28278 + Interview Questions in Maths in Class 12 Page 531 InterviewSolution

26501.	Which of the following propositions is a contradiction ?
Answer» <P>`(~p VV ~Q) vv (p vv ~q)` `(p RARR q) vv (p ^^ ~q)` `(~p ^^ q) ^^ (~q)` `(~p ^^ q) ^^ (~q)` ANSWER :D

Discussion

26502.	Which of the following is not a statement in logic?
Answer» X+0=x, x`in` R 1 - `cos^2theta`=`sin^2theta` for all REAL `THETA` `x^2- 3x-4=0, x `in` R Every SET is a finite set Answer :C

Discussion

26503.	The number of ways can 5 red and 4 white balls be drawn from a bag containing 10 red and 8 white balls is
Answer» `""^(8)C_(5)XX""^(10)C_(4)` `""^(10)C_(5)xx""^(8)C_(4)` `""^(18)C_(9)` `""^(18)C_(5)xx""^(14)C_(4)` Answer :B

Discussion

26504.	Let AP and BQ be two vertical poles at points A and B respectively. If AP = 16 m, BQ = 22m and AB = 20 m, then find the distance of a point R on AB from the point A such that RP^(2)+RQ^(2) is minimum.
Answer» ANSWER :10 METER

Discussion

26505.	What is common amylase, rennin and trypsin ?
Answer» They are produce in stomach These ACT at a PH lower than 7 These all are protein These all are proteolytic enzymes Answer :A

Discussion

26506.	Show that the circles x^(2) + y^(2)-6x -2y + 1=0 , x ^(2) + y^(2) + 2x -8y + 13 =0touch each other find the point of contact and the equation of the common tangent at their point of contact.
Answer» ANSWER :` (5,1) ,4x-7y -13=0 `

Discussion

26507.	If alpha, beta are roots of equation x^(2)-(a-1)x+(a^(2))/(4)=0 and delta, gamma are roots of equation x^(2)-(a+1)x+1=0, A quadrilateral is constructed in real plane by taking vertices as (alpha, beta), (beta, alpha), (delta, gamma), (gamma, delta). Find the value of 'a' such that quadrilateral PQRS never cuts the circle (x-1)^(2)+(y-1)^(2)=a^(2).
Answer» ANSWER :`RARR a LT - 4 - sqrt(26)`

Discussion

26508.	Let Delta = \|{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}\| and the equation px^(3) +qx^(2) +rx+s=0 has roots a,b,c where a,b,c in R^(+) The value of Delta is
Answer» `le 9r^(2)//p^(2)` `ge 27S^(2)//p^(2)` `le 27s^(3)//p^(3)` none of these Solution :Multiplying `R_(1) ,R_(2),R_(3) " by " a,B,c ` respectively ,and thentaking a,b,ccommon from` C_(1),C_(2)" and " C_(3)` we get `Delta = \|{:(-bc,,ab+AC,,ac+ab),(ab+bc,,-ac,,bc+ab),(ac+bc,,bc+ac,,-ab):}\|` Nowusing `C_(2) to C_(2)-C_(1)" and " C_(3) to C_(3) -C_(1) ` and thentaking(ab+bc+ca) common from `C_(2) " and" C_(3)` we get `Delta =\|{:(-bc,,1,,1),(ab+bc,,-1,,0),(ac+bc,,0,,-1):}\|xx (ab +bc +ca)^(2)` Nowapplying `R_(2) to R_(2) +R_(1)` we get `Delta = \|{:(-bc,,1,,1),(ab,,0,,1),(ac+bc,,0,,-1):}\| (ab+bc+ca)^(2)` Expandingalong `c_(2)` we get `Delta =(ab+bc+ca)^(2)[ac+bc+ca)^(2)` `=(ab+bc+ca)^(2)` `=(r//p)^(3) =r^(3)//p^(3)` Now givena,b,care allpositivethen `A.M ge G.M.` `rArr (ab+bc+ac)/(3) ge (abxx bcxx ac)^(1//3)` `" or" (ab+bc+ac)^(3) ge 27A^(2)b^(2)c^(2)` `"or" (ab+bc+ca)^(3) ge 27(s^(2)//p^(2))` if `Delta =27` thenab+bc+ca =3 and GIVENTHAT `a^(2) +b^(2)+c^(2)=3` From `(a+b+c)^(2) =a^(2) +b^(2)+c^(2) +2 (ab+bc+ca)` we have `a+b+c = ne 3` `rArr a+b+c =3` `rArr 3p+q=0`

