28278 + Interview Questions in Maths in Class 12

1.	Evaluate :intsqrt(1+2cotx("cosec"x+cotx))dx
Answer» ANSWER :`log\|2sin^2""(X)/2\|+c`

Discussion

2.	Statement-I : ((x-y)/(x))+(1)/(2)((x-y)/(x))^(2)+(1)/(3)((x-y)/(x))^(3)+….=log_(e )x-log_(e )y Statement-II : (a-1)-((a-1)^(2))/(2)+((a-1)^(3))/(3)-((a-1)^(4))/(4 )+….=log_(e )a Which of the above statements true
Answer» Only I Only II both I & II NEITHER I nor II ANSWER :C

Discussion

3.	Consider a hyperbola: ((x-7)^(2))/(a) -((y+3)^(2))/(b^(2)) =1. The line 3x - 2y - 25 =0, which is not a tangent, intersect the hyperbola at H ((11)/(3),-7) only. A variable point P(alpha +7, alpha^(2)-4) AA alpha in R exists in the plane of the given hyperbola. The eccentricity of the hyperbola is
Answer» `sqrt((7)/(5))` `sqrt(2)` `(sqrt(13))/(2)` `(3)/(2)` Solution :The GIVEN line must be PARALLEL to asymptotes `rArr` SLOPE of asymptotes are `(3)/(2)` and `-(3)/(2)` `rArr (b)/(a) =(3)/(2) rArr e = (sqrt(13))/(2)`

Discussion

4.	Locus of the point of intersection of tangentsto the circle x^(2)+y^(2)+2x+4y-1=0 which include an angle of 60^(2) is
Answer» `X^(2)+y^(2)+2x+4y-19=0` `x^(2)+y^(2)+2x+4y+19=0` `x^(2)+y^(2)-2x-4y-19=0` `x^(2)+y^(2)-2x-4y+19=0` ANSWER :A

Discussion

5.	(1.3)/(1!) +(2.4)/(2!)+(3.5)/(3!) + (4.6)/(4!) + .......oo = .......
Answer» e 2e 3e 4e Answer :D

Discussion

6.	Integrate the following functions: (cos2x-cos 2alpha)/(cosx-cos alpha)
Answer» Solution :`(cos2x-cos2alpha)/(cosx-cos alpha)` `(2cos^2x-1-(2 cos^2 alpha-1))/(cosx-cos alpha)` =`(2cos^x-2 cos^2 alpha)/(cosx-cos alpha)` `(2(cosx-cos alpha)(cosx+cos alpha))/(cosx-cos alpha)` =`2(cosx+cosalpha)` therefore` int(cos2x-cos2 alpha)/9cosx -cos alpha) DX` =`INT2(cosx+cos alpha) dx` =`2[sinx+xcos alpha]+C`

Discussion

7.	If (1)/(1!11!)+(1)/(3!9!)+(1)/(5!7!)=(2^(p))/(q!) and f(x+y)=f(x).f(y) for all x and y, f(1)=1,f'(0)=10, then:
Answer» <P>`F'(p)=Q` `f'(q)=p` `f'(p)nef'(q)` NONE of these Answer :B

Discussion

8.	int(2x^2+1)e^(x^2)dx
Answer»

Discussion

9.	int_0^pisin^3xcosxdx
Answer» SOLUTION :`int_0^pisin^3x.cosxdx` [PUT sinx=t, then cosxdx=dt When x=0,t=0,when `x=pi`,t=0] =`int_0^pit^3dt=0`

