28278 + Interview Questions in Maths in Class 12

1.	intcos^4xdx
Answer» SOLUTION :`intcos^4xdx=INT((1+cos2x)/2)^2dx` =`1/4 int(1+2cos2x+cos^2 2X)DX` =`1/4 int(1+2cos2x+(1+cos4x)/2)dx` =`1/4 int(3/2+2cos2x+1/2cos4x)dx` =3x/8+1/4sin2x+1/32sin4x+C

Discussion

2.	Find (dy)/(dx) in the following : 2x+3y= sin y.
Answer» ANSWER :`(2)/(COS y-3)`

Discussion

3.	Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A= {1, 5}, B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw. based on the above information, answer the following questions. P(B\|C) =
Answer» `(2)/(3)` `(1)/(12)` `(1)/(9)` 0 Answer :A

Discussion

4.	Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A= {1, 5}, B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw. based on the above information, answer the following questions. P(A capB\|C) =
Answer» `(1)/(6)` `(1)/(2)` `(1)/(12)` `(1)/(3)` Answer :D

Discussion

5.	Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A= {1, 5}, B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw. based on the above information, answer the following questions. P(A\|B) =
Answer» `(1)/(6)` `(1)/(3)` `(1)/(2)` `(2)/(3)` Answer :B

Discussion

6.	Check monotonocity at following points for (i) f(x) =x^(3) -3x +1 =-1,2 (ii) f(x) +\|x -1\|+2\|x-3\|-\|x+2\| "at " x=-2,0,3,5 (iii) f(x) =x^(1//3) """at""" x=0 (iv) f(x)=x^(2) n +(1)/(x^(2)) """at""" x=1,2 (v) f(x) ={underset( 3 sin x , ""x ge0)(x^(3) +2x^(2) 5xgt 0)," at " x=0
Answer» Answer :(i) Neither increasingnor decreasingat x=-1 and INCREASING at x=2 (ii) at x=-2 decreasing at x=0 decreasing at x=3 neitherincreasing nor decreasing at x=5 increasing (III) STRICTLY increasing at x=0 (IV) Strictly increasing at x=2 neither 1 nor D at x=1 (v) Strictly increasing at x=0

Discussion

7.	int_(0)^(pi//2)(sin^(1000)xdx)/(sin^(1000)x+cos^(1000)x) is equal to
Answer» 1)1000 2)1 3)`(PI)/(2)` 4)`(pi)/(4)` Answer :D

Discussion

8.	Write the function tan^(-1)((sqrt(1+x^(2))-1)/x)x ne 0, in the simplest form.
Answer» ANSWER :`1/2tan^(-1)X`

Discussion

9.	(85)^(1//4)~~....
Answer» `3.3` `3.034` `3.4` `30.50` ANSWER :B

Discussion

10.	Vertify mean value theorem for the following functions: f(x)= x^(2) + 2x+ 3, x in [4, 6]
Answer» ANSWER :5

Discussion

11.	Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A= {1, 5}, B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw. based on the above information, answer the following questions. P(A\|C) =
Answer» `(1)/(4)` 1 `(2)/(3)` NONE of these Answer :C

Discussion

12.	Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A= {1, 5}, B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw. based on the above information, answer the following questions. P(A cup B\|C) =
Answer» 0 `(1)/(2)` `(2)/(3)` 1 Answer :D

Discussion

13.	{:(" "Lt),(x rarr 0):} (int_(0)^(x) sin^(2)t cost dt)/(x^(3))=
Answer» 1 `1/2` `1/3` `2/3` ANSWER :C

Discussion

14.	Determine whethera** b =ma - nb "on" Q + "where" m "and" n in Noperations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.
Answer» Solution :LET a =1, b=2 `m =1, n=3` ` ma -nb =1-6= -5 !in Q^+` `:. A,b in Q^+ IMPLIES a b in Q^+` ` implies "is not a BINARY operation on" Q^+`

