28278 + Interview Questions in Maths in Class 12

1.	Two cards are drawn from a pack of 52 cards . What is the probability of getting both kings if the card drawn in first draw is replaced before 2^(nd) draw.
Answer» ANSWER :`(1)/(169)`

Discussion

2.	Consider a game played by 10 prople in which each flips a fair coin at the same time. If all but one of the coins comes up the same, then the add persons wing (e.g., if there are nine tails and one head then person having lead wins.) If such a situation does not occur, the players flips again. Find the probability that game is settled on or after nth toss.
Answer» SOLUTION :(i) the probability that ONE GETS tail and nine get head is `""^(10)C_(1)1/2((1)/(2))^(9)` (ii) The probability that one gets head and nine hets tail is `""^(10)C_(1)1/2((1)/(2))^(9)` Hence, probability that the game is settled is `2xx""^(10)C_(1)1/2((1)/(2))^(9)=(5)/(2^(8))` If the game is not settled in the FIRST TOSS, its probability is `1-5//2^(8).` If the game is not settled in the second toss, its probability is `(1-5//2^(8))^(n-1),` whihc isequal to the probability that the game is settled on or after the nth toss.

Discussion

3.	Let I_(1)=int_(0)^(oo)ln(x+1/x)(dx)/(1+x^(2)) and l_(2)=int_(0)^(pi//2)((theta)/(sintheta))^(2)d theta, then which of the following is/are correct?
Answer» `I_(1)gtI_(2)` `I_(2)gtI_(1)` `I_(1)=I_(2)` `I_(1)+I_(2)=0` SOLUTION :`I_(2)=int_(0)^(pi//2)((theta)/(sintheta))^(2) d theta=int_(0)^(pi//2) theta^(2)cosec^(2) theta d theta` `=[-theta^(2) cot theta]_(0)^(pi//2)+int_(0)^(pi//2) 2 theta cot theta d theta` Use: integration by parts `=- int_(0)^(pi//2)L nsin theta=pi l n 2` `I_(1)=int_(0)^(oo)ln(X+1/x)(dx)/(1+x^(2))` Let `tan^(-1)x = theta` `=int_(0)^(pi//2)l n (tan theta+cot theta) d theta` `=-int_(0)^(pi//2)(ln sin theta+l n cos theta) d theta=pi l N2` `:.I_(1)=I_(2)`

Discussion

4.	Let f :R ^(+) to R be a differentiable function with f (1)=3 and satisfying : int _(1) ^(xy) f(t) dt =y int_(1) ^(x) f (t) dt +x int_(1) ^(y) f (t) dt AA x, y in R^(+), then f (e) =
Answer» 3 4 1 None of these Answer :D

Discussion

5.	Findthe valuesof a,b,c,d, if 1,2,3,4are therootsof x^4+ax^3 +bx^2 +cx+d=0
Answer» ANSWER :`-10 ,35,-50,24`

Discussion

6.	Find the number of ways of selecting 5 objects from 9 dissimilar objects such that a particular object is included
Answer» ANSWER :`""^8C_4`

Discussion

7.	Find the number of ways of selecting 5 objects from 9 dissimilar objects such that a particular object is not included
Answer» ANSWER :`""^8C_5`

Discussion

8.	Integrate the following int(dx)/(x{(Inx)^2+25}
Answer» Solution :`int(dx)/(X{(INX)^2+25}` [ put `Inx=5tantheta` then `(1/x)dx=5sec^2thetad theta` `int(5sec^2theta d theta)/(25+25tan^2theta)` `(1/5)int(sec^2theta)/seca^2theta d theta=(1/5)INTD theta` `(1/5)tan^(-1)(Inx/5)+C`

Discussion

9.	If two cards are drawn from a pack, find the probability of getting one king and one queen or both red.
Answer» ANSWER :`(337)/(1326)`

Discussion

10.	If the vectors ai + j + k, I + bj + k, I + j + ck (a != b, c != 1) are coplanar, then the value of (1)/(1 -a) + (1)/(1 -b ) + (1)/(1 - c) =
Answer» 0 3 2 1 Answer :D

Discussion

11.	Let F(x) bet the area bounded by the curve f(t)=(e^(t))/(t) between t=a (a gt 1), t=x and axis of abscissa then the area bounded by g(t)=(e^(t))/(1+t_(0)) (t_(0) gt 0) betweent = a,t = x and axis of abscissais
Answer» `E^(t_(0))[F(x+t_(0))-2F(a+t_(0))]` `e^(t_(0))[F(x+t_(0))-F(a+t_(0))]` `e^(-t_(0))[F(x+t_(0))-F(a+t_(0))]` `e^(-t_(0))[F(x+t_(0))-F(a+t_(0))]` Answer :C

