3)+ . . .+.^(n)C_(n)7^(n)}-7n` `={1+7n+7^(2).^(n)C_(2)+7^(3).^(n)C_(3)+. . . .+7^(n)}-7n` `=1+7^(2)({.^(n)C_(2)+7.^(n)C_(3)+ . . .+7^(n-2)}` `=1+49{.^(n)C_(2)+7.^(n)C_(3)+ . . .+7^(n-2)}` `therefore8^(n)-7n=49k+1,` where `K=.^(n)C_(2)+7.^(n)C_(3)+ . . .+7^(n-2)` this shows that when `8^(n)-7n` is divided by 49 always leaves remainder as 1.

Discussion

13.	If cos2x=(sqrt2+1)(cosx-1/sqrt2), cos x ne 1/2,then xbelong to
Answer» `{2npi pm PI/3: n in I}` `{2npi pm pi/6: n in I}` `{2npi pm pi/2 : n in I}` `{2N pi pm pi/4 : n in I}` ANSWER :D

Discussion

14.	If int(x)=int(a+b), then int_(0)^(2a)int= int =(x)=2int_(0)^(a) int(x)dx
Answer» Answer :`int_(0)^(2)f(x)dx=2int_(0)^(a)f(x)dx`.

Discussion

15.	If "cos"(pi)/(15) "cos"(2pi)/(15) "cos"(4pi)/(15) "cos"(5pi)/(15) "cos"(7pi)/(15) "cos"(30pi)/(15) =x, " then " (1)/(8x) =
Answer» 4 `(1)/(4)` 8 `(4)/(3)` ANSWER :A

Discussion

16.	Find the position vector of the mid point of the vector joiningthe points P(2,3,4) and Q(4,1,-2).
Answer» ANSWER :`3hati+2hatj+hatk`

Discussion

17.	Find the probability of getting a prime number when a die is rolled
Answer» ANSWER :`(1)/(2)`

Discussion

18.	Evaluate the following integrals int e^(x) (1+x^(2))dx
Answer» ANSWER :`E^(x) (x^(2) - 2X + 3) + C`

Discussion

19.	If (9+4sqrt(5))^(n)=p+beta, where n and p are positive integers and beta is a positive proper fraction, prrove that (1-beta)(p+beta)=1 and p is an odd integer.
Answer» <P> Solution :We have, `(9+4sqrt(5))^(n)=p+beta,p` is a +ve integer, `oltbetalt1` LET `(9-4sqrt(5))^(n)lt1` ALSO `(9-4sqrt(4))^(n)gt0` `therefore0ltflt1` now, `p+beta+f=(9+4sqrt(5))^(n)+(9-4sqrt(5))^(n)=2[.^(n)C_(0)9^(n)+.^(n)C_(2)9^(n-2)80+.^(n)C_(4)9^(n-4)(80)^(2)+ . . . .]` `impliesp+beta+f`=even integer `impliesbeta+f=`even integer -p=even integer-integer=an integer. since `0 lt beta lt 1 and 0 lt f lt 1` `implies 0 lt beta+ f lt 2, ` but `beta+f=` an integer `impliesbeta+f=1impliesf=1-beta` Now `p+beta+f=`ven integer `impliesp`=even integer-1= odd integer annd `(p+beta)f=(9+4sqrt(5))^(n)(9-4sqrt(5))^(n)` `=(81-80)^(n)=1` `implies(p+beta)(1-beta)=1`.

Discussion

20.	Let P(n): n^2 + n is an odd integer and P(K) impliesP(k+ 1) is true, then P(n) is true for all
Answer» `n GT 2` ` n gt 1` ` n` Noneof these Answer :D

Discussion

21.	If 1, alpha_(1), alpha_(2),·····, alpha_(n -1) are the n^(th) roots of unity, then (2-alpha_(1)),(2-alpha_(2))…..(2-alpha_(n-1))=
Answer» N `2^(n)` `2^(n)+1` `2^(n)-1` ANSWER :D

Discussion

22.	Find the unit vector in the direction of the vector = veca= hati+hatj+ 2hatk
Answer» Answer :`(1)/(sqrt(6))HATI+(1)/(sqrt(6))hatj+(2)/(sqrt(6))HATK`

Discussion

23.	arg(barz)-arg(-barz) is
Answer» `PI` `pi` `pi/2` `pi/2` ANSWER :A

Discussion

24.	If y==2x^(2)-1 then (1)/(x^(2))+(1)/(2x^(4))+(1)/(2x^(4))+(1)/(3x^(6))+…infty equals to
Answer» `log_(E)((y+1)/(y-1))` `log_(e)((1+y)/(1-y))` `log_(e)((1-y)/(1+y))` `LOG((1+2y)/(1-2y))` ANSWER :B