Discussion

26509.	A(3,2,-1),B(4,1,1,)C(6,2,5) and D(3,3,3) are four points G_(1)G_(2)G_(3)" and "G_(4)respectively are the centroids of the triangle DeltaBCD, DeltaCDA,DeltaDAB,DeltaABC. The point of concurrence of the lines AG_(1),BG_(2),CG_(3)," and "DG_(4) is
Answer» (4,2,2) (2,4,2) (2,2,4) (2,2,2) ANSWER :A

Discussion

26510.	Which of the following is not a statementin logic?
Answer» The SUM of ANGLES of a quadrilateral is `180^@`. EVERY statement has one truth value. `SQRT 3` is an irrational number. x + 5=7, x`in`Q. Answer :D

Discussion

26511.	Match the following
Answer» c,d,a,b b,a,d,c c,a,b,d d,c,b,a Answer :B

Discussion

26512.	Let Delta = \|{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}\| and the equation px^(3) +qx^(2) +rx+s=0 has roots a,b,c where a,b,c in R^(+) the valueof Deltais
Answer» <P>`r^(2)//p^(2)` `r^(3)//p^(3)` `-s//p` NONE of these Solution :Multiplying `R_(1) ,R_(2),R_(3) " by " a,b,c ` respectively ,and thentaking a,b,ccommon from` C_(1),C_(2)" and " C_(3)` we get `Delta = \|{:(-bc,,ab+AC,,ac+ab),(ab+bc,,-ac,,bc+ab),(ac+bc,,bc+ac,,-ab):}\|` Nowusing `C_(2) to C_(2)-C_(1)" and " C_(3) to C_(3) -C_(1) ` and thentaking(ab+bc+ca) common from `C_(2) " and" C_(3)` we get `Delta =\|{:(-bc,,1,,1),(ab+bc,,-1,,0),(ac+bc,,0,,-1):}\|xx (ab +bc +ca)^(2)` Nowapplying `R_(2) to R_(2) +R_(1)` we get `Delta = \|{:(-bc,,1,,1),(ab,,0,,1),(ac+bc,,0,,-1):}\| (ab+bc+ca)^(2)` Expandingalong `c_(2)` we get `Delta =(ab+bc+ca)^(2)[ac+bc+ca)^(2)` `=(ab+bc+ca)^(2)` `=(r//p)^(3) =r^(3)//p^(3)` Now GIVENA,b,CARE allpositivethen `A.M ge G.M.` `rArr (ab+bc+ac)/(3) ge (abxx bcxx ac)^(1//3)` `" or" (ab+bc+ac)^(3) ge 27a^(2)b^(2)c^(2)` `"or" (ab+bc+ca)^(3) ge 27(s^(2)//p^(2))` if `Delta =27` thenab+bc+ca =3 and giventhat `a^(2) +b^(2)+c^(2)=3` From `(a+b+c)^(2) =a^(2) +b^(2)+c^(2) +2 (ab+bc+ca)` we have `a+b+c = ne 3` `rArr a+b+c =3` `rArr 3p+q=0`

Discussion

26513.	Match the following
Answer» C,d,a,B b,a,d,c c,a,b,d d,c,b,a Answer :D

Discussion

26514.	int "log " sqrt(x + 1 ) dx =
Answer» `(1)/(2)` [ (X - 1) LOG (x + 1) - x ] + C `(1)/(2)` [ (x +1) log (x + 1) - x ] + c `(1)/(2) [ x log (x + 1) - (x^(2))/(2) ]+ c` `(1)/(2) [ (x - 1) log (x + 1) - (x^(2))/(2) ]+ c` ANSWER :B

