28278 + Interview Questions in Maths in Class 12

1.	W(x,y,z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u, v inR. Find (del w)/(del u) , (del w)/(del v)and evaluate then at ((1)/(2), 1).
Answer» ANSWER :`((1)(2),1) (delW)/(delv) =2. (1)/(4) -2 =- 3/2`

Discussion

2.	The position vectors of the points A,B,C are given to be 2overset^^i+overset^^j-overset^^k,overset^^3i-overset^^2j+overset^^k"and" 1overset^^i+4overset^^j-3overset^^k respectively.Prove that A,B,C are collinear
Answer» SOLUTION :From `vec(AB)=-vec(AC)` `:. A,B,C` are collinear

Discussion

3.	Find the equation of the circum circle of the triangle formed by the straight lines given in each of the following: x-y-2=0, 2x - 3y + 4 = 0, 3x - y +6 = 0
Answer» ANSWER :`x^(2) + y^(2) - 24 x + 16Y - 52 = 0`

Discussion

4.	Find the equation of the circum circle of the triangle formed by the straight lines given in each of the following: 5x-3y + 4 = 0, 2x+ 3y - 5 = 0, x + y = 0
Answer» ANSWER :`7(X^(2) + y^(2) ) + 40x - 37 y + 35 = 0`

Discussion

5.	If A and B are two events of a sample space then P(A)-P(B)=
Answer» <P>`P(ANNB')+P(A'nnB)` `P(AnnB')-P(A'nnB)` `P(A'nnB')-P(AUUB)` `P(AuuB)-P(AnnB)` ANSWER :B

Discussion

6.	Find the equation of the circum circle of the triangle formed by the straight lines given in each of the following: 2x+y = 4, x+y=6, x+2y = 5
Answer» Answer :`x^(2) + y^(2) - 17 x - 19 y + 50 =0`

Discussion

7.	Differentiate the functions (x+ 3)^(2).(x+ 4)^(3).(x+ 5)^(4)
Answer» `(1)/(y) sum_(i=1)^(3) (i-1)/( (X+1- i) )` `(x)/(y) sum_(i=1)^(3) (i+1)/( ( x+1+i))` `(y)/(x) sum_(i=2)^(4) (i)/( ( x+1-i))` `y sum_(i=2)^(4) ((i)/( (x+1)+i) )` ANSWER :D

Discussion

8.	The sides BC, CA, AB of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these points as vertices is
Answer» 220 217 215 205 Answer :D

Discussion

9.	{:("List- I" , " List - II") , ( "(P) At any point on the curve " y=a^(x) " then lengthof subnormal " , "(1) Varies as cube of the ordinate of the point "),("(Q) At any point on the curve " xy = a^(2) " then length of subnormal" , "(2) Varies as square of the ordinate of the point"),("(R) At any point on the curve " x^(2)y^(3) = a^(5) " the length of subtangent" , "(3) Constant"),("(S) At any point on the curve " Y =be^(x/a) " the length of subtangent " , "(4) Varies as abscissa of the point "):}
Answer» <P>(I) (II) ( R) (III) (i) (Q) (I) (i) (Q) (IV) (iii) (P) ANSWER :B

Discussion

10.	Choose the correct answer int dx/sqrt(9x-4x^2)=
Answer» `1/9 sin^-1((9x-8)/8)+C` `1/2 sin^-1((8x-9)/9)+c` `1/3 sin^-1((9x-8)/8)+c` `1/2 sin^-1((9x-8)/9)+c` ANSWER :B

Discussion

11.	cot theta - tan theta - 2 tan 2theta- 4 tan 4theta=
Answer» `4cot8theta-tan6theta` `COT 8theta + TAN 3theta` `cot8theta+ cot6theta` `8cot8theta` ANSWER :D

Discussion

12.	A letter is known to have come from either 'MAHARASTRA or MADRAS on the post mark only consecutive letters 'RA' can be read clearly. What is the chance that the letter came from 'MAHARASTRA'.
Answer» ANSWER :`(10)/(19)`

