28278 + Interview Questions in Maths in Class 12

1.	The point on y^(2) = 4ax nearest to the focus has its abscissa equal to
Answer» `-a` a `(a)/(2)` `(a)/(2)` ANSWER :D

Discussion

2.	Obtain the equation of straight lines : Passing through (1,-1) and making an angle 150^@.
Answer» SOLUTION :Slope of the line =`tan150^@ = - 1/sqrt3` `therefore` EQUATION of the line `y-y_1 = m(X-x_1)` or, `y+1 = -1/sqrt3 (x-1)` or, `ysqrt3 + sqrt3 = -x + 1` or, `x+ysqrt3 + sqrt3-1 = 0` .

Discussion

3.	Let f : R rarr Rbe the Signum Function defined as f(x) = {(1,xgt0),(0,x = 0),(-1,xlt0):}andg : R rarr Rbe the Greatest Integer Function given by g(x) = [x] , where [x] is greatest integer less than or equal to x. Then , does fog and gof coincide in (0,1] ?
Answer» Yes No Nothing can be SAID COMPOSITE FUNCTION does not EXISTS SOLUTION :N/A

Discussion

4.	If P(A) =(3)/(5) and P (B)=(1)/(5) find P(A cap B) if A and B are independent events.
Answer» ANSWER :`(3)/(25)`

Discussion

5.	Form the differential equation of the family of circles having centre on y - axis and radius 3 units.
Answer» ANSWER :`(X^(2) - 9)(y')^(2) + x^(2) = 0`

Discussion

6.	Consider the hyperbola (X^(2))/(9)-(y^(2))/(a^(2))=1 and the circle x^(2)+(y-3)=9. Also, the given hyperbola and the ellipse (x^(2))/(41)+(y^(2))/(16)=1 are orthogonal to each other. The number of points on the hyperbola and the circle from which tangents drawn to the circle and the hyperbola, respectively, are perpendicular to each other is
Answer» 0 2 4 6 Solution :Ellipse and hyperbola are orthogonal and THEREFORE, THEYARE confocal. `"So,"a_(h)e_(h)=a_(e)e_(e)` `rArr""a^(2)+9=41-16` `rArr""9+a^(2)=25` `rArr""a^(2)=16` Thus, hyperbola is `(x^(2))/(9)-(y^(2))/(16)=1.` So, common tangents to the circle and hyperbola are `x = pm 3`. Director circle of hyperbola does not exist as `a lt B.` Director circle of circle is `x^(2)+(y-3)^(2)=18` `rArr""x^(2)+y^(2)-6y-9=0` This meets the hyperbola `16x^(2)-9y^(2)=144` at four points from where tangents drawn to the circle `x^(2)+(y-3)^(2)=9` are perpendicular to each other. Let MIDPOINT of AB be (h,k). So, equation of line AB is `hx+ky-3(y+k)=h^(2)+k^(2)-6k`. Since tangents at C and D intersect at the directrix, CD is the focal chord of hyperbola. So, AB PASSES through focus of the hyperbola and that is `(pm5,0)`. Therefore, required lacus is `x^(2)+y^(2)pm5x-3y=0`.

Discussion

7.	Consider the hyperbola (X^(2))/(9)-(y^(2))/(a^(2))=1 and the circle x^(2)+(y-3)=9. Also, the given hyperbola and the ellipse (x^(2))/(41)+(y^(2))/(16)=1 are orthogonal to each other. A variable line cuts the circle at point A and B and it cuts the hyperbola atpoints C and D. The locus of midpoint of AB such that tangents at points C and D always intersect each other at the directrix of the hyperbola, is
Answer» `X^(2)+y^(2)pm5x-3y=0` `x^(2)+y^(2)+5xpm3y=0` `x^(2)-y^(2)pm5x-3y=0` `x^(2)-y^(2)+3xpm3y=0` Solution :Ellipse and HYPERBOLA are orthogonal and therefore, theyare confocal. `"So,"a_(h)e_(h)=a_(e)e_(e)` `rArr""a^(2)+9=41-16` `rArr""9+a^(2)=25` `rArr""a^(2)=16` Thus, hyperbola is `(x^(2))/(9)-(y^(2))/(16)=1.` So, common tangents to the circle and hyperbola are `x = pm 3`. Director circle of hyperbola does not EXIST as `a LT b.` Director circle of circle is `x^(2)+(y-3)^(2)=18` `rArr""x^(2)+y^(2)-6y-9=0` This meets the hyperbola `16x^(2)-9y^(2)=144` at four points from where tangents drawn to the circle `x^(2)+(y-3)^(2)=9` are perpendicular to each other. Let midpoint of AB be (h,k). So, equation of LINE AB is `hx+ky-3(y+k)=h^(2)+k^(2)-6k`. Since tangents at C and D intersect at the directrix, CD is the focal chord of hyperbola. So, AB passes through focus of the hyperbola and that is `(pm5,0)`. Therefore, required lacus is `x^(2)+y^(2)pm5x-3y=0`.