Discussion

12.	Verify that\|P({a})\| =2^1
Answer» <P> SOLUTION :LET `A ={a} then P(A)={{a},PHI}` `:.` P(A)=2=2^1`

Discussion

13.	If n be a natural number define polynomial f_(n)(x) of n^(th) degree as follows f_(n)(costheta)cosntheta i.e. f_(2)(x)=2x^(2)-1 f_(3)(x)=4x^(3)-3x_(1) Then f_(6)(x) is equal to
Answer» `36x^(6)-48X^(4)+18X^(2)-5` `32x^(6)-48x^(4)+18x^(2)-1` `36x^(6)-45X^(4)+18x^(2)-8` `36x^(6)-48x^(4)+18x^(2)-7` SOLUTION :As, `f(x)=x` & `cos(n+1)theta+cos(n-1)theta=2cos(ntheta).COSTHETA` `f_((n+1))(x)+f_((n-1))=2x.f_(n)(x)` `f_(n)(x)=1/(2x)[f_(n+1)(x)+f_(n-1)(x)]` Now, Put `x=costheta, impliessqrt(x^(2)-1)=isintheta` `(x+sqrt(x^(2)-1))^(10)+(x-sqrt(x^(2)-1))^(10)=(costheta+isintheta)^(10)+(costheta-isintheta)^(10)` `=2cos(10theta)=2f_(10)(costheta)=2f_(10)(x)`

Discussion

14.	If n be a natural number define polynomial f_(n)(x) of n^(th) degree as follows f_(n)(costheta)cosntheta i.e. f_(2)(x)=2x^(2)-1 f_(3)(x)=4x^(3)-3x_(1) Then (x+sqrt(x^(2)-1))^(10)+(x-sqrt(x^(2)-1s))^(10) is equal to
Answer» `f_(10)(x)` `f_(11)(x)+f_(9)(x)` `f_(11)(x)` `2f_(10)(x)` SOLUTION :As, `f(x)=x` & `cos(N+1)theta+cos(n-1)theta=2cos(ntheta).costheta` `f_((n+1))(x)+f_((n-1))=2x.f_(n)(x)` `f_(n)(x)=1/(2x)[f_(n+1)(x)+f_(n-1)(x)]` Now, Put `x=costheta, impliessqrt(x^(2)-1)=isintheta` `(x+sqrt(x^(2)-1))^(10)+(x-sqrt(x^(2)-1))^(10)=(costheta+isintheta)^(10)+(costheta-isintheta)^(10)` `=2cos(10theta)=2f_(10)(costheta)=2f_(10)(x)`

Discussion

15.	Theproductof all therealrootsofX^2 - 8x+9-(8)/(x)+(1)/(x^2)=0is
Answer» 2 1 3 7 Answer :B

Discussion

16.	1+ 1/(2!)+2/(3!)+4/(2!)+ .... oo =
Answer» `E^2` `e^(2) +1` `e^(2) -1` `1/4 (e^(2) +1)` ANSWER :D

Discussion

17.	If S_n denotes the sum of first n natural number then S_1 + S_2x + S_3 x^2 +……+S_n x^(n-1) + ……oo terms =
Answer» `(1 -X )^(-1)` `(1-x)^(-2)` `(1 -x)^(-3)` `(1 - x)^(-4)` Answer :C

Discussion

18.	Find the coordinates of the vertex and focus, and the equtions of the directrix and axes of the followingparabolas.
Answer» `y^(2)=16X` `x^(2)=-4Y` `3x^(2)-9x+5y-2=0` `y^(2)-x+4y+5=0` ANSWER :B

Discussion

19.	A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.
Answer» ANSWER :`(3sqrt(3))/4r^(2)` SQ. unit

Discussion

20.	Area bounded by the curves y = sin (pi x)/(2) and y = x^(3) is equal to
Answer» `(4 - pi)/(pi)` sq. units `(4 - pi)/(2 pi)` sq. units `(8 - pi)/(pi)` sq. units `(8 - pi)/(2 pi)` sq. units Answer :D

Discussion

21.	Find the number of ways of arranging the letters of the word SINGING so that relative positions of vowels and consonants are not disturbed.
Answer» ANSWER :30

