28278 + Interview Questions in Maths in Class 12

1.	A speaks truth in 60% of the cases and B in 70% of the cases. What is the probability that (i) both speak truth (ii) both speak lie (iii) their statements about an incident do not match.
Answer» ANSWER :`(i) (21)/(50)` `(II) (3)/(25)` `(iii) (23)/(50)`

Discussion

2.	Show that the relation R in the set A={x in Z :0lexle12} is given by R={(a,b) : \|a-b\|" is a multiple of 4"} is an equivalence relation .
Answer» ANSWER :HENCE R is an EQUIVALENCE RELATION

Discussion

3.	If Q is the foot of the perpendicular from a point p on the parabola y^(2)=8(x-3) to its directrix. S is an equilateral triangle then find the lengh of side of the triangle.
Answer» `(4sqrt(3),8)` `(8,4sqrt(3))` `(9,4sqrt(3))` `(4sqrt(3),9)` ANSWER :C

Discussion

4.	Evaluate : intsqrt((x-5)(x-7))dx
Answer» ANSWER :`(1)/(2)(x-6)SQRT(-35+12x-x^(2))+(1)/(2)SIN^(-1)(x-6)+c`

Discussion

5.	Integrate the following functions 1/sqrt((2-x)^2+1)
Answer» SOLUTION :`INT 1/(SQRT(2-x)^2+1) DX` =`1/-1 log\|2-x +sqrt((2-x)+1)\|+c` =`-log\|2-x+sqrt((2-x)^2+1)\|+c` =`log\|2-x+sqrt((4-4x+x^2)+1)\|^-1+c` `log\|1/(2-x+sqrt(x^2-4x+5)\|+c`

Discussion

6.	Find the values of the following integrals (i) int_(0)^(pi/2) sin^(4)x dx
Answer» ANSWER :`(3PI)/(16)`

Discussion

7.	Solve the following equations : i) x^(4)-5x^(2)+6=0 ii) x^(2//3)+x^(1//3)-2=0 iii) 7^(1-x)+7^(1+x)=50 iv) sqrt((3x0/(x+1))+sqrt((x+1)/(3x))=2, when x != 0, -5 v) sqrt((x)/(1-x))+sqrt((1-x)/(x))=(13)/(6), x!=0, x!=1 vi) 2(x+(1)/(x))^(2)-7(x+(1)/(x))+5=0, x != 0 vii) (x^(2)+(1)/(x^(2)))-5(x+(1)/(x))+6=0, x != 0 viii) (x+1)(x+2)(x+3)(x+4)=120 ix) 2x^(4)+x^(3)-11x^(2)+x+2=0
Answer» Answer :i) `+-SQRT2, +-sqrt3` ii) -8, 1 iii) -1, 1 iv) {`1//2`} V) `(4)/(13), (9)/(13)` vi) `(1)/(2), 2, (1+- i sqrt3)/(2)` vii) `(1+- i sqrt3)/(2), 2+-sqrt3` viii) `(-5+- i SQRT(39))/(2)`, -6, 1 ix) `{(-3 +- sqrt5)/(2), (1)/(2), 2}`

Discussion

8.	The sum to n terms of the series 4/3 + 10/9 + 28/27 + ……. is
Answer» `N + 1/2(1 + 3^(-n))` `n - 1/2(1 + 3^(-n))` `n + 1/2(2 + 3^(-n))` `n + 1/2(2 - 3^(-n))` ANSWER :D

Discussion

9.	When the origin is shifted to (-1,2) by the translation of axes, find the transformed equation x^2+y^2+2x-4y+1=0.
Answer» ANSWER :`X^2+Y^2-4=0`

Discussion

10.	Letf(x) ={{:( (a(1-x sinx) + b cccos x+5)/(x^(2)),xlt0),( 3, x=0),( {1+((p(x))/(x^(2)))}^(1//x),xgt0):} wherep(x)is a cubic functionand f is continuous at x=0,if theleading coefficient of p (x)is positive, thentheequation p(x)= b has
Answer» onlyone real , positiveroot only ONE real, NEGATIVE root THREE real roots noneof these Answer :B