Discussion

8.	Consider the hyperbola (X^(2))/(9)-(y^(2))/(a^(2))=1 and the circle x^(2)+(y-3)=9. Also, the given hyperbola and the ellipse (x^(2))/(41)+(y^(2))/(16)=1 are orthogonal to each other. Combined equation of pair of common tangents between the hyperbola and the circle is given be
Answer» `x^(2)-y^(2)=0` `x^(2)-9=0` `9y^(2)-19x^(2)=0` No common tangent. Solution :Ellipse and hyperbola are orthogonal and therefore, theyare confocal. `"So,"a_(h)e_(h)=a_(e)e_(e)` `rArr""a^(2)+9=41-16` `rArr""9+a^(2)=25` `rArr""a^(2)=16` Thus, hyperbola is `(x^(2))/(9)-(y^(2))/(16)=1.` So, common tangents to the CIRCLE and hyperbola are `x = pm 3`. Director circle of hyperbola does not exist as `a lt b.` Director circle of circle is `x^(2)+(y-3)^(2)=18` `rArr""x^(2)+y^(2)-6y-9=0` This meets the hyperbola `16x^(2)-9y^(2)=144` at FOUR points from where tangents drawn to the circle `x^(2)+(y-3)^(2)=9` are PERPENDICULAR to each other. Let midpoint of AB be (h,k). So, equation of line AB is `hx+ky-3(y+k)=h^(2)+k^(2)-6k`. Since tangents at C and D intersect at the directrix, CD is the focal chord of hyperbola. So, AB passes through focus of the hyperbola and that is `(pm5,0)`. Therefore, required lacus is `x^(2)+y^(2)pm5x-3y=0`.

Discussion

9.	Find inte^(x)sinxdx
Answer» ANSWER :`I= (E^(X))/(2)(sinx-cosx)+C`

Discussion

10.	Find the area of the region enclosed by the parabola y=x^(2) + 2, the lines y= -x, x= 0 and x=1
Answer» ANSWER :`(17)/(6)`

Discussion

11.	There. are three boxes, the first one containing 1 white, 2 red -and 3 black balls, the second one containing 2 white, 3 red and 1 black ball and the third one containing 3 white, 1 red and 2 black balls. A box is chosen at random and from it two balls are drawn at random. One ball is red and the other, white. What is the probability that they come from the second box?
Answer» ANSWER :`6/11`

Discussion

12.	Integrate the functions tan(2x-3)
Answer»

Discussion

13.	Area of the region bounded by the curve y^(2) = 4x, y-axis and the line y = 3 is
Answer» 2 `9/4` `9/3` `9/2` ANSWER :B

Discussion

14.	Evaluate underset(0)overset(16)int (x^(1//4))/(1+x^(1//2))dx
Answer» ANSWER :`4((2)/(3) + TAN^(-1)2)`

Discussion

15.	lim_(xrarr 0 ) (x^(2) (tan2x - 2 tan x )^(2))/((1-cos 2 x )^(4)) =
Answer» 4 2 `1/2` `1/4` ANSWER :D

