28278 + Interview Questions in Maths in Class 12

1.	Evaluate the following (iii) int_(0)^(5)x[x]dx
Answer» ANSWER :35

Discussion

2.	For the binomial distribution (a + p)^(n) = ((1)/(2) + (1)/(2))^(10) if A = P (x = 3 ) , B = P (x = 6) , C = P (x = 9) then the ascending order of A , B , C is
Answer» C , A , B A , B , C A , C , B B , C , A ANSWER :A

Discussion

3.	Find the direction cosines of the unit vector perpendicular to the plane vecr.(6hati-3hatj-2hatk)+1=0 passing through the origin.
Answer» ANSWER :`(-6)/(7),3/7,2/7`

Discussion

4.	Evaluate: int _( 0 ) ^(pi) e ^(\|cos x\|)( 2 sin ((1)/(2)cos x )_+ 3 cos ((1)/(2)cos x ))sin x dx
Answer» Answer :`(24)/(5) [ E cos ""1/2 + e SIN ""1/2 -1]`

Discussion

5.	If 15 boys of different ages are distributed into 3 groups of 4,5, and 6 boys randomly then the probability that three youngest boys are in different groups is
Answer» `24/91` `71/91` `67/91` `20/91` ANSWER :A

Discussion

6.	If n is an integer then show that (1 + cos theta +i sin theta)^(n) + (1 + cos theta - i sin theta )^(n)= 2^(n+1) cos ^(n) ( theta//2) cos ((n theta)/2).
Answer»

Discussion

7.	Leta = hati+ hatj+ hatkand letr beavariablevectorsuchthatr . "" hati, r."" hatj , r .""Kare positiveintegersifr.ale12thenthe numberof valuesofr is
Answer» `""^(12)C_(9)-1` `""^(12 )C_(3)` `""^(12)C_(4)` none Answer :B

Discussion

8.	If f (x) =\|sin x-\|cos x \|\|, then f '((7pi)/(6))=
Answer» `(sqrt3+1)/(2)` `(1-sqrt3)/(2)` `(sqrt3-1)/(2)` `(-1-sqrt3)/(2)` ANSWER :C

Discussion

9.	Solve the differential equation : (dy)/(dx) =1 + x^(2)+y^(2)+x^(2)y^(2) .
Answer» ANSWER :`TAN^(-1)y =X+(x^(3))/(3)+c`

Discussion

10.	Prove that :int_(0)^(1) (log x)/(sqrt(1-x^(2)))dx=-(pi)/(2)log 2
Answer»

Discussion

11.	Evaluate the following determinants. [[1,2,3],[1,2,3],[3,4,5]]
Answer» SOLUTION :`[[1,2,3],[1,2,3],[3,4,5]]=0 (becauseR_1=R_2)`

Discussion

12.	(i) ifA=[{:(1,-4),(3,1):}]and b=[{:(1,0,5),(-2,4,3):}],then show that : (AB)'=B'A' implies(1)/(2)(A-A')=[{:(0,3//2,-4),(-3//2,0,7//2),(4,-7//2,0):}] Now A=(1)/(2)(A+A')+(1)/(2)(A-A') implies[{:(2,0,-4),(-3,1,5),(4,-2,3):}]=[{:(2,-3//2,0),(-3//2,1,3//2),(0,3//2,3):}]+[{:(0,3//2,-4),(-3//2,0,7//2),(4,-7//2,0):}](ii) if A=[{:(2,3),(0,1):}]and B=[{:(3,4),(2,1):}],then prove that : (AB)'=B'A'
Answer» SOLUTION :N/a

Discussion

13.	Show that the operation * on Z define by a* b =a+b+1 for all a,b in Z Satisfies (i) the closure property (ii) the associative law and (iii) the commutative law ltrbgt (iv) find the identity element inZ (v) what is the inverse iof an element a in Z ?
Answer» Solution :(i) closure porperty LET a `in` A ,b in Z then ab=a+b+1 Nowa `in` Z , b in Z `RARR` a+b in Z `rarr` a+b +1 `in` Z `therefore` on Z satisfies the closure propety (ii) Associative law For all a,b,c `in` Z we have (ab)c=(a+b+1)c =(a+b+1)+c+1 =a+b+c+2 a(bc)=a(b+c+1) =a+(b+c+1)+1 =a+b+c+2 `therefore` (ab)c=a(bc) (iii) commutative law For all a,b `in` Z we have ab=a+b+1 ltbrgeb+a+1 [`therefore` a+b+=b+1] =ba (iv) Existence of identity element Let e be the identity element in Z Then a * e =a `rarr` a+e+1=a `rarr` =-1 Thus -1 in Z is the identity element for * (V) Existence of inverser Let a in Z and let ISTS iverse e be b then a*b=-1 `rarr` a+b+1=-1 `rarr` b =-(2+a) clearly 2`in` Z a in Z `rarr` -(2+a) `in` Z Thus each a in Z has -(2+a)in Z as its INVERSE

