28278 + Interview Questions in Maths in Class 12

1.	Let S be the set of all real values of a for which the following system of linear equations : ax+2y+5z=1 2x+y+3z=1 3y +7z=1 is consistent . Then the set S is
Answer» an empty set equal to R equal to R - {1} equal to {1} ANSWER :B

Discussion

2.	Let alpha and beta be the roots of x^2-x-1=0, with alpha gt beta. For all positive integers n, define a_n=(alpha^n=beta^n)/(alpha-beta),nge 1. b_1=1 and b_b=a_(n-1)+a_n+1,n ge 2 Then which of the following options is/are correct ?
Answer» `sum_(n=1)^(oo)(b_n)/(10^n)=(8)/(89)` `b_n=ALPHA^n+beta^n " for all " n ge 1` `a_1+a_2+a_3+.......+a_n=a_(n+2)-1" for all " n ge 1` `sum_(n=1)^(oo)(a_n)/(10^n)=(10)/(89)` Solution :Given quadratic EQUATION `x^2-x-1=0` having roots `alpha` and `beta,(alpha gt beta)` So, `alpha=(1+sqrt(5))/(2)and beta =(1-sqrt(5))/(2)` and `alpha +beta =1,alpha beta =-1` `because a_n=(a^n-beta^n)/(alpha-beta),n ge1` So, `a_(n+1)=((alpha^n+1)-beta^(n+1))/(alpha-beta)=alpha^n+alpha^n-1beta+alpha^n-2beta+.......ALPHABETA^(n-1)+beta^(n)` ` =alpha^n-alpha^n-1-alpha^n-3beta......-beta ^n-2+beta^n` `=alpha^n+beta^n-(alpha^n-1+alpha^n-3beta+....+beta^n-2)` `=alpha^n+beta^n-a_(n-1)` `[as a_(n-1)=(alph^n-1 -beta^n-1)/(alpha-beta)=alpha^n-2+alpha^n-3beta+....beta^n-2]` `impliesalpha_(n+1)+alpha_(n-1)=alpha^(n)+beta^(n)=b_(n),AAnge1` So,option (B) is incorrect Now `Sigma_(n=1)^(oo) (b_n)/(10^n)=Sigma_(n=1)^(oo) (a^n+b^n)/(10^n)` ` =Sigma_(n=1)^(oo) ((alpha)/(10))^n+Sigma_(n=1)^(oo) ((beta)/(10))^n [because \|(alpha)/(10)\|lt 1 and \|(beta)/(10)\| lt 1 and \|(beta)/(10)lt 1\|` ` =((alpha)/(10))/(10(alpha)/(10))+((beta)/(10))/(1-(beta)/(10))=(alpha)/(10-alpha)+(beta)/(1-beta)` ` =(10alpha-alpha beta +10beta-alpha beta)/((10-alpha)(10-beta))=(10(alpha+beta)-2alphabeta)/(100-10(alpha+beta)+alpha beta)` `=(10(1)-2(-1))/(100-100(1)-1)` `=(12)/(89)` So, option (a) is not correct. `because alpha^2=alpha +1 and beta^2=beta+1` `rArr alpha^n+1=alpha^n+1+alpha^n and beta^n+2=beta^n+1+beta^n`. `rArr (alpha^n+2+beta^n+2)=(alpha^n+1+beta^n+1)+(alpha^n+beta^n)` `rArr a_(n+1)=a_(n+1)+a_n` SIMILARLY `a_(n+1)=a_n+a_n-1` `a_n=a_n-1+a_(n-2)` `.............` `...................` On adding , we get `a_(n+2)=(a_n+a_n-1+a_n-2+....+a_2+a_1)+a_2` `[because a_2=(alpha^2-beta^2)/(alpha-beta)=alpha+beta=1]` So, ` a_n+2-1=a_1=a_1+a_2+a_3+......+a_n` So, option (c) is aslo correct. And, now `Sigma_(n =1)^(oo) (a_n)/(10^n)=Sigma_(n=1)^(oo) (alpha^n-beta^n)/((alpha-beta) 10^n)` ` =(1)/(alpha-beta)[Sigma _(n=1)^(oo) ((alpha)/(10))^n-Sigma _(n=1)^(oo) ((beta)/(10))^n]` ` =(1)/(alpha-beta) [((alpha)/(10))/(1-(alpha)/(10))-((beta)/(10))/(1-(beta)/(10))],[as \|(alpha)/(10)\|lt 1 and \|(beta)/(10)\|lt 1]` `=(1)/(alpha-beta) ((alpha)/(10-alpha)-(beta)/(10-beta))=(1)/(alpha-beta)[(10alpha-alphabeta-10beta+alpha beta)/(100-10(alpha +beta)+alpha beta)]` ` =(10(alpha -beta))/((alpha -beta)[100-10(alpha+beta )+alpha beta])=(10)/(100-10-1)=(10)/(89)` Hence options, (b),(c) and (d) are correct.