Discussion

15.	A tangenttotheellipsedistancedistancefromthe centreoftheellipsex^(2) +2y^(2) =6 P andQprovethatthetangent at P andQof theellipsex^(2)+ 2y^(2) =6areat rightangles .
Answer» Solution :GIVEN , `x^(2)+ 4y^(2)=4 or(x^(2))/(4)+(y^(2))/(1)=1` Equation of anytangentto theellipseon(i) can bewritten as `(x)/(2)costheta + y sintheta =1` EQUATIONOF second ellipse is `x^(2)+2y^(2)=6` `implies (x^(2))/(6)+(y^(2))/(3)=1` Supposethe tangent at P andQ meetsat (h,K)Equationof thechordof contactof thetangentsthrough A (h,k) is `(hx)/(6)+(ky)/(3)=1` ButEqs . (iv)and (ii)represent the same STRAIGHTLINE , socomparing Eqs. (iv)adn (ii)we get `(h//6)/( cos theta//2)=(k//3)/( sintheta)=(1)/(1)` `implies h= 3 cos theta andk=3 sintheta` therefore , coordinates of A are( 3 cos,`theta,3 sin theta)` Now , thejointequationof thetangents At A isgivenby `T^(2)=SS_(1)`, `i.e., ((hx)/(6)+(ky)/(3)-1)^(2)=((x^(2))/(6)+(y^(2))/(3)-1)((h^(2))/(6)+(h^(2))/(3)-1)` in Eq. (V)Coefficient of `x^(2)=(h^(2))/(36)-(1)/(6)((h^(2))/(6)+(h^(2))/(3)-1))` `=(h^(2))/(36)-(h^(2))/(36)-(k^(2))/(18)+(1)/(6)=(1)/(6)-(k^(2))/(18)` andcoefficient of `y^(2)=(k^(2))/(9)-(1)/(3)((h^(2))/(6)+(k^(2))/(3)-1)` `=(k^(2))/(9)-(h^(2))/(18)-(k^(2))/(9)+(1)/(3)=-(h^(2))/(18+(1)/(3)` Again , coefficient of `x^(2)+` coefficient of `y^(2)` `=-(1)/(18)(h^(2)+k^(2))+(1)/(6)+(1)/(3)` `=-(1)/(18)( 9 cos ^(2)theta+ 9sin ^(2)theta+(1)/(2)` `=-(9)/(18)+(1)/(2)=0` whichshowsthattwo linesrepresent by Eq. (v)are atrightanglesto eachother.

Discussion

16.	Number of quadratic equations which are unchanged by squaring their roots is p and the sum of roots of all those quadratic equations is q then
Answer» p = q = 4 p = 4, q = 2 p = 2, q=4 p = q = 2 Answer :B

Discussion

17.	If a, b,c in {1, 2, 3, 4} the number of quadratic equation of the form ax^(2) + bx + c = 0 which have non-real complex roots is :
Answer» 27 35 52 56 Answer :C

Discussion

18.	Find the coefficient of x^4 in (1 - 4x)^(-3//5)
Answer» ANSWER :`(234 XX 256)/(625)`

Discussion

19.	The slope of a chord of the parabola y^(2)=4ax which is normal at one end and which subtends a right angle at the origin is
Answer» `1//SQRT(2)` `sqrt(2)` 2 none Answer :B

Discussion

20.	Method of integration by parts : int [sin(logx)+cos(logx)]dx=....
Answer» `X COS (LOGX)+c` `SIN(logx)+c` `cos(logx)+c` `x sin(logx)+c` ANSWER :D

Discussion

21.	The transformed equation with integer coefficients and unity for the coefficient of first term, whose roots are multiplied by some constant of those of 3x^(4) - 5x^(3)+ x^(2) - x + 1 = 0 is
Answer» `y^(4) - y^(3) + 3y^(2) - 10y+ 1 = 0` `y^(3) - 24y^(2) - 9y - 24 = 0` `y^(4) - 2Y^(3) + 6y - 6 = 0` `y^(4) - 5y^(3) + 3y^(2) - 9y + 27 = 0` ANSWER :4

