28278 + Interview Questions in Maths in Class 12 Page 57 InterviewSolution

2801.	Evaluate int_(0)^(oo) log ( x + 1/x) . (1)/(1+x^(2))dx
Answer» ANSWER :`PI LOG 2`

Discussion

2802.	((costheta+isintheta)^5(cos3theta-isin3theta)^6)/((cos2theta+isin2theta)^3(cos4theta-isin4theta)^5)=
Answer» 1 `CIS(THETA)` i -1 Answer :B

Discussion

2803.	Show that if R is an equivalence relation on X then dom R=rngR =X.
Answer» Solution :LET R is an EQUIVALENCE RELATION on X . `implies R "is REFLEXIVE"` `implies(x,x)in R AA x in X ` `implies "Dom" R="Rng" R=X`

Discussion

2804.	Let phi(x)=(x)/(x^(2)+1)(x+1) if a, band c are the roots of the equation x^(3) -3x+lambda=0(lambda ne 0) Then phi(a)phi(b)phi(c ) =
Answer» `LAMBDA` `(-lambda)/((lambda + 2) (lambda^(2) + 16))` `(lambda)/((lambda+ 2))` `(lambda )/((lambda + 2) (lambda^(2) + 16))` ANSWER :D

Discussion

2805.	Refers to question 14. How many sweaters of each type of should the company make in an day to get a maximum profit? What is the maximum profit?
Answer» Solution :Referring to solution 14, `3x+y le 600, x-y ge 100, x ge 0, y ge 0` On solving x+y=300 and 3x+y=600 we get x=150, y=150 On solving x-y=100 and x+y=300 we get x=100, y=200 From the SHADED FEASIBLE region it is CLEAR that corrdingates of corner points are (0,0),(200,0),(150,150),(100,200), and (0,100) Hence 150 sweaters of each type made by company and MAXIMUM profit=48000.

Discussion

2806.	intsqrt((a-x)/(a+x))dx=....+c
Answer» `(a)/(2)SIN^(-1)((X)/(a))-sqrt(a^(2)-x^(2))` `(1)/(a)sin^(-1)((x)/(a))-sqrt(a^(2)-x^(2))` `sin^(-1)((x)/(a))-sqrt(a^(2)-x^(2))` `a sin^(-1)((x)/(a))-sqrt(a^(2)-x^(2))` Answer :D

Discussion

2807.	Find the area bounded by the curves y = log_(2) x and x = 2 y - y^(2) and the lines y = 0, y = 2
Answer» Answer :`(3)/(In 2) - (4)/(3)` SQ. UNITS

Discussion

2808.	Consider the function f(x)cosx^2. Then
Answer» f is of PERIOD `2pi` f is of period `sqrt(2pi)` f is not periodic f is of period `PI` Answer :C

Discussion

2809.	Rewrite the following sets by the set builder method. (i) {1,3,5,7,9}, (ii){(1)/(9),(1)/(25),(1)/(49),(1)/(81)}.
Answer» ANSWER :`(i)` `{x : x=2n+1, N in N+{0} 0 LE n le 4}`. `(II) {x:x=(1)/((2n+1)^(2)), n in N, n le 4}`.

Discussion

2810.	int(tan(x-theta)tan(x+theta)tan2xdx
Answer» ANSWER :`(1)/(20)log\|sec2x\|-log\|sec(x+theta)\|-log\|sec(x-theta)\|+C`

Discussion

2811.	One root of the equation (12x -1)(6x -1)(4x-1)(3x-1)=5 is
Answer» `1/2` `(-1)/12` `7/24` `24/7` ANSWER :D

Discussion

2812.	Find derivatives of the following functions(e^(sin x) - a^(cos x))
Answer» SOLUTION :`y = E^(SIN X)- a^ (COS x) dy/dx = d/dx(e^(sin x))-d/dx(a^(cos x) = e^(sin x).d/dx(sin x) - a^(cos x). In a. d/dx(cos x) e^(sin x).cos x + a^(cos x). In a. sin x`

Discussion

2813.	If (2,3,-3) is one end of a diameter of the sphere x^(2)+y^(2)+z^(2) - 6x - 12y - 2z +20 = 0then the other end of the diameteris
Answer» `(4,9,-1)` `(4,9,5)` `(-8,-15,1)` `(8,15,5)` ANSWER :B

Discussion

2814.	Given that the events E and F are such that P(E) = (1)/(2), P(E cup F) = (3)/(5) and P(F) = p. Find the p, if E and F are independent events.
Answer» ANSWER :`(1)/(5)`

Discussion

2815.	From a the point P(h, k) three normals are drawn to the parabola x^2=8y and m_1m_2" and "m_3 are the slopes of three normals, if the two normals from P are such that they make complementary angles with the axis then the directrix of the locus of point P (conic) is :
Answer» `2x-3=0` `2x-5=0` `2y-3=0` `2y-5=0` ANSWER :C