Discussion

3.	If x=1/2 then (1-2x)/(1-x+x^(2))+ (2x-4x^(3))/(1-x+x^(4)) + (4x^(3)-8x^(7))/(1-x^(4)+x^(6))+.....infty is
Answer» 1 `8/7` `4/7` `7/16` ANSWER :B

Discussion

4.	Which of the following is a statement in logic?
Answer» what a great fall it is ! Please MIND your own business. Let us GO for a walk. The quadratic EQUATION `x^2` -3X + 2 = 0 has two REAL roots. Answer :D

Discussion

5.	Evaluate the following integrals. (1)/(2x^(2)+3x-(11)/(4))dx
Answer» Answer :`(1)/(SQRT(31))log\|(4x+3-sqrt(31))/(4x+3+sqrt(31))\|+C`

Discussion

6.	If alpha, beta and gamma are direction cosines of the vector vec(x) then 1+cos2alpha+cos 2beta+cos 2lambda= …………..
Answer» 0 1 `-1` 2 Answer :A

Discussion

7.	int3^xdx
Answer» SOLUTION :`int3^xdx=3^x/(IN3) +C`

Discussion

8.	How many 4 letter words can be formed using the letters of the word ARTICLEsuch that the word contains A but not E
Answer» ANSWER :240

Discussion

9.	Ifthe equation fo two circles whose radii are a, a ' areS = 0 and S^' = 0, then show that circles S/a + (S^')/(a^') = 0 and S/a - (S^')/(a') = 0 intersect orthogonally.
Answer» ANSWER :`= (d^2 - AA') + (d^2 + aa') = 2d^2`

Discussion

10.	If the sum of the coefficients in the expansion of (1 -3x + 10x^2)^(n) is a and the sum of the coefficients in the expansion of (1 - x^(2))^(n) is b, then
Answer» `a = 3B` `B = a^(3)` `a = b^(3)` `a = b` ANSWER :C

Discussion

11.	Formthe differentialequationof family ofcurcesy= ae^(2x) +be^(-2x)by eliminatingthe arbitaryconstantsa & b.
Answer» ANSWER :`THEREFORE(d^2 y)/(DX^2)- (DY)/(dx) - 6y =0`

Discussion

12.	Find the area of the surface formed by revolving the astroid x^(2//3) + y^(2//3) = a^(2//3) about the x-axis
Answer» ANSWER :`(12)/(5) PI a^(2)`

Discussion

13.	intsin( logx) dx=………+c
Answer» `X/2[ COS( logx) - SIN( log x )]` `x/2[ sin(logx) + cos( logx)]` `x/2[ sin (log x)- cos ( log x)]` `x[ sin (logx)- cos(log x)]` Answer :C

Discussion

14.	If vec(a) is vector perpendicular to both vec(b) and vec(c ), then
Answer» `VEC(a)+(vec(B)+vec(c ))=vec(0)` `vec(a)xx(vec(b)+vec(c ))=vec(0)` `vec(a)xx(vec(b)xx vec(c ))=vec(0)` `vec(a).(vec(b)xx vec(c ))=vec(0)` ANSWER :C