Discussion

22.	Find the centre, eccentricity, foci, vertices of ellipse x^(2)+4y^(2)-8x-16y-68=0
Answer» ANSWER :`5SQRT3`

Discussion

23.	Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If triangle S'BS is a right angled triangle with right at B and area (triangle S'BS)= 8 sq. units, then the length of a latus rectum of the ellipse is
Answer» `2sqrt(2)` `4sqrt(2)` `2` `4` ANSWER :D

Discussion

24.	If y=1/(x(x+1))then what is y_3?
Answer» SOLUTION :`y=1/(X(x+1))=1/x-1/(x+1)rArry_1=(-1)/x^2+1/((x+1))^2y_2=2/x^3-2/((x+1))^4thereforey_3=(-6)/x^4+6/((x+1))^4`

Discussion

25.	The integral int_(2)^(4)(logx^(2))/(logx^(2)+log(36-12x+x^(2)))dx is equal to
Answer» 2 4 1 6 Answer :3

Discussion

26.	If n is a positive integer then 2^(4n) - 15n - 1 is divisible by
Answer» 64 196 225 256 Answer :C

Discussion

27.	Let a, b, c be such that b(a+c) ne 0. If\|[a,a+1,a-1],[-b,b+1,b-1],[c,c-1,c+1]\|+ \|[a+1,b+1,c-1],[a-1,b-1,c+1],[(-1)^(n+2) a ,(-1)^(n+1) b ,(-1)^(n) c]\|=0 then the value of n is
Answer» zero any EVEN integer any ODD integer any integer Answer :C

Discussion

28.	If A, B, C are the angle of a triangle and \|(sinA,sinB,sinC),(cosA,cosB,cosC),(cos^(3)A,cos^(3)B,cos^(3)C)\|=0, then show that DeltaABC is an isosceles.
Answer» ANSWER :

Discussion

29.	If the probability of chossing an interger "k" out of 2m integers 1,2,3,….,2m is inversely proportional to k^(4)(1leklem).Ifx_(1) is the probability that chosen number is odd and x^(2) is the probability that chosen number is even, then
Answer» `x_(1)gt1//2` `x_(1)gt2//3` `x_(2)lt1//2` `x_(2)lt2//3` SOLUTION :Let one probabilitt of choosing one INTEGER K be P (K) `=lamda//k^(4).` (`lamda` is one constant of proportionality). Then `underset(k=1)overset(2M)sum(lamda)/(k^(4))=1` `or underset(k=1)overset(2m)sum(1)/(k^(4))=1` Let `x_(1)` be the probability of choosing the odd numer. Then `x_(1)underset(k=1)overset(m)sumP(2k-1)=lamdaunderset(k=1)overset(m)sum(1)/((2k-1)^(4))` Also, `1-x_(1)=underset(k=1)overset(m)sumP(2k)` `=lamdaunderset(k=1)overset(m)sum (1)/((2k)^(4))` `ltlamdaunderset(k=1)overset(m)sum(1)/((2k-1)^(4))` `implies1-x_(1)ltx_(1)` `impliesx_(1)gt1//2` `impliesx_(2)lt1//2`

Discussion

30.	Find (dy)/(dx)," if "x=a(cos t+log tan""t/2), y=a sin t.
Answer» ANSWER :`TAN t`

Discussion

31.	Integrate the following functions : sin2xlog(cosx)
Answer» Answer :`-cos^(2)x.log(cosx)+(cos^(2)x)/(2)+C`

Discussion

32.	x = (sin^3t)/(sqrt(cos 2t)), y = (cos^3 t)/(sqrt(cos 2t)).
Answer» ANSWER :`-COT 3T`

Discussion

33.	If x=a(theta-sin theta) and y=a(1+cos theta), then proe that (dy)/(dx)=-cot""(theta/2).
Answer» ANSWER :`-COT (THETA)/(2)`

Discussion

34.	Find int(x^(2))/((x^(2)+1)(x^(2)+4))dx
Answer» Answer :`(-1)/(3) TAN^(-1) x+(2)/(3) tan^(-1)((x)/(2))+C`

Discussion

35.	x = cos theta - cos 2 theta, y = sin theta - sin 2 theta. then find dy/dx
Answer» ANSWER :`(cos theta- 2 cos 2 theta)/(2 SIN 2theta- sin theta)`