Discussion

22.	If hat(i)+2hat(j)+3hat(k)and2hat(i)-hat(j)+4hat(k) are the position vectors of the points A and B, then the position vector of the points of trisection of AB are
Answer» `(4)/(3)hat(i)+hat(J)+(10)/(3)hat(k),(5)/(3)hat(i)+(11)/(3)hat(k)` `-(4)/(3)hat(i)-hat(j)-(10)/(3)hat(k),-(5)/(3)hat(i)-(11)/(3)hat(k)` `(4)/(3)hat(i)-hat(j)-(10)/(3)hat(k),(-5)/(3)hat(i)+(11)/(3)hat(k)` `-(4)/(3)hat(i)+hat(j)-(10)/(3)hatk,(5)/(3)hat(i)-(11)/(3)hat(k)` ANSWER :A

Discussion

23.	Integrate the functions (sin^(-1)x)^(2)
Answer» ANSWER :`(SIN^(-1)X)^(2)=2sqrt(1-x^(2))sin^(-1)x-2x+C`

Discussion

24.	Integrate the following functions : intx^(3)cosx^(4)dx
Answer» ANSWER :`(1)/(4)sin(X^(4))+C`

Discussion

25.	The value of ((-1+sqrt(-3))/(2))^(26)+((-1-sqrt(-3))/(2))^(26) is
Answer» `-1` `1` `0` `2` ANSWER :A

Discussion

26.	Two circles which cut each other orthogonally, pass through the points, (a, 0) and (-a, 0). If both of them touch the line y=mx+c, then c^(2)=
Answer» `a^(2)+m^(2)` `a^(2)(1+m)^(2)` `a^(2)(1+m^(2))` `a^(2)(1+2m^(2))` ANSWER :B

Discussion

27.	The plane 2x + 3y -2sqrt(3)z + 25 = 0 makes an angle.......with X-axis.
Answer» `sin^(-1) ""(2)/(sqrt(21))` `TAN^(-1)""(2)/(sqrt(21))` `sin^(-1)""(1)/(sqrt(21))` `cos^(-1)""(1)/(21)` ANSWER :B

Discussion

28.	A: If f (x) = x sin ""((1)/(x)) (x ne 0) and f (0) =0 then f '(0) =0 R: Lt_(x to 0) sin ""((1)/(x)) does not exist
Answer» Both A and R are TRUE R is correct reason of A Both A and R are TREU R is not correct reason of A A is true but R is false A is false but R is true ANSWER :D

Discussion

29.	Evaluate Lt_(n rarr oo) ((1)/(n+1) + (1)/(n+2) +..+(1)/(3n))
Answer» ANSWER :`log_(E) 3`

Discussion

30.	The root of the equation : \|(a-x,b,c),(0,b-x,0),(0,b,c-x)\|=0 are :
Answer» a and B b and C a and c a,b,c ANSWER :D

Discussion

31.	Solve ""^(2n)C_3:""^nC_3=44:5
Answer» Solution :A COMMITTEE of 4 GENTLEMEN and 3 LADIES is to be formed out 7 gentlemen and 6 ladies. `:.` "The NUMBER of ways in which the committee can be formed".""^7C_4xx""^6C_3=(765)/(32)xx(654)/(32)=700`

Discussion

32.	If ""^(n)C_(r) denotes the number of combinations of n things taken r at a time, then the expression ""^(n)C_(r+1)+""^(n)C_(r-1)+2xx""^(n)C_(r), equals
Answer» `""^(n+2)C_(R+1)` `""^(n+1)C_(r)` `""^(n+1)C_(r+1)` `""^(n+2)C_(r)` ANSWER :A

Discussion

33.	int_(0)^(1)Tan^(-1)((2x)/(1-x^(2)))dx=
Answer» `PI/2 - LN 2` `pi - ln 2` `pi + ln 2` `pi/4` ANSWER :A