Discussion

15.	Does a^(2)+a^(4)+a^(6) for all value of a?
Answer» ANSWER :No

Discussion

16.	The two lines x=ay+b,z=cy+d and x=a'y+b', z=c'y +d' are pendicular to each other if
Answer» `AA' + ALPHA =1` `(a)/( a') + (C )/( c') = -1` `(a)/( a') + ( c )/( c') =1` `aa'+ c c'=-1` ANSWER :D

Discussion

17.	If z_(1)=2+3i, z_(2)=3-2i and z_(3)=-1-2sqrt3i, then which of the following is true? (where, i^(2)=-1)
Answer» `ARG((z_(2))/(z_(3)))=arg((z_(2)-z_(1))/(z_(3)-z_(1)))` `arg((z_(2))/(z_(3)))=arg((z_(3))/(z_(1)))` `(1)/(2)arg((z_(2))/(z_(3)))=arg((z_(2)-z_(1))/(z_(3)-z_(1)))` `2arg((z_(3))/(z_(2)))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))` ANSWER :C

Discussion

18.	Write the differential equation whose general solution is y=ce^(2x)
Answer» SOLUTION :Given equation is `y=ce^(2x)rArrdy/dx=2ce^(2x)=2Y`THEREFORE The required DIFFERENTIAL equationis `dy/dx-2y=0`

Discussion

19.	int_(0)^(pi//2) e^(x) (cos x - sin x ) dx =
Answer» 1 -1 0 `1/2` ANSWER :B

Discussion

20.	Find the number of words with or without meaning which can be made using all letters of the word 'AGAIN'. If these words are written as in a dictionary, what will be the 50^(th) word?
Answer» ANSWER :48

Discussion

21.	If (-1/3-1) is a centre of similitude for the circles x^(2)+y^(2)=1 and x^(2)+y^(2)_2x-6y-6=0 then the length of common tangent of the circles is
Answer» `1/3` `4/3` `1` cannot be determined Answer :C

Discussion

22.	Find the second order derivatives of the functions tan^(-1)x
Answer» ANSWER :`(-2X)/((1 + X^(2))^(2))`

Discussion

23.	Let X be a random varibale such that P(X = -2) = P(X = -1) = P(X = 2) = P(X = 1) = (1)/(6) and P(X = 0) = (1)/(3).
Answer» ANSWER :`0, (5)/(3)`

Discussion

24.	Write the locusof a point p which moves in space such that its distance from origin is 4 units.
Answer» SOLUTION :The CENTROID is `((1+2+3)/3,(2+1+0)/3,(3+2+1)/3)=(2,12)`

Discussion

25.	Which of the following is not correct combination?
Answer» ANSWER :D

Discussion

26.	From the origin chords are drawn to the circle (x-1)^(2)+y^(2)=1 then equation of locus of mid ponts of these chords is
Answer» `X^(2)+y^(2)+x=0` `x^(2)+y^(2)-x=0` `x^(2)+y^(2)-2x=0` `x^(2)+y^(2)-2x-2y=0` ANSWER :B

Discussion

27.	Find the Binomial probability distribution whose mean is 3 red veriance is 2 .
Answer» `((2)/(3)+ (1)/(3))^(9)` `((5)/(3) + (2)/(3))^(9)` `((3)/(2) + (1)/(2))^(9)` None of these ANSWER :A

Discussion

28.	If a^(logb^c)=3.3^(log4^3) .3^((log 4^3)^(log 4^3)).3^((log4^3)^((log4^3)^log4^3))...oo where a,b,cin Q the value of abc is -+
Answer» 9 12 16 20 Answer :C

Discussion

29.	Sum of coefficients of terms of odd powers of x in (1 - x + x^2 - x^3)^9 is
Answer» `-2^17` `2^17` `2^13` `2^12` ANSWER :A

Discussion

30.	Find the equation of a curve passing through the point (0,1).Ifthe slope of the tangent to the curve at any point (x,y) is equal to the sum of the xcoordinate(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.
Answer» ANSWER :`y = -1 + 2E^((X^(2))/(2))`

