28278 + Interview Questions in Maths in Class 12 Page 70 InterviewSolution

3451.	d, e, and f are in GP
Answer» d, e and f are in GP `(d)/(a), (e)/(B) and (f)/(c)` are in AP d,e and f are in AP `(d)/(a), (e)/(b) and (f)/(c)` are in GP Solution :Given, three distinct numbers a, b and c are in GP `:. ""b^(2) = ac`..(i) and the given quadratic equations `AX^(2) + 2b x + c = 0`...(i) `dx^(2) + 2ex + f = 0`...(III) For quadratic Eq. (ii) , the DISCRIMINANT `D = (2b)^(2) - 4ac` `= 4 (b^(2) - ac) = 0` [from Eq. (i)] `rArr` Quadratic Eq. (ii) have equal roots, and it is equal to `x = -(b)/(a)`, and it is given that quadratic Eqs. (ii) and (iii) have a common root, so `d (-(b)/(a))^(2) + 2e (-(b)/(a)) + f = 0` `rArr db^(2) - 2eba + a^(2) f = 0` `rArr d (ac) - 2eab + a^(2) f = 0 "" [ :' b^(2) = ac]` `rArr d (ac) - 2eab + a^(2) f = 0 "" [ :' a != 0]` `rArr 2eb = dc + af` `rArr 2 (e)/(b) = (dc)/(b^(2)) + (af)/(b^(2))` [dividing each term by `b^(2)`] `rArr 2 ((e)/(b)) = (d)/(a) + (f)/(c) "" [ :'b^(2) = ac]` So, `(d)/(a), (e)/(b), (f)/(c)` are in AP. Alternate Solution Given, three distinct numbers a, b and c are in GP. Let `a = a, b = ar, c = ar^(2)` are in GP, which satisfies `ax^(2) + 2bx + c = 0` `:. ax^(2) + 2(ar) x + ar^(2) = 0` `rArr x^(2) + 2rx + r^(2) = 0 "" [ :' a != 0]` `rArr (x + r)^(2) = 0 rArr x = -r` According to the equation, `ax^(2) + 2bx + c = 0 and dx^(2) + 2ex + f = 0` have a common root. So, `x = -r` satisfies `dx^(2) + 2ex + f = 0` `:. d(-r)^(2) + 2e(-r) + f = 0` `rArr dr^(2) - 2er + f = 0` `rArr d((c)/(a)) - 2e ((c)/(b)) + f = 0` `rArr (d)/(a) - (2e)/(b) + (f)/(c) = 0` `rArr (d)/(a) + (f)/(c) = (2e)/(b) "" [ :' c != 0]`

Discussion

3452.	Consider f:[0,1]toR has a continuous derivative and int_(0)^(1)f(x)dx=0, then for ever alpha epsilon(0,1) \|int_(0)^(1)f(x)dx\|lex. "Max"_(0le x le 1)\|f^(')(x)\| then
Answer» `[1/k]gt 5` `1/8 [1/k]EPSILON[0,1]` `kepsilon(0,1/5)` `1/(2k)+1` is an odd number Solution :`\|int_(0)^(9)f(x)dx\|=\|a int_(0)^(1)f(at)dt\|le 1/8 M`

Discussion

3453.	Show that the line whose vector equation isvec r=(2 hat i-2 hat j+3)+lambda( hat i- hat j+4 hat k)is parallel to the plane whose vector equationvec r( hat i+5 hat j+ hat k)=5. Also, find thedistance between them.
Answer» ANSWER :1.92 UNITS

Discussion

3454.	If vec(a),vec(b),vec(c) are the position vectors of points A, B, C respectively such that 5vec(a)-3vec(b)-2vec(c)=vec(0), then
Answer» C divides BA INTERNALLY in ratio 5 : 3 C divides BA externally in ratio 5 : 3 C divides AB internally in ratio 5 : 3 C divides AB externally in ratio 5 : 3 Answer :B

Discussion

3455.	""^20C_10.""^15C_0 + ""^20C_9.""^15C_1 + ""^20C_8.""^15C_2 + …..+""^20C_0.""^15C_10 =
Answer» `""^20C_10` `""^25C_10` `""^35C_10` `""^40C_10` ANSWER :C

