28278 + Interview Questions in Maths in Class 12 Page 235 InterviewSolution

11701.	If a function is invertible the graph of its inverse is the mirror image of the function in y=x. If f^(-1)(x) is the inverse function of f(x), then f(f^(-1)(x))=f^(-1)(f(x))=x. If f(x)=ln[((100+x)e^(x^(3)))/((100-x))], then the graph of f^(-1)(x) will be symmetrical about x+y=1
Answer» `x+y=1` `y=x+1` `y=-x` can not say Answer :C

Discussion

11702.	Show that the function f defined by f(x)={(x if "x is rational"),(-x if "x is irrational"):} is continuous at x=0 AAxne0inR
Answer» Solution :f(0)=0 L.H.L=`lim_(xto0)f(x)=lim_(hto0)f(-h)` `=lim_(hto0){(-h if "h is RATIONAL" =0),(h if "h is irrational"):}` Similarly R.H.L.=0 THUS L.H.L.=R.H.L.=f(0) Hence f(x) is continuous at x=0 We can easily show that f(x) is discontinuous at all real POINTS `xne0`

Discussion

11703.	A box contain 6 tickets. Two of the tickets carry a price of Rs 5/- each and the other 4 are the price of Rs 1 each. If one ticket is drawn at random, what is the mean price.
Answer» ANSWER :`7//3`

Discussion

11704.	A : The value of (Tan^(-1)(4//3))/(Tan^(-1)(1//2)) is equal to 2. R : AAx in[0,1],Tan^(-1)((2x)/(1-x^(2)))=2Tan^(-1)x
Answer» Both A and R are TRUE and R is the CORRECT EXPLANATION of A Both A and R are true but R is not correct explanation of A A is true but R is FALSE A is false but R is true Answer :A

Discussion

11705.	A box has 100 pens of which 10 are defective. The probability that out of a sample of 5 pens drawn one by one with replacement and atmost one is defective is
Answer» `a.(9)/(10)` b. `(1)/(2)((9)/(10))^(4)` c.`((9)/(10))^(5)+(1)/(2)((9)/(10))^(4)` d.`(1)/(2)((9)/(10))^(5)` ANSWER :C

Discussion

11706.	The radius of the director circle of the hyperbola x^2//25-y^(2)//9=1 is
Answer» 3 4 5 8 Answer :B

Discussion

11707.	Find the roots of the following cubic equations 2x^(3)-3x^(2) cos (A-B)-2 x cos^(2) (A+B)+ sin 2A sin 2 B cos (A-B)=0 .
Answer» ANSWER :`2 COS A cos B, 2 SIN A sin B "and" -(1)/(2) cos (A-B)`.

Discussion

11708.	If A={a,b,c,d} mention the type of relations on A given below, which of them are equivalence relations?{(b,c),(b,d),(c,d)}
Answer» SOLUTION :Only TRANSITIVE

Discussion

11709.	The radical axis of two circles whose centres are ( 3,4), (-1,2) and each passing through the centre of the other is
Answer» ` 2X+ y +5=0 ` ` x-2y +5=0` ` 2x+ y-1=0 ` ` 2x+ y-5=0` ANSWER :D

Discussion

11710.	Evaluate the following integrals int(logx^(2))dx
Answer» ANSWER :`X((LOGX)^(2)-2logx+2)+C`

Discussion

11711.	When the coordinate axes ar rotated about the origin in the positive direction through an angle pi/4, IF the equation 25x^2+9y^2=225 is transformed to ax^2+betaxy+ygamma^2=delta, then (alpha+beta+gamma-sqrtdelta)^2=
Answer» 3 9 4 16 Answer :B

Discussion

11712.	Let ABCD is a rectangle with AB=a & BC=b & circle is drawn passing through A & B and touching side CD. Another circle is drawn passing through B & C and touching side AD. Let r_(1) & r_(2) be the radii of these two circle respectively. Minimum value of (r_(1)+r_(2) equals
Answer» `5/8 (a-b)` `5/8 (a+b)` `3/8 (a-b)` `3/8 (a+b)` Solution :`r_(1)=b-x_(1)=OP=OA` `AP_(1)=a//2` `r_(1)^(2)=x_(1)^(2)+(a/2)^(2)=(b-x_(1))^(2)` `x_(1)^(2)+(a^(2))/4=b^(2)+x_(1)^(2)-2bx_(1)` `x_(1)=(4b^(2)-a^(2))/(8b)` `r_(1)=b-x_(1)=(4b^(2)+a^(2))/(8b)` SIMILARLY `r_(2)=(4A^(2)+b^(2))/(8a)` `r_(1)+r_(2)=(a^(3)+b^(3)+4ab(a+b))/(8AB)` `implies((a+b)(a^(2)+3ab+b^(2)))/(8ab)` `=((a+b)/8)([(a-b)^(2)+5ab])/(AB)` But `(a-b)^(2)ge0` `r_(1)+r_(2)GE((a+b))/8.(5ab)/(ab)` `impliesr_(1)+r_(2)ge(5(a+b))/8`