Discussion

36.	x = sin t, y = cos 2 t.
Answer» ANSWER :`-4 SIN t`

Discussion

37.	x = 4t, y = 4/t then find dy/dx
Answer» ANSWER :`-(1)/(t^2)`

Discussion

38.	x = a cos theta, y = b sin theta.
Answer» ANSWER :`(B)/(a)`

Discussion

39.	If x and y are connected parametrically by the equationswithout eliminating the parameter, Find (dy)/(dx). x= 2at^(2), y= at^(4).
Answer» ANSWER :`t^2`

Discussion

40.	The probability of a man hitting a target is 0.25. He shoots 7 times, then what is the probability of his hitting atleast twice ?
Answer» ANSWER :`(4547)/(8192)`

Discussion

41.	The radius of an air bubble is increasing at the rate of (1)/(2) cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm ?
Answer» ANSWER :`= 2PI CM^(3)//SEC`.

Discussion

42.	Obtain following definite integrals : overset(1)underset(0)int (dx)/(e^(x)+e^(-x))
Answer» ANSWER :`:.I=tan^(-1)(E)-(PI)/(4)`

Discussion

43.	Find the number of positive intergral solutions of x_1x_2x_3=54 where x_1,x_2,x_3 ne 1
Answer» ANSWER :9

Discussion

44.	x = a sec theta, y = b tan theta.
Answer» ANSWER :`(B)/(a)" COSEC "THETA`

Discussion

45.	x = a (cos theta + theta sin theta), y = a (sin theta - theta cos theta). then find dy/dx
Answer» ANSWER :`TAN THETA`

Discussion

46.	The proposition (pto q ) harr (~p to~q)is a
Answer» tautology contradiction contingency none of these Answer :C

Discussion

47.	R is thesum of squareof 50 consectiveevenintegers statingwith1.S is whatpercentageless thanR ?
Answer» 0.25 0.33 0.5 0.75 Answer :D

Discussion

48.	Evalute the following integrals int (3x+ 1)/((x + 1)^(2) (x + 3))dx
Answer» Answer :`2 LOG \|(x + 1)/(x + 3)\| + (1)/(x + 1) + C `

Discussion

49.	Let N denotes the set of all natural numbers. Define two binary relations on N as R_(1) ={(x,y) in N xx N : 2x + y=10} R_(2) = {(x,y) in N xx N , x+2y = 10} Then
Answer» range of `R_(1)` is {2,4,8} range of `R_(2)` is {1,2,3,4} both `R_(1)`and `R_2` are symmetric relations both `R_(1)` and `R_2` are TRANSITIVE relations Answer :B

Discussion

50.	Expand in ascending powers of x.2^x,
Answer» SOLUTION :`2x=e^x(log_e2)=1+x(log_e2)+(x^2(log_e2)^2)/(2!)+(x^3(log_e2)^3)/(3!)+…`

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Two cards are drawn from a pack of 52 cards . What is the probability of getting both kings if the card drawn in first draw is replaced before 2^(nd) draw.

Let I_(1)=int_(0)^(oo)ln(x+1/x)(dx)/(1+x^(2)) and l_(2)=int_(0)^(pi//2)((theta)/(sintheta))^(2)d theta, then which of the following is/are correct?

Let f :R ^(+) to R be a differentiable function with f (1)=3 and satisfying : int _(1) ^(xy) f(t) dt =y int_(1) ^(x) f (t) dt +x int_(1) ^(y) f (t) dt AA x, y in R^(+), then f (e) =

Findthe valuesof a,b,c,d, if 1,2,3,4are therootsof x^4+ax^3 +bx^2 +cx+d=0

Find the number of ways of selecting 5 objects from 9 dissimilar objects such that a particular object is included

Find the number of ways of selecting 5 objects from 9 dissimilar objects such that a particular object is not included

Integrate the following int(dx)/(x{(Inx)^2+25}

If two cards are drawn from a pack, find the probability of getting one king and one queen or both red.