Discussion

3456.	For x in R, the least value of (x^(2)-6x+5)/(x^(2)+2x+1) is
Answer» `-1` `-1/2` `-1/4` `-1/3` ANSWER :D

Discussion

3457.	Ground state electronic configuration of 'C' written as [He] 2s^2 2p^2, it is not as [He], 2s^1,2p^3 , because
Answer» EXCHANGE ENERGY GT EXCITATION energy excitation energy gt exchange energy excitation energy = enchange energy All of these

Discussion

3458.	The value of lim_(xto oo)x^6/6^x is
Answer» 1 0 `-1` not a finite NUMBER ANSWER :B

Discussion

3459.	If f.R rarr R,f(x)=(x^(2)+bx+1)/(x^(2)+2x+b),(b gt 1) " and " f(x), 1/f(x) have the same bounded set as their range, the value of b is
Answer» `2sqrt(3)-2` `2sqrt(3)+2` `2sqrt(2)-2` `2sqrt(2)+2` ANSWER :A

Discussion

3460.	If z = x +iy and if the point P in the argand plane represents z then find the locus of P satisfying the equation Re ((z + 1)/(z +i)) = 1
Answer» ANSWER :A

Discussion

3461.	If z = x +iy and if the point P in the argand plane represents z then find the locus of P satisfying the equation \|z+ 4i\| + \|z -4i\| = 10
Answer» ANSWER :B

Discussion

3462.	Let P (alpha, beta, gamma) and Q(1,-1,0) be points such that the mid-point of PQ is R (x,y,z). If x is the AM of alpha and beta, y is the G.M. of beta and gamma and z is the H.M . Of gamma and alphathen :
Answer» Answer :`A to 1, B TO3, C to 2`

Discussion

3463.	A hyperbola whose transverse axis , centre (0,0) and foci (+- sqrt(10), 0) passes thorugh the point (3,2). i. Find the equation of the hyperbola. ii. Find its eccentricity .
Answer» ANSWER :`(x^2)/5 - (y^2)/5 - 1`. II. `SQRT(2)`.

Discussion

3464.	The set of values of 'a' for which(13 x-1 ) ^(2) + (13y -2) ^(2)=a( 5x+ 12y -1)^(2)represents an ellipse if
Answer» ` 1 LT a lt 2 ` ` 2 lt a lt 3 ` ` 0 lt alt 1 ` ` 3 lt a lt 4 ` ANSWER :D

Discussion

3465.	I : If r alpha a + beta b represents a line passing through the points a, b then alpha + beta = 1. II : If r = alpha a + beta b + gamma c represents a plane passing through the points a, b, c then alpha + beta + gamma = 1.
Answer» only I is TRUE only II is true both I and II are true NEITHER I nor II are true Answer :C

Discussion

3466.	Let w(x, y, z)=(1)/(sqrt(x^(2)+y^(2)+z^(2)))(x, y, z) != (0, 0, 0). Show that (del^(2)w)/(delx^(2))+(del^(2)w)/(dely^(2))+(del^(2)w)/(delz^(2))=0
Answer» Answer :`(DEL ^(2)W)/(del y ^(2)) + (del ^(2) w)/(del Z ^(2))=0.`

Discussion

3467.	If [(1,1),(-1,1)][(x),(y)]=[(2),(4)],then the values of x and y respectively are
Answer» -3,-1 1,3 3,1 -1, 3 Answer :D

Discussion

3468.	If 2vecavecb = \|veca\| \|vecb\| then the angle between veca & vecb is
Answer» a.`30^(@)` B.`0^(@)` c.`90^(@)` d.`60^(@)` Answer :D

Discussion

3469.	Find an anti derivative (or integral) of the following functions . e^(2x)
Answer» ANSWER :`1/2e^(2X)`

Discussion

3470.	A hyperbola passing through a focus of the ellipse x^(2)/(169)+y^(2)/(25)=1. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of eccentricities is 1. Then, the equation of the hyperbola is,
Answer» `X^(2)/(144)-y^(2)/(9)=1` `x^(2)/(169)-y^(2)/(25)=1` `x^(2)/(144)-y^(2)/(25)=1` `x^(2)/(25)-y^(2)/(9)=1` Answer :C