Discussion

10.	Consider a hyperbola: ((x-7)^(2))/(a) -((y+3)^(2))/(b^(2)) =1. The line 3x - 2y - 25 =0, which is not a tangent, intersect the hyperbola at H ((11)/(3),-7) only. A variable point P(alpha +7, alpha^(2)-4) AA alpha in R exists in the plane of the given hyperbola. Which of the following are not the values of alpha for which two tangents can be drawn one to each branch of the given hyperbola is
Answer» `(2,oo)` `(-oo,-2)` `(-(1)/(2),(1)/(2))` None of these Solution :Equation of asymptotes to hyperbola is `((x-7)^(2))/(a^(2))- ((y+3)^(2))/(b^(2)) =0` or `(b^(2)(x-7)^(2))/(a^(2)) -(y+3)^(2) =0` or `(9(x-7)^(2))/(4) -(y+3)^(2) =0` Now point `P(alpha + 7, alpha^(2)-4)` is such that two tangents can be drawn ONE to each BRANCH of the given hyperbola, then `(9(alpha+7-7)^(2))/(4) - (alpha^(2)-1)^(2) lt 0` `rArr (alpha^(2)-1)^(2) GT (9alpha^(2))/(4)` `rArr (2alpha^(2)-3alpha -2) (2alpha^(2)+3alpha-2) gt 0` `rArr (2alpha +1) (alpha-2) (2alpha-1) (alpha+2) gt 0` `rArr alpha in (-oo,-2) uu (-1//2,1//2) uu(2,oo)`

Discussion

11.	int (1-x^(4))/(1-x)dx=
Answer» ANSWER :`x+(x^(2))/(2)+(x^(3))/(3)+(x^(4))/(4)+c`

Discussion

12.	Choose the correct answer The value of int_((-pi)/2)^(pi/2)(x^(3)+xcosx+tan^(5)x+1)dx is
Answer» 0 2 `PI` 1 Answer :C

Discussion

13.	Let f : R rarr Rbe the function defined by f(x) = 2x - 2 , AA x in R . Write f^(-1) .
Answer» SOLUTION :N/A

Discussion

14.	Let X ={1,2,3,4}Determine whether f:X rarr Xdefined as given below have inverses. Find f^(-1) if it exist f={(1,2),(2,2),(3,2),(4,2)}
Answer» SOLUTION :F is not INJECTIVE HENCE not INVERTIBLE.

Discussion

15.	If r = alpha a + beta b + gamma c represents a plane passing through the points a, b, c then
Answer» `ALPHA = beta = GAMMA` `alpha + beta + gamma = 0` `alpha + beta + gamma = 1` `alpha - beta = beta - gamma = gamma - alpha` ANSWER :C

Discussion

16.	Let (1+x)^(10)=underset(r=0)overset(10)sumc_(r)x^(r)and(1+x)^(7)=underset(r=0)overset(7)sumd_(r)x^(r). " if "p=underset(r=0)overset(5)sumandQ=underset(r=0)overset(3)sumd_(2r+1)," then " (p)/(q)is equal to -
Answer» 4 8 16 32 Answer :B

Discussion

17.	Find the equation of a circle which passes through (2,-3) and (-4,5) and having the centre on 4x + 3y + 1 =0
Answer» Answer :` 3(X^(2) +y^(2)) -14x -67 =0`

Discussion

18.	Examine the differentiability of f, where f is defined by f(x) = {(x.[x]",","if " 0 le x lt 2),((x-1)x,"if " 2 le x lt 3):} at x= 2
Answer» ANSWER :x=2

Discussion

19.	If A,B are symmetric matrices of same order then AB-BA is always a …………….
Answer» Skew-symmetric MATRIX Symmetric matrix ZERO matrix IDENTIFY matrix Answer :A

Discussion

20.	Find the number of positive divisors of 1080
Answer» ANSWER :32

Discussion

21.	{:(" "Lt),(n rarr oo):} [ (1)/(n^(2)) sec^(2). (1)/(n^(2)) + (2)/(n^(2))sec^(2). (4)/(n^(2))+...+1/n sec^(2)1]=
Answer» `1/2 ` SEC 1 `1/2 COSEC 1` TAN 1 `1/2 tan 1` ANSWER :D

Discussion

22.	If f(x)f(x),{{:(,x-5,"for x"le 1),(,4x^(2)-8,"for "1 lt x lt 2"then "f(2+)-f(2-)=),(,3x+4,"for "x ge 2):}" then" f(2+)-f(2-)=
Answer» 0 2 3 4 Answer :C