Discussion

13.	IF2x^2 -10 xy+2 lamda y^2 +5x-16y-3=0 representsa pairof straightlines, thenpointof intersectionof those linesis
Answer» `(2,-3)` `(5,-16)` `(-10 ,(-7)/(2))` `(-10,(-3)/(2))` Answer :C

Discussion

14.	Assertion(A ):theequationwhoseroosare multipled ofby 2 ofthoseofx^5 - 2x^4 +3x^3 -2x^2 +4x +3=0isx^5 - 4x^4 +12 x^3 -16 x^2 +64 x+96=0 Reason (R ) :theequationwhoserootsare mulipliedby k of thoseoff(x ) =0isf((x )/(k ))=0
Answer» bothA and RaretrueR isthe correctexplanationof A both AandRare trueand RIS notcorrectexplanationof A AIS trueand Risfalse A isfalseand Ristrue Answer :A

Discussion

15.	If cosalpha + cosbeta + cosgamma =0= sinalpha + sinbeta + singamma, then
Answer» `sin^(2)alpha + sin^(2)BETA + sin^(2)GAMMA = 3/2` `sin^(2)alpha+ sin^(2)beta+ sin^(2)gamma =3/4` `cos^(2)alpha + cos^(2)beta + cos^(2)gamma =3/2` `cos2alpha + cos2beta + cos2gamma =-1` ANSWER :A::C

Discussion

16.	Match the following. I. int (sin x)/(1+cos 2x)dx= "" a) tan x-(1)/(2) x+c II int (1)/(1-cos2x)dx= "" b)- cot x-(1)/(2)x+c III. int (1+cos^(2)x)/(1-cos2x)dx= "" c) (1)/(2)secx+c IV. int (1+sin^(2)x)/(1+cos2x)dx= "" -d) (1)/(2)cotx+c
Answer» a,b,C,d b,c,a,d d,a,b,c a,b,c,a Answer :4

Discussion

17.	Evaluate the following:lim_(xtoinfty)(x^4-5x+2)/(x^3-3x+1)
Answer» SOLUTION :`lim_(xtoinfty)(x^4-5x+2)/(x^3-3x+1)` `lim_(xtoinfty)(x-5/x^2+2/x^3)/(1-3/x^2+1/x^3)=INFTY`

Discussion

18.	Let n be positive integer such that, (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+….+a_(2n)x^(2n), then (n-r)a_(r)+(2n-r+1)a_(r-1),0ltrlt2n is :
Answer» `(r+1)a_(r+1)` `(r-1)a_(r+1)` `(2r+1)a_(r+1)` NONE of these ANSWER :A

Discussion

19.	f(theta) =sin 2 theta in (0, pi)
Answer» ANSWER :`((PI)/(2), pi)`

Discussion

20.	int e^(x)(1+sin x)/(1+cos x) dx is equal to
Answer» `E^(x) sin x + C` `e^(x) + C` `TAN ((x)/(2)) + C` `e^(x) tan ((x)/(2)) + C` ANSWER :D

Discussion

21.	Write the simplest form of tan^(-1)((cosx-sinx)/(cosx+sinx)), 0 lt x lt (pi)/(2)
Answer» ANSWER :`(PI)/(4)-X`

Discussion

22.	A point on the hypotenuse of a right - angled triangle is at distance a and b from sides of the triangle . Showthat the minimum length of the hypotenuse is(a^((2)/(3))+b^((2)/(3)))^((3)/(2))
Answer» ANSWER :`=(a^((2)/(3))+B^((2)/(3)))^((2)/(3))`

Discussion

23.	If k and l respectively are the order and degree of the differentialequation whose general solution represents the family of circles of constant radius, then k^(2)+ l^(2)=
Answer» 2 6 8 10 Answer :C