Discussion

14.	Radius of the circle vec(r_(2))+vecr(2hati-2hatj-4hatk)-19=0. vecr.(hati-2hatj+2hatk)+8=0, is
Answer» 5 4 3 2 Answer :B

Discussion

15.	Illustrate the use of all connectives and the modified 'not' in five separate examples of propositions
Answer» Solution :Conjunction :HARI and Madhu are two friends. DISJUNCTION :Hari or Madhu STAND first in their CLASS. Implication : If I will be the king , then I will you my queen. Double Implication . I will love you If you will love me. Negation. I don.tlike her.

Discussion

16.	A particle performing S.H.M undergoes displacement of A//2 (where A = amplitude of S.H.M.) in one second. At t = 0 be the particle was located at either position or mean position. The time period of S.H.M. can be : (consider all possible cases)
Answer» 12s 2.4 6s 1.2s Answer :A::B::C::D

Discussion

17.	A survey shows that in a city that 63% of the citizens like tea where as 76% like coffee. If x% like both tea and coffee, then
Answer» `x=39` `x=63` `39le x le 63` `30 le x le 80` ANSWER :C

Discussion

18.	If omega is an imaginary cube root of unity, then the value of (2 - omega )(2 - omega^(2) ) + 2(3 - omega )(3 - omega^(2)) + .... .... + (n - 1)(n - omega)(n -omega^(2)) is
Answer» `(N^(2))/4(n+1)^(2)-n` `(n^(2))/4(n+1)^(2)+n` `(n^(2))/4(n+1)^(2)` `(n^(2))/4(n+1)^(2)-n` ANSWER :A::D

Discussion

19.	A plane pi makes intercepts 3 and 4respectivelyon z - axisand x- axis . If pi is parallel to y - axis , then its equation is
Answer» `3x+4Z = 12` `3Z + 4X = 12` `3y + 4z = 12` `3z +4y = 12` ANSWER :A

Discussion

20.	Find the values of each of the following : If "tan"^(-1)(x-1)/(x-2)+"tan"^(-1)(x+1)/(x+2)=(pi)/4, then find the value of x.
Answer» ANSWER :`+-1/(SQRT(2))`

Discussion

21.	When 9th termof AP is divided by its 2nd term then quotient is 5 and when 13th term is divided by 6th term then quotient is 2 and Remainder is 5 then find first term of AP.
Answer» Solution :given `impliesT_(0)=5T_(2)` &` T_(13)=2T_(6)+5` so `a+8d=5(a+d)` & `a+12d=2(a+5d)+5` `impliesa=3`

Discussion

22.	Differentiate(a^x-b^x)/x
Answer» SOLUTION :`y=(a^X-b^x)/x` `(DY)/(DX)=(d/dx(a^x-b^x)cdotx-(a^x-b^x)cdotd/(dx)(x))/x^2` `((a^xlna-b^xlnb)x-(a^x-b^x))/x^2`

Discussion

23.	Determine the truth of falsity of the If A is a proper subset of B and B is a subset of C Then A is a proper subset of C propositions with reasons.
Answer» SOLUTION :If A is a proper subset of B and B is a subset of C, then A is a proper subset of C.It is TRUE, as `X in A impliesx in B (:.A sub B)`and x in `B implies x in C(:.B sub C) x in A impliesx in C MEANS A sub C`.