Discussion

25.	A man motors from A to B. A large part of the distace is uphill and he gets a mileage of only 10 miles per gallon of gasoline. ON the return trip, he makes 15 miles per gallon, then the average of his mileage (assuming that the distance from A to B is 60 miles ) is
Answer» 12 11 10 20 Answer :A

Discussion

26.	Let veca=hati+hatj+hatk,vecb=hati+4hatj-hatk and vec c =hati+hatj+2hatk. If vecS be a unit vector, then the magnitude of the vector (veca.vecS)(vecbxxvecc)+(vecb.vecS)(veccxxveca)+(vecc.vecS)(vecaxxvecb) is equal to
Answer» 1 2 3 4 Answer :C

Discussion

27.	Find the number of necklaces that can be prepared using 7 different coloured beads.
Answer» ANSWER :`(6!)/(2!)`

Discussion

28.	consider the function f(X) =x+cosx -a values of a which f(X) =0 has exactly one positive root are
Answer» (0,1) `(-oo,1)` (-1,1) `(1,oo)` SOLUTION :`F(x)=x+cosx-a or f(X)=1 -sinx ge 0 forall x in R` Thus f(X) is increasing in `(-oo,oo)` as for f(x) =0 x is not FORMING an interval also f(X) =-COS x =0 or `x =(2n+1)(pi)/(2),nin Z` Hence there are INFINITE points of inflection Now f(x) =1-a For positive root `1-alt0 or agt1` for negative root `1-agt 0 or a lt1`

Discussion

29.	{:("Column-I","Column-II"),(A.lim_(x to 0) [("sin"[x])/([x])],p.0),(B.lim_(x to 0)) [(tan [x])/([x])],q.10),(C. lim_(x to 0) [x],r.9),(lim_(x + (pi)/(2)) [cos x],s.-1),( , t.lim_(x to (pi)/(2))(e^(cosx) - 1)/(cos X)):}
Answer» Answer :A(p),B(p),C(q,R),D(s,t)

Discussion

30.	Find int(3x-2)/((x+1)^(2)(x+3))dx
Answer» ANSWER :`(11)/(4)\|(x+1)/(x+3)\|+(5)/(2(x+1))+C`

Discussion

31.	int_(0)^(pi//2)x sin^2 x dx=
Answer» `pi^2/4 + 1/16` `pi^2/16 + 1/4` `pi^2` 0 Answer :B

Discussion

32.	If p: It rains today, q :I go to school, r: I shall meet any friend and s : I shall go for a movie , then which of the following is the proposition. If it does not rain or if I do not go to school, then I shall meet any friend and go for a movie ?
Answer» <P>`~~(p^Q) rarr(R^^s)` `~~(p^^q) rarr(r^^s)` `~~(p ^^ q) rarr (r vv s)` NONE of these Answer :A

Discussion

33.	Find int x e^(x) dx
Answer» ANSWER :`XE^(X)-E^(x)+C`

Discussion

34.	Identical dielectric ...................
Answer» Solution :When SLAB is removed capacitance of B decreases due to which overall capacitance and part of the charge in steady state flows into the BATTERY and work is done on the battery. As capacitors A and B are in SERIES thus charge on both capacitors are always equal. In the process of removal of slab external agent will do some work but it will not APPEAR as heat as in slow removal or insertion of dielectric slab no heat is PRODUCED. Thus option (D) is correct.

Discussion

35.	Evaluate : (i) intcos^(-1)xdx (ii) inttan^(-1)xdx (iii) intsec^(-1)xdx
Answer» SOLUTION :(i) Put `COS^(-1)x=t" so that "x=cost anddx=-sintdt`. `:.intcos^(-1)xdx=-inttsintdt` `=-[t(-cost)-int(-cost)DT]` [integrating by parts] `=tcost-intcostdt=tcost-sint+C` `=XCOS^(-1)x-sqrt(1-x^(2))+C` `[becausecost=xrArrsint=sqrt(1-x^(2))]`. (II) Put `tan^(-1)x=t" so that "x=tant anddx=sec^(2)tdt`. `:.inttan^(-1)xdx=inttsec^(2)tdt` `=ttant-int1tantdt""` [integrating by parts] `=ttant+log\|cost\|+C` `=(tan^(-1)x)x+log\|(1)/(sqrt(1+x^(2)))\|+C` `[becausetant=xrArrcost=(1)/(sqrt(1+x^(2)))]` `=x(tan^(-1)x)-(1)/(2)log\|1+x^(2)\|+C`. (iii) Put `sec^(-1)x=t" so that"x=sectanddx=sec t tantdt`. `:.intsec^(-1)xdx=intt(sec t tan t)dt` `=t(sect)-int1*sect dt""` [integrating by parts] `=t(sect)-log\|sec t+tant\|+C` `=t(sect)-log\|sect+sqrt(sec^(2)t-1)\|+C` `=x(sec^(-1)x)-log\|x+sqrt(x^(2)-1)\|+C`.