Discussion

26515.	Integrate the following intsin3xdx
Answer» SOLUTION :`INTSIN3XDX` [PUT 3x=t then 3dx=dt or dx=1/3dt `intsintcdot(1/3)dt` 1/3(-cost+C =1/3(COS3X)+C

Discussion

26516.	Construct the switching circuit for the following statement : [p vv (~p^^q)]vv[(~q^^r) vv ~ p].
Answer» ANSWER :`(##NVT_21_MAT_XII_C01_E07_001_A01##)` `(##NVT_21_MAT_XII_C01_E07_001_A02##)`

Discussion

26517.	Find a vector in the direction of the vector (3, -2, 2) which has magnitude 2sqrt(17) units.
Answer» ANSWER :`6hati-4hatj-4hatk`

Discussion

26518.	A ladder rests against the side of a wall and reaches a point that is h meters above the ground. The angle formed by the ladder and the ground is theta^(@). A point on the ladder is k meters from the wall.What is the vertical distance, in meters, from this point on the ladder to the ground?
Answer» `(h-k)TAN THETA^(@)` `(h-k)COS theta^(@)` `h-k SIN theta^(@)` `h-k tan theta^(@)` ANSWER :D

Discussion

26519.	LetA and B be obtuse angles such thatsinA=(4)/(5)andcosB=-(12)/(13).
Answer» `-(63)/(65)` `-(33)/(65)` `(33)/(65)` `(63)/(35)` Solution :`sinA=(4)/(5)andcosB=-(12)/(13)` It is given that A and B are obtuse angle `rArrcosA=+-sqrt(1-sin^(2))A=+-sqrt(1-(16)/(25))=-(3)/(5)` Negative sign is taken for cos A SINCE A being obtuse lies in second quadrant. `sinB=+-sqrt(1-cos^(2)B)=+-sqrt(1-((-12)/(13))^(2))` `=sqrt((169-144)/(169))=(5)/(13)` Positive sign is taken since , sin B is positive in second quadrant . `rArrcosA=(-3)/(5)andsinB=(3)/(13)` `thereforesin(A+B)=sinAcosB+cosAsinB` `=(4)/(5)XX((-12)/(13))+(-3)/(5)xx((5)/(13))=-(48)/(65)-(15)/(65)` `=(-48-15)/(65)=(-63)/(65)`

Discussion

26520.	S is sample space if P(A)= (1)/(3)P(B) and S= A cup B Then P(A) = ....... Where A and B are mutually exclusive events.
Answer» `(1)/(4)` `(1)/(2)` `(3)/(4)` `(3)/(8)` ANSWER :A

Discussion

26521.	The solution of (dy)/(dx) + y = e^(x) is
Answer» `E^(y//x) = CX` `e^(x//y) = cx` `e^(y//x) = CY` `e^(x//y) = cy` ANSWER :B

Discussion

26522.	Differentiate.5^(ln sinx)
Answer» SOLUTION :`y=5^(In SIN X)` `dy/dx=5^(In sin xcdot)In5d/dx(In sin x)`CDOT)In 51/sinxcdotcosx` `5^(In sin x).cdot In 5cdot x`

Discussion

26523.	Find the equation of the circle for which the point given below are the end points of a diameter. (1,1) (2,-1)
Answer» ANSWER :` X^(2) + y^(2) -3X +1 =0`

Discussion

26524.	Find the number of seven digited palindromes that can be formed using 0,1,2,3,4.
Answer» ANSWER :`4xx5^(3)`

Discussion

26525.	C_(1) is a circle with centre at the origin and radius equal to r and C_(2) is a circle with centre at (3r, 0)and radius t 2r. The number of common tangents that can be drawn to the two circles are
Answer» 1 2 3 4 Answer :C