Discussion

23.	A :cos 24^(@) cos 48^(@) cos 96^@ cos 168^(@) = 1//16 R : cos xcos 2 x cos 4x ........ cos (2^n x)=( sin (2^(n+1)x))/(2^(n+1) sin x )
Answer» A is true , R is true and R is correct EXPLANATION of A A is true , R is TRUEAND R is not correct explanation of A A is true , R is false A is false , R is true Answer :A

Discussion

24.	If a, b, c, d be real numbers andl=int(dx)/(ax^(2)+bx+c) then match the various forms of l with values of a, b, c {:(,P,Q,R,S),((A),3,4,1,2),((B),4,3,1,2),((C),1,2,3,4),((C),2,1,4,3):}
Answer» ANSWER :4

Discussion

25.	int(1+cosalphacos x)/(cosalpha+cosx)dx=
Answer» `xcosalpha-sinalpha LOG\|(COT((ALPHA)/(2))+tan((x)/(2)))/(cot((alpha)/(2))-tan((x)/(2)))\|+C` `xcosalpha+sinalpha log\|(cot((alpha)/(2))+tan((x)/(2)))/(cot((alpha)/(2))-tan((x)/(2)))\|+c` `xcosalpha-sinalpha log\|(tan((x)/(2))+tan((alpha)/(2)))/(tan((x)/(2))-tan((alpha)/(2)))\|+c` `xcosalpha+sinalpha log\|(tan((x)/(2))+tan((alpha)/(2)))/(tan((x)/(2))-tan((alpha)/(2)))\|+c` ANSWER :B

Discussion

26.	Integrate the functions (x^(2))/((2+3x^(3))^(3))
Answer» ANSWER :`( :.t=2+3x^(2))`

Discussion

27.	a and b are the roots of the quadratic equation x^2 + lambdax - 1/(2lambda^(2)) =0 where x is the unknown and lambdais a real parameter. The minimum value of a^(4) + b^(4) is:
Answer» ANSWER :`2+sqrt(2)`

Discussion

28.	Let the incircle of DeltaABC touches the sides BC, CA, AB at A_(1), B_(1),C_(1) respectively. The incircle ofDeltaA_(1)B_(1)C_(1) touches its sides of B_(1)C_(1), C_(1)A_(1) and A_(1)B_(1)" at " A_(2), B_(2), C_(2) respectively and so on. Q. In DeltaA_(4)B_(4)C_(4), the value of angleA_(4) is:
Answer» `(3pi+A)/(6)` `(3pi-A)/(8)` `(5pi-A)/(16)` `(5pi+A)/(16)` ANSWER :D

Discussion

29.	Let the incircle of DeltaABC touches the sides BC, CA, AB at A_(1), B_(1),C_(1) respectively. The incircle ofDeltaA_(1)B_(1)C_(1) touches its sides of B_(1)C_(1), C_(1)A_(1) and A_(1)B_(1)" at " A_(2), B_(2), C_(2) respectively and so on. Q. lim_(n to oo) angleA_(n)=
Answer» 0 `(PI)/(6)` `(pi)/(4)` `(pi)/(3)` Answer :D

Discussion

30.	If the length of three sides of a trapezium other than base are equal to 10 cm then find the area of the trapeziumwhen it is maximum.
Answer» ANSWER :`75sqrt(3)`

Discussion

31.	Which one of the following industry groups employs the maximum number of people?
Answer» UTILITIES Information State GOVERNMENT Natural RESOURCES and Mining Answer :C

Discussion

32.	Evaluate the definite integrals int_(0)^(pi/2)(cos^(2)xdx)/(cos^(2)x+4sin^(2)x)
Answer» ANSWER :`pi/6`

Discussion

33.	I=int sin^(3)x cos^(3)x
Answer»

Discussion

34.	Evalute the following integrals int "cosec"^(4) xdx
Answer» `Cotx+ (1)/(3) Cot^(3) X + (2x)/(3) ` + C `Cotx - (1)/(3) Cot^(3)x` + c `Cotx+ (1)/(3) Cot^(3) x` + c `-Cotx- (1)/(3) Cot^(3) x +` + c ANSWER :D