Discussion

11.	The pair of lines lx^(2) + 2(l + m) xy + my^(2) = 0 lies along two diameters of a circle and divides the circle into 4 sectors. If the area of bigger sector is 5 times the area of smaller sector, then (l m)/((l+m)^(2)) =
Answer» `(1)/(2)` `(2)/(sqrt(3))` `(11)/(12)` `(13)/(12)` Answer :C

Discussion

12.	Letf(x) ={{:( (a(1-x sinx) + b cccos x+5)/(x^(2)),xlt0),( 3, x=0),( {1+((p(x))/(x^(2)))}^(1//x),xgt0):} wherep(x)is a cubic functionand f is continuous at x=0,thevalueofp''(0)is
Answer» `LOG _(E) 9` `log_(e)2` 2 1 Answer :A

Discussion

13.	Integrate the function is exercise. sqrt(x^(2)+4x+6)
Answer» ANSWER :`(X+2)/(2)SQRT(x^(2)-4x+6) log\|(x+2)+sqrt(x^(2)+4x+6)\|+c`

Discussion

14.	Letf(x) ={{:( (a(1-x sinx) + b cccos x+5)/(x^(2)),xlt0),( 3, x=0),( {1+((p(x))/(x^(2)))}^(1//x),xgt0):} wherep(x)is a cubic functionand f is continuous at x=0,the range of functiong(X) =3a sinx-b cosx is
Answer» [-10,10] [-5,5] [-12,12] NONEOF these Answer :B

Discussion

15.	Two numbers are selected at random from 1,2,3,……100 without replacement. Find the probability that the minimum of the two numbers is less than 70.
Answer» ANSWER :`1-(.^(31)C_(2))/(.^(100)C_(2))`

Discussion

16.	The points collinear (1,-2,-3) and (2,0,0) among the following is
Answer» (0,4,6) (0,-4,-5) (0,-4,-6) (0,-4,6) Answer :C

Discussion

17.	Find the real values of theta , such that (3 + 2i sin theta)/(1 - 2i sin theta) is purely Imaginary
Answer» `n PI , n in Z ` `n pi PM (pi)/(2) , n in Z ` `n pi pm (pi)/(3) , n in Z ` `n pi pm (pi)/(6) , n in Z ` Answer :C

Discussion

18.	f(x) = {(3",","if" 0 le x le 1),(4",","if" 1 lt x lt 3),(5",","if" 3 le x le 10):}
Answer» ANSWER :x=1

Discussion

19.	Find the sum of the degree and the order of the differential equation: y = (x-(2y)/((dy)/(dx)))(((2y)/(dy))/(dx))^(2).
Answer» 2 3 4 6 Answer :C

Discussion

20.	Statement -1: If a,b,c are the integers such that a+b+cle8, then the number of possible value of the ordered triplets (a,b,c,) is 56. because Statement -2: The number of ways in which n identical things can be distributed into r different groups is .^(n-1)C_(r-1).
Answer» STATEMENT -1 is True, Statement-2 is True, Statement -2 is a CORRECT explanation for Statement -1 Statement -1 is True, Statement -2 is True, Statemetn -2 is NOT a correct explanation for Statement -1. Statement -1 is True, Statement -2 is False. Statement -1 isFalse, Statement -2 is True Answer :a

Discussion

21.	If int_(1)^(0) (x)/(x+1+e^(x))dx is equal to -lnk then find the value of k.
Answer» SOLUTION :N//A

Discussion

22.	Let a=hati+hatj+hatk, b=hati-hatj+2hatk and c=x hati+(x-2) hatj-hatk. If the vector c lies in the plane of a and b, then x equal to
Answer» 0 1 `-4` `-2` SOLUTION :SINCE, givenvectorsa, b and c are coplanar. `:. \|{:(1,1,1),(1,-1,2),(x,x-2,-1):}\| = 0` `rArr 1{1-2(x-2)} -1(-1-2X)+ 1(x-2 +x) = 0` `rArr1 -2x + 4 + 1 + 2x + 2x- 2= 0` `rArr 2x = - 4` `x = - 2`