Discussion

24.	int(sinx)/(sqrt(9-sin^(2)x))dx=
Answer» `LOG(sinx+SQRT(9-sin^(2)x))+C` `-log(cosx-sqrt(9-sin^(2)x))+c` `log(sinx+sqrt(8+cos^(2)x))+c` `-log(sqrt(1-sin^(2)x)+sqrt(9-sin^(2)x))+c` Answer :D

Discussion

25.	If y=3 cos(log x)+4sin(logx) show that x^(2)y_(2)+xy_(1)+y=0
Answer» Answer :`rArr X^(2)y_(2)+xy_(1)+y=0`

Discussion

26.	The numberof trianglewhosevertices are atthe verticesof anoctagonbutnoneofwhosesideshappento comefrom the sidesof theoctagonis
Answer» 24 52 48 16 Answer :D

Discussion

27.	If arg(z - 1) = pi/6 and arg(z + 1) = 2 pi/3 , then prove that \|z\| = 1.
Answer» ANSWER :1

Discussion

28.	State which of the following statements are true (T) or false(F) The line (x+3)/(-1)=(y-2)/3=(z-1)/4 is perpendicular to the plane 3x-3y+3z-1=0
Answer» ANSWER :F

Discussion

29.	The number of circles that touch all the 3 lines 2x+y=3, 4x-y=3, x+y=2 is
Answer» 1 2 3 4 Answer :D

Discussion

30.	Evaluate : int (3-x^(2))/(1-2x+x^(2))e^(x)dx
Answer» ANSWER :`E^(X)((1+x)/(1-x))+C`

Discussion

31.	Solution set of the inequality 2x+y gt 5 is ………
Answer» The half PLANE containing origin The open half plane not containing origin xy - plane excepts the POINTS on the LINE 2x + y = 5 None of these Answer :B

Discussion

32.	Prove that the sequence with the general term x_(n)=(1)/(3+1) +(1)/(3^(2)+2)+...+(1)/(3^(n)+n) has a finite limit.
Answer» ANSWER :ESTABLISH the boundedness of the sequence by COMPARING `x_n` with the sum of some GEOMETRIC progression.

Discussion

33.	The random variable X has probability distribution P(X) of the following form. P(X)= {(k, if X=0),(2k,if X=1),(3k, if X=2),(0, "otherwise"):} Find P(X lt 2)
Answer» ANSWER :`(1)/(2)`

Discussion

34.	The random variable X has probability distribution P(X) of the following form. P(X)= {(k, if X=0),(2k,if X=1),(3k, if X=2),(0, "otherwise"):} Determine value of K
Answer» ANSWER :`(1)/(6)`

Discussion

35.	If Delta_(k)=\|(2(3^(k-1)),3(4^(k-1)),4(5^(k)-1)),(alpha, beta, gamma),(3^(n)-1,4^(n)-1,5^(n)-1)\| then the value of sum_(k=1)^(n)Delta_(k) depends
Answer» only on `ALPHA` and `BETA` not on `GAMMA` on all `alpha, beta` and `gamma` onnone of `alpha, beta` and `gamma` only on `alpha`, no on `beta` and `gamma` ANSWER :C

Discussion

36.	I :tan 20^@+ tan 40^(@) + sqrt(3) tan 20^(@) tan 40^(@) =1 II : ((1+ tan 21^(@))(1+ tan 24^(@)))/((1+ tan 22^(@) )(1+ tan 23^(@)))=2
Answer» only I is TRUE only II is true both I & II are true NEITHER I nor II are true Answer :D

Discussion

37.	If (a^(2) , a-2) be a point interior to the region of the parabola y^(2) = 2x bounded by the chord joining the points (2,2) and (8,-4) then find the set of all possible real values of a.
Answer» ANSWER :`a in (-2,-2+ SQRT(2) )`