Discussion

11713.	A bag contains 10 white and 15 black balls. The balls are drawn one at a time until only those of the same colour are left. Show that the probability that they are all black is 3/5.
Answer» ANSWER :`(3)/(5)`

Discussion

11714.	Find mean of following probability distribution.
Answer» ANSWER :2.75

Discussion

11715.	The solution of (x+y)^(2)(dy)/(dx) = a^(2) is
Answer» `y = TAN^(-1)((X+y)/(a)) + C` `a y = tan^(-1)(x+y) + c` `y = a tan^(-1)((x+y)/(a)) + c` `x = a tan^(-1)((x+y)/(a)) + c` ANSWER :C

Discussion

11716.	Let A=[{:(1,-2,1),(-2,3,1),(1,1,5):}] Verify that (A^(-1))^-1=A
Answer» ANSWER :A

Discussion

11717.	Statement:1 In triangleABC, if a lt b sinA, then the triangle is possible. And Statement:2 In triangleABC a/(sinA)= b/(sinB)
Answer» Statement-1 is TRUE, Statement-2 is true, Statement-2 is a CORRECT EXPLANATION for statement-16 Statement-1 is true, Statement-2 is true, Statement-2 is not a correct expanation for statement-16 Statement-1 is True, Statement-2 is FALSE Statement-1 is False, Statement-2 is true ANSWER :D

Discussion

11718.	The volume of the tetrahedron formed by the coterminous edges veca, vecb, vec c is 3. Then the volume of the parallelopiped formed by the coterminous edges veca+vecb, vecb+vec c, vec c+ veca is
Answer» 36 34 32 30 Solution :`(1)/(6)[VECA vecb vec C]=3` (GIVEN) `[veca vecb vec c]=18` `"VOL. of parallelopiped"=[veca+vecb""vecb+vec c""vec c + a]` `=2[veca vecb vec c]=2xx18=36`

Discussion

11719.	Let A(2sec theta, 3 tan theta ) and B( 2sec phi ,3 tan phi ) where theta + phi =(pi)/(2)be two point on the hyperbola(x^(2))/( 4) -( y^(2))/( 9) =1 . If (alpha , beta )is the point of intersection of normals to the hyperbola at A and B ,then beta =
Answer» `- (13)/(3)` `(13)/(3)` `(3)/(13)` `- (3)/(13)` ANSWER :a

Discussion

11720.	Let the smallest positive value of x for which the functionf(x)=sin""(x)/(3)+sin""(x)/(11), ( x in R ) achieves its maximum value bex_(0). Express x_(0) in degree i.e.x_(0)=alpha^(0). Then , the sum of the digits inalphais
Answer» 15 17 16 18 Answer :D

Discussion

11721.	If1 , omega , omega^(2) are the cube roots of unity , then find the values of the following . (a+ b)^(3) + ( a omega + bomega ^(2))^(3) + ( a omega^(2) + b omega)^(3)
Answer» ANSWER :`3(a^3+b^3)`

Discussion

11722.	int(x^(2))/(sqrt(1+x^(2))(1+sqrt(1+x^(2))))dx
Answer» ANSWER :`x-sinh^(-1)x+c`

Discussion

11723.	If the area of the parallelogram whose adjacent sides are (3i + 4j + lambda K) and (2j - 4k) is sqrt(436) square units, and lambda ge 0, then lambda =
Answer» 0 4 1 3 Answer :A

Discussion

11724.	Find the equation of a curve passing through the point (-2,3), given that the slope of the tangent to the curve at any point (x,y) is (2x)/(y^(2)).
Answer» ANSWER :`y = (3X^(2) + 15)^(1/3)`