Discussion

26526.	If sin alpha/2 + cos alpha/2=1.4, then: sin alpha cos alpha=
Answer»

Discussion

26527.	Find the order and degree of the following D.E's (i) (d^(2)y)/(dx^(2)) + 2((dy)/(dx))^(2) + 5y = 0 (ii) 2(d^(2)y)/(dx^(2)) = (5+(dy)/(dx))^((5)/(3)) (iii) 1+((d^(2)y)/(dx^(2)))^(2) = [2+((dy)/(dx))^(2)]^((3//2)) (iv) [(d^(2)y)/(dx^(2))+((dy)/(dx))^(3)]^((6/(5)) = 6y (v) [((dy)/(dx))^(2) + (d^(2)y)/(dx^(2))]^((7)/(3)) = (d^(3y))/(dx^(3)) (vi) [((dy)/(dx))^((1)/(2)) + ((d^(2)y)/(dx^(2)))^((1)/(2))]^((1)/(4)) = 0 (vii) (d^(2)y)/(dx^(2)) + p^(2)y = 0 (viii) ((d^(3)y)/(dx^(3)))^(2) -3((dy)/(dx))^(2) - e^(x) = 4
Answer» ANSWER :(i) ordered 2 degree 1 (II) ORDER 2 , degree 3 (iii) order 2, degree 4 (iv) order 2, degree 6 (v) order 3, degree 3 (VI) order 2, degree 1/2 (vii) order 2, degree 1 (viii) order 3, degree 2

Discussion

26528.	Using differentials, find the approximate value of each of the up to 3 places of decimal. (0.009)^((1)/(3))
Answer» ANSWER :`=0.208`

Discussion

26529.	The number of values of x int [-2pi, 2pi] satisfying tanx + cotx2" cosec x" is
Answer» 2 4 6 8 Answer :B

Discussion

26530.	If bar(a),bar(b) and bar( c ) are unit vectors such that bar(a)+bar(b)+bar( c )=bar(0), then the value of bar(a).bar(b)+bar(b).bar( c )+bar( c ).bar(a)=…………
Answer» 1 3 `-(3)/(2)` NONE of these ANSWER :C

Discussion

26531.	The right hand derivative of f(x)= [x] tan (pi x) at a point x=7 is k pi then find the value of k. where [.] is the greatest integer function.
Answer» ANSWER :`K= PI`

Discussion

26532.	Find the area of the triangle with vertices A(3,-1,2),B(1,-1,-3) and C(4,-3,1).
Answer» ANSWER :`(1)/(SQRT(165))(-10hati-7hatj+4hatk)`

Discussion

26533.	int(x^(2//3) + 1)dx
Answer» ANSWER :`(3)/(5)X^(5//3)+x+c`

Discussion

26534.	Show that the opeation * on Q -{1} defined gby a*b=a+b-ab for all a, b in Q- {1} Satifies (i) the closure property (ii) the associative law (iii) the commutative alw (iv) what is theidentity element ? (v)for each a in Q-{1} find the inverse of a
Answer» Solution :(i) closure porperty Let a `in` Q-{1} and b in Q-{1} We know that Q is closed for addition substraction and MULTIPLICATION `therefore`a+b-ab in Q but ab =1 `rarr` a+b-ab=1 `rarr` a(1-b)=(1-b)`rarra`=1 which is a contradication since 1 `in` Q-{1} therefore ab `ne` 1 Thus a `in` Q-{1} ,b `in` Q-{1} `rarr` ab in Q-{1} `therefore` is a binary operation on Q-{1} (II) Associative law Let a,b,c =(a+b-ab)c =(a+b-ab)+c-(a+b-ab)c =(a+b+c)-(ab+BC+ac)+bc And a(bc)=a(b+c-bc) =a(b+c-bc)-a(b+c-bc) =(a+b+c)-(ab+bc+ac)+abc `therefore` (ab)c=a(bc) Hence * is associative (iii) commutative Law Let a,b `in`Q-{1} Then ab=a+b-ab ltrbgt =b+a-ab =ba (iv)Existence of identity element Let e be the identity element Then for all a in Q-{1} we have a* e =a `rarr` a + e - ae =a `rarr` e(1-a)=x `rarre=0 in` Q-{1} Nowa0=a+0-ax0=a And 0a=0+a-0xa=a ltrbgt Thus o is the identity element in Q-{1} Existence of inverse Let a in Q-{1} and `le t a^(-1)=b`then a*b=0 `rarra`+b-ab=0 `rarr` a=ab-b=(a-1)b `RARRB=(a)/(a-1)in Q-{1}` `therefore a^(-1)=(a)/(a-1)in Q-{1}` Thus each a in Q-{1} has its iverse in Q-{1}