Discussion

3471.	On Z, define a relation R as follows: aRb if 5\|(a-b)\| Equivalent class [3] is equal to,
Answer» {……….,`-13, -8, -3,0, 3,8,`……} {…,-7,-2,3,8,13,….} {……,-8,-2,3,7,11,……} {…..,-7,-2,3,7,12,……} ANSWER :B

Discussion

3472.	If a line makes an angle of pi//4with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is
Answer» `(PI)/(6)` `(pi)/(3)` `(pi)/(4)` `(pi)/(2)` Answer :D

Discussion

3473.	f(x)= sin^(2)x + sin^(2) (x + (pi)/(3)) + cos x cos (x + (pi)/(3)) then f'(x) = ………
Answer» 1 `COS^(2)X` 0 sin 2x Answer :C

Discussion

3474.	A ray of light comes light comes along the line L = 0 and strikes the plane mirror kept along the plane P = 0 at B. A(2, 1, 6)is a point on the line L = 0 whose image about P = 0 is A'. It is given that L = 0 is (x-2)/(3)= (y-1)/(4)= (z-6)/(5) and P =0is x+y-2z=3. The coordinates of B are
Answer» `(5, 10, 6)` `(10, 15, 11)` `(-10, -15, -14)` none of these Solution :Let `(x-2)/(3)=(y-1)/(4)= (z-6)/(5)=lamda` `""x=2+ 3lamda, y=1 + 4 lamda, z=6+ 5lamda` LIES on PLANE `""x+y-2z=3` `rArr""2+3lamda +1 +4lamda- 2 (6+5lamda)=3` or `"" 3+7lamda-12-10lamda=3 or -3lamda=12 or lamda= -4` Point `B-= (-10, -15, -14)`

Discussion

3475.	A ray of light comes light comes along the line L = 0 and strikes the plane mirror kept along the plane P = 0 at B. A(2, 1, 6)is a point on the line L = 0 whose image about P = 0 is A'. It is given that L = 0 is (x-2)/(3)= (y-1)/(4)= (z-6)/(5) and P =0is x+y-2z=3. The coordinates of A' are
Answer» `(6, 5, 2)` `(6, 5, -2)` `(6, -5, 2)` NONE of these Solution :Let `Q(x_(2), y_(2), z_(2))` bethe IMAGE of `A(2, 1, 6)` about mirror`x +y -2z =3`. Then, `""(x_(2)-2)/(1)= (y_(2)-1)/(1)= (z_(2)-6)/(-2)` `""=(-2(2+1-12-3))/(1^(2)+1^(2)+2^(2))=4` `rArr ""(x_(2), y_(2), z_(2))-= (6, 5, -2)`.

Discussion

3476.	Value of "sin"^(-1)(1)/(sqrt(5))+cot^(-1)3 ''is
Answer» `(PI)/(2)` `(pi)/(4)` `(pi)/(3)` None of these Solution :N/A

Discussion

3477.	A ray of light comes light comes along the line L = 0 and strikes the plane mirror kept along the plane P = 0 at B. A(2, 1, 6)is a point on the line L = 0 whose image about P = 0 is A'. It is given that L = 0 is (x-2)/(3)= (y-1)/(4)= (z-6)/(5) and P =0is x+y-2z=3. If L_(1) =0is the reflected ray, then its equation is
Answer» `(X+10)/(4)= (y-5)/(4)= (z+2)/(3)` `(x+10)/(3)= (y+15)/(5)= (z+14)/(5)` `(x+10)/(4)= (y+15)/(5)=(z+14)/(3)` none of these Solution :The EQUATION of the reflected ray `L_(1)=0` is the line joining `Q(x_(2), y_(2), z_(2)) and B(-10, -15, -14)`. `""(x+10)/(16)= (y+15)/(20) = (z+14)/(12)` or `(x+10)/(4)= (y+15)/(5)= (z+14)/(3)`

Discussion

3478.	A normal to the hyperbola (x^(2))/(4)-(y^(2))/(1)=1 has equal interceptsonpositivex and positivey-axes. If this normaltouches the ellipse(x^(2))/(a^(2))+(y^(2))/(b^(2))=1, then3(a^(2)+b^(2)) is equal to :
Answer» 5 25 16 None of these Answer :B