Discussion

16.	Compute the following: [(a,b),(-b,a)]+[(a,b),(b,a)] (ii) [(a^(2)+b^(2),b^(2)+c^(2)),(a^(2)+c^(2),a^(2)+b^(2))]+[(2ab,2bc),(-2ac,-2ab)] (iii)[(-1,4,-6),(8,5,16),(2,8,5)]+[(12,7,6),(8,0,5),(3,2,4)] (iv) [(cos^(2)x, sin^(2)x),(sin^(2)x, cos^(2)x)]+[(sin^(2)x, cos^(2)x),(cos^(2)x, sin^(2)x)]
Answer» ANSWER :`(i) [(2a,2b),(0,2a)] (II) [((a+b)^(2),(b+c)^(2)),((a-c)^(2),(a-b)^(2))]` `(III)[(11,11,0),(16,5,21),(5,10,9)] (IV) [(1,1),(1,1)]`

Discussion

17.	Find the area of the region in the first quadrant method enclosed by the x-axis, the line y=x and the circle x^2+y^2=32.
Answer» ANSWER :8

Discussion

18.	Solvetheequation x^4- 9x^3 + 27 x^2 - 29 x + 6 =0giventhat2 - sqrt(3)is a root.
Answer» ANSWER :`2+- SQRT(3) ,2,3`

Discussion

19.	The measure of the angle between the curves y=2 sin^(2)x and y = cos 2 x at x=(pi)/(6) is ……….
Answer» `(pi)/(2)` `(pi)/(3)` `(pi)/(4)` `(pi)/(6)` ANSWER :B

Discussion

20.	If 3 dice are rolled then the probability of getting different numbers or sum 16 is
Answer» `(7)/(12)` `(4)/(9)` `(2)/(9)` `(1)/(9)` Answer :A

Discussion

21.	int x^2 e^(-2x) dx
Answer» ANSWER :`(-E^(-2X))/4`

Discussion

22.	If the equation x^(2) - y^(2) - 2x+2y+lambda =0 represents a degenerate conic then find the value of lambda.
Answer» ANSWER :`THEREFORE LAMBDA =0`

Discussion

23.	A pair of biased diced is rolled until sum 10 appears for the first time. Given that the probability of getting sum 10 on the biased dice is p(0 lt p lt 1). If the probability that the number of trails required to get sum 10 is odd is 5/9 then find the value of p.
Answer» ANSWER :`(1)/(5)`

Discussion

24.	If the vectors phati+hatj+hatk, hati, qhatj+hatk and hati+hatj+rhatk(p ne q ne r ne1) are coplanar, then the value of pqr-(p+q+r) is
Answer» `-2` 2 0 `-1` SOLUTION :Let `a = phati+ hatj + HATK, b = HATI +hatj + hatk` andc = `hati + hatj + rhatk`are coplanar and `( p ne q ne R ne 1)`. Since, a, b and c are coplanar . `:. [a,b,c]= 0 rArr \|{:(p,1,1),(1,q,1),(1,1,r):}\| = 0` `rArr p(qr- 1) - 1 (r-1) + 1 (1-q) = 0` `rArr PQR - p -r + 1 + 1-q = 0` `:. pqr - (p+q+r) = - 2`

Discussion

25.	Find the scalar components and magnitude of the vector joining the points P(x_(1),y_(1),z_(1))andQ(x_(2),y_(2),z_(2)).
Answer» ANSWER :`=sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)+(z_(2)-z_(1))^(2))`

Discussion

26.	If a function f:RtoR is defined by f(x)=x^(2)+1, then pre-images of 17 and -3 respectively, are
Answer» ANSWER :C

Discussion

27.	A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.
Answer» ANSWER :`(25)/(216)`

Discussion

28.	Two cards are drawn successively with replacement for a well shuffled pack of 52 cards. Find the probabilility distribution of the number of kings.
Answer» ANSWER :`(##NCERT_TAM_MAT_XII_P2_C13_SLV_024_A01##)`

Discussion

29.	Fundamental theorem of definite integral : If int_(0)^(k)(dx)/(1+4x^(2))=(pi)/(8) then k =……..
Answer» `(1)/(2)` `(PI)/(2)` 1 `pi` ANSWER :A

Discussion

30.	A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female.
Answer» ANSWER :(i) `(1)/(3)` (II) `(1)/(2)`

Discussion

31.	Find a vector perpendicular to both the vectors hati-2hatj+3hatk and hati+2hatj-hatk.
Answer» ANSWER :`(1)/(SQRT(3))(-hati+hatj+hatk)`