Discussion

34.	int_(0)^((pi)/(2)) sin^(6)x cos^(6)x dx
Answer» ANSWER :`(5PI)/(2048)`

Discussion

35.	Evalute the following integrals int (1)/(sqrt(x^(2) + 2 x + 10))dx
Answer» ANSWER :`SINH^(-1)((x+1)/(3))+C`

Discussion

36.	Solve the following problem graphically: Maximum and minimize Z=10500x+9000y Subject to the constraints x+y le 50 2x+y le 80 x ge 0, y ge 0
Answer»

Discussion

37.	Find the value (s) of r satisfying the equation .^(69)C_(3r-1)-.^(69)C_(r^(2)-1)-.^(69)C_(3r)
Answer» Solution :`.^(69)C_(3R-1)+ .^(69)C_(r^(2)-1)+ .^(69)C_(r^(2))` `implies .^(70)C_(3r)= .^(70)C_(r^(2))` `implies 3r=r^(2) " or" r=0,3` or `3r+r^(2)-70=0` or r=7, -10 Hence, POSSIBLE values of r and 3 and 7 as for these values all the terms in equation are defined.

Discussion

38.	There are 20 pairs of shoes in a closet. Out of them 4 shoes are selected at random. The probability that there is exactly one pair among the 4 shoes is
Answer» `(.^(20)C_(1)XX.^(38)C_(2))/(.^(20)C_(2))` `(.^(20)C_(1)(.^(38)C_(2)-.^(19)C_(1)))/(.^(40)C_(4))` `(.^(20)C_(1)(.^(38)C_(2)-.^(9)C_(1)))/(.^(20)C_(2))` `(.^(20)C_(1)xx.^(38)C_(2))/(.^(40)C_(4))` Answer :B

Discussion

39.	Let f : [a,b] to Rbe differentiable on [a,b]& k in R. Let f (a) =0 = f (b). Also let J (x) =f '(x) + kf (x). Then
Answer» `J (x) GT 0` for all `x in [a,b]` `J(x) gt 0` for all `x in [a,b]` `J(x) =0`haa at least ONE ROOT in (a,b) `J(x) =0` through (a,b) ANSWER :C

Discussion

40.	Simplify(3^(n-2)*9^(2-n))/(3^(2-n))
Answer» Solution :In order to combine exponents USING the PROPERTIES above, the BASE of each factor must be the same `(3^(n-2)9^(2-n))/(3^(2-n))=(3^(n-2)(3^(2))^(2-n))/(3^(2-n))=(3^(n-2)*3^(4-2n))/(3^(2-n))` `=3^(n-2+4-2n-(2-n))=3^(0)=1`

Discussion

41.	Let f:RtoR be a function defined as f(x)={(5,"if", xle1),(a+bx,"if", 1ltxlt3),(b+5x,"if",3lexlt3),(30,"if",xge5):} Then f is :
Answer»

Discussion

42.	Find the probability of getting 2 tails and 1 head when 3 coins are tossed.
Answer» ANSWER :`3//8`

Discussion

43.	Let I be any interval disjoint from [-1,1]. Prove that the function f given by f(x)=x+(1)/(x) is increasing on I.
Answer» ANSWER :`F'(X)GT 0`

Discussion

44.	For x in R,\|\|x\|\| is defined as follows,
Answer» ANSWER :`X={0,2}`

Discussion

45.	If a=cisalpha,b=cisbeta,c=cisgamma, and a/b+b/c+c/a-1=0 then cos(alpha-beta)+cos(beta-gamma)+cos(gamma-alpha)=
Answer» 0 1 -1 2 Answer :B

Discussion

46.	Solve the equation z^6=-i
Answer» Solution :`Z^6=-i=COS(3pi)/2+ISIN(3pi)/2` `=cos((3pi)/2+2kpi)+isin((3pi)/2+2kpi)` `cos(PI(4k+3))/2+isin(pi(4k+3))/2` `z=[(cos(4k+3)pi)/2=isin((4k+3)pi)/2]^(1/6)` `cos((4k+3)pi)/(12)=isin((4k+3)pi)/(12)` `"where "k=0,2,3,4,5`