Discussion

24.	Differentiate 2sqrtcot(x^2) w.r.t.x
Answer» SOLUTION :`d/dx(2sqrtcot(x^2))=2d/dx[sqrtcot(x^2)]=2XX1/(2sqrtcot(x^2))d/dx(COT(x^2))=1/sqrtcot(x^2)xx-cosec^2(x^2)=(-2xcosec^2(x^2))/(sqrtcot(x^2)`

Discussion

25.	If theequationwhoserootsarep timestherootsofx^4 +2x^3 +46x^2 +8x+16 =0isareciprocalequationthen p=
Answer» 2 3 `+- 1/2` `+-1/3` ANSWER :C

Discussion

26.	Find the equation of line passing through the given points using determinants (5,-1),(5,3)
Answer» ANSWER :x=5

Discussion

27.	Integrate the rational functions x/((x-1)^(2)(x+2))
Answer» ANSWER :`(2)/(9) log\|x-1\|+(1)/(3)((x-1)^(-2+1))/(-1)-(2)/(9)log\|x+2\|+c`

Discussion

28.	Constract the composition table/multiplication table for the binary operation * defined on {0,1,2,3,4}by a**b =axxb ("mod" =5). Find the identity element if any. Also find the inverse elements of 2 and 4.
Answer» SOLUTION :`A ={0,1,2,3,4}` `a**b =axx b "mod" 5` As 3rd row is identical to the FIRST row we have 1 is the IDENTITY CLEARLY `2^(-1) =3 "and" 4^(-1) =4.`

Discussion

29.	Match the following {:("I.","The approximate value of"(2001)^4,"a",04983),("II".,"The approximate value of"(10002)^(3000),b,2.02),("III.","The approximate value of"sqrt(408),c,16),("IV.","The approximate value of" 1/(3sqrt(808)),d,16*032):}
Answer» C,B,d,a d,c,b,a a,c,b,d c,b,d,a Answer :C

Discussion

30.	y = cos^(-1)((2x)/(1+x^2)), -1 lt x lt 1.
Answer» ANSWER :`(-2)/(1+x^(2))`

Discussion

31.	A person observes the top of a tower from a point A on the ground. The elevation of the tower from this point is 60^(@). He moves 60 m in the direction perpendicular to the line joining A and base of the tower. The angle of elevation of the tower from this point is 45^(@). Then, the height of the tower (in meters) is
Answer» `60sqrt(3/2)` `60sqrt(2)` `60sqrt(3)` `60sqrt(2/3)` ANSWER :A

Discussion

32.	1+(1+5)/(2!) +(1+5+5^2)/(3!) + ......oo =
Answer» `1/4 E(e^4 +1)` `1/4 e(e^4 -1)` `1/3 e(e^4 +1)` `1/3 e(e^4 -1)` ANSWER :B

Discussion

33.	From the top of a tower 100 m heigh, the angles of depression of two objects 200 m apart on the horizontal plane and in a line passing through the foot of the tower and on the same side of the tower are 45^(@) - A and 45^(@) + A, then angle A is equal to
Answer» `15^(@)` `22.5^(@)` `30^(@)` `35^(@)` ANSWER :B

Discussion

34.	Find the derivative of the following functions 'ab initio', that is, using the definition.t(t-1)
Answer» Solution :LET `y=t(t-1)=t^2-t` Then `dy/dt=lim_(DELTAT TO0)([(t+deltat)^2-(t-deltat)]-[t^2-t])/(deltat)` `=lim_(deltat to0)(t^2+2tdeltat+deltat^2-t-deltat-t^2+t)/(deltat)` `=lim_(deltat to0)[(2t-1)+deltat]=2t-1`

Discussion

35.	Evaluate the integral (a)int_(-pi)^(pi) f(x) cos nx dx (b) int_(-pi)^(pi) f(x) sin nx dxf : (1) f(x) is an evenfunction ,(2) f(x) is an odd function.
Answer» ANSWER :(1)if f (x)is an even function, then `int_(-pi)^(pi) f(x) cos NX DX = 2 int_(0)^(pi) f(x) cos nx dx, and int_(-pi)^(pi) f(x) sin x dx = 0 ` (2)If f(x) is an odd function, then `int_(-pi)^(pi) f(x) cos nx dx = 0`, and`int_(-pi)^(pi)f(x) sin nx dx = 2 int_(0)^(pi) f(x)`sin n dx

Discussion

36.	Find the product of the perpendiculars drawn from the point (x_1,y_1) on the lines ax^2+2hxy+by^2=0
Answer» ANSWER :`=(\|ax_1^2+2hx_1y_1+by_1^2\|)/SQRT({(a-b)^2+4h^2})`

Discussion

37.	Find dy/dx,x=a cos'theta',y=a sin theta.
Answer» SOLUTION :`x=a cos theta,y=a SIN theta ` Then `DX/(d theta)=-a sin theta ` `DY/(d theta)=a cos theta fracdy/d theta` `dy/dx=(dy/(d theta))/((dx)/(d theta))=(a COSTHETA)/(-a sin theta) =-cot theta`