Discussion

23.	A bag contains 4 brown and 5 white balls. A man pulls two balls at random without replacement. The probability that the man gets both the balls of the same colour is
Answer» `(5)/(108)` `(1)/(6)` `(5)/(18)` `(4)/(9)` Answer :D

Discussion

24.	The solution of the differential equation xy' = 2xe^(-y//x) + y is
Answer» `E^(y//x) \| LN \|CX\| = 0` `e^(-y//x) = x+c` `e^(y//x) = ln \|cx\|` `e^(y//x) = 2lx \|cx\|` ANSWER :D

Discussion

25.	For the decomposition of NH_(3)(g) at standard state. 2NH_(3)(g) rarr N_(2)(g) + 3H_(2)(g) [Assume that Delta H^(@) and DeltaS^(@) of reaction remain constant, 'T' represents temperature in Kelvin scale & K_(P) represents equilibrium constant.] Choose the correct statement(s).
Answer» Graph between `lnK_(P)` VS `1//T` is a straiht line with POSITIVE slope and positive intercept. Graph between `lnK_(P)` vs `1//T` is a straiht line with negative slope and positive intercept. `K_(P)` increases with increases in temperature At very low temperature REACTION is non-spontaneus. Answer :B::C::D

Discussion

26.	Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Answer» ANSWER :`(1)/(3)`

Discussion

27.	(ii) Find the coefficient of x^(n) in the expansion of (x)/((x-1)^(2)(x-2)) in which the expansion is valid.
Answer» ANSWER :`1-N- (1/2)^(n)`

Discussion

28.	A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^(3) . If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
Answer» ANSWER :RS 1000

Discussion

29.	obtain the equation of hyperbola in each of the following cases: centre at (0,0) conjugate axis along x-axis of length 6 and eccentricity 2.
Answer» SOLUTION :Here 2b=6 `IMPLIES` b=3 , c/a=2 or ca=2 We have `a^2+b^2=c^2` or `a^2+9=4a^2` OR `a^2=3` `THEREFORE` EQN of HYPERBOLA is `y^2/a^2-x^2/b^2=1 or `y^2/3-x^2/9=1`

Discussion

30.	Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = C tan^(-1) x
Answer» SOLUTION :`y = C tan^(-1) x implies y/tan^(-1) x = C implies d/dx (y/(tan^(-1) x)) = 0

Discussion

31.	Evaluation of definite integrals by subsitiution and properties of its : If F(x)=f(x)+f((1)/(x)) then where f(x)=int_(1)^(x)(logt)/(1+t)dt,F(e)=...........
Answer» `(1)/(2)` 0 1 2 Answer :A

Discussion

32.	Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = A sec x
Answer» SOLUTION :`y = A SEC X` Then dy/dx = A sec x TAN x = y tan x

Discussion

33.	Evalute the following integrals int x^(2)cos xdx
Answer» ANSWER :`(x^(2) - 2)`SINX + 2X cos x + C

Discussion

34.	Iff : R to Ris defind by f(X)= {:{((x-2)/(x^(2) -3x +2 ) , " if x in R - {1,2}) , ( 2, ifx =1) , (1 , if x =2 ):}
Answer» 0 `-1` 1 `1/2` ANSWER :B

Discussion

35.	If the normals to x^(2)/a^(2)+y^(2)/b^(2)=1 at the ends of the choeds lx+my=1 and l' x+m'y=1.
Answer» ANSWER :A::B

Discussion

36.	Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = a cos x + b sin x
Answer» SOLUTION :y=a COS X+bsin x dy/dx=-a SIN x+b cos x `(d^2y)/dx^2=-acosx-b sinx=-y` `RARR(d^2y)/dx^2+y=0`

Discussion

37.	(C_0)/(1) + (C_2)/(3) + (C_4)/(5) + ……+(C_16)/(17) =
Answer» `(2^15)/(14)` `(2^16)/(17)` `(2^15)/(16)` `(2^20)/(22)` ANSWER :B

Discussion

38.	Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = A x^2 + B x
Answer» SOLUTION :`y=Ax^2+Br` `RARR y/x=Ax+B` `rArr(xy_i-y)/x^2=A`rArr x^2((xy_2+y_1-y_1)-(xy_1-y)cdot2x)/x^4=0` `x^2y_2-2xy_1+2y=0`