Discussion

35.	Find the area of the region bounded by y= sin x and the x-axis in the interval [0, 2pi]
Answer» ANSWER :4

Discussion

36.	Theradicalcentreof thecirclesx^2+y^2-4x-6y+5=0, x^2+y^2-2x-4y-1=0 andx^2 +y^2-6x-2y=0=0 lieson theline
Answer» `x+y-5=0` `2x-4y+7=0` `4x-6y+5=0` `18X -12y+1=0` ANSWER :D

Discussion

37.	Find the equation of tangent to y^(2) = 16x inclined at an angle 60^(@) with its axis also find its point of contact.
Answer» ANSWER :`3x - sqrt( 3y) + 4 =0 ((4)/(3) , (8)/( sqrt3) )`

Discussion

38.	If the vectors bara=x i+yj+zk,barb=j and barc are such that bara,barc,barb form a right-handed system, then : barc=
Answer» ZI-XK `BAR0` YJ `- zi + xk` ANSWER :A

Discussion

39.	The statement prarr ( q rarr p )is equivalent
Answer» `p RARR ( p rarr Q ) ` `p rarr ( p vee q) ` ` p rarr (p ^^ q )` `p rarr ( p harr q )` Answer :B

Discussion

40.	If vec(a),vec(b) and vec(c) are unit vectors such that vec(a)+vec(b)+vec(c)=0, then 3vec(a).vec(b)+2vec(b).vec(c)+vec(c).vec(a)=
Answer» 3 -3 1 -1 Answer :B

Discussion

41.	The table below lists the number (to the nearest 1,000) of book club members in the United States for 2001 through 2004. of the following expressions with x representing the number of years after 2001, which best models the number of book club members (in thousands) in the United States ?
Answer» 539x+2,001 `3/8x+2,001` `8/3x+539` 547x+2,001 Answer :C

Discussion

42.	Integrate the following function : int(dx)/(1-6x-9x^(2))
Answer» ANSWER :`(-1)/(6sqrt2)LOG\|(3x+1-sqrt2)/(3x+1+sqrt2)\|+C`

Discussion

43.	Let S={(a,b)\|a,b in Z, 0 le a,b le 18} . The number of linesin R^(2) passing though (0,0) and exactlu one other points in S is-
Answer» 16 22 28 32 Solution :N//A

Discussion

44.	For the function y=f(x)=(x^(2)+bx+c)e^(x), which of the following holds?
Answer» If `f(X)GT0` for all real `xcancelrArrf'(x)gt0` If `f(x)gt0` for all real `xrArrf'(x)gt0` If `f'(x)gt0` for all real `xrArrf(x)gt0` If `f'(x)gt0` for all real `xcancelrArrf(x)gt0` ANSWER :A::C

Discussion

45.	Match the statements in List I with those in List II
Answer» Solution :`tan^(-1) (x + 3) - tan^(-1) (x -3) = sin^(-1) (3//5)` `rArr tan^(-1).((x +3)-(x -3))/(1 + (x^(2) -9)) = tan^(-1).(3)/(4)` `rArr (6)/(x^(2) -8) = (3)/(4)` `:. X^(2) -8 = 8` or `x = +- 4` CLEARLY, both values satisfy the equation Note : Solution of the remaining parts are given in their respective chapters.

Discussion

46.	veca xx [veca xx (veca xx vecb)] equals :
Answer» `(VECA.veca) (veca XX VECB)` `(veca.veca) (vecb xx veca)` `(vecb.vecb) (veca xx vecb)` `(vecb . Vecb) (vecb xx veca).` ANSWER :B

Discussion

47.	If ABCDEF is a regular hexagon with AB = a and BC = b, then CE equals
Answer» b - a `- b` b - 2a None of these Solution :In `DELTAABC, AB + BC = AC` IMPLIES AC = a + b AD is PARALLEL to BC and AD = 2BC `therefore AD = 2b` In `DeltaACD`, AC + CD = AD `implies CD=2b-(a+b)=b-a` Now, `CE = CD + DE = (b-a)-a=b-2a`

