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

551.	Which of the following function f:R rarr R are bijective
Answer» `F(x)=x SIN x` `f(x)=x-sin^(2)x` `f(x)=x +sqrt(x^(2))` `f(x)=-x+cos^(2)x` ANSWER :B,D

Discussion

552.	Letthe couble ordinate pp' of the hyperbola(x^(2))/(4)-(y^(2))/(3)=1 is producedboth sides to meetasymptotes of hyperbola in Q and Q'. The product (PQ)(PQ)' is equal to :
Answer» 3 4 1 5 Answer :A

Discussion

553.	Find the point R in which the line AB cuts the plane CDE, where position vectors of points A, B, C, D, E are respectively . vec(a) = hat(i) + 2hat(j) + hat(k), vec(b) = 2hat(i)+ hat(j) + 2hat(k) , vec(c) = - 4hat(j) + 4hat(j) , vec(d) = 2hat(i) - 2hat(j)+ 2hat(k) and vec(e) = 4hat(i) + hat(j) + 2hat(k) .
Answer» ANSWER :`((19)/(8),(11)/(8),(19)/(8))`

Discussion

554.	Let a, b, c be real number (a ne 0). If alpha is a root of a^(2)x^(2)+bx +c=0, beta is a root of a^(2)x^(2) -bx-c=0 and 0 lt alpha beta, then the root of the equation (say gamma) a^(2) x^(2)+2bx+2c=0 always satisfies
Answer» `gamma=(ALPHA+beta)/(2)` `F(gamma)=0` `gamma=alpha` `alpha LT gamma lt beta` ANSWER :b,d

Discussion

555.	Write minors and cofactors of the elements of following determinant \|{:(5,-10),(0,3):}\|
Answer» Answer :`M_(11)=3,M_(12)=0,M_(21)=-10,M_(22)=5` `A_(11)=3,A_(12)=0,A_(21)=10,A_(22)=5`

Discussion

556.	If Delta={:\|(1,2,3),(2,0,1),(5,3,8)\| Write the minor of the element a_(22).
Answer» ANSWER :-7

Discussion

557.	Assertion (A) : The least length of the focal chord ofy^(2) =4axis 4a Reason (R) : Length of the focal chord of y^(2) =4axmakes an angle thetawith axis is 4a cosec ^(2) theta
Answer» Both A and R are true and R `rArr A ` Both A and R are true and` R CANCEL rArr A ` A is true but R is false A is false but R is true. ANSWER :A

Discussion

558.	Let a curve y=f(x) pass through (1,1), at any point p on the curve tangent and normal are drawn to intersect the X-axis at Q and R respectively. If QR = 2 , find the equation of all such possible curves.
Answer» ANSWER :`logy-x=pm(LOG-\|(1sqrt(1-y^(2)))/(y)\|+SQRT(1-y^(2)))`

Discussion

559.	If A^(2)-A+I=O," then "A^(-1)=
Answer» `I-A` `A-I` A `A+I` ANSWER :A

Discussion

560.	The maximum value of Z=2x+y subject to 3x+5yle26 and 5x+3yle30,xge0,yge0 is
Answer» 12 11.5 10 17.33 Solution :We have, Maximize Z=2x+y SUBJECT to CONSTRAINTS 3x+5y`le26` `5x+3yle30` `xge0,YGE0` the graph off INEQUALITY are Maximum value of z=12.

Discussion

561.	A manufacturer produces two models A and B of a product. Each piece of model A requires9 labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 Labour hours for fabricatingand finishing the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs 8000 on each piece of model A and Rs 12000 on each piece of model B . Formulate an L.P.P. So as to maximize his profit.
Answer» ANSWER :`XGE0,YGE0`

Discussion

562.	r(f) nn R (g)=
Answer» `[-2,(1)/(2)]` `[-(1)/(2),(1)/(2)]-{0}` `[-1,0]` NONE of these ANSWER :b

Discussion

563.	A fair coin and an unbiased die are tossed. Let A be the event 'head appears on the coin' and B be the event '3 on the die' . Check whether A and Bare independent events or not.
Answer» ANSWER :A and B are INDEPENDENT