Discussion

38.	Write the following functions in the simplest form : tan^(-1)((cos x -sin x)/(cos x + sin x)), -pi/4 lt x lt (3pi)/4
Answer» SOLUTION :`tan^(-1)((cos X - sin x)/(cos x + sin x)) = tan^(-1)((cos x(1-tanx))/ (cos x(1+tanx)))` `tan^(-1)((1-tanx)/(1+tanx))= tan^(-1)tan(pi/4-x) = pi/4-x`

Discussion

39.	If int_(-3)^(2) f(x)dx= 7/3 and int_(-3)^(9) f(x)dx= - 5/6 then int_(2)^(9) f(x)dx=
Answer» `3/2` `- 3/2` `- 19/6` `19/6` ANSWER :C

Discussion

40.	If Delta ABC, A' B' C' are such that B=B', A+A'=180^@, then aa'=
Answer» `aa'+bb'` bb'+cc' aa'+cc' None ANSWER :B

Discussion

41.	There are (n + 1) white and (n + 1) black balls, each set numbered 1 to n + 1. The number of ways in which the balls can be arranged in a row so that adjacent balls are of different colours, is
Answer» (2N + 1) 2(2n)! 2[(N + 1)!] `2[(n+1)!]^(2)` ANSWER :D

Discussion

42.	Which one of the following vectors of magnitude sqrt(51) makes equal angles with three vectors vec(a)=(hat(i)-2hat(j)+2hat(k))/(3), vec(b)=(-4hat(i)-3hat(k))/(5) and vec(c)=hat(j)?
Answer» `5hat(i)-hat(j)-5hat(K)` `5hat(i)+hat(j)+5hat(k)` `-5hat(i)-hat(j)+5hat(k)` `5hat(i)+5hat(j)-k` Solution :Let `vec(p)=XHAT(i)+yhat(j)+zhat(k)` is a vector of magnitude `sqrt(51)` `:.sqrt(x^(2)+y^(2)+z^(2))=sqrt(51)` `rArr x^(2)+y^(2)+z^(2)=51 " " ...(i)` Let `vec(p)` makes equal angle `theta` with and `vec(a), vec(B) and vec(c)`. `:.vec(p).vec(a)=\|vec(p)\|.\|vec(a)\|cos theta` `:. cos theta = (vec(p).vec(a))/(\|vec(p)\|\|vec(a)\|)` Similarly, `cos theta = (vec(p).vec(b))/(\|vec(p)\|\|vec(b)\|)` and `cos theta = (vec(p).vec(c))/(\|vec(p)\|\|vec(c)\|)` `:. (vec(p).vec(a))/(\|vec(p)\|\|vec(a)\|)=(vec(p).vec(b))/(\|vec(p)\|\|vec(b)\|)=(vec(p).vec(c))/(\|vec(p)\|\|vec(c)\|)` `rArr ((1)/(3)(x-2y+2z))/(sqrt(x^(2)+y^(2)+z^(2))(1)/(3)sqrt(1+4+4))=((1)/(5)(-4x-3z))/(sqrt(x^(2)+y^(2)+z^(2))(1)/(5)sqrt(16+9))=(y)/(sqrt(x^(2)+y^(2)+z^(2))sqrt(1))` `rArr (x-2y+2z)/(3sqrt(x^(2)+y^(2)+z^(2)))=(-4x-3z)/(5sqrt(x^(2)+y^(2)+z^(2)))=(y)/(sqrt(x^(2)+y^(2)+z^(2)))` `rArr (x-2y+2z)/(3)=(-4x-3z)/(5)=y` `:. 5(x-2y+2z)=-3(4x+3z)=15Y` `:. 5x-10y+10z=15y and -12x-9z=15y` `rArr 5x-25y+10z=0 and -12x-15y-9z=0` `rArr x-5y+2z=0 and 4x+5y+3z=0` `x-5y+2z=0` `4x+5y+3z=0` `\|(x,y,z),(1,-5,2),(4,5,3)\|=(x)/(-15-10)=(y)/(8-3)=(z)/(5+20)` `(x)/(-25)=(y)/(5)=(z)/(25)` `(x)/(-5)=(y)/(1)=(z)/(5)=k`(let) `:. x=-5k, y=k, z=5k` Now, `x^(2)+y^(2)+z^(2)=51` `:. (-5k)^(2)+k^(2)+(5k)^(2)=51` `rArr 25k^(2)+k^(2)+25k^(2)=51` `rArr 51k^(2)=51` `:. k=pm1` When, k=1, then `x=-5, y=1, z=5` `and vec(p)=5hat(i)+hat(j)+5hat(k)` when k=-1 , then x=5, y=-1, z=-5 `and vec(p)=5hat(i)-hat(j)-5hat(k)`