Discussion

32.	IF a,b,care inA. Pand if( b-c )^2+(c-a)x+ (c-a ) x+(a-b) =0and2(c+a) x^2 +(b+c) x=0havea commonrootthen
Answer» `a^2 , B^2 ,c^2 ` are inA.P `a^2,c^2 ,b^2` are in A.P `a^2, ,c^2,b^2` are in G.P NONEOF these Answer :B

Discussion

33.	Let be a binary operation on the set Q of rational numbers as followsab=a-b. Check whether * is commutative and associative
Answer» SOLUTION :SINCE `1-2ne2-1, 12 ne 21` `therefore ` is not COMMUTATIVE Since `(2-4)-5ne2-(4-5), `(24)5 ne 2(45)` `therefore ` is not associative `**` is not commutative, not associative

Discussion

34.	The maximum volume of the cylinder which can be inscribed in a sphere of radius a
Answer» `(4pia^3)/(3SQRT3)` CUBIC UNIT `4pi a^3` cubic unit `(4pia^3)/(sqrt3)` cubic unit none Answer :A

Discussion

35.	Three bags contain : (i) 4 red and 4 black, 2 red and 6 black balls (ii) 6 red and 3 balck, 5 red and 5 black balls (iii) 6 red and 4 black, 3 red and 3 black balls. One ball is drawn at random from one of the bags and found to be red. Find the probability that it was drawn from the second bag.
Answer» Answer :(i) `(1)/(3)` (ii) `(3)/(7)` (iii) `(5)/(11)`.

Discussion

36.	Find the area of the region bounded by the ellipse x^(2)/4+y^(2)/9=1.
Answer» ANSWER :`6PI`

Discussion

37.	Find the number of ways of selecting cricket eleven from 20 players such that Exactly one of Sachin and Dravid must be included
Answer» ANSWER :`2XX""^(18)C_(8)`

Discussion

38.	Let f(x)=(x^(5)-1)(x^(3)+1),g(x)=(x^(2)-1)(x^(2)-x+1) and let h(x) be such that f(x)=g(x)h(x). Then lim_(xto1)h(x) is
Answer» 0 1 3 5 Answer :D

Discussion

39.	If\|x+1\|-\|x\|+3\|x-1\|-2\|x-2\|=x+2,thenx=
Answer» `x=2 orx LE2` ` x=-5 orx gt5` `x=- 2 orx GE2 ` NONEOF these Answer :C

Discussion

40.	Let A be a 2xx2 matrix such that det(A^(2)+4I_(2))=0 then det(A) + tr(A)= _____
Answer» ANSWER :`4.00`

Discussion

41.	Find\|vecb\|if( veca +vecb) .( veca - vecb) =8 and \|veca\|= 8\|vecb\|
Answer» ANSWER :`= SQRT((8)/(63))`

Discussion

42.	The probabilities of new purchased television and refrigerator will be working 10 years hence are (7)/(15) and (7)/(10) respectively. The probability that both will not be working 10 years hence is ………..
Answer» `(21)/(150)` `(24)/(150)` `(49)/(150)` `(56)/(150)` Answer :B

Discussion

43.	Given: the cycloid x=a (t- sin t),y = a (1- cos t), 0 le t le 2pi Compute. (a) the areas of the surface formed by revolving the arc OBA about the x and y-axis, (b) the volumes of the solids generated by revolving the figure OBAO about the y-axis and the axis BC, (c ) the area of the surface generated by revolving the are BA about the axis BC, (d) the volume of the solid generated by revolving the figure ODBEABO about the tangent line DE touching the figure at the vertex B, (e ) the area of the surface formed by revolving the are of the cycloid (see item (d)]
Answer» Answer :(a) `16pi^(2) a^(2)` (b) `(PI a^(3))/(6) (9pi^(2)-16)` (c ) `4 (2pi - (8)/(3))^(2) pi a^(2)` (d) `pi^(2) a^(3)` (e ) `(32)/(3) pi a^(2)`