Discussion

48.	Evaluate: inte^(x)(x^(3)-x+2)/(x^(2)+1)^(2)dx
Answer» Solution :`E^(X)(x+1)/(x^(2)+1)+C`

Discussion

49.	a_(1),a_(2),….a_(n) be in arithmeticalprogression , show that a_(1)^(2)a_(2)^(2)…….a_(n)^(2) gt a_(1)^(n) a_(n)^(n)
Answer» ANSWER :`a_(1)^(2).a_(2)^(2)…..a_(N)^(2)GT (a_(1).a_(n))^(n)`

Discussion

50.	If functions f : A rarr B and g : B rarr A satisfy gof = I_A , then show that f is one one and g is onto.
Answer» SOLUTION :N/A

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Evaluate :intsqrt(1+2cotx("cosec"x+cotx))dx

Statement-I : ((x-y)/(x))+(1)/(2)((x-y)/(x))^(2)+(1)/(3)((x-y)/(x))^(3)+….=log_(e )x-log_(e )y Statement-II : (a-1)-((a-1)^(2))/(2)+((a-1)^(3))/(3)-((a-1)^(4))/(4 )+….=log_(e )a Which of the above statements true

Consider a hyperbola: ((x-7)^(2))/(a) -((y+3)^(2))/(b^(2)) =1. The line 3x - 2y - 25 =0, which is not a tangent, intersect the hyperbola at H ((11)/(3),-7) only. A variable point P(alpha +7, alpha^(2)-4) AA alpha in R exists in the plane of the given hyperbola. The eccentricity of the hyperbola is

Locus of the point of intersection of tangentsto the circle x^(2)+y^(2)+2x+4y-1=0 which include an angle of 60^(2) is

(1.3)/(1!) +(2.4)/(2!)+(3.5)/(3!) + (4.6)/(4!) + .......oo = .......

Integrate the following functions: (cos2x-cos 2alpha)/(cosx-cos alpha)

If (1)/(1!11!)+(1)/(3!9!)+(1)/(5!7!)=(2^(p))/(q!) and f(x+y)=f(x).f(y) for all x and y, f(1)=1,f'(0)=10, then:

int(2x^2+1)e^(x^2)dx

int_0^pisin^3xcosxdx

int (1-x^(4))/(1-x)dx=

Choose the correct answer The value of int_((-pi)/2)^(pi/2)(x^(3)+xcosx+tan^(5)x+1)dx is

Let f : R rarr Rbe the function defined by f(x) = 2x - 2 , AA x in R . Write f^(-1) .

Let X ={1,2,3,4}Determine whether f:X rarr Xdefined as given below have inverses. Find f^(-1) if it exist f={(1,2),(2,2),(3,2),(4,2)}

If r = alpha a + beta b + gamma c represents a plane passing through the points a, b, c then

Let (1+x)^(10)=underset(r=0)overset(10)sumc_(r)x^(r)and(1+x)^(7)=underset(r=0)overset(7)sumd_(r)x^(r). " if "p=underset(r=0)overset(5)sumandQ=underset(r=0)overset(3)sumd_(2r+1)," then " (p)/(q)is equal to -

Find the equation of a circle which passes through (2,-3) and (-4,5) and having the centre on 4x + 3y + 1 =0

Examine the differentiability of f, where f is defined by f(x) = {(x.[x]",","if " 0 le x lt 2),((x-1)x,"if " 2 le x lt 3):} at x= 2

If A,B are symmetric matrices of same order then AB-BA is always a …………….