Discussion

43.	Evalute the following integrals int (1)/(sqrt(5x - 6- x^(2)))dx
Answer» ANSWER :`sin^(-1) (2X - 5)` + C

Discussion

44.	If ""^(n)C_(3)=10, then n is
Answer»

Discussion

45.	Ifalpha, beta are the roots of the equationx^(2) + x + 1 = 0 then prove thatalpha^(4) + beta^(4) + alpha^(-1) beta^(-1) = 0 .
Answer»

Discussion

46.	If radii of two circles are 4 and 3 and distance between centres is sqrt37, then angle between the circles is
Answer» `30^@` `45^@` `60^@` `90^@` ANSWER :C

Discussion

47.	Find the sum of all five digited numbers that can be formed using the digits 1, 2, 3, 7, 9.
Answer» ANSWER :5866608

Discussion

48.	1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+....=
Answer» 1 `log_(e )2` `log_(e )3` `log_(e )4` ANSWER :C

Discussion

49.	If a + b + c = 0, then the equation 3ax^(2) + 2bx + c = 0 has, in the interval (0, 1)
Answer» at leastone ROOT atmost ONE root no root exactly one root ANSWER :A

Discussion

50.	In thegridshownbelow, eachsmallsquarehasa side lengthof 1 unit. Intheshadedregion , eachvertex lieson avertexof a small square. Whatis the area, in squareunits , of theshaded region ?
Answer» 35 25 24 19 Answer :D

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Evaluate the following (iii) int_(0)^(5)x[x]dx

For the binomial distribution (a + p)^(n) = ((1)/(2) + (1)/(2))^(10) if A = P (x = 3 ) , B = P (x = 6) , C = P (x = 9) then the ascending order of A , B , C is

Find the direction cosines of the unit vector perpendicular to the plane vecr.(6hati-3hatj-2hatk)+1=0 passing through the origin.

Evaluate: int _( 0 ) ^(pi) e ^(|cos x|)( 2 sin ((1)/(2)cos x )_+ 3 cos ((1)/(2)cos x ))sin x dx

If 15 boys of different ages are distributed into 3 groups of 4,5, and 6 boys randomly then the probability that three youngest boys are in different groups is

If n is an integer then show that (1 + cos theta +i sin theta)^(n) + (1 + cos theta - i sin theta )^(n)= 2^(n+1) cos ^(n) ( theta//2) cos ((n theta)/2).

Leta = hati+ hatj+ hatkand letr beavariablevectorsuchthatr . "" hati, r."" hatj , r .""Kare positiveintegersifr.ale12thenthe numberof valuesofr is

If f (x) =|sin x-|cos x ||, then f '((7pi)/(6))=

Solve the differential equation : (dy)/(dx) =1 + x^(2)+y^(2)+x^(2)y^(2) .

Prove that :int_(0)^(1) (log x)/(sqrt(1-x^(2)))dx=-(pi)/(2)log 2

Evaluate the following determinants. [[1,2,3],[1,2,3],[3,4,5]]

Show that the operation * on Z define by a* b =a+b+1 for all a,b in Z Satisfies (i) the closure property (ii) the associative law and (iii) the commutative law ltrbgt (iv) find the identity element inZ (v) what is the inverse iof an element a in Z ?

Radius of the circle vec(r_(2))+vecr(2hati-2hatj-4hatk)-19=0. vecr.(hati-2hatj+2hatk)+8=0, is

Illustrate the use of all connectives and the modified 'not' in five separate examples of propositions

A particle performing S.H.M undergoes displacement of A//2 (where A = amplitude of S.H.M.) in one second. At t = 0 be the particle was located at either position or mean position. The time period of S.H.M. can be : (consider all possible cases)

A survey shows that in a city that 63% of the citizens like tea where as 76% like coffee. If x% like both tea and coffee, then

If omega is an imaginary cube root of unity, then the value of (2 - omega )(2 - omega^(2) ) + 2(3 - omega )(3 - omega^(2)) + .... .... + (n - 1)(n - omega)(n -omega^(2)) is

A plane pi makes intercepts 3 and 4respectivelyon z - axisand x- axis . If pi is parallel to y - axis , then its equation is

Find the values of each of the following : If "tan"^(-1)(x-1)/(x-2)+"tan"^(-1)(x+1)/(x+2)=(pi)/4, then find the value of x.