Discussion

44.	Let f(x)=sqrt(1-cos (x-2))/(x-2), x ne 2. The lim_(x to 2) f(x)
Answer» exists and is EQUAL to `SQRT2` does not EXIST because `lim_(x to 2+)f(x)` doesn’t exist equal to 1 doesn't exist because `lim_(x to 2+) f(x)ne lim_(x to 2-) f(x)` Answer :D

Discussion

45.	Evaluate the following determinates \|{:(log_ab,1),(1,log_ba):}\|
Answer» ANSWER :1

Discussion

46.	By examining the chest X-ray probability that T.B. is detected when person is actually suffering is 0.99. The probability that doctor diagnoses incorrectly that person has T. B. on the basis of X-ray is 0.001. In certain city 1 to 1000 persons suffering from T.B. A person is selected at random is diagnosed to have a T.B. what is the chance he has actually T.B. ?
Answer» ANSWER :`(110)/(221)`

Discussion

47.	Find the equation of the locus of a point, which forms a triangle of area 2 with the points A(1, 1) and B (-2, 3).
Answer» ANSWER :`(2x-3y-9)(2x+3y-1)=0`

Discussion

48.	Two cards are drawn at randrom from a pack of 52 cards. The probability of these two being "Aces" is
Answer» 1)`(1)/(26)` 2)`(1)/(221)` 3)`(1)/(2)` 4)`(1)/(13)` ANSWER :B

Discussion

49.	If the lines 2x+y+12=0, 4x-3y-10=0 are conjugate w.r.t the circle with centre (2,(-3)/2) then r=
Answer» `(SQRT(29))/2` `sqrt(29)` `5/2` `5` ANSWER :A

Discussion

50.	If{x:p_1 (x)} ={x:p_2(x)}, show each x,p_1(x) and p_2(x) have the same truth value.
Answer» Solution :GIVEN that `{X:p_1(x)}={x:p_2(x)}` `:.The TWO sets have the same element. So `P_1(x)!=P_2(x)` i.e., they have same truth VALUE.

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

The point on y^(2) = 4ax nearest to the focus has its abscissa equal to

Obtain the equation of straight lines : Passing through (1,-1) and making an angle 150^@.

Let f : R rarr Rbe the Signum Function defined as f(x) = {(1,xgt0),(0,x = 0),(-1,xlt0):}andg : R rarr Rbe the Greatest Integer Function given by g(x) = [x] , where [x] is greatest integer less than or equal to x. Then , does fog and gof coincide in (0,1] ?

If P(A) =(3)/(5) and P (B)=(1)/(5) find P(A cap B) if A and B are independent events.﻿

Form the differential equation of the family of circles having centre on y - axis and radius 3 units.

Consider the hyperbola (X^(2))/(9)-(y^(2))/(a^(2))=1 and the circle x^(2)+(y-3)=9. Also, the given hyperbola and the ellipse (x^(2))/(41)+(y^(2))/(16)=1 are orthogonal to each other. Combined equation of pair of common tangents between the hyperbola and the circle is given be

Find inte^(x)sinxdx

Find the area of the region enclosed by the parabola y=x^(2) + 2, the lines y= -x, x= 0 and x=1

Integrate the functions tan(2x-3)

Area of the region bounded by the curve y^(2) = 4x, y-axis and the line y = 3 is

Evaluate underset(0)overset(16)int (x^(1//4))/(1+x^(1//2))dx

lim_(xrarr 0 ) (x^(2) (tan2x - 2 tan x )^(2))/((1-cos 2 x )^(4)) =

Compute the following: [(a,b),(-b,a)]+[(a,b),(b,a)] (ii) [(a^(2)+b^(2),b^(2)+c^(2)),(a^(2)+c^(2),a^(2)+b^(2))]+[(2ab,2bc),(-2ac,-2ab)] (iii)[(-1,4,-6),(8,5,16),(2,8,5)]+[(12,7,6),(8,0,5),(3,2,4)] (iv) [(cos^(2)x, sin^(2)x),(sin^(2)x, cos^(2)x)]+[(sin^(2)x, cos^(2)x),(cos^(2)x, sin^(2)x)]

Find the area of the region in the first quadrant method enclosed by the x-axis, the line y=x and the circle x^2+y^2=32.

Solvetheequation x^4- 9x^3 + 27 x^2 - 29 x + 6 =0giventhat2 - sqrt(3)is a root.