Discussion

564.	[( p rarr q) ^^ ~ r ] vee r vee ( p ^^ ~q ^^- r )is equivalent to
Answer» `(p rarr q ) vee r ` `~ ( p rarr q ) vee r ` `~ ( q rarr p) vee r ` `(q rarr p ) vee r ` ANSWER :C

Discussion

565.	Let f be a function continuous on R, then int_(-pi//2)^(pi//2) (x + sin x) {f(x) + f(-x)}dx is :
Answer» `0` `PI + 2` `pi` NONE of these ANSWER :A

Discussion

566.	The variance of observation x_(1), x_(2),x_(3),…,x_(n) is sigma^(2) then the variance of alpha x_(1), alpha x_(2), alpha x_(3),….,alpha x_(n), (alpha != 0) is
Answer» `SIGMA^(2)` `ALPHA sigma^(2)` `alpha^(2) sigma^(2)` `(sigma^(2))/(alpha^(2))` ANSWER :C

Discussion

567.	The middle term in the expansion of (10/x + x/10)^(10) is
Answer» `""^(10)C_(5)` `""^(10)C_(6)` `""^(10)C_(5) 1/x^(10)` `""^(10)C_(5)x^(10)` Answer :D

Discussion

568.	Foreachn inN, 2^(3n )-1is divisibleby
Answer» 7 8 6 16 Answer :A

Discussion

569.	The coordinates of a point, which lies on y-axis and is at a distance of 4 units above x-axis is ______.
Answer» `(0,-4)` `(4,4)` `(4,0)` `(0,4)` ANSWER :D

Discussion

570.	int_(0)^(pi//2) ( f ( co x )) /( f (sinx ) + f(co sx) ) dx is equal to
Answer» 0 `(pi)/(2)` `(pi)/( 4)` NONE of these Answer :C

Discussion

571.	Let f:[a,b] to R be a function such that , for c in (a,b), f.(c ) = f..( c) = f...( c) = f.... (c ) = f.....( c) = 0 . Then :
Answer» f has local extremum at x = C f has NEITHER local maximum nor local minimum at x = c f is necessarily a constant function It is difficult to say whether (A) or (B) Solution :`f.(C) = f..(C) = f...(C) = f^(iv)(C) = f^(v) (C) = 0` Now if n is the least positive integer such that `f^n (C) NE 0`, then it is not clear whether n is even or ODD. So nothing can be said whether `f(x)` has local EXTREMA at x= c or not .

Discussion

572.	State which of the followingstatementis true?
Answer» the constraints in a linear programming problem does not karise dueto limitation of RESOURCES the objective function of an LPP MAY be linear or nonlinear in DECISIONS varibales if one can get two optimal solution then one can obtain an infintie number of optimal solution the constraints of a linear programming problem are not linearequation or inequalites arising out of practicallimitations ANSWER :C

Discussion

573.	If alpha, beta , gamma are the roots of x^(3) + 3x + 2 = 0then the equation whose roots (beta - gamma)^(2), (gamma - alpha)^(2), (alpha - beta)^(2)is
Answer» `y^(3) - 29y^(2) - 50y + 625 = 0 ` `y^(3) + 18y^(2) + 81y + 216 = 0 ` `y^(4) - 24y^(2) + 65y - 55 = 0 ` `5y^(5) - 12y^(2) + 68y - 155 = 0 ` Answer :2

Discussion

574.	The locus of the point of intersection of two tangents to the parabola y^(2)=4ax which make the angles theta_(1) and theta_(2) with the axis so that tan theta_(1) tan theta_(2) =k is
Answer» KX-y=0 kx-a=0 y=ka=0 `kx^(2)+2ax-y^(2)=0` ANSWER :D

Discussion

575.	If f(x) = (2x)/(4 + 3\|x\|), x in R, then f'(0) =
Answer» A.0 B.`(1)/(4)` C.`(1)/(2)` D.`(3)/(4)` ANSWER :C

Discussion

576.	Ore overset("Calcination")(rarr)Residue overset("dil".HNO_(3))(rarr)underset("Metal(M)+Zn^(2+)(aq.))underset(darr zn dust)(aq. solution) above metallurgy is possible when ore is :-
Answer» `ZnCO_(3)` `CaCO_(3). MgCO_(3)` `CuCO_(3).Cu(OH_(2))` PbS Answer :C

Discussion

577.	A die is thrown. E is the event "the number appearing us a multiple of 3" and F be the event "the number appearing is even". Find whether E and F are independent.
Answer» ANSWER :E and F are INDEPENDENT EVENTS.