Discussion

3479.	int_(1)^(4) ln [x] dx=
Answer» `LN 4` `ln 6` `ln 8` `ln 3` ANSWER :B

Discussion

3480.	Using the property of determinants andd without expanding in following exercises 1 to 7 prove that \|{:(a^2+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1):}\|=1+a^2+b^2+c^2
Answer» ANSWER :`1+a^2+b^2+c^2`

Discussion

3481.	If z_(1) , z_(2) are conjugate complex numbers and z_(3) , z_(4) are also conjugate , then Arg ((z_(3))/(z_(2)))=
Answer» ARG `((z_(1))/(z_(4)))` Arg `((z_(4))/(z_(1)))` Arg `((z_(2))/(z_(4)))` Arg `((z_(1))/(z_(3)))` ANSWER :A

Discussion

3482.	Let S be the set of 2xx2 matrices given by S={A=[[a,b],[c,d]],"where" a,b,c,d,in I},such that A^(T)=A^(-1) Then
Answer» number of matrices in set s is equal to 6 number of matricesin set S such that `\|A-I_(2)\| NE 0` is equal to 3 symmetric matrices are more than skew -symmetric matrices in set S all matrices in set S are singular Solution : As, `A A^(T)=I_(2)` `implies [[a,b],[c,d]],[[a,c],[b,d]]=[[1,0],[0,1]]` `impliesa=0,b=pm1,d=0,c=pm1` therefore Total 8 matrices are POSSIBLE They are `[[1,0],[0,1]],[[1,0],[0,-1]],[[-1,0],[0,1]],[[-1,0],[0,-1]]` `[[0,1],[1,0]],[[0,-1],[1,0]],[[0,1],[-1,0]],[[0,-1],[-1,0]]` ALSO `\|A-I_(2)\|=\|A-AA^(T)\|=\|A\|\|I_(2)-A^(T)\|` `=\|A\|\|(I_(2)-A^(T))^(T)\|=\|A\|\|I_(2)-A\|` `=\|A\|\|A-I_(2)\|` `implies\|A\|=1(As,\|A-I_(2)\|ne 0)` except `A=1=[[1,0],[0,1]]` where `\|A\|=1 but` `\| A-I_(2)\|=0`

Discussion

3483.	For any acute angled triangleABC,(sinA)/(A)+(sinB)/(B)+(sinC)/(C ) can
Answer» 1 2 3 4 Solution :Let `f(x)=(sinx)/(x)` `rarr f(x)=(xcosx-sinx)/(x^(2))=(cosx(x-tanx))/(x^(2))` `rarr f(X)=(-x^(2)sinx-2xcosx+2 sinx)/(x^(3))` g(x)=`-x^(2)sinx-2xcosx+2 sinx` `g(X)=-x^(2)cosxlt0 forall x in (0,pi//2)` `rarr for xgt0` we have `g(X) lt g(0)i.e g(x)lt0` `rarr f(X)lt0andf(x)lt0,x in (0,(pi)/(2))` `rarr f(a+B+C)/(3)gtf(A)+f(B)+f(C )/(3)` `rarr (SINA)/(A)+(sinB)/(B)+(sinC)/(C )LE(9sqrt(3))/(2PI)=2.5`

Discussion

3484.	inte^(3x)cos4xdx=.........+c
Answer» `(E^(3x))/(25)(3cos4x-4sin4x)` `(e^(3x))/(5)(3cos4x-4sin4x)` `(e^(3x))/(5)(3cos4x+4sin4x)` `(e^(3x))/(5)cos(4x-tan^(-1)""(4)/(3))` Answer :D

Discussion

3485.	vec(a)=hati+hatj+sqrt(2)hatk,vec(b)=b_(1)hati+b_(2)hatj+sqrt(2)hatk and vec( c )=5hati+hatj+sqrt(2)k are three vectors. The projection of the vector vec(b) on vec(a) is \|vec(a)\|. If vec(a)+vec(b) is perpendicular to vec( c ) then \|vec(b)\|= ...........
Answer» `SQRT(32)` `sqrt(22)` 4 6 Answer :D