Discussion

31.	If the equatio of one tangent to the circle with centre (2,-1) from the origin is 3x+y=0, then the equation of the other tangent through the origin is
Answer» `3x-y=0` `x+3y=0` `x-3y=0` `x+2y=0` ANSWER :C

Discussion

32.	If A is any square matrix of order 3x3 such that absA=3, then the value of abs(adjA)
Answer» 3 (1/3) 9 27 Answer :C

Discussion

33.	Using the result of the preceding problem obtain the following formula : 1 - (C_(n)^(1))/(3) + (C_(n)^(2))/(5) - (C_(n)^(3))/(7) + . . . . . + (-1)^(n) (C_(n)^(n))/(2 n + 1) = ((2n)!!)/((2n + 1)!!),where C_(n)^(k)are binomial coefficients.
Answer» ANSWER :1 we GET

Discussion

34.	A player tosses two coins. Hewins Rs. 1 if 1 head appears, Rs 2 if 2 heads appear. But he lose Rs 5 if no head appears. The mean of the prized money is
Answer» `(1)/(2)` `(1)/(4)` `-(1)/(4)` `(1)/(5)` Answer :C

Discussion

35.	A ladder 5 cm long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 cm away from the wall ?
Answer» ANSWER :`(8)/(3)` CM /s

Discussion

36.	If f and g are defined on [0, oo) by f(x) = underset(n rarr oo)(lim) (x^(n)-1)/(x^(n)+1) and g(x) = int_(0)^(x) f(t) dt. Then
Answer» G has local MAXIMUM at x=1 g has local minimum at x=1 g is an INCREASING function on `(0, oo)` g is a DECREASING function on `(0,oo)` Answer :B

Discussion

37.	Equation of plane equidistant form planes 3x+4y+5z-6=0 and 3x+4y+5z+6=0 is
Answer» 3x+4y+5z=0 3x+4y+5z=3 3x+4y+5z=12 4x+5y+3z=0 Answer :A

Discussion

38.	In a class 30% of the student fail in Mathematics,20% of the student fail in English and 10% fail in both.What is the probability that he has failed in both ?
Answer» <P> SOLUTION :Probability that he has FAILED in both `i.e.,P(A cap B)=10/100=1/10`

Discussion

39.	Prove the tan75^@ + cot75^@ = 4
Answer» SOLUTION :L.H.S `tan75^@+cot75^@=tan75^@+(1)/(tan75^@) = (tan^2 75^@+1)/(tan75^@) = (2)/(((2tan75^@)/(1+tan^2 75^@)))= (2)/sin(2xx75^@) = (2)/sin150^@ = (2)/(((1)/2) = 4` R.H.S.

Discussion

40.	Find the projection of the vector hati-hatj on the vector hati+hatj.
Answer» ANSWER :VECTOR `VEC(a)` on the vector `vec(B)` is ZERO.

Discussion

41.	Show that A'A and A A ' are both symmetric matrices for any matrix A .
Answer» ANSWER :`thereforeA*A`'

Discussion

42.	Differentiate x^8+x^7
Answer» SOLUTION :LET `y=x^8+x^7` Then `dy/dx=8x^7+7x^6`

Discussion

43.	If (3sqrt(3)+5)^(2n+1)=x" and "f=x-[x] where ([x] is the integral part of x), find the value of x.f.
Answer» ANSWER :`:." XF"=2^(2n+1)`

Discussion

44.	If P(A) = 0.8, P(B) = 0.5 and P(B/A)= 0.4, find P (A/B)
Answer» <P> SOLUTION :P(A/B)=(P`(ANNB))/(P(B))` =0.32/0.5 =32/50=16/25=0.64

Discussion

45.	If a ne b, x ne n pi n in Z and y^(2) = a^(2) cos^(2) x +b^(2) sin^(2)x, " then " (d^(2)y)/(dx^(2))+y =
Answer» `((ab)/(y))^(2)` `(1)/(y)((ab)/(y))^(2)` `((ab)^(2))/(y)` `(ab)/(y^(3))` Answer :B