If the vectors ai + j + k, I + bj + k, I + j + ck (a != b, c != 1) are coplanar, then the value of (1)/(1 -a) + (1)/(1 -b ) + (1)/(1 - c) =

Let F(x) bet the area bounded by the curve f(t)=(e^(t))/(t) between t=a (a gt 1), t=x and axis of abscissa then the area bounded by g(t)=(e^(t))/(1+t_(0)) (t_(0) gt 0) betweent = a,t = x and axis of abscissais

Verify that|P({a})| =2^1

If n be a natural number define polynomial f_(n)(x) of n^(th) degree as follows f_(n)(costheta)cosntheta i.e. f_(2)(x)=2x^(2)-1 f_(3)(x)=4x^(3)-3x_(1) Then f_(6)(x) is equal to

If n be a natural number define polynomial f_(n)(x) of n^(th) degree as follows f_(n)(costheta)cosntheta i.e. f_(2)(x)=2x^(2)-1 f_(3)(x)=4x^(3)-3x_(1) Then (x+sqrt(x^(2)-1))^(10)+(x-sqrt(x^(2)-1s))^(10) is equal to

Theproductof all therealrootsofX^2 - 8x+9-(8)/(x)+(1)/(x^2)=0is

1+ 1/(2!)+2/(3!)+4/(2!)+ .... oo =

If S_n denotes the sum of first n natural number then S_1 + S_2x + S_3 x^2 +……+S_n x^(n-1) + ……oo terms =

Find the coordinates of the vertex and focus, and the equtions of the directrix and axes of the followingparabolas.

A point P is given on the circumference of a circle of radius r. Chord QR is parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

Area bounded by the curves y = sin (pi x)/(2) and y = x^(3) is equal to

Find the number of ways of arranging the letters of the word SINGING so that relative positions of vowels and consonants are not disturbed.

Find the centre, eccentricity, foci, vertices of ellipse x^(2)+4y^(2)-8x-16y-68=0

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If triangle S'BS is a right angled triangle with right at B and area (triangle S'BS)= 8 sq. units, then the length of a latus rectum of the ellipse is

If y=1/(x(x+1))then what is y_3?

The integral int_(2)^(4)(logx^(2))/(logx^(2)+log(36-12x+x^(2)))dx is equal to

If n is a positive integer then 2^(4n) - 15n - 1 is divisible by

Let a, b, c be such that b(a+c) ne 0. If|[a,a+1,a-1],[-b,b+1,b-1],[c,c-1,c+1]|+ |[a+1,b+1,c-1],[a-1,b-1,c+1],[(-1)^(n+2) a ,(-1)^(n+1) b ,(-1)^(n) c]|=0 then the value of n is

If A, B, C are the angle of a triangle and |(sinA,sinB,sinC),(cosA,cosB,cosC),(cos^(3)A,cos^(3)B,cos^(3)C)|=0, then show that DeltaABC is an isosceles.

If the probability of chossing an interger "k" out of 2m integers 1,2,3,….,2m is inversely proportional to k^(4)(1leklem).Ifx_(1) is the probability that chosen number is odd and x^(2) is the probability that chosen number is even, then

Find (dy)/(dx)," if "x=a(cos t+log tan""t/2), y=a sin t.

Integrate the following functions : sin2xlog(cosx)

x = (sin^3t)/(sqrt(cos 2t)), y = (cos^3 t)/(sqrt(cos 2t)).

If x=a(theta-sin theta) and y=a(1+cos theta), then proe that (dy)/(dx)=-cot""(theta/2).

Find int(x^(2))/((x^(2)+1)(x^(2)+4))dx

x = cos theta - cos 2 theta, y = sin theta - sin 2 theta. then find dy/dx

x = sin t, y = cos 2 t.

x = 4t, y = 4/t then find dy/dx

x = a cos theta, y = b sin theta.

If x and y are connected parametrically by the equationswithout eliminating the parameter, Find (dy)/(dx). x= 2at^(2), y= at^(4).

The probability of a man hitting a target is 0.25. He shoots 7 times, then what is the probability of his hitting atleast twice ?

The radius of an air bubble is increasing at the rate of (1)/(2) cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm ?

Obtain following definite integrals : overset(1)underset(0)int (dx)/(e^(x)+e^(-x))

Find the number of positive intergral solutions of x_1x_2x_3=54 where x_1,x_2,x_3 ne 1

x = a sec theta, y = b tan theta.

x = a (cos theta + theta sin theta), y = a (sin theta - theta cos theta). then find dy/dx

The proposition (pto q ) harr (~p to~q)is a

R is thesum of squareof 50 consectiveevenintegers statingwith1.S is whatpercentageless thanR ?

Evalute the following integrals int (3x+ 1)/((x + 1)^(2) (x + 3))dx

Let N denotes the set of all natural numbers. Define two binary relations on N as R_(1) ={(x,y) in N xx N : 2x + y=10} R_(2) = {(x,y) in N xx N , x+2y = 10} Then

Expand in ascending powers of x.2^x,