Discussion

11725.	A point is taken at random from inside of the circumcircle of an equilateral triangle. The probability that it lies inside the circumcircle but outside the incircle is
Answer» `1//4` `3//4` `1//2` `1//3` ANSWER :B

Discussion

11726.	Let A=NxxN Define * on A by (a,b)(c,d)=(a+c,b+d) Show that (i) A is closed for (ii) * is commutative (iii) * is associative (iv) identify element does not exist in A
Answer» Solution :(i) Let (a,b) in A and (c,d) inA Then a,b,c d in N `(a,b)(c,d)=(a+c,b+d)in A`[before a+c in N, b +d `in` N] `therefore` A is CLOSED for (a,b)(c,d)=(a+c,b+d) `=(c+a,d+b) ["before" a+c=c+a and b+d=d+b]` =(c,d)(a,b) (III)(a,b)(c,d)(e,F)=(a+c,b+d)(e,f) =[(a+c)+e,(b+d)+f] =([a+(c+e),b+(d+f)] =(a,b)[(c,de,d+f) `=(a,b)[(c,d)(e,f)]` (IV) (a,b)*(0,0)=(a+0,b+0)=(a,b) But ,(0,0)`ne` A since one N So identity element does not BELONG to A

Discussion

11727.	Let I =int _(0)^(1 ) sqrt((1+sqrtx)/(1-sqrtx))dx and J = int _(0)^(1 ) sqrt((1-sqrtx)/(1+sqrtx))dxthen correct statement (s) is/are:
Answer» `I+J=2` `I -J =pi` `I =(2+pi)/(2)` `J =(4-pi)/(2)` Answer :B::C

Discussion

11728.	A body is thrown horizontally with a velocity of v m/s from the top of a tower of height 2h reaches the ground in 't' seconds. If another body double the mass is thrown horizontally with a velocity 5v m/s from the top of another tower of height 8h. In the above problem if the first reaches the ground at a horizontal distance 'x' the second body reaches the ground at a horizontal distance
Answer» 1 2 3 `1.5` ANSWER :B

Discussion

11729.	IfDelta = {:[( a_11, a_12, a_13),( a_21,a_22,a_23) ,(a_31,a_32, a_33) ]:} andis Cofactors of a_ijthen value of Deltais given by
Answer» ` a_11 A_31+ a_12A_32+a_13A_33` ` a_11A_11+a_12A_21+a_13A_31` ` a_21A_11+a_22A_12+a_23A_13` `a_11A_11+a_21A_21+a_31A_31` ANSWER :D

Discussion

11730.	Let k denote the number of ways in n boys sit in a row such that three particular boys are repeated. Then
Answer» 3! Divides K `(n-2) !` Divide k `.^(n-2)C_(3)` divides k `(n-3)^(2)(n-4)^(2)` divides k Answer :a,b,C,d

Discussion

11731.	Evaluate the following integrals : int_(0)^(pi/2)(dx)/(4sin^(2)x+5cos^(2)x)
Answer» ANSWER :`(PI)/(4sqrt5)`

Discussion

11732.	Show that \|{:(a,b,c),(a^(2),b^(2),c^(2)),(a^(2),b^(3),c^(3)):}\|=abc(a-b)(b-c)(c-a)
Answer» ANSWER :`(ABC)(a-b)(b-c)(c-a)=R.H.S`

Discussion

11733.	A straight line through the point (2, 2) intersects the lines sqrt3 x + y = 0 and sqrt3 x - y = 0 at the points A and B. The equation of AB so that the triangle OAB is equilateral, where O is the origin.
Answer» X - 2 = 0 y - 2 = 0 x + y - 4 = 0 x+y+z = 0 Answer :B

Discussion

11734.	findthe areaboundedby thecurvey=2 cos xandtheX- axisfromx=0tox = 2 pi
Answer» ANSWER :`=8 ` SEQ . UNIT

Discussion

11735.	The number of values of x in [0,4pi] satisfying \|sqrt(3cosx - sinx )\| gt=2 is :
Answer» 2 0 4 8 Answer :C

Discussion

11736.	Two adjacent sides of a parallelogram are 4x+5y=0,and 7x+2y=0. Area of the parallelogram is
Answer» 3 `3/2` 6 `(2sqrt85)/(3)` ANSWER :A