Find the number of positive divisors of 1080

{:(" "Lt),(n rarr oo):} [ (1)/(n^(2)) sec^(2). (1)/(n^(2)) + (2)/(n^(2))sec^(2). (4)/(n^(2))+...+1/n sec^(2)1]=

If f(x)f(x),{{:(,x-5,"for x"le 1),(,4x^(2)-8,"for "1 lt x lt 2"then "f(2+)-f(2-)=),(,3x+4,"for "x ge 2):}" then" f(2+)-f(2-)=

A :cos 24^(@) cos 48^(@) cos 96^@ cos 168^(@) = 1//16 R : cos xcos 2 x cos 4x ........ cos (2^n x)=( sin (2^(n+1)x))/(2^(n+1) sin x )

If a, b, c, d be real numbers andl=int(dx)/(ax^(2)+bx+c) then match the various forms of l with values of a, b, c {:(,P,Q,R,S),((A),3,4,1,2),((B),4,3,1,2),((C),1,2,3,4),((C),2,1,4,3):}

int(1+cosalphacos x)/(cosalpha+cosx)dx=

Integrate the functions (x^(2))/((2+3x^(3))^(3))

a and b are the roots of the quadratic equation x^2 + lambdax - 1/(2lambda^(2)) =0 where x is the unknown and lambdais a real parameter. The minimum value of a^(4) + b^(4) is:

Let the incircle of DeltaABC touches the sides BC, CA, AB at A_(1), B_(1),C_(1) respectively. The incircle ofDeltaA_(1)B_(1)C_(1) touches its sides of B_(1)C_(1), C_(1)A_(1) and A_(1)B_(1)" at " A_(2), B_(2), C_(2) respectively and so on. Q. In DeltaA_(4)B_(4)C_(4), the value of angleA_(4) is:

Let the incircle of DeltaABC touches the sides BC, CA, AB at A_(1), B_(1),C_(1) respectively. The incircle ofDeltaA_(1)B_(1)C_(1) touches its sides of B_(1)C_(1), C_(1)A_(1) and A_(1)B_(1)" at " A_(2), B_(2), C_(2) respectively and so on. Q. lim_(n to oo) angleA_(n)=

If the length of three sides of a trapezium other than base are equal to 10 cm then find the area of the trapeziumwhen it is maximum.

Which one of the following industry groups employs the maximum number of people?

Evaluate the definite integrals int_(0)^(pi/2)(cos^(2)xdx)/(cos^(2)x+4sin^(2)x)

I=int sin^(3)x cos^(3)x

Evalute the following integrals int "cosec"^(4) xdx

Find the area of the region bounded by y= sin x and the x-axis in the interval [0, 2pi]

Theradicalcentreof thecirclesx^2+y^2-4x-6y+5=0, x^2+y^2-2x-4y-1=0 andx^2 +y^2-6x-2y=0=0 lieson theline

Find the equation of tangent to y^(2) = 16x inclined at an angle 60^(@) with its axis also find its point of contact.

If the vectors bara=x i+yj+zk,barb=j and barc are such that bara,barc,barb form a right-handed system, then : barc=

The statement prarr ( q rarr p )is equivalent

If vec(a),vec(b) and vec(c) are unit vectors such that vec(a)+vec(b)+vec(c)=0, then 3vec(a).vec(b)+2vec(b).vec(c)+vec(c).vec(a)=

The table below lists the number (to the nearest 1,000) of book club members in the United States for 2001 through 2004. of the following expressions with x representing the number of years after 2001, which best models the number of book club members (in thousands) in the United States ?

Integrate the following function : int(dx)/(1-6x-9x^(2))

Let S={(a,b)|a,b in Z, 0 le a,b le 18} . The number of linesin R^(2) passing though (0,0) and exactlu one other points in S is-

For the function y=f(x)=(x^(2)+bx+c)e^(x), which of the following holds?

Match the statements in List I with those in List II

veca xx [veca xx (veca xx vecb)] equals :

If ABCDEF is a regular hexagon with AB = a and BC = b, then CE equals

Evaluate: inte^(x)(x^(3)-x+2)/(x^(2)+1)^(2)dx

a_(1),a_(2),….a_(n) be in arithmeticalprogression , show that a_(1)^(2)a_(2)^(2)…….a_(n)^(2) gt a_(1)^(n) a_(n)^(n)

If functions f : A rarr B and g : B rarr A satisfy gof = I_A , then show that f is one one and g is onto.