28278 + Interview Questions in Maths in Class 12 Page 93 InterviewSolution

4601.	Prove that (.^(n)C_(1)sin2x+.^(n)C_(2)sin4x+.^(n)C_(3)sin6x+"…..")/(1+.^(n)C_(1)cos2x+.^(n)C_(2)cos4x+.^(n)C_(3)cos 6x+"……")
Answer» SOLUTION :L.H.S. `=(underset(r=0)overset(n)SUM.^(n)C_(r)sin2rx)/(underset(r=0)overset(n)sum.^(n)C_(r)cos2x)""(1)` `0=(underset(r=0)overset(n)sum.^(n)C_(n-r)sin2(n-r)x)/(underset(r=0)overset(n)sum.^(n)C_(n-r)COS2(n-r)x)` `=(underset(r=0)overset(n)sum.^(n)C_(r)sin(2n-2r)x)/(underset(r=0)overset(n)sum.^(n)C_(r)cos(2n-2r)x)""(2)` `=(underset(r=0)overset(n)sum.^(n )C_(r)sin2rx+underset(r=0)overset(n)sum.^(n)C_(r)sin(2n-2r)x)/(underset(r=0)overset(n)sum.^(n)C_(r)cos2rx+underset(r=0)overset(n)sum.^(n)C_(r)cos(2n-2r)x)`. (USING `a/b = c/d = (a+c)/(b+d)` and using (1) and (2)) `=(underset(r=0)overset(n)sum.^(n)C_(r)[sin2rx+sin(2n-2r)x])/(underset(r=0)overset(n)sum.^(n)C_(r)[cos2rx+cos(2n-2r)x])` `= (2sin nx underset(r=0)overset(n)sum.^(n)C_(r)cos(2r-n)x)/(2cos nx underset(r=0)overset(n)sum.^(n)C_(r)cos(2r-n)x) = tan nx`

Discussion

4602.	a, b, c are digits of a 3-digit number such that 64a + 8b + c = 403, then the value of a + b + c + 2013 is
Answer» ANSWER :2024

Discussion

4603.	Find the inverse of the following using elementary transformations. A=[[3,1],[5,2]]
Answer» SOLUTION :`A^(-1) = [(2,-1),(-5,3)]`

Discussion

4604.	Determine the truth of falsity of the"Every set has a proper subset" propositions with reasons.
Answer» SOLUTION :EVERY set. has a PROPER SUBSET is false as `PHI` has no proper subset.

Discussion

4605.	Evalute the following integrals int x^(3) e^(2x) dx
Answer» Answer :`(E^(2x))/(16) [ (2x)^(3) - 3 (2x)^(2) + 6 (2x) - 6 ] + c `

Discussion

4606.	Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation :y = x^2 + 2x +c:y' - 2x - 2 = 0
Answer» Solution :`y= x^2 + 2x + C` `rArr y. = 2x + 2` `rArr y. - 2x - 2 = 0` `THEREFORE y = x^2 + 2x + c` is a solution of y. - 2x -2= 0

Discussion

4607.	Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation : y = e^x + 1: y'' - y' = 0
Answer» SOLUTION :` y = e^x + 1` `RARR y. = e^x` `rArr y.. = e^x = y.` `THEREFORE y.. - y. = 0 ` Therefore, `y = e^x + 1` is a solution of y.. - y. = 0

Discussion

4608.	Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation : y = sqrt(1 + x^2): y' = (xy)/(1 + x^2)
Answer» SOLUTION :`y = sqrt(1 + x^2)` `y.= 1/(2 sqrt(1+x^2)) (0+2x)` `=x/sqrt(1+x^2) = (xsqrt(1+x^2))/(1+x^2) = (xy)/(1 + x^2) ` `THEREFORE y = sqrt(1 + x^2)` is a solution of `y. = (xy)/(1+x^2)`

Discussion

4609.	Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation : y = cos x + c: y' + sin x = 0
Answer» SOLUTION :y = cos x + C y. = -SIN x `rArr y. + sin x = 0` ` THEREFORE` y = cos x + c is a solution of y. + sin x = 0