Discussion

578.	The probability distribution of x is
Answer» `0.14` `0.3` `0.7` 1 Answer :A

Discussion

579.	If (4,0) and (1, -1) are two vertices of a triangle of area 4 square units, then its third vertex lies on
Answer» X = y 5x + 4 + 12 = 0 x + 5Y = 4 5x – y + 5 = 0 Answer :C

Discussion

580.	Find (dy)/(dx) given x^(y) + y^(x) = 1
Answer» ANSWER :`-(YX^(y-1)+y^(X)LOG y)/(x^(y)log x+xy^(x-1))`

Discussion

581.	Which of the following relations are both odd and even? I. x^(2)+y^(2)=1 II. x^(2)-y^(2)=0 III. x+y=0
Answer» only III only I and II only I and III only II and III Solution :The ANALYSIS of RELATION III in the above examples INDICATES that I and II are both even and ODD. Since -x+y`NE0` when x+y=0 unless x=0, III is not even, and is therefore not both even and odd.

Discussion

582.	The manufacturing cost of an item consists of rupes 100 as overheads , material cost rupes 2 per item and the labour cost( x^(2))/(90)for x items , produced . Find the average cost.
Answer» ANSWER :` AC = ( 100)/(x) +(x)/(90)+2`

Discussion

583.	Find the area of the parallelogram whose adjacent sides are determined by the vectors vec(a)=hati-hatj+3hatk and vec(b)=2hati-7hatj+hatk.
Answer» ANSWER :`=\|VEC(a)xx vec(b)\|=15sqrt(2)` SQ. unit

Discussion

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

Discussion

585.	If alpha , beta , gamma are the roots of x^(3) + 3x + 2 =0 then the equation whose roots alpha (beta + gamma), beta( gamma + alpha), gamma(alpha + beta) is
Answer» `x^(3) + 6X^(2) + 9X + 4= 0` `x^(3) - 6x^(2) + 9x + 4 = 0` `x^(3) + 6x^(2) - 9x + 4 = 0` `x^(3) - 6x^(2) - 9x - 4 = 0` ANSWER :2

Discussion

586.	Let A, B and C be ntimesn matrices, which one of the following is a correct statement
Answer» If `A^(2)=0`, then A=0 If AB=AC, then B=C If `A^(3)+2A^(2)+3A+5I=0`, then A is invertible If AB=AC, then B=C Answer :C

Discussion

587.	Obtain the following integrals : int(1+cosx)/(x+sin x)dx
Answer» ANSWER :`I=log\|x+sinx\|+C`

Discussion

588.	If A={1,2,5} and B={3,4,5,9}, then ADeltaB is equal to
Answer» `{1,2,5,9}` `{1,2,3,4,9}` `{1,2,3,4,5,9}` NONE of these Answer :B

Discussion

589.	Draw the graph of y = (x^(2))/(10) sin x.
Answer» Solution :We have `y = g(x) = (x^(2))/(10)sin x` FIRST DRAW the GRAPH of `y = +-(x^(2))/(10).` The graph of the function is as shown in the following FIGURE.

Discussion

590.	IF 5+n=9-1/3n,what is the value of n?
Answer» 3 4 6 `21/2` ANSWER :A

Discussion

591.	Assertion (A): A line through the point P(5,10) cut the line x+2y=5 at Q and the circle x^(2)+y^(2)=25 at A B. PA, PQ, PB are in H.P. Reason (R): A line through the point P cuts the polar of P w.r.t circle S=0 at Q and the circle S=0 at A and B then PA, PQ, PB are in H.P. The correct statement among the following is
Answer» A is TRUE, R is false A is false, R is true A is true, R is true, `RtoA` A is false, R is false Answer :C

Discussion

592.	An angle between the curves x^2 - y^2= 4and x^2 + y^2= 4 sqrt2 is
Answer» `pi/2` `pi/4` `pi/3` `pi/6` ANSWER :B

Discussion

593.	Find the maximum and minimum values in the following functions :(x-8)^(4)(x-9)^(5)
Answer» ANSWER :Maximum VALUE 0 at X = 8 Minimum value `(18)^(4)(22)^(10)` at `(76)/(6)`

Discussion

594.	int (1+ x^(2) )^(2) cos x dx.
Answer» Answer :`(X^(4) - 10x^(2) + 21) sin x + x (4X^(2) -20) COS x+ C`.