Discussion

2816.	Verify that \|P(phi)\|=2^@
Answer» SOLUTION :LET `A =phi , then P(A)={phi}` `:.`P(A) =1 =2^@`

Discussion

2817.	Find the angle between two vectors a and b with magnitudes sqrt(3) and 2 respectively, having a.b=sqrt(6).
Answer» ANSWER :`(PI)/(4)`

Discussion

2818.	Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values x, has the following form, where k is some unknown constant. P(X=x)={{:(0.1,",",x=0),(kx,",","x=1 or 2"),(k(5-x),",","X=3 or 4 "),(0,",","otherwise"):} i. Find the value of k ii. What is the probability that you study atleast two hours? Exactly 2 hours? Atmost 2 hours?
Answer» ANSWER :(a) k=0.15, (B) 0.55

Discussion

2819.	Let P={(x,y)//x in R, y in R, x^(2)+y^(2)=1}, then P is
Answer» REFLEXIVE SYMMETRIC anti-symmetric EQUIVALENCE ANSWER :B

Discussion

2820.	Sum to infininty of the series 1 + 4/5 + 7/5^(2) + 10/5^(3) + ...... is
Answer» `16/35` `11/8` `35/16` `8/11` ANSWER :C

Discussion

2821.	Find the order and the degree of the following differential equation (d^2y)/(dx^2) + 5x ((dy)/(dx))^2 - 6y = log x
Answer» SOLUTION :ORDER = 2, DEGREE = 1

Discussion

2822.	Find a vector in the direction of vector veca=hati-2hatj that has magnitude7 units.
Answer» ANSWER :`hata=(2)/(sqrt(14))hati+(3)/(sqrt(14))hatj+(1)/(sqrt(14))hatk`

Discussion

2823.	Evaluate the limit . underset(n to 00)("lim")(1)/(n) [tan (pi)/(4n) + tan (2pi)/(4n) +………..tan (npi)/(4n)]
Answer» ANSWER :`(2)/(PI) LOG 2`

Discussion

2824.	Determine P(E\|F) A dice is thrown three times, E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses
Answer» ANSWER :`(1)/(6)`

Discussion

2825.	Integrate thefunction in Exercise. (x^(3) sin(tan^(-1)x^(4)))/(1+x^(8))
Answer» ANSWER :`-(1)/(4) COS (TAN^(-1)x^(4))+c`

Discussion

2826.	If (2^(200)-2^(192).31+2^n) is the perfect square of a natural number , then find the sum of digits of 'n' .
Answer» ANSWER :18

Discussion

2827.	If ""^(n)C_(r-1)=36 ,""^(n)C_(r)= 84 and ""^(n)C_(r+1)=126,then the value of ""^(n)C_(8) is
Answer» 10 7 9 8 Answer :C

Discussion

2828.	Let g(x)=e^f(x)), g(x+1)=xg(x), then (g'((2n+1)/(2))g((1)/(2))-g'((1)/(2))g((2n+1)/(2)))/(g((2n+1)/(2))g((1)/(2))), where n in N, is
Answer» `2 (1+(1)/(2)+(1)/(3)+….+(1)/(n))` `2(1+(1)/(3)+(1)/(5)+….+(1)/(2n-1))` 0 none of these Answer :B

Discussion

2829.	Prove that : int_(9)^(a) (dx)/(x+sqrt(a^(2)-x^(2)))=(pi)/(4)
Answer»

Discussion

2830.	Evaluate the following definite integrals as limit of sums : int_(1)^(2)3^(x)dx
Answer» ANSWER :`log_(3)^(E)`

Discussion

2831.	Construct Collection of all the nationalised political parties in the form of set and describe it with the help of proposition.
Answer» SOLUTION :C ={BJP, Congress, CPI, CPI (M), JU (U), JD(S),JP}

Discussion

2832.	int_(2)^(e) ((1)/(ln x) - (1)/((ln x)^(2)))dx=
Answer» 0 E `e-2log_(2)e` `2 log_(2)e` ANSWER :C

Discussion

2833.	Prove that the circles x^(2) + y^(20 - 8x - 6y + 21 =0 and x^(2) + y^(2) - 2y - 15 = 0have exactly two common tangents. Also find the point of intersection of those tangents.
Answer» ANSWER :`=( 8, 5)`

Discussion

2834.	The pint of extremum of f(x)=int_0^x(t-2)^2(t-1)dtis a
Answer» a)MAXIMUM at x=1 b)maximum at x=2 c)MINIMUM at x=1 d)minimum at x=2