Discussion

4610.	Derivative of e^(3 log x)w.r.t.x is 3x^2.
Answer» ANSWER :T

Discussion

4611.	If \|z+i\|^2-\|z-i\|^2=3then the locus of z is
Answer» `4Y = 0` `4y= 2` `4y = 3` `4y = 4` ANSWER :C

Discussion

4612.	The volume of a sphere is increasing at the rate of pi cm^(2)//sec. The rate at which the radius is increasing is ……….., when the radius is 3 cm.
Answer» `(1)/(36)` cm/sec 36 cm/sec 9 cm/sec 27 cm/sec Answer :A

Discussion

4613.	The normal at a point P to the parabola y^(2) = 4x is parallel to the tangent at Q sqrt(2,2) to the hyperbola nd meets the axis of the parabola at If 5 is the focus of the parabola, area of the triangle PSR in sq. units is
Answer» `9sqrt(2)` `10sqrt(2)` `18sqrt(2)` `20sqrt(2)` ANSWER :C

Discussion

4614.	If A^T=[[-2,3],[1,2]] and B^T=[[-1,0],[1,2]] then find (A+2B)^T
Answer» SOLUTION :we have `(A+2B)^T=A^T+(2B)^T=A^T+2B^T` ` B^T=[[-1,0],[0,2]] IMPLIES 2B^T=[[-2,0],[0,4]]` THUS, `(A+2B)^T=A^T+2B^T=[[-4,5],[1,6]]`

Discussion

4615.	C_0-2. C_1+3 . C_2 …..+ (-1)^n (n+1). C_n=
Answer» `-1` 1 0 none Answer :C

Discussion

4616.	If a+b+c=0 and \|a\|=5, \|b\|=3 and \|c\|=7, then angle between a and b is
Answer» `pi/2` `pi/3` `pi/4` `pi/6` Solution :Given, `\|a\|=5, \|b\|=3, \|c\|=7 and a+b+c=0` `impliesa+b=-c` On squaring both SIDES, we get `(a+b)^(2)=(-c)^(2)IMPLIES\|a+b\|^(2)=\|c\|^(2)` `implies\|a\|^(2)+\|b\|^(2)+2a*b=\|c\|^(2)` `[because theta` be the ANGLE between a and b] `implies(5)^(2)+(3)^(2)+2\|a\|\|b\|cos theta =(7)^(2)` `implies25+9+2.5.3cos theta =49` `implies30 cos theta =15` `implies cos theta =1/2=cos 60^(@)` `impliestheta =(pi)/(3)`

Discussion

4617.	If n is an integer other than a multiple of 3, then the value of 1 + omega^(n) + omega^(2n) is
Answer» 1 `-1` 0 3 Answer :C

Discussion

4618.	If a times b=c" and "b times c=a, then
Answer» a, B, c are orthogonal in PAIRS but `ABS(a) ne abs(c)` a, b, c are orthogonal in pairs but `abs(b) ne 1` a, b, c are not orthogonal to each other in pairs a, b, c are orthogonal in pairs and `abs(a)=abs(c), abs(b)=1` Answer :D

Discussion

4619.	The sum of series Sigma_(n=1)^(oo) (2n)/(2n+1)! is
Answer» E `e^(-1)` 2E `2e^(-1)` ANSWER :B

Discussion

4620.	Find the number of non negative integral solutions of x_(1)+x_(2)+x_(3)+x_(4)le20
Answer» ANSWER :`""^(23)C_3-"^(14)C_3`

Discussion

4621.	The value of x such that 3^(2x)-2(3^(x+2))+81=0 is
Answer» `1` `2` `3` `4` ANSWER :B

Discussion

4622.	The orthogonal trajectories of the family of curves a^(n-1)y=x^(n) are give by
Answer» `X^(n)+n^(2)y=` constant `NY^(2)+x^(2)=` constant `n^(2)x+y^(n)=` constant `n^(2)x-y^(n)=` constant ANSWER :2

Discussion

4623.	If x+iy=sqrt(phi+iy), where i=sqrt(-1)" and "phi and Psi are non-zero real parameters, then phi" and "Psi are constants, represents two system of rectangular hyperbola which intersect at an angle of :
Answer» `(pi)/(6)` `(pi)/(3)` `(pi)/(4)` `(pi)/(2)` Answer :D