When 9th termof AP is divided by its 2nd term then quotient is 5 and when 13th term is divided by 6th term then quotient is 2 and Remainder is 5 then find first term of AP.

Differentiate(a^x-b^x)/x

Determine the truth of falsity of the If A is a proper subset of B and B is a subset of C Then A is a proper subset of C propositions with reasons.

Differentiate 2sqrtcot(x^2) w.r.t.x

If theequationwhoserootsarep timestherootsofx^4 +2x^3 +46x^2 +8x+16 =0isareciprocalequationthen p=

Find the equation of line passing through the given points using determinants (5,-1),(5,3)

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

Constract the composition table/multiplication table for the binary operation * defined on {0,1,2,3,4}by a**b =axxb ("mod" =5). Find the identity element if any. Also find the inverse elements of 2 and 4.

Match the following {:("I.","The approximate value of"(2*001)^4,"a",0*4983),("II".,"The approximate value of"(1*0002)^(3000),b,2.02),("III.","The approximate value of"sqrt(4*08),c,1*6),("IV.","The approximate value of" 1/(3sqrt(8*08)),d,16*032):}

y = cos^(-1)((2x)/(1+x^2)), -1 lt x lt 1.

1+(1+5)/(2!) +(1+5+5^2)/(3!) + ......oo =

From the top of a tower 100 m heigh, the angles of depression of two objects 200 m apart on the horizontal plane and in a line passing through the foot of the tower and on the same side of the tower are 45^(@) - A and 45^(@) + A, then angle A is equal to

Find the derivative of the following functions 'ab initio', that is, using the definition.t(t-1)

Evaluate the integral (a)int_(-pi)^(pi) f(x) cos nx dx (b) int_(-pi)^(pi) f(x) sin nx dxf : (1) f(x) is an evenfunction ,(2) f(x) is an odd function.

Find the product of the perpendiculars drawn from the point (x_1,y_1) on the lines ax^2+2hxy+by^2=0

Find dy/dx,x=a cos'theta',y=a sin theta.

Write the following functions in the simplest form : tan^(-1)((cos x -sin x)/(cos x + sin x)), -pi/4 lt x lt (3pi)/4

If int_(-3)^(2) f(x)dx= 7/3 and int_(-3)^(9) f(x)dx= - 5/6 then int_(2)^(9) f(x)dx=

If Delta ABC, A' B' C' are such that B=B', A+A'=180^@, then aa'=

There are (n + 1) white and (n + 1) black balls, each set numbered 1 to n + 1. The number of ways in which the balls can be arranged in a row so that adjacent balls are of different colours, is

Which one of the following vectors of magnitude sqrt(51) makes equal angles with three vectors vec(a)=(hat(i)-2hat(j)+2hat(k))/(3), vec(b)=(-4hat(i)-3hat(k))/(5) and vec(c)=hat(j)?

Evalute the following integrals int (1)/(sqrt(5x - 6- x^(2)))dx

If ""^(n)C_(3)=10, then n is

Ifalpha, beta are the roots of the equationx^(2) + x + 1 = 0 then prove thatalpha^(4) + beta^(4) + alpha^(-1) beta^(-1) = 0 .

If radii of two circles are 4 and 3 and distance between centres is sqrt37, then angle between the circles is

Find the sum of all five digited numbers that can be formed using the digits 1, 2, 3, 7, 9.

1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+....=

If a + b + c = 0, then the equation 3ax^(2) + 2bx + c = 0 has, in the interval (0, 1)

In thegridshownbelow, eachsmallsquarehasa side lengthof 1 unit. Intheshadedregion , eachvertex lieson avertexof a small square. Whatis the area, in squareunits , of theshaded region ?

Match the following {:("I.","The approximate value of"(2001)^4,"a",04983),("II".,"The approximate value of"(10002)^(3000),b,2.02),("III.","The approximate value of"sqrt(408),c,16),("IV.","The approximate value of" 1/(3sqrt(808)),d,16*032):}