Discussion

36.	Find two positive numbers x and y such that x + y = 60 and xy^(3) is maximum.
Answer» ANSWER :45,45

Discussion

37.	If f(theta) = 64 sec theta + 27 cosec theta " when " theta lies in (0, pi//4) then min f(theta) is equal to
Answer» ANSWER :125

Discussion

38.	Between stations A and B there 10 intermediate stations. Find the number of ways in which a train can be stopped at 4 of these intermediate stations so that, no two stopping stations are adjacent ? (Hint: Stop rarr S. Non stop rarr N we have to arrange SSSSNNNNNN letters so that no two S's are
Answer» ANSWER :`("^7P_4)/(4!)`

Discussion

39.	A vertical tower OP of height h subtends angle alpha, beta, gamma respectively at the point A, B, C on the horizontal plane through the foot O of the tower. A is due west of the tower. B is due east of A and on the same side of the tower as A. C is due south of B, then AC =
Answer» `h(COT ALPHA - cot beta)` `h sqrt(cot^(2)GAMMA -cot^(2)beta)` `h sqrt(cot^(2)alpha+cot^(2)gamma-2cot alpha cot beta)` `h sqrt(cot^(2)gamma + cot^(2) beta - 2 cot alpha cot beta)` Answer :C

Discussion

40.	If H(x,y)=0 represents the equation of a hyperbola and A( x,y ) =0 ,C( x,y)=0 the joint equation of its asymptotes and the conjugate hyperbola respectively , then for any point (a,b) in the plane H(a,b) ,A( a,b) and C( a,b)are in
Answer» A.P. G.P. H.P. A.G.P ANSWER :A

Discussion

41.	The perpendicular distancebetween the planes x + 2y- 3z = 2 and 2x+4y-6z = -2 is.........
Answer» `(3)/(sqrt(14))` `(1)/(sqrt(14))` `(2)/(sqrt(14))` `(4)/(sqrt(14))` ANSWER :A

Discussion

42.	Let a, b, c be positive real numbers. The following system of equations in x, y and z (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1, (x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1,(-x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1 has
Answer» no SOLUTION unique solution FINITELY MANY SOLUTIONS infinitely many solutions Answer :D

Discussion

43.	If 1,2,3 and 4are therootsof theequationx^4 + ax^3 + bx^2 +cx +d=0thena+ 2b +c=
Answer» `-25` 0 10 24 Answer :C

Discussion

44.	The rate of change of volume of a sphere with respect to its surface area when the radius is 4 cm is
Answer» `4cm^(3)//CM^(2)` `2 cm^(3)//cm^(2)` `6cm^(3)//cm^(2)` `8 cm^(3)//cm^(2)` Answer :B

Discussion

45.	The midpoint of the chord 2x-y-2=0 of the parabola y^(2)=8x is
Answer» `((5)/(2),-1)` `(-1,(5)/(2))` `((3)/(2),-1)` `(-1,(3)/(2))` ANSWER :A

Discussion

46.	IF f(x), f'(x) f''(x) are positive functions and f(0)=1, f')0)=2 then the solution of the differential equation {:\|(f(x)f'(x)),(f'(x) f''(x))\|=0 is
Answer» ANSWER :A

Discussion

47.	Find the value of lambda so that the vectors veca and vecb are perpendicular to each other. veca = (6,2,-3), vecb = (1,-4,lambda)
Answer» SOLUTION :If `veca` and `vecb` are perpendicular `veca.vecb = 0` `IMPLIES (6,2,-3).(1,-4,lambda) = 0` `implies 6-8-3lambda = 0` `implies -2-3lambda = 0 implies lambda = -2/3