Discussion

4624.	If ""^9P_5+"5 "^9P_4=^(10)P_r then find r
Answer» ANSWER :5

Discussion

4625.	If A is a skew Hermitian matrix then the main diagonal elements of A are all
Answer» PURELY REAL POSITIVE NEGATIVE purely imaginary Answer :D

Discussion

4626.	If the second and fifth terms of a G.P. are 24 and 3 respectively, then the sum of first six terms is
Answer» 181 `181/2` 189 `189/2` ANSWER :D

Discussion

4627.	If the circles x^2+y^2-2lambda x-2y-7=0 and 3(x^2+y^2)-8x+29y=0 are orthogonal then lambda=
Answer» 4 3 2 1 Answer :D

Discussion

4628.	Differentiate the following w.r.t.x 2sin^-1x+tan^-1x+1
Answer» SOLUTION :`d/dx(2sin^-1x+tan^-1x+1)=2XX1/(SQRT(1-x^2))+1/(1+x^2)+0=2/(sqrt1-x^2)+1/(1+x^2)`

Discussion

4629.	If (1+x-2x^2)^8 = 1 + a_1x + a_2x^2 + ……+ a_16 x^16, then a_1 +a_3 + a_5 + ……+a_15 =
Answer» `2^7` `-2^7` `3^2` `4^6` ANSWER :B

Discussion

4630.	If z_1 & z_2 are two complex number & if arg (z_1+z_2)/(z_1-z_2)=pi/2 but \|z_1+z_2\|ne\|z_1-z_2\| thenthe figure formed by the points represented by 0, z_1, z_2 & z_1 + z_2is
Answer» a parallelogram but not a rectangle or a RHOMBUS a rectangle but not a square a rhombus but not a square a square ANSWER :C

Discussion

4631.	Find the equation of the circle whose diameter is the common chord of the circles x^2+y^2+2x+3y+1=0 and x^2+y^2+4x+3y+2=0
Answer» ANSWER :`2(x^2+y^2)+2x+6y+1=0`

Discussion

4632.	Evalute the following integrals int sqrt(1 + sec x )dx on ((2n - (1)/(2)) pi, (2n + (1)/(2)) pi, (n in z ))
Answer» Answer :`- sin^(-1) X (2 COS x - 1) + C `

Discussion

4633.	Area of the triangle with vertices (a, b ), (x_(1), y_(1)) and (x_(2), y_(2)), where a, x_(1), x_(2) are in G.P. with common ratio r and b, y_(1), y_(2) are in G.P. with common ratio s is :
Answer» `AB(r-1) (s-1) (s-r)` `(1)/(2) ab (r+1) (s+1) (s-r)` `(1)/(2) ab (r-1)(s-1) (s-r)` `ab(r+1)(s+1) (r-s)` ANSWER :C

Discussion

4634.	Let f (x,y) = (2xy)/(x ^(2)+2y ^(2)) (x,y) ne (0,0) =0 if (x,y) = (0,0) Show that f (x,y) is not continuous at (0,0) through continuous at all other points of R ^(2).
Answer» ANSWER :`THEREFORE ` F is CONTINUOUS at `(0,0)`

Discussion

4635.	Eleven apples are distributed between a girl and a boy.Then which one of the following statements is true?
Answer» ATLEAST one of them will receive 7 apples. The girl receives atleast 4 apples or the BOY receives at least 9 apples. The girl receives atleast 5 apples or the boy receives atleast 8 apples The girl receives atleast 4 apples or the boy receives atleast 8 apples. Answer :D

Discussion

4636.	Minimize and Maximize z=600x+400y Subject to the constraints : x+2y le 12 2x+y le 12 4x+5y ge 21 and xge0,y ge 0 graphical method.
Answer» ANSWER :6

Discussion

4637.	If a=sin(Cot^(-1)x)andb=cot(Sin^(-1)x) where xgt0, then 1//x^(2)-x^(2)=
Answer» `B^(2)/a^(2)` `a^(2)/b^(2)` `(a^(2)+1)/(b^(2)-1)` NONE of these Answer :A