Discussion

46.	There are 3 bags, each containing 5 white and 3 black balls. Also, there are 2 bags, each containing 2 white and 4 black balls. A white ball is drawn at random. Find the proqability that this ball is from a bag of the first group.
Answer» <P> Solution :Let `E_1` = event of selecting a bag from the first GROUP, and `E_2` = event of selecting a bag from the SECON group. Then, `P(E_1)=3/5 and P(E_2)=2/5` Let E = event that the ball drawn is WHITE. Then, `P(E//E_1)=5/8,P(E//E_2)=2/6=1/3`. `:. P(E_1//E)=(P(E//E_1).P(E_1))/(P(E//E_1).P(E_1)+P(E//E_2).P(E_2))`

Discussion

47.	A = ({:(1),(2),(3):}),"find " A * A^(T)
Answer» ANSWER :`[(1,2,3),(2,4,6),(3,6,9)]`

Discussion

48.	Integrate the following functions 1/sqrt(9-25x^2)
Answer» Solution :`INT 1/sqrt(9-25x^2)DX = int 1/sqrt(3^2-(5x)^2) dx` =`1/5 sin^-1 ((5x)/3)+C`

Discussion

49.	Iff(theta)=sum_(n=1)^(6)cos ec(theta+((n-1)pi)/(4))cos ec(theta+(npi)/(4)), where 0ltthetalt(pi)/(2), then find the minimum valueof f(theta).
Answer» ANSWER :`2sqrt(2)`

Discussion

50.	Assertion(A) : if x^4 -x^3 -6x^2 +4x +8=0hasamultiplerootthentheequationhavingthesamerootis4x^3-3x^2- 12 x +4=0 Reason (R ):Ifalphais repeatedrootoff(x)=0thenalphaisalsoa rootoff ^1(x ) =0
Answer» BOTHA and RaretrueR isthe correctexplanationof A both AandRare TRUEAND RIS notcorrectexplanationof A AIS trueand Risfalse A isfalseand Ristrue Answer :A

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Let S be the set of all real values of a for which the following system of linear equations : ax+2y+5z=1 2x+y+3z=1 3y +7z=1 is consistent . Then the set S is

Let alpha and beta be the roots of x^2-x-1=0, with alpha gt beta. For all positive integers n, define a_n=(alpha^n=beta^n)/(alpha-beta),nge 1. b_1=1 and b_b=a_(n-1)+a_n+1,n ge 2 Then which of the following options is/are correct ?

If x=1/2 then (1-2x)/(1-x+x^(2))+ (2x-4x^(3))/(1-x+x^(4)) + (4x^(3)-8x^(7))/(1-x^(4)+x^(6))+.....infty is

Which of the following is a statement in logic?

Evaluate the following integrals. (1)/(2x^(2)+3x-(11)/(4))dx

If alpha, beta and gamma are direction cosines of the vector vec(x) then 1+cos2alpha+cos 2beta+cos 2lambda= …………..

int3^xdx

How many 4 letter words can be formed using the letters of the word ARTICLEsuch that the word contains A but not E

Ifthe equation fo two circles whose radii are a, a ' areS = 0 and S^' = 0, then show that circles S/a + (S^')/(a^') = 0 and S/a - (S^')/(a') = 0 intersect orthogonally.

If the sum of the coefficients in the expansion of (1 -3x + 10x^2)^(n) is a and the sum of the coefficients in the expansion of (1 - x^(2))^(n) is b, then

Formthe differentialequationof family ofcurcesy= ae^(2x) +be^(-2x)by eliminatingthe arbitaryconstantsa & b.

Find the area of the surface formed by revolving the astroid x^(2//3) + y^(2//3) = a^(2//3) about the x-axis

intsin( logx) dx=………+c

If vec(a) is vector perpendicular to both vec(b) and vec(c ), then

Does a^(2)+a^(4)+a^(6) for all value of a?

The two lines x=ay+b,z=cy+d and x=a'y+b', z=c'y +d' are pendicular to each other if

If z_(1)=2+3i, z_(2)=3-2i and z_(3)=-1-2sqrt3i, then which of the following is true? (where, i^(2)=-1)

Write the differential equation whose general solution is y=ce^(2x)

int_(0)^(pi//2) e^(x) (cos x - sin x ) dx =

Find the number of words with or without meaning which can be made using all letters of the word 'AGAIN'. If these words are written as in a dictionary, what will be the 50^(th) word?