The measure of the angle between the curves y=2 sin^(2)x and y = cos 2 x at x=(pi)/(6) is ……….

If 3 dice are rolled then the probability of getting different numbers or sum 16 is

int x^2 e^(-2x) dx

If the equation x^(2) - y^(2) - 2x+2y+lambda =0 represents a degenerate conic then find the value of lambda.

A pair of biased diced is rolled until sum 10 appears for the first time. Given that the probability of getting sum 10 on the biased dice is p(0 lt p lt 1). If the probability that the number of trails required to get sum 10 is odd is 5/9 then find the value of p.

If the vectors phati+hatj+hatk, hati, qhatj+hatk and hati+hatj+rhatk(p ne q ne r ne1) are coplanar, then the value of pqr-(p+q+r) is

Find the scalar components and magnitude of the vector joining the points P(x_(1),y_(1),z_(1))andQ(x_(2),y_(2),z_(2)).

If a function f:RtoR is defined by f(x)=x^(2)+1, then pre-images of 17 and -3 respectively, are

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find﻿ the probability of two successes.﻿

Two cards are drawn successively with replacement for a well shuffled pack of 52 cards. Find the probabilility distribution of the number of kings.

Fundamental theorem of definite integral : If int_(0)^(k)(dx)/(1+4x^(2))=(pi)/(8) then k =……..

A couple has two children,﻿ (i) Find the probability that both children are males, if it is known that at least﻿ one of the children is male.﻿ (ii) Find the probability that both children are females, if it is known that the﻿ elder child is a female.﻿

Find a vector perpendicular to both the vectors hati-2hatj+3hatk and hati+2hatj-hatk.

IF a,b,care inA. Pand if( b-c )^2+(c-a)x+ (c-a ) x+(a-b) =0and2(c+a) x^2 +(b+c) x=0havea commonrootthen

Let ** be a binary operation on the set Q of rational numbers as followsa**b=a-b. Check whether * is commutative and associative

The maximum volume of the cylinder which can be inscribed in a sphere of radius a

Three bags contain : (i) 4 red and 4 black, 2 red and 6 black balls (ii) 6 red and 3 balck, 5 red and 5 black balls (iii) 6 red and 4 black, 3 red and 3 black balls. One ball is drawn at random from one of the bags and found to be red. Find the probability that it was drawn from the second bag.

Find the area of the region bounded by the ellipse x^(2)/4+y^(2)/9=1.

Find the number of ways of selecting cricket eleven from 20 players such that Exactly one of Sachin and Dravid must be included

Let f(x)=(x^(5)-1)(x^(3)+1),g(x)=(x^(2)-1)(x^(2)-x+1) and let h(x) be such that f(x)=g(x)h(x). Then lim_(xto1)h(x) is

If|x+1|-|x|+3|x-1|-2|x-2|=x+2,thenx=

Let A be a 2xx2 matrix such that det(A^(2)+4I_(2))=0 then det(A) + tr(A)= _____

Find|vecb|if( veca +vecb) .( veca - vecb) =8 and |veca|= 8|vecb|

The probabilities of new purchased television and refrigerator will be working 10 years hence are (7)/(15) and (7)/(10) respectively. The probability that both will not be working 10 years hence is ………..

Let f(x)=sqrt(1-cos (x-2))/(x-2), x ne 2. The lim_(x to 2) f(x)

Evaluate the following determinates |{:(log_ab,1),(1,log_ba):}|

Find the equation of the locus of a point, which forms a triangle of area 2 with the points A(1, 1) and B (-2, 3).

Two cards are drawn at randrom from a pack of 52 cards. The probability of these two being "Aces" is

If the lines 2x+y+12=0, 4x-3y-10=0 are conjugate w.r.t the circle with centre (2,(-3)/2) then r=

If{x:p_1 (x)} ={x:p_2(x)}, show each x,p_1(x) and p_2(x) have the same truth value.

If P(A) =(3)/(5) and P (B)=(1)/(5) find P(A cap B) if A and B are independent events.

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.

A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female.

Let be a binary operation on the set Q of rational numbers as followsab=a-b. Check whether * is commutative and associative