Discussion

39.	Find the equation of a curve passing through the origin given that the slope of thetangent to the curve at any point (x, y) is equal to the sum of the coordinates ofthe point.
Answer» ANSWER :`X + y + 1= E^(x)`

Discussion

40.	int_(0)^(1) x sin^(-1) x dx =
Answer» `PI/8` `pi/4` `pi` `pi/2` ANSWER :A

Discussion

41.	Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. a x^2 + b y = 1
Answer» SOLUTION :`ax^2+by=1rArry=-a/bx^2+1/b` `rArr dy/dx=-(2A)/bxrArr(dy/dx)/x=-(2a)/b` Again DIFFERENTIATING we GET `(d^2y)/dx^2cdotx-dy/dx=0`

Discussion

42.	Differentiate sqrt((x-3) (x^(2) + 4))/sqrt((3x^(2) + 4x + 5)) w.r.t x.
Answer» Answer :`(1)/(2) [(1)/((x-3)) + (2X)/(x^(2) + 4)-(6X + 4)/(3x^(2) + 4x + 5)]`

Discussion

43.	The resultant of forces vecP and vec Q is vecR . If vec Q is doubled the vecR is doubled . If the direction of vec Q is reversed , then vec R is again doubled , then P^2 : Q^2 : R^2is
Answer» `2:3:1` `3:1:1` `2:3:2` `1:2:3` ANSWER :C

Discussion

44.	If y=sin^(-1)x+sin^(-1)sqrt(1-x^2), find (dy)/(dx)in each of thefollowing cases: (i) x in (0,\ 1)(ii) x in (-1,\ 0)
Answer» ANSWER :`(DY)/(DX) = 0`

Discussion

45.	Consider two point A(2,5) and B(3,4) in the XY plane . P is a point divides the line segment AB externally in the ratio 2:5 .Find the co ordinate of P.
Answer»

Discussion

46.	Number of binary opertions on the set {a,b} are
Answer» 10 16 20 8 Answer :B

Discussion

47.	The solution of the differential equation y dx + (x+ x^(2)y) dy = 0 is
Answer» `(-1)/(XY) = C` log y = CX `(1)/(xy) + log y = c` `(-1)/(xy) + log y = c` ANSWER :D

Discussion

48.	{:(,"List-I",,"List-II",),((P),"The number of integral values of k for which the ",(1),1,),(,"equation" sin^(01)x^(-2)+tan^(-1)x^(2)k+1 "has a solution",,,),((Q),"Let" sum_(K=1)^(oo)cot^(-1)(k^(2)/(8))=(p)/(q)pi where" (p)/(q) "is rational in its lowest",(2),2,),(,"form then find" \|q-p\|,,,),((R),"If" tan^(-1)(sin^(2)theta-2 sin theta +3)+cot^(-1) (5^(sec^(2)_(y)+1)=(pi)/(2), then the",(3),3,),(,"value of" cos^(2)y-sin theta,,,),((S),"Minimum positive integral value of x such that f(x)",(4),0,),(,"is defined" f(x)=sqrt(sec^(-1)((1-\|x\|)/(2))),(4),0,):}
Answer» `{:(P,Q,R,S),(3,1,2,2):}` `{:(P,Q,R,S),(4,3,1,4):}` `{:(P,Q,R,S),(1,3,2,3):}` `{:(P,Q,R,S),(1,4,3,2):}` Solution :Let `f(x) =sin^(-1)x^(2)+TAN^(-1)x^(2)`is `UARR` even function and domain is `[-1.1]` `f(x)\|._(min) =f(0)=0` `f(x)\|_(min) =f(1) =(pi)/(2)+(pi)/(4)=(3pi)/(4)` `f(x)in[0.(3pi)/(4)]` `if K=0,2k+1=1`, possible if `k=1, 2k +1=3gt(3pi)/(4)` Not possible i.e only one integral value of k is possible (Q) `T_(K)tan^(-1)((8)/(k^(2)))=tan^(-1),(2)/(k^(2)/(4))=tan^(-1),(2)/(1+((k)/(2)-1)((k)/(2)+1))` `T_(K)=tan^(-1)((k)/(2)+1)-tan^(-1)((k)/(2)-1)` `T_(1)=tan^(7).(3)/(2)-tan^(-1)((-1)/(2))` `T_(2)=tan^(-1).(4)/(2)-tan^(-1)0` `T_(3)=tan^(-1).(5)/(2)-tan^(-1).(1)/(2)` `T_(4)=tan^(-1).(6)/(2)-tan_(-1),(2)/(2)` `T_(5)=tan^(-1).(7)/(2)-tan^(-1).(3)/(2)` : : `Sum =` `2pi-(-tan^(-1).(1)/(2)+tan_(-1)/(2)+tan^(-1)1)=(7pi)/(4)impliesp=7,q=4` `(2pi us sum of last four term)` we know `tan^(-1)x + cot^(-1) y=(pi)/(2)impliesx=y` according to question `underset([2.6])ubrace((sintheta-1)^(2)+2)=underset(ge6)ubrace(5sec^(2)y+1)` Only possible `implies sintheta -1 =-1` `sin theta=-1` ` sec^(2)y=1implies cos^(2)y=1` `cos^(2)y-sin theta=2` `(S) sec^(-1)( (1-\|x\|)/(2))ge0implies (1-\|x\|)/(2)ge1` or `(1-\|x\|)/(2)le-1` `1\|x\|ge2 or 3le\|x\|` `-1ge\|x\|0r x epsilon (-oo,-3]uu[3,oo)` Not possible least positve integer is 3