Discussion

47.	Consider the random experiment of tossing dice. If number obtained on it is multiple of 3 then toss dice again, and if any other number is obtained toss a coin. Find probability of an event that coin shows tail given that at least once number 2 comes up.
Answer» ANSWER :`(3)/(8)`

Discussion

48.	If 2^(3)+4^(3)+6^(3)+….+(2n)^(3)=h n^(2)(n+1)^(2) then h is equal to
Answer» `(1)/(2)` 1 `(3)/(2)` 2 Answer :D

Discussion

49.	Consider f(x)=(cos^(-1)(cos(sinx))-\|x-pi\|)/(sin^(3)x), then
Answer» `underset(xto0)Limf(X)` =does not exist f(x) has REMOVABLE discontinuity at `x=pi`. JUMP of discontinuity at `x=pi" is "(1)/(3)` f(x) is discontinuous at `x=npi,n in I` Solution :`underset(xto0)Lim(COS^(-1){cos(sinx)}-\|x-pi\|)/(sin^(3)x)=underset(xto pi^(-))Lim(sinx+(x-pi))/(sin^(3)x)""(becausesinxgt0asxto pi^(-))` Let `x-pi=t=underset(t TO0^(-1))Lim(-sint+t)/(-sin^(3)t)=(-1)/(6)` R.H.L. `atx=pi` `underset(x to pi^(+))Lim(-sinx-(x-pi))/(sin^(3)x)""(becausesingt0asxto pi^(+))` Let `x-pi=t` `underset(t to0)Lim(sint-t)/(sin^(3)t)=(1)/(6)`.

Discussion

50.	If each pair of three equations x^(2)-p_(1)x+2=0", "x^(2)-p_(2)x+3=0, and x^(2)-p_(3)x+6=0 have a common root, then positive values of p_(1),p_(2),p_(3) are, respectively :
Answer» 3,4,5 2,3,6 1,3,6 3,5,6 Answer :A

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

intcos^4xdx

Find (dy)/(dx) in the following : 2x+3y= sin y.

Check monotonocity at following points for (i) f(x) =x^(3) -3x +1 =-1,2 (ii) f(x) +|x -1|+2|x-3|-|x+2| "at " x=-2,0,3,5 (iii) f(x) =x^(1//3) """at""" x=0 (iv) f(x)=x^(2) n +(1)/(x^(2)) """at""" x=1,2 (v) f(x) ={underset( 3 sin x , ""x ge0)(x^(3) +2x^(2) 5xgt 0)," at " x=0

int_(0)^(pi//2)(sin^(1000)xdx)/(sin^(1000)x+cos^(1000)x) is equal to

Write the function tan^(-1)((sqrt(1+x^(2))-1)/x)x ne 0, in the simplest form.

(85)^(1//4)~~....

Vertify mean value theorem for the following functions: f(x)= x^(2) + 2x+ 3, x in [4, 6]

{:(" "Lt),(x rarr 0):} (int_(0)^(x) sin^(2)t cost dt)/(x^(3))=

Determine whethera** b =ma - nb "on" Q + "where" m "and" n in Noperations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

A tangenttotheellipsedistancedistancefromthe centreoftheellipsex^(2) +2y^(2) =6 P andQprovethatthetangent at P andQof theellipsex^(2)+ 2y^(2) =6areat rightangles .

Number of quadratic equations which are unchanged by squaring their roots is p and the sum of roots of all those quadratic equations is q then

If a, b,c in {1, 2, 3, 4} the number of quadratic equation of the form ax^(2) + bx + c = 0 which have non-real complex roots is :

Find the coefficient of x^4 in (1 - 4x)^(-3//5)

The slope of a chord of the parabola y^(2)=4ax which is normal at one end and which subtends a right angle at the origin is

Method of integration by parts : int [sin(logx)+cos(logx)]dx=....