Discussion

38.	If a=hati+hatj+hatk, b=hati-hatj+2hatk and c=xhati+(x-2)hatj-hatk and if the vector c lies in the plane of vectors a and b, then x equals
Answer» 0 1 `-2` 2 Solution :GIVEN, `a=hati+hatj+hatk, B=hati-hatj+2hatk` and `c=xhati+(x-2)hatj-hatk` Since, c lies in the plane of VECTORS a and b, THEREFORE a, b and c are coplanar. `therefore \|(1,1,1),(1,-1,2),(x,(x-2),-1)\|=0` `implies 1(1-2x+4)-1(-1-2x)+1(x-2+x)=0` `implies 5-2x+1+2x+2x-2=0` `implies x = - 2 `

Discussion

39.	Evaluate the following definite integrals : int_(0)^(pi/4)sqrt(1+sin2x)dx
Answer» ANSWER :`(2)/(3)`

Discussion

40.	Let vecam vecvm vecc be the position vectors of the vertices A,B,C respectively of triangleABC. The vector area of triangleABC is
Answer» `1/2 {veca XX (vecb xx VECC)+vecb xx (vecc xx veca)+vecc xx (veca xx vecb)}` `1/2 {veca xx vecb +vecb xx vecc +vecc xx veca }` `1/2 {veca +vecb +vecc }` `1/2 {(vecb. vecc) veca+(vecc. Veca)vecb+(veca.vecb)vecc}` ANSWER :B

Discussion

41.	Find the Principle values of the following : cost^(-1)(sqrt(3))
Answer» ANSWER :`(PI)/6`

Discussion

42.	If alpha is an n^(th) root of unity then prove that 1+2alpha+3alpha^2+4alpha^3+….+nalpha^(n-1)=-(n)/(1-alpha)
Answer» `(N)/(1-alpha)` `(-n)/(1-alpha)` `(-n)/((1-alpha)^2)` `(n)/((1-alpha)^2)` ANSWER :B

Discussion

43.	Does the following graph pass the vertical or horizontal line test ?
Answer» SOLUTION :CLEARLY, the graph passes the vertical line test as any vertical line, including x = 1 andx = 2, meets the graph only once. HOWEVER, it does not pass the horizontal test as the LINEY = 2 meets the graph at two POINTS(1, 2) and (2, 2).

Discussion

44.	Let f(x)=int_(0)^(x)f(t) dt equals
Answer» `5int_(X+5)^(5)g(t)dt` `int_(5)^(x+5)F(t)dt` `2int_(5)^(x+5)f(t)dt` `int_(x+5)^(5)g(t)dt` Solution :Given , f (x) `=int_(0)^(x)f(t)dt` On replacingx by (-x) , we GET `f(x)=int_(0)^(-x)g (t)dt` Now , PUT t =- u , so `int(-x)=-int_(0)^(x)g(-u)du=-int_(0)^(x)g(u)=-f(x)` `[:' `g is an even function] `rArrf(-x)=-f(xrArr f` an odd function. Now , it is given that f (x +5) = g (x) `:.f(5-x)=g(-x)=g(x)=f(x+5)` [ `:.` g is an even function] `rArrf(5-x)=f(x+5)`. . .(i) Let `I=int_(0)^(x)f(t)dt` Putt = u +5 `rArrt -5=u rArrdt = du` `:.u=-trArrdu=-dt ` , we get `I=-int_(5)^(5-x)g(-t)dt=int_(5-x)^(5)f (t) dt` `[:'-int_(a)^(b)f(x)dx=int_(b)^(a)f (x)` dx andg is an even function] `I=int_(5-x)^(5)f'(t)dt` [ by Leibnitz RULE f ' (x) = g (x) ] `=f(5)-f(5-x)=f(5)-f(5+x)`[ from Eq . (i)] `=int_(5+x)^(5) f ' (t) dt = int_(5+x)^(5)g (t) dt`

Discussion

45.	Compute the integralint _(1)^(0) I n ^(3) x dx
Answer» ANSWER :`-2E`

Discussion

46.	Let S(M) denote the sum of the digits of a positive integer M written in base 10. Let N be the smallest positive integer such that S(N) = 2017. Find the value of S(5N + 2017)
Answer» ANSWER :6