Discussion

49.	The number of ways in which 6 different nuts can be used in machine left without the nut is
Answer» 540 180 270 None of these Answer :A

Discussion

50.	The triangle formed by the points 1 , (1 +i)/(sqrt2) and ias vertices in the Argand diagram is
Answer» scalane EQUILATERAL ISOSCELES RIGHT angled ANSWER :C

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

A speaks truth in 60% of the cases and B in 70% of the cases. What is the probability that (i) both speak truth (ii) both speak lie (iii) their statements about an incident do not match.

Show that the relation R in the set A={x in Z :0lexle12} is given by R={(a,b) : |a-b|" is a multiple of 4"} is an equivalence relation .

If Q is the foot of the perpendicular from a point p on the parabola y^(2)=8(x-3) to its directrix. S is an equilateral triangle then find the lengh of side of the triangle.

Evaluate : intsqrt((x-5)(x-7))dx

Integrate the following functions 1/sqrt((2-x)^2+1)

Find the values of the following integrals (i) int_(0)^(pi/2) sin^(4)x dx

The sum to n terms of the series 4/3 + 10/9 + 28/27 + ……. is

When the origin is shifted to (-1,2) by the translation of axes, find the transformed equation x^2+y^2+2x-4y+1=0.

Letf(x) ={{:( (a(1-x sinx) + b cccos x+5)/(x^(2)),xlt0),( 3, x=0),( {1+((p(x))/(x^(2)))}^(1//x),xgt0):} wherep(x)is a cubic functionand f is continuous at x=0,if theleading coefficient of p (x)is positive, thentheequation p(x)= b has

The pair of lines lx^(2) + 2(l + m) xy + my^(2) = 0 lies along two diameters of a circle and divides the circle into 4 sectors. If the area of bigger sector is 5 times the area of smaller sector, then (l m)/((l+m)^(2)) =

Letf(x) ={{:( (a(1-x sinx) + b cccos x+5)/(x^(2)),xlt0),( 3, x=0),( {1+((p(x))/(x^(2)))}^(1//x),xgt0):} wherep(x)is a cubic functionand f is continuous at x=0,thevalueofp''(0)is

Integrate the function is exercise. sqrt(x^(2)+4x+6)

Letf(x) ={{:( (a(1-x sinx) + b cccos x+5)/(x^(2)),xlt0),( 3, x=0),( {1+((p(x))/(x^(2)))}^(1//x),xgt0):} wherep(x)is a cubic functionand f is continuous at x=0,the range of functiong(X) =3a sinx-b cosx is

Two numbers are selected at random from 1,2,3,……100 without replacement. Find the probability that the minimum of the two numbers is less than 70.