Discussion

4638.	Evaluate the following definite integrals . int_(-1)^(1)e^(x)dx
Answer» ANSWER :`e-1/e`

Discussion

4639.	Lat l_(1)=int_(0)^(3)(sinx)/([(x)/(pi)]+(1)/(2))dx and l_(2)=int_(-3)^(0)(sinx)/([(x)/(pi)]+(1)/(2))dx, then (where[.] represent G.l.F.)
Answer» `l_(2)+l_(2)=0` `l_(2)=l_(2)` `l_(1)=3l_(2)` `l_(2)=3l_(1)` Solution :`l_(1)=underset(0)OVERSET(3)(int)(sinx)/([(x)/(pi)]+(1)/(2))DX=underset(0)overset(3)(int)(sinx)/(0+(1)/(2))dx=underset(0)overset(3)(int)2sinx dx` and `l_(2) =underset(-3)overset(0)(int)(sinx)/([(x)/(pi)]+(1)/(2))dx=underset(-3)overset(0)(int)(sinx)/(-1+(1)/(2))dx=underset(-3)overset(0)(int)-2sinx dx = underset(0)overset(3)(int)2sin x dx""thereforel_(1)=l_(2)`

Discussion

4640.	If f is a continuous and differentiable funciton in x in (0, 1) such that Sigma_(r=0)^(10)(f(x+r)-\|e^(x)-r-1\|)=0, then int_(0)^(11)f(x)dx is
Answer» `65+4ln 2 -7E` `63+4ln 2-9e` `69-9e` `29-23e` ANSWER :A

Discussion

4641.	If S-= x^(2) + y^(2) + 2gx + 2fy + c= 0repre- sents a circle then show that the straight line lx + my + n = 0 (i) touches the circle S = 0 if (g^(2)+f^(2)-c) = ((gl+mf-n)^(2))/((l^(2)+m^(2)))
Answer» ANSWER :`g^(2)+ f^(2) - C LT ((gl+mf-n)^(2))/((L^(2)+M^(2))

Discussion

4642.	Delta=\|{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}\|and Delta'\|{:(A_1,B_1,C_1),(A_2,B_2,C_2),(A_3,B_3,C_3):}\| where A_1,B_1,C_1,A_2,B_2,……… are respectively the cofactors of the elements a_1,b_1,c_1,a_2,b_2,….. of the determinant then Delta'=........
Answer» 0 `2DELTA` `DELTA^3` `Delta` ANSWER :C

Discussion

4643.	For eachof the following differential equations , determine its order , degree ( if exist ) ((d^(2)y)/(dx^(2))) + ((dy^(2))/(dx))^(2)= x sin ((d^(2)y)/(dx^(2)))
Answer» ANSWER :the DEGREE is not DEFINED

Discussion

4644.	int(dx)/((1+x)^(5)sqrt(x^2+2x))=(-1)/(8)(3theta-2sin2theta+(sin4theta)/(4))+C, where :
Answer» `theta=sin^(-1)(x+1)` `theta=sin^(-1)((1)/(1+x))` `theta=2sin^(-1)((1)/(1+x))` NONE of these Answer :B

Discussion

4645.	Iftheta is the angle between the pair of linesax^(2)+2hxy+by^(2)=0 then prove thatcostheta=(a+b)/(sqrt((a-b))+4h^(2))
Answer» ANSWER :`(a+b)/(SQRT((a+b)^(2)+4H^(2)))`

Discussion

4646.	The product of all nth roots of unity (n gt 1) is
Answer» 0 `(-1)^(n-1)` -1 1 Answer :B

Discussion

4647.	Integrate the following rational functions : int(5x)/((x+1)-(x^(2)+9))dx
Answer» ANSWER :`-(1)/(2)log\|X+1\|+(1)/(4)log\|x^(2)+9\|+(3)/(2)tan^(-1)((x)/(3))+c`

Discussion

4648.	Solution the differential equation (dy)/(dx) + 1 = e^(x+y) is
Answer» `E^(-(x+y))+y = C` `e^(x+y)+x+y = c` `e^(-(x+y))+x+y = c` `e^(-(x+y))+y + 2X = c` ANSWER :C

Discussion

4649.	If1 , omega , omega^(2)are the cube roots of unity prove that x^(2) + 4x + 7 = 0 " where " x = omega - omega^(2) - 2.
Answer»