Discussion

595.	Let P=[{:((-z)^r,z^(2r)),(z^(2r),z^r):}] and Ibe the identitymatrix of order 2. Then the total number of ordered pairs (r, s) for which P^2 = – Iis
Answer» ANSWER :A

Discussion

596.	Find the number of all one-one functions from set A = {1,2,3}to itself.
Answer» ANSWER :`{1,2,3}`

Discussion

597.	Show that the differential equation (x^(3)-3xy^(2))dx=(y^(3)-3x^(2)y)dy is homogenous and solve it.
Answer» Solution :The given differential EQUATION may be written as `(DY)/(DX)=(x^(3)-3xy^(2))/(y^(3)-3x^(2)y)`………..(i) On dividing the Nr and Dr of RHS of (i) by `x^(3)`, we get `(dy)/(dx)=(1-3(y/x)^(2))/((y/x)^(2)-3(y/x))=f(y/x)`. Thus, the given differential equation is homogenous. Putting, `v=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `(dy)/(dx)=(1-3(y/x)^(2))/((y/x)^(3)-3(y/x))=f(y/x)`. Thus, the given differential equation is homogeneous. Putting, `y=vx` and `(dy)/(dx)=v+x(dv)/(dx)` in (i), we get `=v+x(dv)/(dx)=(x^(3)-3v^(2)x^(3))/(v^(3)x^(3)-3vx^(3))` `rArr v+x(dv)/(dx)=(1-3v^(2))/(v^(3)-3v)` `rArr x(dv)/(dx)=((1-3v^(2))/(v^(3)-3v)-v)` `rArr x(dv)/(dx)=(1-v^(4))/(v^(3)-3v)` `rArr (3v-v^(3))/(v^(4)-1)dv=int(dx)/x`.....................(ii) Let `((3v-v^(3))/(v^(4)-1)=A/(v-1)+B/(v+1)+(Cv+D)/(v^(2)+1))`. Then, `(3v-v^(3))-=A(v+1)(v^(2)+1)+B(v-1)(v^(2)+1)+(Cv+D)(v-1)(v+1))`............(iii) Putting v=1 on each side of (iii), we get `A=1/2`. Putting v`=-1` on each side of (iii), we get `B=1/2`. Comparing coefficients of `v^(3)` on both sides of (iii), we get `A+B+C=-1 rArr 1/2+1/2+C=-1 rArr C=-2`. Comparing the independent terms on both sides of (iii), we get `A-B-D=0 rArr D=(A-B)=(1/2-1/2)=0`. `therefore ((3v-v^(3))/(v^(4)-1)=1/(2(v-1))+1/(2(v+1))-(2v)/(v^(2)+1))`. Putting this value in (ii), we get `1/2int(dv)/(v-1)+1/2int(dv)/(v+1)-int(2v)/(v^(2)+1)dv=int(dx)/x` `rArr 1/2log\|v-1\|+1/2log\|v+1\|-LOG\|v^(2)+1\|=log\|x\|+log\|C\|`, where C is an arbitrary constant. `rArr log\|v-1\|+log\|v+1\|-2log\|v^(2)+1\|=2log\|x\|+2log\|C\|` `rArr log\|((v-1)(v+1))(v^(2)+1)^(2)\|=log\|C^(2)x^(2)\| rArr log\|(v^(2)-1)/(v^(2)+1)^(2)\|=log\|C^(2)x^(2)\|` `rArr (y^(2)-x^(2))=C^(2)(y^(2)+x^(2))^(2)[therefore v=y/x]`. Hence, `(y^(2)-x^(2))=C^(2)(y^(2)+x^(2))^(2)` is the required solutions.