Discussion

11737.	If y=f(x) is the solution of differential equation , e^y((dy)/(dx)-2)=e^(3x) such that f(0)=0 , then f(2) is equal to :
Answer» 3 5 6 None of these Solution :Put `e^y =t rArr e^y DY= dtrArr (dt)/(dx)-2t=e^(3X)` I.F. =`e^(INT -2dx) =e^(-2x)` `t.e^(-2x) = int e^(3x). E^(-2x) dx` `t.e^(-2x) = inte^X dx= e^x +c , e^y e^(-2x) =e^x + c` Put x=0 ,y=0 we get `e^0 .e^0 =1+c` `rArr e^y e^(-2x)= e^x` `e^y =e^(3x) rArr y=3x rArr f(x)=3x`

Discussion

11738.	If tanA+sinA=p and tanA-sinA=q, then the value of ((p^(2)-q^(2))^(2))/(pq) is :
Answer» 16 22 18 42 Answer :A

Discussion

11739.	A company manufactures the bicycle with gears and without gears in two different factories. The factory A produces 16 bicycles without gears and 20 bicycles with gears in a day. The factory B produces 12 bicycles without gears and 20 bicycles with gears in a day. The expenditure of factory A is Rs. 50,000 daily and the expenditure of factory B is Rs. 40,000 daily. A company has ordered for 96 bicycles without gears and 140 bicycles with gears. To complete these order, how many days the factories will work so that there is minimum expenditure ?
Answer» ANSWER :FACTORY A for 3 DAYS and Factory B for 4 days.

Discussion

11740.	Which one of the following statements is not equiva- lent to p rarr( q vee r ) ?
Answer» `( p rarr Q ) VEE ( p rarr R ) ` `P ^^ ( ~ q ) rarr r ` `p ^^ ( ~ r) rarr q ` `p ^^ q rarr ( p ^^ r) vee ( q ^^ r)` Answer :D

Discussion

11741.	(1^(3))/(1!)+(2^(3))/(2!)+(3^(3))/(3!)+(4^(3))/(4!)+....=
Answer» e 3e 5e `e//5` ANSWER :C

Discussion

11742.	If joint equation of two lines through the origin, each making an angle theta with the line x+y=0 is x^(2)+2hxy+y^(2)=0 then h=
Answer» `SEC 2 theta` `-sec 2 theta` `TAN 2 theta` `-TAN2 theta` Answer :A

Discussion

11743.	If alpha, beta, gamma in {1,omega,omega^(2)} (where omega and omega^(2) are imaginery cube roots of unity), then number of triplets (alpha,beta,gamma) such that \|(a alpha+b beta+c gamma)/(a beta+b gamma+c alpha)\|=1 is
Answer» `3` `6` `9` `12` Solution :`(C )` As `\|(aalpha+bbeta+cgamma)/(a beta+bgamma+calpha)\|=1` `implies` When `ALPHA`, `beta`, `gamma` are different, then number of triplet `(alpha,beta,gamma)=` permutation of `1`, `omega` and `omega^(2)=6` and when `alpha-beta=gamma`, number of TRIPLETS `=3`.

Discussion

11744.	Evalute the following integrals int tan (x - theta) tan x (x + theta)tan 2x dx
Answer» Answer :`(1)/(2)` log \|sec 2X \| - log \| sec `(X + theta)` \| - log \| sec `(x - theta)` \| + C

Discussion

11745.	Consider the system of equations x cos^(3) y+3x cos y sin^(2) y=14 x sin^(3) y+3x cos^(2) y sin y=13 The number of values of y in [0, 6pi] is
Answer» 5 3 4 6 Solution :The given equations are `x cos^(3)y+3x cos y sin^(2) y=14` ...(i) and `x sin^(2) y+3x cos^(2) y sin y=13` ...(ii) Adding EQS. (i) and (ii), we have `x(cos^(3) y+3 cos y sin^(2) y+3 cos^(2) y sin y+ sin^(3) y)=27` or `x(cos y+ sin y)^(3)=27` or `x^(1//3) (cos y + sin y) =3` ...(iii) Subtracting Eq. (ii) from Eq. (i), we have `x(cos^(3)y+3 cos y sin^(2) y-3 cos^(2) y sin y- sin^(3) y)=1` or `x(cos y- sin y)^(3)=1` or `x^(1//3) (cos y- sin y)=1` ...(iv) Dividing Eq. (iii) by (iv), we get `cos y+sin y=3 cos y-3 sin y` or `tan y=1//2` Case I : `sin y=1//SQRT(5) and cos y =2//sqrt(5)` `y=2n pi +alpha`, where `0 lt alpha lt pi//2` and `sin alpha =1//sqrt(5)` i.e., y lies in the first quadrant From Eqs. (iii) `x^(1//3) (3//sqrt(5))=3 or x=5 sqrt(5)` Case II : `sin y=-1//sqrt(5) and cos y=-2//sqrt(5)` `y=2npi+(pi+alpha)`, where `0 lt alpha lt pi//2` and `sin alpha = -1 //sqrt(5)` i.e., y lies in the third quadrant. Therefore, from Eq. (iii), `x^(1//3) (-3//sqrt(5))=3 or x=-5sqrt(5)`. Thus, `sin^(2) y+2 cos^(@) y=1//5+8//5=9//5`. ALSO there are exactly six VALUES of `y in [0, 6pi]`, there in 1ST quadrant and three in 3rd quadrant.