Discussion

47.	If f(x) = (1)/(x^(2)) int_(3)^(x)(2t-3f'(t)dt, then f'(3) is equal to
Answer» `-(1)/(2)` `-(1)/(3)` `(1)/(2)` `(1)/(3)` ANSWER :C

Discussion

48.	Integrate the functions x/(sqrt(x+4)),xgt0
Answer» ANSWER :`2/3sqrt(x+4)(x-8)+C`

Discussion

49.	The matrices P[(u_(1),v_(1),w_(1)),(u_(2),v_(2),w_(2)),(u_(3),v_(3)w_(3))] and Q=(1)/(9)[(2,2,1),(12,-5,m),(-8,1,5)] are such that PQ=l, an identify matrix. Solving the equation [(u_(1),v_(1),w_(1)),(u_(2),v_(2),w_(2)),(u_(3),v_(3),w_(3))][(x),(y),(z)]=[(1),(1),(5)], the value of y comes out to be -3, then the value of m is equal to
Answer» 27 7 `-27` `-7` ANSWER :D

Discussion

50.	The trajectories orthogonal to x^(2)+y^(2) = 2ax is
Answer» `y = x^(2)` `y = c(x-a)` `y = c(x^(2)+2Y^(2))` `y^(2) = x` ANSWER :B

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

W(x,y,z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u, v inR. Find (del w)/(del u) , (del w)/(del v)and evaluate then at ((1)/(2), 1).

The position vectors of the points A,B,C are given to be 2overset^^i+overset^^j-overset^^k,overset^^3i-overset^^2j+overset^^k"and" 1overset^^i+4overset^^j-3overset^^k respectively.Prove that A,B,C are collinear

Find the equation of the circum circle of the triangle formed by the straight lines given in each of the following: x-y-2=0, 2x - 3y + 4 = 0, 3x - y +6 = 0

Find the equation of the circum circle of the triangle formed by the straight lines given in each of the following: 5x-3y + 4 = 0, 2x+ 3y - 5 = 0, x + y = 0

If A and B are two events of a sample space then P(A)-P(B)=

Find the equation of the circum circle of the triangle formed by the straight lines given in each of the following: 2x+y = 4, x+y=6, x+2y = 5

Differentiate the functions (x+ 3)^(2).(x+ 4)^(3).(x+ 5)^(4)

The sides BC, CA, AB of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these points as vertices is

Choose the correct answer int dx/sqrt(9x-4x^2)=

cot theta - tan theta - 2 tan 2theta- 4 tan 4theta=

A letter is known to have come from either 'MAHARASTRA or MADRAS on the post mark only consecutive letters 'RA' can be read clearly. What is the chance that the letter came from 'MAHARASTRA'.

IF2x^2 -10 xy+2 lamda y^2 +5x-16y-3=0 representsa pairof straightlines, thenpointof intersectionof those linesis

Assertion(A ):theequationwhoseroosare multipled ofby 2 ofthoseofx^5 - 2x^4 +3x^3 -2x^2 +4x +3=0isx^5 - 4x^4 +12 x^3 -16 x^2 +64 x+96=0 Reason (R ) :theequationwhoserootsare mulipliedby k of thoseoff(x ) =0isf((x )/(k ))=0

If cosalpha + cosbeta + cosgamma =0= sinalpha + sinbeta + singamma, then

Match the following. I. int (sin x)/(1+cos 2x)dx= "" a) tan x-(1)/(2) x+c II int (1)/(1-cos2x)dx= "" b)- cot x-(1)/(2)x+c III. int (1+cos^(2)x)/(1-cos2x)dx= "" c) (1)/(2)secx+c IV. int (1+sin^(2)x)/(1+cos2x)dx= "" -d) (1)/(2)cotx+c

Evaluate the following:lim_(xtoinfty)(x^4-5x+2)/(x^3-3x+1)