Discussion

598.	A particle is acted upon constant forces vec(F)_(1)=4hati+hatj-3hatk and vec(F)_(2)=3hati+hatj-hatk which displace it from a point A=hati+2hatj+3hatk to the point B=5hati+4hatj+hatk. The work done in standard units by the forces is given by ……….......
Answer» 40 30 25 15 Answer :A

Discussion

599.	Mr. Martinson is building a concrete patio in his backyard and deciding where to buy the materials and rent the tools needed for the project. The table below shows the materials’ cost and daily rental costs for three different stores. The total cost, y, for buying the materials and renting the tools in terms of the number of days, x, is given by y=M+(W+K)x . For what number of days, x, will the total cost of buying the materials and renting the tools from Store B be less than or equal to the total cost of buying the materials and renting the tools from Store A ?
Answer» `xle6` `xge6` `xle7.3` `xge7.3` Solution :The total cost y, in dollars, of buying the materials and RENTING the tools for x days from Store A and Store B is found by substituting the respective values for these stores from the table into the GIVEN equation, y = M + (W + K)x, as shown below. Store A :` y =750+( 15 +65 )x= 750+80 x` Store B:`y =600 + ( 25 + 80 ) x= 600 + 105 x ` Thus, the number of days, x, for which the total cost of buying the materials and renting the tools from Store B is less than or equal to the total cost of buying the materials and renting the tools from Store A can be found by solving the inequality `600 + 105x le 750 + 70x`. Subtracting 80x and 600 from each side of `600 + 105x le 750 + 70x` and combining like terms YIELDS `25X le 150`. Dividing each side of `25x le 150` by 25 yields `x le 6`. Choice B is incorrect. The inequality x ge 6 is the number of days for which the total cost of buying the materials and renting the tools from Store B is greater than or equal to the total cost of buying the materials and renting the tools from Store A. Choices C and D are incorrect and may be the result of an error in setting up or simplifying the inequality.

Discussion

600.	Mr. Martinson is building a concrete patio in his backyard and deciding where to buy the materials and rent the tools needed for the project. The table below shows the materials’ cost and daily rental costs for three different stores. The total cost, y, for buying the materials and renting the tools in terms of the number of days, x, is given by y=M+(W+K)x . If the relationship between the total cost, y, of buying the materials and renting the tools at Store C and the number of days, x, for which the tools are rented is graphed in the xy-plane, what does the slope of the line represent?
Answer» The total cost of the project The total cost of the materials The total daily cost of the project The total daily rental costs of the TOOLS Solution :The total cost, y, of buying the materials and renting the tools in terms of the number of days, x, is GIVEN as y = M + (W + K)x. If this relationship is graphed in the xy-plane, the slope of the graph is equal to W + K, which is the daily rental cost of the wheelbarrow plus the daily rental cost of the CONCRETE mixer, that is, the total daily rental costs of the tools. Choice A is incorrect because the total cost of the project is y. Choice B is incorrect because the total cost of the materials is M, which is the y-intercept of the graph of y = M + (W + K)x. Choice C is incorrect because the total daily cost of the project is the total cost of the project divided by the total number of days the project took and, since materials cost more than 0 dollars, this is not the same as the total daily rental costs.

Discussion

Explore topic-wise InterviewSolutions in .

Which of the following function f:R rarr R are bijective

Letthe couble ordinate pp' of the hyperbola(x^(2))/(4)-(y^(2))/(3)=1 is producedboth sides to meetasymptotes of hyperbola in Q and Q'. The product (PQ)(PQ)' is equal to :

Let a, b, c be real number (a ne 0). If alpha is a root of a^(2)x^(2)+bx +c=0, beta is a root of a^(2)x^(2) -bx-c=0 and 0 lt alpha beta, then the root of the equation (say gamma) a^(2) x^(2)+2bx+2c=0 always satisfies

Write minors and cofactors of the elements of following determinant |{:(5,-10),(0,3):}|

If Delta={:|(1,2,3),(2,0,1),(5,3,8)| Write the minor of the element a_(22).

Assertion (A) : The least length of the focal chord ofy^(2) =4axis 4a Reason (R) : Length of the focal chord of y^(2) =4axmakes an angle thetawith axis is 4a cosec ^(2) theta

Let a curve y=f(x) pass through (1,1), at any point p on the curve tangent and normal are drawn to intersect the X-axis at Q and R respectively. If QR = 2 , find the equation of all such possible curves.

If A^(2)-A+I=O," then "A^(-1)=

The maximum value of Z=2x+y subject to 3x+5yle26 and 5x+3yle30,xge0,yge0 is

r(f) nn R (g)=

A fair coin and an unbiased die are tossed. Let A be the event 'head appears on the coin' and B be the event '3 on the die' . Check whether A and Bare independent events or not.