Discussion

48.	Find the sum of the vectorsveca=hati-2hatj+hatk,vecb=-2hati+4hatj+5hatkandvecc=hati-6hatj--7hatk.
Answer» ANSWER :`=-4hatj-hatk`

Discussion

49.	If ax^(2)+bx+c =0 where ane0 is satisfied by alpha,beta,alpha^(2)andbeta^(2) where alphabetane0. Let set S be the set of all possible unordered pairs (alpha,beta). Then match the following lists:
Answer» `{:(,a,B,c,d),((1),q,s,s,r):}` `{:(,a,b,c,d),((2),r,s,q,p):}` `{:(,a,b,c,d),((3),q,s,r,p):}` `{:(,a,b,c,d),((4),r,s,p,q):}` Solution :EQUATION is satisfied by `alpha,beta,alpha^(2)andbeta^(2)`. So, we have following POSSIBILITIES: (1) Let `alpha^(2)=alphaandbeta^(2)=beta`. `IMPLIES(alpha,beta)-=(1,1)` (ii) `alpha^(2)=betaandbeta^(2)=alpha` `impliesalpha^(4)=alpha` `impliesalpha(alpha^(3)-1)=0` `impliesalpha=0,1,(-1pmsqrt3i)/(2)` `implies(alpha,beta)-=((-1-sqrt3i)/(2),(-1+sqrt3i)/(2))` (iii) `alpha^(2)betaandbeta^(2)=beta(oralpha^(2)=alphaandbeta^(2)=alpha)` `impliesalpha^(2)=beta^(2)impliesalpha=pmbeta` `implies(alpha,beta)=(-1,1),(1,1)` Thus, possible unordered pairs `(alpha,beta)` such that that`alphabetane0` is `(1,1),(-1,1)or((-1-sqrt3i)/(2),(-1+sqrt3i)/(2))`.

Discussion

50.	The radius of an air bubble is increasing at the rate of (1)/(2) cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Answer» ANSWER :`2PI CM^(3)` /s

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

[veca, veca+vecb, veca+vecb+vecc] is :

State which of the following are positive ?cos 315^@

According to the table above , for what value of x does g(f(x))=-1 ?

(a)Maximise z = 4x + y subject to constraints : x + y le 50 3x + y le90 x ge0 y ge 0 by graphical method. (b) Find the value of K, if f(x) = {{:(Kx + 1, "if"x le pi),(cos x, "if " x gt pi):} is continuous at x = pi.

If veca, vecb and vecc are the position vectors of the vertices A,B and C. respectively , of triangleABC. Prove that the perpendicualar distance of the vertex A from the base BC of the triangle ABC is (|vecaxxvecb+vecbxxvecc+veccxxveca|)/(|vecc-vecb|)

Ifthetais the angle between any two vectors veca and vecb, then |veca.vecb|=|veca xx vecb| when theta is equal to

int_(0)^(pi//4) [sqrt(tan x)+sqrt(cot x)]dx

Show that the line lx+my+n=0 is a normal to the circles S=0 iff gl+mf=mn.

Let S = {(x,y) in N xx N, x^(2) -y^(2) = 10,21,954}, then

If f(x)=|{:(2cos^(2)x,sin2x,-sinx),(sin2x,2sin^(2)x,cosx),(sinx,-cosx,0):}|" then "int_(0)^(pi//2)[f(x)+f'(x)]dx=

Using binomial theorem, prove that 8^(n)-7n always leaves remainder 1 when divided by 49.

If cos2x=(sqrt2+1)(cosx-1/sqrt2), cos x ne 1/2,then xbelong to

If int(x)=int(a+b), then int_(0)^(2a)int= int =(x)=2int_(0)^(a) int(x)dx

If "cos"(pi)/(15) "cos"(2pi)/(15) "cos"(4pi)/(15) "cos"(5pi)/(15) "cos"(7pi)/(15) "cos"(30pi)/(15) =x, " then " (1)/(8x) =

Find the position vector of the mid point of the vector joiningthe points P(2,3,4) and Q(4,1,-2).

Find the probability of getting a prime number when a die is rolled

Evaluate the following integrals int e^(x) (1+x^(2))dx

If (9+4sqrt(5))^(n)=p+beta, where n and p are positive integers and beta is a positive proper fraction, prrove that (1-beta)(p+beta)=1 and p is an odd integer.