Let n be positive integer such that, (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+….+a_(2n)x^(2n), then (n-r)a_(r)+(2n-r+1)a_(r-1),0ltrlt2n is :

f(theta) =sin 2 theta in (0, pi)

int e^(x)(1+sin x)/(1+cos x) dx is equal to

Write the simplest form of tan^(-1)((cosx-sinx)/(cosx+sinx)), 0 lt x lt (pi)/(2)

A point on the hypotenuse of a right - angled triangle is at distance a and b from sides of the triangle . Showthat the minimum length of the hypotenuse is(a^((2)/(3))+b^((2)/(3)))^((3)/(2))

If k and l respectively are the order and degree of the differentialequation whose general solution represents the family of circles of constant radius, then k^(2)+ l^(2)=

int(sinx)/(sqrt(9-sin^(2)x))dx=

If y=3 cos(log x)+4sin(logx) show that x^(2)y_(2)+xy_(1)+y=0

The numberof trianglewhosevertices are atthe verticesof anoctagonbutnoneofwhosesideshappento comefrom the sidesof theoctagonis

If arg(z - 1) = pi/6 and arg(z + 1) = 2 pi/3 , then prove that |z| = 1.

State which of the following statements are true (T) or false(F) The line (x+3)/(-1)=(y-2)/3=(z-1)/4 is perpendicular to the plane 3x-3y+3z-1=0

The number of circles that touch all the 3 lines 2x+y=3, 4x-y=3, x+y=2 is

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

Solution set of the inequality 2x+y gt 5 is ………

Prove that the sequence with the general term x_(n)=(1)/(3+1) +(1)/(3^(2)+2)+...+(1)/(3^(n)+n) has a finite limit.

The random variable X has probability distribution P(X) of the following form. P(X)= {(k, if X=0),(2k,if X=1),(3k, if X=2),(0, "otherwise"):} Find P(X lt 2)

The random variable X has probability distribution P(X) of the following form. P(X)= {(k, if X=0),(2k,if X=1),(3k, if X=2),(0, "otherwise"):} Determine value of K

If Delta_(k)=|(2(3^(k-1)),3(4^(k-1)),4(5^(k)-1)),(alpha, beta, gamma),(3^(n)-1,4^(n)-1,5^(n)-1)| then the value of sum_(k=1)^(n)Delta_(k) depends

I :tan 20^@+ tan 40^(@) + sqrt(3) tan 20^(@) tan 40^(@) =1 II : ((1+ tan 21^(@))(1+ tan 24^(@)))/((1+ tan 22^(@) )(1+ tan 23^(@)))=2

If (a^(2) , a-2) be a point interior to the region of the parabola y^(2) = 2x bounded by the chord joining the points (2,2) and (8,-4) then find the set of all possible real values of a.

If a=hati+hatj+hatk, b=hati-hatj+2hatk and c=xhati+(x-2)hatj-hatk and if the vector c lies in the plane of vectors a and b, then x equals

Evaluate the following definite integrals : int_(0)^(pi/4)sqrt(1+sin2x)dx

Let vecam vecvm vecc be the position vectors of the vertices A,B,C respectively of triangleABC. The vector area of triangleABC is

Find the Principle values of the following : cost^(-1)(sqrt(3))

If alpha is an n^(th) root of unity then prove that 1+2alpha+3alpha^2+4alpha^3+….+nalpha^(n-1)=-(n)/(1-alpha)

Does the following graph pass the vertical or horizontal line test ?

Let f(x)=int_(0)^(x)f(t) dt equals

Compute the integralint _(1)^(0) I n ^(3) x dx

Let S(M) denote the sum of the digits of a positive integer M written in base 10. Let N be the smallest positive integer such that S(N) = 2017. Find the value of S(5N + 2017)

If f(x) = (1)/(x^(2)) int_(3)^(x)(2t-3f'(t)dt, then f'(3) is equal to

Integrate the functions x/(sqrt(x+4)),xgt0

The trajectories orthogonal to x^(2)+y^(2) = 2ax is