Discussion

26535.	If A= {1,2,3...,n}and B ={ a,b}Then , the number of subjection from A into B is .........
Answer» `""^(N)P_2` `2^(n)-2` `2^(n)-1` NONE of these SOLUTION :N/A

Discussion

26536.	If x = (2)/(5) + (1.3)/(2!)((2)/(5))^(2) + (1.3.5)/(3!)((2)/(5))^(3) +…, then x + (1)/(x) =
Answer» `(1+sqrt(5))/(4)` 3 `(5sqrt(5) + 3)/(4)` `(5sqrt(5) - 3)/(4)` ANSWER :D

Discussion

26537.	Find the probability that a leap year will have 53 sundays.
Answer» ANSWER :`(2)/(7)`

Discussion

26538.	If f(x) and g(x) are of degrees 7 and 4 respectively such that f(x)= g(x) q(x)+ r(x) then find possible degrees of q(x) and r(x).
Answer» ANSWER :i.e., `r(x)` is of the form `PX^(3)+qx^(2)+rx+s`

Discussion

26539.	Let be the binary opertion on N defined by a b=H.C.F. of a and b. Is commutative ? Is associative ? Does there exist identity for this binary opertion on N ?
Answer» ANSWER :`` is both COMMUTATIVE and ASSOCIATIVE;` ` does not have any IDENTITY in N

Discussion

26540.	(i) Find the equation of the circle passing through (0,0) and making intercepts 4,3 on X axis and Y-axis respectively. (ii) Find the equation of the circle passing through (0,0) and making intercepts 6 units on X-axis and intercepts 4 units on Y-axis.
Answer» Answer :`X^(2) + y^(2) +- 6x+- 4Y =0`

Discussion

26541.	Evaluate the following integrals: int_0^(pi/4) sin2xdx
Answer» Solution :`int_0^(pi/4) SIN 2x DX = ((-COS 2x)/2)_0^(pi/4)` = `-1/2(cos2x)_0^(pi/4)` =`-1/2 (cos pi/2 - cos 0)` =-1/2(0-1) = 1/2

Discussion

26542.	On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the guests leaving one by one, put on at random, any pair of shoes big enough for their feet, (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there ?
Answer» ANSWER :5

Discussion

26543.	If a, b. c in R and 3b^(2)-8ac lt 0, then the equation ax^(4)+bx^(3)+cx^(2)+5x-7=0 has
Answer» all real roots all IMAGINARY roots EXACTLY two real and two imaginary roots none Solution :`(c )` `f(x)=ax^(4)+bx^(3)+cx^(2)+5x-7` `f'(x)=4ax^(3)+3BX^(2)+2cx+5` `f''(x)=12ax^(2)+6bx+2c=0` `implies "disc"=36b^(2)-4(12a)(2c)` `=12(3b^(2)-8ac) lt 0` `(:' 3b^(2) lt 8ac)` `f'(x)=0` has only one real root `implies f(x)=0` has exactly two real roots.