[( p rarr q) ^^ ~ r ] vee r vee ( p ^^ ~q ^^- r )is equivalent to

Let f be a function continuous on R, then int_(-pi//2)^(pi//2) (x + sin x) {f(x) + f(-x)}dx is :

The variance of observation x_(1), x_(2),x_(3),…,x_(n) is sigma^(2) then the variance of alpha x_(1), alpha x_(2), alpha x_(3),….,alpha x_(n), (alpha != 0) is

The middle term in the expansion of (10/x + x/10)^(10) is

Foreachn inN, 2^(3n )-1is divisibleby

The coordinates of a point, which lies on y-axis and is at a distance of 4 units above x-axis is ______.

int_(0)^(pi//2) ( f ( co x )) /( f (sinx ) + f(co sx) ) dx is equal to

Let f:[a,b] to R be a function such that , for c in (a,b), f.(c ) = f..( c) = f...( c) = f.... (c ) = f.....( c) = 0 . Then :

State which of the followingstatementis true?

If alpha, beta , gamma are the roots of x^(3) + 3x + 2 = 0then the equation whose roots (beta - gamma)^(2), (gamma - alpha)^(2), (alpha - beta)^(2)is

The locus of the point of intersection of two tangents to the parabola y^(2)=4ax which make the angles theta_(1) and theta_(2) with the axis so that tan theta_(1) tan theta_(2) =k is

If f(x) = (2x)/(4 + 3|x|), x in R, then f'(0) =

Ore overset("Calcination")(rarr)Residue overset("dil".HNO_(3))(rarr)underset("Metal(M)+Zn^(2+)(aq.))underset(darr zn dust)(aq. solution) above metallurgy is possible when ore is :-

A die is thrown. E is the event "the number appearing us a multiple of 3" and F be the event "the number appearing is even". Find whether E and F are independent.

The probability distribution of x is

If (4,0) and (1, -1) are two vertices of a﻿ triangle of area 4 square units, then its third﻿ vertex lies on﻿

Find (dy)/(dx) given x^(y) + y^(x) = 1

Which of the following relations are both odd and even? I. x^(2)+y^(2)=1 II. x^(2)-y^(2)=0 III. x+y=0

The manufacturing cost of an item consists of rupes 100 as overheads , material cost rupes 2 per item and the labour cost( x^(2))/(90)for x items , produced . Find the average cost.

Find the area of the parallelogram whose adjacent sides are determined by the vectors vec(a)=hati-hatj+3hatk and vec(b)=2hati-7hatj+hatk.

If A={a,b,c,d} mention the type of relations on A given below, which of them are equivalence relations? {(a,b),(b,a),(b,d),(d.b)}

If alpha , beta , gamma are the roots of x^(3) + 3x + 2 =0 then the equation whose roots alpha (beta + gamma), beta( gamma + alpha), gamma(alpha + beta) is

Let A, B and C be ntimesn matrices, which one of the following is a correct statement

Obtain the following integrals : int(1+cosx)/(x+sin x)dx

If A={1,2,5} and B={3,4,5,9}, then ADeltaB is equal to

Draw the graph of y = (x^(2))/(10) sin x.

IF 5+n=9-1/3n,what is the value of n?

An angle between the curves x^2 - y^2= 4and x^2 + y^2= 4 sqrt2 is

Find the maximum and minimum values in the following functions :(x-8)^(4)(x-9)^(5)

int (1+ x^(2) )^(2) cos x dx.

Let P=[{:((-z)^r,z^(2r)),(z^(2r),z^r):}] and Ibe the identitymatrix of order 2. Then the total number of ordered pairs (r, s) for which P^2 = – Iis

Find the number of all one-one functions from set A = {1,2,3}to itself.

Show that the differential equation (x^(3)-3xy^(2))dx=(y^(3)-3x^(2)y)dy is homogenous and solve it.

A particle is acted upon constant forces vec(F)_(1)=4hati+hatj-3hatk and vec(F)_(2)=3hati+hatj-hatk which displace it from a point A=hati+2hatj+3hatk to the point B=5hati+4hatj+hatk. The work done in standard units by the forces is given by ……….......

If (4,0) and (1, -1) are two vertices of a triangle of area 4 square units, then its third vertex lies on