If (-1/3-1) is a centre of similitude for the circles x^(2)+y^(2)=1 and x^(2)+y^(2)_2x-6y-6=0 then the length of common tangent of the circles is

Find the second order derivatives of the functions tan^(-1)x

Let X be a random varibale such that P(X = -2) = P(X = -1) = P(X = 2) = P(X = 1) = (1)/(6) and P(X = 0) = (1)/(3).

Write the locusof a point p which moves in space such that its distance from origin is 4 units.

Which of the following is not correct combination?

From the origin chords are drawn to the circle (x-1)^(2)+y^(2)=1 then equation of locus of mid ponts of these chords is

Find the Binomial probability distribution whose mean is 3 red veriance is 2 .

If a^(logb^c)=3.3^(log4^3) .3^((log 4^3)^(log 4^3)).3^((log4^3)^((log4^3)^log4^3))...oo where a,b,cin Q the value of abc is -+

Sum of coefficients of terms of odd powers of x in (1 - x + x^2 - x^3)^9 is

Find the equation of a curve passing through the point (0,1).Ifthe slope of the tangent to the curve at any point (x,y) is equal to the sum of the xcoordinate(abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

If the equatio of one tangent to the circle with centre (2,-1) from the origin is 3x+y=0, then the equation of the other tangent through the origin is

If A is any square matrix of order 3x3 such that absA=3, then the value of abs(adjA)

Using the result of the preceding problem obtain the following formula : 1 - (C_(n)^(1))/(3) + (C_(n)^(2))/(5) - (C_(n)^(3))/(7) + . . . . . + (-1)^(n) (C_(n)^(n))/(2 n + 1) = ((2n)!!)/((2n + 1)!!),where C_(n)^(k)are binomial coefficients.

A player tosses two coins. Hewins Rs. 1 if 1 head appears, Rs 2 if 2 heads appear. But he lose Rs 5 if no head appears. The mean of the prized money is

A ladder 5 cm long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 cm away from the wall ?

If f and g are defined on [0, oo) by f(x) = underset(n rarr oo)(lim) (x^(n)-1)/(x^(n)+1) and g(x) = int_(0)^(x) f(t) dt. Then

Equation of plane equidistant form planes 3x+4y+5z-6=0 and 3x+4y+5z+6=0 is

In a class 30% of the student fail in Mathematics,20% of the student fail in English and 10% fail in both.What is the probability that he has failed in both ?

Prove the tan75^@ + cot75^@ = 4

Find the projection of the vector hati-hatj on the vector hati+hatj.

Show that A'A and A A ' are both symmetric matrices for any matrix A .

Differentiate x^8+x^7

If (3sqrt(3)+5)^(2n+1)=x" and "f=x-[x] where ([x] is the integral part of x), find the value of x.f.

If P(A) = 0.8, P(B) = 0.5 and P(B/A)= 0.4, find P (A/B)

If a ne b, x ne n pi n in Z and y^(2) = a^(2) cos^(2) x +b^(2) sin^(2)x, " then " (d^(2)y)/(dx^(2))+y =

There are 3 bags, each containing 5 white and 3 black balls. Also, there are 2 bags, each containing 2 white and 4 black balls. A white ball is drawn at random. Find the proqability that this ball is from a bag of the first group.

A = ({:(1),(2),(3):}),"find " A * A^(T)

Integrate the following functions 1/sqrt(9-25x^2)

Iff(theta)=sum_(n=1)^(6)cos ec(theta+((n-1)pi)/(4))cos ec(theta+(npi)/(4)), where 0ltthetalt(pi)/(2), then find the minimum valueof f(theta).

Assertion(A) : if x^4 -x^3 -6x^2 +4x +8=0hasamultiplerootthentheequationhavingthesamerootis4x^3-3x^2- 12 x +4=0 Reason (R ):Ifalphais repeatedrootoff(x)=0thenalphaisalsoa rootoff ^1(x ) =0