Discussion

26544.	If y = tan^(-1) [(5 cos x - 12 sin x)/(12 cos x + 5 sin x)] , then (dy)/(dx) is equal to
Answer» 1 `-1` `-2` `(1)/(2)` ANSWER :B

Discussion

26545.	Prove the following: int_0^1 x e^x dx = 1
Answer» SOLUTION :`int_0^1 X E^x DX = [x e^x]_0^1 - int_0^1 1xxe^x dx` =`(1e^1-0)-(e^x)_0^1` = `e-(e^1-e^0) = e^0 = 1`

Discussion

26546.	If a gt 0" then " underset(x to oo)"Lt" ([ax+b])/(x)=
Answer» 0 1 a b Answer :C

Discussion

26547.	The set of all real values of lambdafor which exactly two common tangents can be drawn to the circlex^(2) +y^(2) -4x -4y +6=0and x^(2) +y^(2) -10 x-10 y +lambda =0 is the interval
Answer» `( 12,32) ` ` ( 18, 42) ` ` ( 12,24)` `( 18, 48 )` ANSWER :B

Discussion

26548.	If f(x) = (4x+3)/(6x-4) , x ne 2/(3) , show that fof (x) = x, for all x ne 2/(3) . What is the inverse of f ?
Answer» SOLUTION :N/A

Discussion

26549.	Write the value of inte^x(In sin x+cotx)dx.
Answer» SOLUTION :`inte^x(In SIN x+COTX)dx=e^xIn(SINX)+C(1d/dx(Insinx)=cotx)`

Discussion

26550.	IFalpha, betaand gammaareanglessuchthattan alpha+ tanbeta+ tangamma=tanalphatan betatangammaandx=cosalpha+ i sinalpha, y = cosbeta +i sinbetaandz= cosgamma+ i sin gamma, thenxyz =
Answer» 1, butnot -1 `-1 ` but NOT1 `+1` or -1 `0` ANSWER :B

Discussion

Explore topic-wise InterviewSolutions in .

Which of the following propositions is a contradiction ?

Which of the following is not a statement in logic?

The number of ways can 5 red and 4 white balls be drawn from a bag containing 10 red and 8 white balls is

Let AP and BQ be two vertical poles at points A and B respectively. If AP = 16 m, BQ = 22m and AB = 20 m, then find the distance of a point R on AB from the point A such that RP^(2)+RQ^(2) is minimum.

What is common amylase, rennin and trypsin ?

Show that the circles x^(2) + y^(2)-6x -2y + 1=0 , x ^(2) + y^(2) + 2x -8y + 13 =0touch each other find the point of contact and the equation of the common tangent at their point of contact.

Let Delta = |{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}| and the equation px^(3) +qx^(2) +rx+s=0 has roots a,b,c where a,b,c in R^(+) The value of Delta is

A(3,2,-1),B(4,1,1,)C(6,2,5) and D(3,3,3) are four points G_(1)G_(2)G_(3)" and "G_(4)respectively are the centroids of the triangle DeltaBCD, DeltaCDA,DeltaDAB,DeltaABC. The point of concurrence of the lines AG_(1),BG_(2),CG_(3)," and "DG_(4) is

Which of the following is not a statementin logic?

Match the following

Let Delta = |{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}| and the equation px^(3) +qx^(2) +rx+s=0 has roots a,b,c where a,b,c in R^(+) the valueof Deltais

Match the following

int "log " sqrt(x + 1 ) dx =

Integrate the following intsin3xdx

Construct the switching circuit for the following statement : [p vv (~p^^q)]vv[(~q^^r) vv ~ p].

Find a vector in the direction of the vector (3, -2, 2) which has magnitude 2sqrt(17) units.

A ladder rests against the side of a wall and reaches a point that is h meters above the ground. The angle formed by the ladder and the ground is theta^(@). A point on the ladder is k meters from the wall.What is the vertical distance, in meters, from this point on the ladder to the ground?

LetA and B be obtuse angles such thatsinA=(4)/(5)andcosB=-(12)/(13).