Discussion

3486.	1+(1)/(2). (3)/(5) + (1.3)/(2.4) . (9)/(25)+(1.3.5)/(2.4.6).(27)/(125)+…...oo=
Answer» `SQRT(5/2)` `sqrt(3/5)` `sqrt(2/5)` `sqrt(5/3)` ANSWER :A

Discussion

3487.	The value of the expression (int_(0)^(a)x^(4)sqrt(a^(2)-x^(2))dx)/(int_(0)^(a)x^(2)sqrt(a^(2)-x^(2))dx)=
Answer» `(a^(2))/(6)` `(3A^(2))/(2)` `(3a^(2))/(4)` `(a^(2))/(2)` ANSWER :4

Discussion

3488.	Find an approximate value of the following correctedto 4 decimal places. root(7)(127)
Answer» ANSWER :`1.9978`

Discussion

3489.	Statement Isin ^(-1)(1/sqrte) gt tan^(-1)(1/sqrtpi) Statement II sin^(-1) x gt tan^(-1) y " for " xgt y , AA x, y in (0,1)
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

3490.	If y=tan^-1(cotx)+cot^-1(tanx), find (dy)/dx.
Answer» SOLUTION :`y=tan^-1(COTX)+cot^-19tanx)=tan^-1tan(pi/2-x)+cot^-1cot(pi/2-x)=pi/2-x+pi/2-x=pi-2xrArr(DY)/dx=-2`

Discussion

3491.	Consider all the 5 digit numbers where each of the digits is chosen from the set { 1, 2, 3, 4} . Then the number of numbers, which contain all the four digits is :
Answer» 240 244 586 781 Answer :A

Discussion

3492.	Product of roots of the equation sqrt(13 - x^(2)) = x + 5 is
Answer» `-6` 7 6 `-7` ANSWER :C

Discussion

3493.	If Sin^(-1)(1)-"Sin"^(-1)sqrt(3/x)=pi/6 and x is a root of the equation x^(2)+kx-12=0, then value of k is
Answer» -2 -1 1 2 Answer :B

Discussion

3494.	Find derivatives of the following functions.sin^4x
Answer» SOLUTION :`y = SIN^4 X dy/dx = d/dx(sin x)^4 = 4(sin x)^3.d/dx(sin x) = 4 sin^3 x. COS x`

Discussion

3495.	The value of hati.(hatjxxhatk)+hatj.(hatkxxhati)+hatk.(hatixxhatj)=…….
Answer» 1 3 -3 0 Answer :B

Discussion

3496.	If the vectors 2 hati - hatj + lamda hatk , hati - hatj + 2 hatk and 3 hati - 2 hatj + hatk are coplanar, then the value of lamdais :
Answer» `-1` `-2` `-3` `-4.` ANSWER :A

Discussion

3497.	Five ordinary dice are rolled at random and sum of the numbers shown on them is 16. What is the probability that the numbers shown on each is any one from 2, 3, 4, 5.
Answer» ANSWER :`(9)/(49)`

Discussion

3498.	CF is the internal bisector of angle C of angle ABC, then CF is equal to
Answer» `(2AB)/(a + B) cos.(C)/(2)` `(a +b)/(2ab) cos.(C)/(2)` `(b sin A)/(sin(B + (C)/(2)))` none of these Solution : `DELTA = Delta_(1) + Delta_(2)` `rArr (1)/(2) ab sin C = (1)/(2) b (CF) sin .(C)/(2) + (1)/(2) a (CF) sin.(C)/(2)` or `CF = (ab sin C)/((a + b) sin.(C)/(2)) = (2ab cos.(C)/(2))/(a+b)` Again in `DeltaCFB`, by the sine rule, we have `(CF)/(sin B) = (a)/(sin (pi -theta)) = (a)/(sin theta) = (a)/(sin (B+(C)/(2))) ""( :' theta + B + (C)/(2) = pi)` or `CF = (a sin B)/(sin (B +(C)/(2))) = (b sin A)/(sin (B +(C)/(2)))` [by sine rule]