Let P(n): n^2 + n is an odd integer and P(K) impliesP(k+ 1) is true, then P(n) is true for all

If 1, alpha_(1), alpha_(2),·····, alpha_(n -1) are the n^(th) roots of unity, then (2-alpha_(1)),(2-alpha_(2))…..(2-alpha_(n-1))=

Find the unit vector in the direction of the vector = veca= hati+hatj+ 2hatk

arg(barz)-arg(-barz) is

If y==2x^(2)-1 then (1)/(x^(2))+(1)/(2x^(4))+(1)/(2x^(4))+(1)/(3x^(6))+…infty equals to

A man motors from A to B. A large part of the distace is uphill and he gets a mileage of only 10 miles per gallon of gasoline. ON the return trip, he makes 15 miles per gallon, then the average of his mileage (assuming that the distance from A to B is 60 miles ) is

Let veca=hati+hatj+hatk,vecb=hati+4hatj-hatk and vec c =hati+hatj+2hatk. If vecS be a unit vector, then the magnitude of the vector (veca.vecS)(vecbxxvecc)+(vecb.vecS)(veccxxveca)+(vecc.vecS)(vecaxxvecb) is equal to

Find the number of necklaces that can be prepared using 7 different coloured beads.

consider the function f(X) =x+cosx -a values of a which f(X) =0 has exactly one positive root are

{:("Column-I","Column-II"),(A.lim_(x to 0) [("sin"[x])/([x])],p.0),(B.lim_(x to 0)) [(tan [x])/([x])],q.10),(C. lim_(x to 0) [x],r.9),(lim_(x + (pi)/(2)) [cos x],s.-1),( , t.lim_(x to (pi)/(2))(e^(cosx) - 1)/(cos X)):}

Find int(3x-2)/((x+1)^(2)(x+3))dx

int_(0)^(pi//2)x sin^2 x dx=

If p: It rains today, q :I go to school, r: I shall meet any friend and s : I shall go for a movie , then which of the following is the proposition. If it does not rain or if I do not go to school, then I shall meet any friend and go for a movie ?

Find int x e^(x) dx

Identical dielectric ...................

Evaluate : (i) intcos^(-1)xdx (ii) inttan^(-1)xdx (iii) intsec^(-1)xdx

Find two positive numbers x and y such that x + y = 60 and xy^(3) is maximum.

If f(theta) = 64 sec theta + 27 cosec theta " when " theta lies in (0, pi//4) then min f(theta) is equal to

Between stations A and B there 10 intermediate stations. Find the number of ways in which a train can be stopped at 4 of these intermediate stations so that, no two stopping stations are adjacent ? (Hint: Stop rarr S. Non stop rarr N we have to arrange SSSSNNNNNN letters so that no two S's are

A vertical tower OP of height h subtends angle alpha, beta, gamma respectively at the point A, B, C on the horizontal plane through the foot O of the tower. A is due west of the tower. B is due east of A and on the same side of the tower as A. C is due south of B, then AC =

If H(x,y)=0 represents the equation of a hyperbola and A( x,y ) =0 ,C( x,y)=0 the joint equation of its asymptotes and the conjugate hyperbola respectively , then for any point (a,b) in the plane H(a,b) ,A( a,b) and C( a,b)are in

The perpendicular distancebetween the planes x + 2y- 3z = 2 and 2x+4y-6z = -2 is.........

Let a, b, c be positive real numbers. The following system of equations in x, y and z (x^2)/(a^2)+(y^2)/(b^2)-(z^2)/(c^2)=1, (x^2)/(a^2)-(y^2)/(b^2)+(z^2)/(c^2)=1,(-x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1 has

If 1,2,3 and 4are therootsof theequationx^4 + ax^3 + bx^2 +cx +d=0thena+ 2b +c=

The rate of change of volume of a sphere with respect to its surface area when the radius is 4 cm is

The midpoint of the chord 2x-y-2=0 of the parabola y^(2)=8x is

IF f(x), f'(x) f''(x) are positive functions and f(0)=1, f')0)=2 then the solution of the differential equation {:|(f(x)f'(x)),(f'(x) f''(x))|=0 is

Find the value of lambda so that the vectors veca and vecb are perpendicular to each other. veca = (6,2,-3), vecb = (1,-4,lambda)

Find the sum of the vectorsveca=hati-2hatj+hatk,vecb=-2hati+4hatj+5hatkandvecc=hati-6hatj--7hatk.

If ax^(2)+bx+c =0 where ane0 is satisfied by alpha,beta,alpha^(2)andbeta^(2) where alphabetane0. Let set S be the set of all possible unordered pairs (alpha,beta). Then match the following lists:

The radius of an air bubble is increasing at the rate of (1)/(2) cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?