Discussion

4650.	Let three matrices A = [(-1,3,2),(2,-1,3),(1,2,3)], B = , C = [(-2,2,-1),(3,-5,4),(5,-6,4)] The value of log_(sqrt(8/7))[Tr^(2)(A) + +Tr^(3) ((ABC)/(2))+ Tr^(3) ((A(BC)^(2))/(2))+Tr^(3) ((A(BC)^(2))/(2))+.....+ oo] is [Note : Tr. (P) denote trace of matrix P] and Tr^(3)(A) = (Tr(A))^(3)
Answer» `-4` `2` `6` `4` Solution :`BC = [(4,-2,3),(8,-3,5),(7,-2,4)][(-2,2,-1),(3,-5,4),(5,-6,4)] = I` `TR(A) = -1 - 1 + 3 = 1` `Tr((ABC)/2) = Tr(A/2) = 1/2` `Tr^(2)(A) + +Tr^(3) ((ABC)/(2))+ Tr^(3) ((A(BC)^(2))/(2))+Tr^(3) ((A(BC)^(2))/(2))+.....+ oo` `= 1^(3) + (1/2)^(3) + (1/4)^(3) + (1/8)^(3) + oo` `= 1/(1 - 1/8) = 1/(7/8) = 8/7`.

Discussion

Explore topic-wise InterviewSolutions in .

Prove that (.^(n)C_(1)sin2x+.^(n)C_(2)sin4x+.^(n)C_(3)sin6x+"…..")/(1+.^(n)C_(1)cos2x+.^(n)C_(2)cos4x+.^(n)C_(3)cos 6x+"……")

a, b, c are digits of a 3-digit number such that 64a + 8b + c = 403, then the value of a + b + c + 2013 is

Find the inverse of the following using elementary transformations. A=[[3,1],[5,2]]

Determine the truth of falsity of the"Every set has a proper subset" propositions with reasons.

Evalute the following integrals int x^(3) e^(2x) dx

Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation :y = x^2 + 2x +c:y' - 2x - 2 = 0

Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation : y = e^x + 1: y'' - y' = 0

Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation : y = sqrt(1 + x^2): y' = (xy)/(1 + x^2)

Verify that the given function (explicit or implicit) isa solution of the correseponding differential equation : y = cos x + c: y' + sin x = 0

Derivative of e^(3 log x)w.r.t.x is 3x^2.

If |z+i|^2-|z-i|^2=3then the locus of z is

The volume of a sphere is increasing at the rate of pi cm^(2)//sec. The rate at which the radius is increasing is ……….., when the radius is 3 cm.

The normal at a point P to the parabola y^(2) = 4x is parallel to the tangent at Q sqrt(2,2) to the hyperbola nd meets the axis of the parabola at If 5 is the focus of the parabola, area of the triangle PSR in sq. units is

If A^T=[[-2,3],[1,2]] and B^T=[[-1,0],[1,2]] then find (A+2B)^T

C_0-2. C_1+3 . C_2 …..+ (-1)^n (n+1). C_n=

If a+b+c=0 and |a|=5, |b|=3 and |c|=7, then angle between a and b is

If n is an integer other than a multiple of 3, then the value of 1 + omega^(n) + omega^(2n) is

If a times b=c" and "b times c=a, then

The sum of series Sigma_(n=1)^(oo) (2n)/(2n+1)! is

Find the number of non negative integral solutions of x_(1)+x_(2)+x_(3)+x_(4)le20

The value of x such that 3^(2x)-2(3^(x+2))+81=0 is

The orthogonal trajectories of the family of curves a^(n-1)y=x^(n) are give by

If x+iy=sqrt(phi+iy), where i=sqrt(-1)" and "phi and Psi are non-zero real parameters, then phi" and "Psi are constants, represents two system of rectangular hyperbola which intersect at an angle of :

If ""^9P_5+"5 "^9P_4=^(10)P_r then find r

If A is a skew Hermitian matrix then the main diagonal elements of A are all

If the second and fifth terms of a G.P. are 24 and 3 respectively, then the sum of first six terms is

If the circles x^2+y^2-2lambda x-2y-7=0 and 3(x^2+y^2)-8x+29y=0 are orthogonal then lambda=