The points collinear (1,-2,-3) and (2,0,0) among the following is

Find the real values of theta , such that (3 + 2i sin theta)/(1 - 2i sin theta) is purely Imaginary

f(x) = {(3",","if" 0 le x le 1),(4",","if" 1 lt x lt 3),(5",","if" 3 le x le 10):}

Find the sum of the degree and the order of the differential equation: y = (x-(2y)/((dy)/(dx)))(((2y)/(dy))/(dx))^(2).

Statement -1: If a,b,c are the integers such that a+b+cle8, then the number of possible value of the ordered triplets (a,b,c,) is 56. because Statement -2: The number of ways in which n identical things can be distributed into r different groups is .^(n-1)C_(r-1).

If int_(1)^(0) (x)/(x+1+e^(x))dx is equal to -lnk then find the value of k.

Let a=hati+hatj+hatk, b=hati-hatj+2hatk and c=x hati+(x-2) hatj-hatk. If the vector c lies in the plane of a and b, then x equal to

A bag contains 4 brown and 5 white balls. A man pulls two balls at random without replacement. The probability that the man gets both the balls of the same colour is

The solution of the differential equation xy' = 2xe^(-y//x) + y is

For the decomposition of NH_(3)(g) at standard state. 2NH_(3)(g) rarr N_(2)(g) + 3H_(2)(g) [Assume that Delta H^(@) and DeltaS^(@) of reaction remain constant, 'T' represents temperature in Kelvin scale & K_(P) represents equilibrium constant.] Choose the correct statement(s).

Two dice are thrown simultaneously. If X denotes the number of sixes, find the﻿ expectation of X.﻿

(ii) Find the coefficient of x^(n) in the expansion of (x)/((x-1)^(2)(x-2)) in which the expansion is valid.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^(3) . If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

obtain the equation of hyperbola in each of the following cases: centre at (0,0) conjugate axis along x-axis of length 6 and eccentricity 2.

Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = C tan^(-1) x

Evaluation of definite integrals by subsitiution and properties of its : If F(x)=f(x)+f((1)/(x)) then where f(x)=int_(1)^(x)(logt)/(1+t)dt,F(e)=...........

Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = A sec x

Evalute the following integrals int x^(2)cos xdx

Iff : R to Ris defind by f(X)= {:{((x-2)/(x^(2) -3x +2 ) , " if x in R - {1,2}) , ( 2, ifx =1) , (1 , if x =2 ):}

If the normals to x^(2)/a^(2)+y^(2)/b^(2)=1 at the ends of the choeds lx+my=1 and l' x+m'y=1.

Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = a cos x + b sin x

(C_0)/(1) + (C_2)/(3) + (C_4)/(5) + ……+(C_16)/(17) =

Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. y = A x^2 + B x

Find the equation of a curve passing through the origin given that the slope of thetangent to the curve at any point (x, y) is equal to the sum of the coordinates ofthe point.

int_(0)^(1) x sin^(-1) x dx =

Form the defferentialequation by eliminating the arbitrary constants in each of the following cases. a x^2 + b y = 1

Differentiate sqrt((x-3) (x^(2) + 4))/sqrt((3x^(2) + 4x + 5)) w.r.t x.

The resultant of forces vecP and vec Q is vecR . If vec Q is doubled the vecR is doubled . If the direction of vec Q is reversed , then vec R is again doubled , then P^2 : Q^2 : R^2is

If y=sin^(-1)x+sin^(-1)sqrt(1-x^2), find (dy)/(dx)in each of thefollowing cases: (i) x in (0,\ 1)(ii) x in (-1,\ 0)

Consider two point A(2,5) and B(3,4) in the XY plane . P is a point divides the line segment AB externally in the ratio 2:5 .Find the co ordinate of P.

Number of binary opertions on the set {a,b} are

The solution of the differential equation y dx + (x+ x^(2)y) dy = 0 is

The number of ways in which 6 different nuts can be used in machine left without the nut is

The triangle formed by the points 1 , (1 +i)/(sqrt2) and ias vertices in the Argand diagram is

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.