Discussion

11746.	Evaluate the following integral int (cos 7x - cos 8x )/(1 + 2 cos 5x) dx
Answer» Answer :`(sin 2X)/(2) - (sin 3x)/(3) + C `

Discussion

11747.	Solve x^4-5x^3+5x^2+5x-6 =0 given that the product of two of its roots is 3
Answer» ANSWER :`+- 1,2,3`

Discussion

11748.	(1+3)log_e3+(1+3^2)/(2!) (log_e3)^2+(1+3^3)/(3!) (log_e3)^3+.....oo =
Answer» 27 28 32 57 Answer :B

Discussion

11749.	Integrate : int (logsqrt(x))^(2)dx
Answer» Answer :`(X)/(4)[(LOGX)^(2)-2logx+2]+C`

Discussion

11750.	if A_(1) ,B_(1),C_(1) ……. arerespectivelythecofactorsof theelementsa_(1) ,b_(1),c_(1)…… ofthe determinant Delta = \|{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}\|, Delta ne 0then the value of\|{:(B_(2),,C_(2)),(B_(3),,C_(3)):}\| is equal to
Answer» `a_(1)^(2)Delta` `a_(1)Delta` `a_(1)Delta^(2)` `a_(1)^(2)Delta^(2)` SOLUTION :`B_(2) =a_(1)c_(3) -a_(3)c_(1),C_(2)=-(a_(1)b_(3)-a_(3)b_(1))` `B_(3) =-(a_(1)C_(2) -a_(2)c_(1)) ,C_(3)=a_(1)b_(2)-a_(2)b_(1)` ` :. \|{:(B_(2),,C_(2)),(B_(3),,C_(2)):}\|= \|{:(a_(1)C_(3)-a_(3)c_(1),,-a_(1)b_(3)+a_(3)b_(1)),(-a_(1)c_(2)+a_(2)c_(1),,a_(1)b_(1)-a_(2)b_(1)):}\|` `=\|{:(a_(1)c_(3),,-a_(1)b_(3)),(-a_(1)c_(2),,a_(1)b_(2)):}\|+ \|{:(a_(1)C_(3),,a_(3)b_(1)),(-a_(1)c_(2),,-a_(2)b_(1)):}\|` `+\|{:(-a_(3)C_(1),,-a_(1)b_(3)),(-a_(1)C_(2),,a_(1)b_(2)):}\|+ \|{:(-a_(3)C_(1),,a_(3)b_(1)),(a_(2)c_(1),,-a_(2)b_(1)):}\|` ` =a_(1)^(2) \|{:(C_(3),,-b_(3)),(-c_(2),,b_(2)):}\|+a_(1)b_(1) \|{:(c_(3),,a_(3)),(-c_(2),,-a_(2)):}\|` `+a_(1)c_(1) \|{:(-a_(3),,-b_(3)),(a_(2),,b_(2)):}\|+b_(1)c_(1) \|{:(-a_(3),,a_(3)),(a_(2),,-a_(2)):}\|` `=a_(1){a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))` `+c_(1)(a_(2)b_(3)-a_(3)b_(2))}` ` =a_(1)\|{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}\| =a_(1) Delta`

Discussion

Explore topic-wise InterviewSolutions in .