Discussion

3499.	A unit vector is coplanner with bar(i)+bar(j)+2bar(k) and bar(i)+2bar(j)+bar(k) and it is perpendicular to the vector bar(i)+bar(j)+bar(k). Then the vector …………..
Answer» `(BAR(i)-bar(J))/(sqrt(2))` `+-((bar(j)-bar(K))/(sqrt(2)))` `(bar(k)-bar(i))/(sqrt(2))` `(bar(i)+bar(j)+bar(k))/(sqrt(3))` ANSWER :B

Discussion

3500.	Statement-1: pH of buffer solution always increases on increasing dilution. Statement-2: pH ofany acidic solution always increases on increasing dilution.
Answer» Statement-1 is true, statement-2 is true and statement-2 is CORRECT explanation for statement-1 . Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. Statement-1is true, statement-2 is FALSE. Statement-1 is false, statement-2 is true. Solution :PH of any acidic buffer solution may have any value & ALWAYS tends TOWARDS 7 on dilution, hence it may increase of decrease with dilution.

Discussion

Explore topic-wise InterviewSolutions in .

d, e, and f are in GP

Consider f:[0,1]toR has a continuous derivative and int_(0)^(1)f(x)dx=0, then for ever alpha epsilon(0,1) |int_(0)^(1)f(x)dx|lex. "Max"_(0le x le 1)|f^(')(x)| then

Show that the line whose vector equation isvec r=(2 hat i-2 hat j+3)+lambda( hat i- hat j+4 hat k)is parallel to the plane whose vector equationvec r( hat i+5 hat j+ hat k)=5. Also, find thedistance between them.

If vec(a),vec(b),vec(c) are the position vectors of points A, B, C respectively such that 5vec(a)-3vec(b)-2vec(c)=vec(0), then

""^20C_10.""^15C_0 + ""^20C_9.""^15C_1 + ""^20C_8.""^15C_2 + …..+""^20C_0.""^15C_10 =

For x in R, the least value of (x^(2)-6x+5)/(x^(2)+2x+1) is

Ground state electronic configuration of 'C' written as [He] 2s^2 2p^2, it is not as [He], 2s^1,2p^3 , because

The value of lim_(xto oo)x^6/6^x is

If f.R rarr R,f(x)=(x^(2)+bx+1)/(x^(2)+2x+b),(b gt 1) " and " f(x), 1/f(x) have the same bounded set as their range, the value of b is

If z = x +iy and if the point P in the argand plane represents z then find the locus of P satisfying the equation Re ((z + 1)/(z +i)) = 1

If z = x +iy and if the point P in the argand plane represents z then find the locus of P satisfying the equation |z+ 4i| + |z -4i| = 10

Let P (alpha, beta, gamma) and Q(1,-1,0) be points such that the mid-point of PQ is R (x,y,z). If x is the AM of alpha and beta, y is the G.M. of beta and gamma and z is the H.M . Of gamma and alphathen :

A hyperbola whose transverse axis , centre (0,0) and foci (+- sqrt(10), 0) passes thorugh the point (3,2). i. Find the equation of the hyperbola. ii. Find its eccentricity .

The set of values of 'a' for which(13 x-1 ) ^(2) + (13y -2) ^(2)=a( 5x+ 12y -1)^(2)represents an ellipse if

I : If r alpha a + beta b represents a line passing through the points a, b then alpha + beta = 1. II : If r = alpha a + beta b + gamma c represents a plane passing through the points a, b, c then alpha + beta + gamma = 1.

Let w(x, y, z)=(1)/(sqrt(x^(2)+y^(2)+z^(2)))(x, y, z) != (0, 0, 0). Show that (del^(2)w)/(delx^(2))+(del^(2)w)/(dely^(2))+(del^(2)w)/(delz^(2))=0

If [(1,1),(-1,1)][(x),(y)]=[(2),(4)],then the values of x and y respectively are

If 2veca*vecb = |veca| * |vecb| then the angle between veca & vecb is

Find an anti derivative (or integral) of the following functions . e^(2x)

A hyperbola passing through a focus of the ellipse x^(2)/(169)+y^(2)/(25)=1. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of eccentricities is 1. Then, the equation of the hyperbola is,

On Z, define a relation R as follows: aRb if 5|(a-b)| Equivalent class [3] is equal to,

If a line makes an angle of pi//4with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is

f(x)= sin^(2)x + sin^(2) (x + (pi)/(3)) + cos x cos (x + (pi)/(3)) then f'(x) = ………