Differentiate the following w.r.t.x 2sin^-1x+tan^-1x+1

If (1+x-2x^2)^8 = 1 + a_1x + a_2x^2 + ……+ a_16 x^16, then a_1 +a_3 + a_5 + ……+a_15 =

If z_1 & z_2 are two complex number & if arg (z_1+z_2)/(z_1-z_2)=pi/2 but |z_1+z_2|ne|z_1-z_2| thenthe figure formed by the points represented by 0, z_1, z_2 & z_1 + z_2is

Find the equation of the circle whose diameter is the common chord of the circles x^2+y^2+2x+3y+1=0 and x^2+y^2+4x+3y+2=0

Evalute the following integrals int sqrt(1 + sec x )dx on ((2n - (1)/(2)) pi, (2n + (1)/(2)) pi, (n in z ))

Area of the triangle with vertices (a, b ), (x_(1), y_(1)) and (x_(2), y_(2)), where a, x_(1), x_(2) are in G.P. with common ratio r and b, y_(1), y_(2) are in G.P. with common ratio s is :

Let f (x,y) = (2xy)/(x ^(2)+2y ^(2)) (x,y) ne (0,0) =0 if (x,y) = (0,0) Show that f (x,y) is not continuous at (0,0) through continuous at all other points of R ^(2).

Eleven apples are distributed between a girl and a boy.Then which one of the following statements is true?

Minimize and Maximize z=600x+400y Subject to the constraints : x+2y le 12 2x+y le 12 4x+5y ge 21 and xge0,y ge 0 graphical method.

If a=sin(Cot^(-1)x)andb=cot(Sin^(-1)x) where xgt0, then 1//x^(2)-x^(2)=

Evaluate the following definite integrals . int_(-1)^(1)e^(x)dx

Lat l_(1)=int_(0)^(3)(sinx)/([(x)/(pi)]+(1)/(2))dx and l_(2)=int_(-3)^(0)(sinx)/([(x)/(pi)]+(1)/(2))dx, then (where[.] represent G.l.F.)

If f is a continuous and differentiable funciton in x in (0, 1) such that Sigma_(r=0)^(10)(f(x+r)-|e^(x)-r-1|)=0, then int_(0)^(11)f(x)dx is

If S-= x^(2) + y^(2) + 2gx + 2fy + c= 0repre- sents a circle then show that the straight line lx + my + n = 0 (i) touches the circle S = 0 if (g^(2)+f^(2)-c) = ((gl+mf-n)^(2))/((l^(2)+m^(2)))

Delta=|{:(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3):}|and Delta'|{:(A_1,B_1,C_1),(A_2,B_2,C_2),(A_3,B_3,C_3):}| where A_1,B_1,C_1,A_2,B_2,……… are respectively the cofactors of the elements a_1,b_1,c_1,a_2,b_2,….. of the determinant then Delta'=........

For eachof the following differential equations , determine its order , degree ( if exist ) ((d^(2)y)/(dx^(2))) + ((dy^(2))/(dx))^(2)= x sin ((d^(2)y)/(dx^(2)))

int(dx)/((1+x)^(5)sqrt(x^2+2x))=(-1)/(8)(3theta-2sin2theta+(sin4theta)/(4))+C, where :

Iftheta is the angle between the pair of linesax^(2)+2hxy+by^(2)=0 then prove thatcostheta=(a+b)/(sqrt((a-b))+4h^(2))

The product of all nth roots of unity (n gt 1) is

Integrate the following rational functions : int(5x)/((x+1)-(x^(2)+9))dx

Solution the differential equation (dy)/(dx) + 1 = e^(x+y) is

If1 , omega , omega^(2)are the cube roots of unity prove that x^(2) + 4x + 7 = 0 " where " x = omega - omega^(2) - 2.

Let three matrices A = [(-1,3,2),(2,-1,3),(1,2,3)], B = , C = [(-2,2,-1),(3,-5,4),(5,-6,4)] The value of log_(sqrt(8/7))[Tr^(2)(A) + +Tr^(3) ((ABC)/(2))+ Tr^(3) ((A(BC)^(2))/(2))+Tr^(3) ((A(BC)^(2))/(2))+.....+ oo] is [Note : Tr. (P) denote trace of matrix P] and Tr^(3)(A) = (Tr(A))^(3)