S is sample space if P(A)= (1)/(3)P(B) and S= A cup B Then P(A) = ....... Where A and B are mutually exclusive events.

The solution of (dy)/(dx) + y = e^(x) is

Differentiate.5^(ln sinx)

Find the equation of the circle for which the point given below are the end points of a diameter. (1,1) (2,-1)

Find the number of seven digited palindromes that can be formed using 0,1,2,3,4.

C_(1) is a circle with centre at the origin and radius equal to r and C_(2) is a circle with centre at (3r, 0)and radius t 2r. The number of common tangents that can be drawn to the two circles are

If sin alpha/2 + cos alpha/2=1.4, then: sin alpha cos alpha=

Using differentials, find the approximate value of each of the up to 3 places of decimal. (0.009)^((1)/(3))

The number of values of x int [-2pi, 2pi] satisfying tanx + cotx2" cosec x" is

If bar(a),bar(b) and bar( c ) are unit vectors such that bar(a)+bar(b)+bar( c )=bar(0), then the value of bar(a).bar(b)+bar(b).bar( c )+bar( c ).bar(a)=…………

The right hand derivative of f(x)= [x] tan (pi x) at a point x=7 is k pi then find the value of k. where [.] is the greatest integer function.

Find the area of the triangle with vertices A(3,-1,2),B(1,-1,-3) and C(4,-3,1).

int(x^(2//3) + 1)dx

Show that the opeation * on Q -{1} defined gby a*b=a+b-ab for all a, b in Q- {1} Satifies (i) the closure property (ii) the associative law (iii) the commutative alw (iv) what is theidentity element ? (v)for each a in Q-{1} find the inverse of a

If A= {1,2,3...,n}and B ={ a,b}Then , the number of subjection from A into B is .........

If x = (2)/(5) + (1.3)/(2!)((2)/(5))^(2) + (1.3.5)/(3!)((2)/(5))^(3) +…, then x + (1)/(x) =

Find the probability that a leap year will have 53 sundays.

If f(x) and g(x) are of degrees 7 and 4 respectively such that f(x)= g(x) q(x)+ r(x) then find possible degrees of q(x) and r(x).

Let ** be the binary opertion on N defined by a **b=H.C.F. of a and b. Is ** commutative ? Is ** associative ? Does there exist identity for this binary opertion on N ?

(i) Find the equation of the circle passing through (0,0) and making intercepts 4,3 on X axis and Y-axis respectively. (ii) Find the equation of the circle passing through (0,0) and making intercepts 6 units on X-axis and intercepts 4 units on Y-axis.

Evaluate the following integrals: int_0^(pi/4) sin2xdx

If a, b. c in R and 3b^(2)-8ac lt 0, then the equation ax^(4)+bx^(3)+cx^(2)+5x-7=0 has

If y = tan^(-1) [(5 cos x - 12 sin x)/(12 cos x + 5 sin x)] , then (dy)/(dx) is equal to

Prove the following: int_0^1 x e^x dx = 1

If a gt 0" then " underset(x to oo)"Lt" ([ax+b])/(x)=

The set of all real values of lambdafor which exactly two common tangents can be drawn to the circlex^(2) +y^(2) -4x -4y +6=0and x^(2) +y^(2) -10 x-10 y +lambda =0 is the interval

If f(x) = (4x+3)/(6x-4) , x ne 2/(3) , show that fof (x) = x, for all x ne 2/(3) . What is the inverse of f ?

Write the value of inte^x(In sin x+cotx)dx.

IFalpha, betaand gammaareanglessuchthattan alpha+ tanbeta+ tangamma=tanalphatan betatangammaandx=cosalpha+ i sinalpha, y = cosbeta +i sinbetaandz= cosgamma+ i sin gamma, thenxyz =

Let be the binary opertion on N defined by a b=H.C.F. of a and b. Is commutative ? Is associative ? Does there exist identity for this binary opertion on N ?