The transformed equation with integer coefficients and unity for the coefficient of first term, whose roots are multiplied by some constant of those of 3x^(4) - 5x^(3)+ x^(2) - x + 1 = 0 is

If hat(i)+2hat(j)+3hat(k)and2hat(i)-hat(j)+4hat(k) are the position vectors of the points A and B, then the position vector of the points of trisection of AB are

Integrate the functions (sin^(-1)x)^(2)

Integrate the following functions : intx^(3)cosx^(4)dx

The value of ((-1+sqrt(-3))/(2))^(26)+((-1-sqrt(-3))/(2))^(26) is

Two circles which cut each other orthogonally, pass through the points, (a, 0) and (-a, 0). If both of them touch the line y=mx+c, then c^(2)=

The plane 2x + 3y -2sqrt(3)z + 25 = 0 makes an angle.......with X-axis.

A: If f (x) = x sin ""((1)/(x)) (x ne 0) and f (0) =0 then f '(0) =0 R: Lt_(x to 0) sin ""((1)/(x)) does not exist

Evaluate Lt_(n rarr oo) ((1)/(n+1) + (1)/(n+2) +..+(1)/(3n))

The root of the equation : |(a-x,b,c),(0,b-x,0),(0,b,c-x)|=0 are :

Solve ""^(2n)C_3:""^nC_3=44:5

If ""^(n)C_(r) denotes the number of combinations of n things taken r at a time, then the expression ""^(n)C_(r+1)+""^(n)C_(r-1)+2xx""^(n)C_(r), equals

int_(0)^(1)Tan^(-1)((2x)/(1-x^(2)))dx=

int_(0)^((pi)/(2)) sin^(6)x cos^(6)x dx

Evalute the following integrals int (1)/(sqrt(x^(2) + 2 x + 10))dx

Solve the following problem graphically: Maximum and minimize Z=10500x+9000y Subject to the constraints x+y le 50 2x+y le 80 x ge 0, y ge 0

Find the value (s) of r satisfying the equation .^(69)C_(3r-1)-.^(69)C_(r^(2)-1)-.^(69)C_(3r)

There are 20 pairs of shoes in a closet. Out of them 4 shoes are selected at random. The probability that there is exactly one pair among the 4 shoes is

Let f : [a,b] to Rbe differentiable on [a,b]& k in R. Let f (a) =0 = f (b). Also let J (x) =f '(x) + kf (x). Then

Simplify(3^(n-2)*9^(2-n))/(3^(2-n))

Let f:RtoR be a function defined as f(x)={(5,"if", xle1),(a+bx,"if", 1ltxlt3),(b+5x,"if",3lexlt3),(30,"if",xge5):} Then f is :

Find the probability of getting 2 tails and 1 head when 3 coins are tossed.

Let I be any interval disjoint from [-1,1]. Prove that the function f given by f(x)=x+(1)/(x) is increasing on I.

For x in R,||x|| is defined as follows,

If a=cisalpha,b=cisbeta,c=cisgamma, and a/b+b/c+c/a-1=0 then cos(alpha-beta)+cos(beta-gamma)+cos(gamma-alpha)=

Solve the equation z^6=-i

Consider the random experiment of tossing dice. If number obtained on it is multiple of 3 then toss dice again, and if any other number is obtained toss a coin. Find probability of an event that coin shows tail given that at least once number 2 comes up.

If 2^(3)+4^(3)+6^(3)+….+(2n)^(3)=h n^(2)(n+1)^(2) then h is equal to

Consider f(x)=(cos^(-1)(cos(sinx))-|x-pi|)/(sin^(3)x), then

If each pair of three equations x^(2)-p_(1)x+2=0", "x^(2)-p_(2)x+3=0, and x^(2)-p_(3)x+6=0 have a common root, then positive values of p_(1),p_(2),p_(3) are, respectively :