If a function is invertible the graph of its inverse is the mirror image of the function in y=x. If f^(-1)(x) is the inverse function of f(x), then f(f^(-1)(x))=f^(-1)(f(x))=x. If f(x)=ln[((100+x)e^(x^(3)))/((100-x))], then the graph of f^(-1)(x) will be symmetrical about x+y=1

Show that the function f defined by f(x)={(x if "x is rational"),(-x if "x is irrational"):} is continuous at x=0 AAxne0inR

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Let ABCD is a rectangle with AB=a & BC=b & circle is drawn passing through A & B and touching side CD. Another circle is drawn passing through B & C and touching side AD. Let r_(1) & r_(2) be the radii of these two circle respectively. Minimum value of (r_(1)+r_(2) equals

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Find mean of following probability distribution.

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The volume of the tetrahedron formed by the coterminous edges veca, vecb, vec c is 3. Then the volume of the parallelopiped formed by the coterminous edges veca+vecb, vecb+vec c, vec c+ veca is

Let A(2sec theta, 3 tan theta ) and B( 2sec phi ,3 tan phi ) where theta + phi =(pi)/(2)be two point on the hyperbola(x^(2))/( 4) -( y^(2))/( 9) =1 . If (alpha , beta )is the point of intersection of normals to the hyperbola at A and B ,then beta =

Let the smallest positive value of x for which the functionf(x)=sin""(x)/(3)+sin""(x)/(11), ( x in R ) achieves its maximum value bex_(0). Express x_(0) in degree i.e.x_(0)=alpha^(0). Then , the sum of the digits inalphais

If1 , omega , omega^(2) are the cube roots of unity , then find the values of the following . (a+ b)^(3) + ( a omega + bomega ^(2))^(3) + ( a omega^(2) + b omega)^(3)

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Find the equation of a curve passing through the point (-2,3), given that the slope of the tangent to the curve at any point (x,y) is (2x)/(y^(2)).

A point is taken at random from inside of the circumcircle of an equilateral triangle. The probability that it lies inside the circumcircle but outside the incircle is

Let A=NxxN Define * on A by (a,b)*(c,d)=(a+c,b+d) Show that (i) A is closed for * (ii) * is commutative (iii) * is associative (iv) identify element does not exist in A

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findthe areaboundedby thecurvey=2 cos xandtheX- axisfromx=0tox = 2 pi

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If y=f(x) is the solution of differential equation , e^y((dy)/(dx)-2)=e^(3x) such that f(0)=0 , then f(2) is equal to :

If tanA+sinA=p and tanA-sinA=q, then the value of ((p^(2)-q^(2))^(2))/(pq) is :

Which one of the following statements is not equiva- lent to p rarr( q vee r ) ?

(1^(3))/(1!)+(2^(3))/(2!)+(3^(3))/(3!)+(4^(3))/(4!)+....=

If joint equation of two lines through the origin, each making an angle theta with the line x+y=0 is x^(2)+2hxy+y^(2)=0 then h=

If alpha, beta, gamma in {1,omega,omega^(2)} (where omega and omega^(2) are imaginery cube roots of unity), then number of triplets (alpha,beta,gamma) such that |(a alpha+b beta+c gamma)/(a beta+b gamma+c alpha)|=1 is

Evalute the following integrals int tan (x - theta) tan x (x + theta)tan 2x dx

Consider the system of equations x cos^(3) y+3x cos y sin^(2) y=14 x sin^(3) y+3x cos^(2) y sin y=13 The number of values of y in [0, 6pi] is

Evaluate the following integral int (cos 7x - cos 8x )/(1 + 2 cos 5x) dx

Solve x^4-5x^3+5x^2+5x-6 =0 given that the product of two of its roots is 3

(1+3)log_e3+(1+3^2)/(2!) (log_e3)^2+(1+3^3)/(3!) (log_e3)^3+.....oo =

Integrate : int (logsqrt(x))^(2)dx

if A_(1) ,B_(1),C_(1) ……. arerespectivelythecofactorsof theelementsa_(1) ,b_(1),c_(1)…… ofthe determinant Delta = |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|, Delta ne 0then the value of|{:(B_(2),,C_(2)),(B_(3),,C_(3)):}| is equal to

Let A=NxxN Define * on A by (a,b)(c,d)=(a+c,b+d) Show that (i) A is closed for (ii) * is commutative (iii) * is associative (iv) identify element does not exist in A

A straight line through the point (2, 2) intersects the lines sqrt3 x + y = 0 and sqrt3 x - y = 0 at the points A and B. The equation of AB so that the triangle OAB is equilateral, where O is the origin.