A ray of light comes light comes along the line L = 0 and strikes the plane mirror kept along the plane P = 0 at B. A(2, 1, 6)is a point on the line L = 0 whose image about P = 0 is A'. It is given that L = 0 is (x-2)/(3)= (y-1)/(4)= (z-6)/(5) and P =0is x+y-2z=3. The coordinates of B are

A ray of light comes light comes along the line L = 0 and strikes the plane mirror kept along the plane P = 0 at B. A(2, 1, 6)is a point on the line L = 0 whose image about P = 0 is A'. It is given that L = 0 is (x-2)/(3)= (y-1)/(4)= (z-6)/(5) and P =0is x+y-2z=3. The coordinates of A' are

Value of "sin"^(-1)(1)/(sqrt(5))+cot^(-1)3 ''is

A normal to the hyperbola (x^(2))/(4)-(y^(2))/(1)=1 has equal interceptsonpositivex and positivey-axes. If this normaltouches the ellipse(x^(2))/(a^(2))+(y^(2))/(b^(2))=1, then3(a^(2)+b^(2)) is equal to :

int_(1)^(4) ln [x] dx=

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(a^2+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1):}|=1+a^2+b^2+c^2

If z_(1) , z_(2) are conjugate complex numbers and z_(3) , z_(4) are also conjugate , then Arg ((z_(3))/(z_(2)))=

Let S be the set of 2xx2 matrices given by S={A=[[a,b],[c,d]],"where" a,b,c,d,in I},such that A^(T)=A^(-1) Then

For any acute angled triangleABC,(sinA)/(A)+(sinB)/(B)+(sinC)/(C ) can

inte^(3x)cos4xdx=.........+c

vec(a)=hati+hatj+sqrt(2)hatk,vec(b)=b_(1)hati+b_(2)hatj+sqrt(2)hatk and vec( c )=5hati+hatj+sqrt(2)k are three vectors. The projection of the vector vec(b) on vec(a) is |vec(a)|. If vec(a)+vec(b) is perpendicular to vec( c ) then |vec(b)|= ...........

1+(1)/(2). (3)/(5) + (1.3)/(2.4) . (9)/(25)+(1.3.5)/(2.4.6).(27)/(125)+…...oo=

The value of the expression (int_(0)^(a)x^(4)sqrt(a^(2)-x^(2))dx)/(int_(0)^(a)x^(2)sqrt(a^(2)-x^(2))dx)=

Find an approximate value of the following correctedto 4 decimal places. root(7)(127)

Statement Isin ^(-1)(1/sqrte) gt tan^(-1)(1/sqrtpi) Statement II sin^(-1) x gt tan^(-1) y " for " xgt y , AA x, y in (0,1)

If y=tan^-1(cotx)+cot^-1(tanx), find (dy)/dx.

Consider all the 5 digit numbers where each of the digits is chosen from the set { 1, 2, 3, 4} . Then the number of numbers, which contain all the four digits is :

Product of roots of the equation sqrt(13 - x^(2)) = x + 5 is

If Sin^(-1)(1)-"Sin"^(-1)sqrt(3/x)=pi/6 and x is a root of the equation x^(2)+kx-12=0, then value of k is

Find derivatives of the following functions.sin^4x

The value of hati.(hatjxxhatk)+hatj.(hatkxxhati)+hatk.(hatixxhatj)=…….

If the vectors 2 hati - hatj + lamda hatk , hati - hatj + 2 hatk and 3 hati - 2 hatj + hatk are coplanar, then the value of lamdais :

Five ordinary dice are rolled at random and sum of the numbers shown on them is 16. What is the probability that the numbers shown on each is any one from 2, 3, 4, 5.

CF is the internal bisector of angle C of angle ABC, then CF is equal to

A unit vector is coplanner with bar(i)+bar(j)+2bar(k) and bar(i)+2bar(j)+bar(k) and it is perpendicular to the vector bar(i)+bar(j)+bar(k). Then the vector …………..

Statement-1: pH of buffer solution always increases on increasing dilution. Statement-2: pH ofany acidic solution always increases on increasing dilution.

If 2vecavecb = |veca| |vecb| then the angle between veca & vecb is