Discussion

2835.	int (1)/((x^(2) + 1)(x^(2)+ 4))dx =
Answer» `(1)/(3) [ TAN^(-1) X - (1)/(2) tan^(-1)""(x)/(2)] + c` `(1)/(3) [ tan^(-1) x + (1)/(2) tan^(-1)""(x)/(2)] + c` `(1)/(3) [tan^(-1) x - (x)/(2) + 2 tan^(-1) x] + c` `(2)/(3) [ tan^(-1) x - (3)/(2) tan^(-1)""(x)/(2)] + c` Answer :A

Discussion

2836.	Find derivatives of the following functions.sec^(-1)(2x+1)
Answer» Solution :`y=sec^(-1)(2x+1)` `dy/DX=1/((2x+1)sqrt(2x+1)^2-1).d/dx(2x+1)` `[becaused/dx(sec^(-1)u)=1/(\|u\|sqrt(u^2-1)).(DU)/dx` `2/((2x+1)sqrt(4x^2+4x))=1/((2x+1)sqrt(x^2+x))`

Discussion

2837.	If the line y = x cuts the curve x^(3) + 3y^(3) - 30xy + 72x - 55 = 0 in points A,B and C, then the value of(4sqrt(2))/(55) OA.OB.OC (where O is the origin), is
Answer» 55 `1/(4sqrt(2))` 2 4 Answer :A

Discussion

2838.	Evaluate the following integrals: int_2^3 xdx/(x^2+1)
Answer» Solution :`int_2^3 (X dx)/(x^2+1) = 1/2 int_2^3 (2X dx)/(x^2+1)` =`1/2[log\|x^2+1\|]_0^3` =`1/2[log10-log5]` `= 1/2 log ((10)5) = 1/2 LOG2`

Discussion

2839.	thevalueof tan 40^@+ tan20^@+ sqrt(3) tan 20^@tan 40^@isequalto
Answer» `SQRT(12)` `(1)/( sqrt(3))` `1` `sqrt(3)` ANSWER :D

Discussion

2840.	If lim_(x to 0) (2^(sin x^(2)) - 2^(tan x^2) - 2x^(7))/(x^n) exists finitely and equal to a non-zero number then n is
Answer» 3 5 6 7 Solution :`lim_(x to 0)(2^(sin x^(2)) - 2^(tan x^2) - 2X^(7))/(x^n) = lim_(x to 0) (2^(tan x^2)(2^(sin x^(2)tan x^2) -1)- 2x^(7))/(x^n)` =`lim_(x to 0) ((2^(tan x^2)(2^(sin x^(2)tan x^2) -1))/(SINX^(2)-tanx^(2)) xx (sin x^(2) - tan x^(2))- 2x^(7))/(x^n)` =`lim_(x to 0) 1/(x^n)[(2^(tan x^2)(2^(sin x^(2)tan x^2) -1))/(sinx^(2)-tanx^(2)) ((x^(2) - (x^6)/(3 !) + (x^10)/(5!)....)-(x^(2) + (x^6)/3 + (2x^(10))/(15)+....))-2x^(7)]` `= lim_(x to 0) (x^6)/(x^n)[(2^(tan x^2)(2^(sin x^(2)tan x^2) -1))/((sinx^(2)-tanx^(2))) (-1/2 + "term containing"x)-2x^(1)]` `implies n = 6`.

Discussion

2841.	If lim_(xto0)f(x)=l in R, then
Answer» `lim_(xto0)F(X^(2))=L^(2)` `lim_(xto0)f((x)/(l))=1` `lim_(xto0)f(2x)=2l` `lim_(xto0)f(-x)=l` Answer :D

Discussion

2842.	Evaluate the following integral : int(x+sqrt(a^(2)+x^(2)))^(2007)dx
Answer» Solution :Let us substitute `x+SQRT(a^(2)+x^(2))=t` so that `sqrt(a^(2)+x^(2)-x=(a^(2))/(t)` `implies2x=t-(a^(2))/(t)` `impliesdx=(1)/(2)(1+(a^(2))/(t^(2)))DT` under given substitution, the given integral reduces to `int(x+sqrt(a^(2)+x^(2)))^(2007)dx=intt^(2007)(1)/(2)(1+(a^(2))/(t^(2)))dt` `=(1)/(2)int(t^(2007)+a^(2)t^(2005))dt` `=(1)/(2)[(t^(2008))/(2008)+a^(2)(t^(2006))/(2006)]+k` `=(1)/(2)(x+sqrt(a^(2)+x^(2)))^(2006)[((x+sqrt(a^(2)+x^(2)))^(2))/(2008)+(a^(2))/(2006)]+k`

Discussion

2843.	Find derivatives of the following functions. In (sqrtx + 1)
Answer» SOLUTION :`y = In (SQRT X + 1) dy/dx = d/dx in(sqrt x + 1)= 1/(sqrt x + 1).d/dx(sqrt x + 1) 1/(sqrt x + 1).1/(2sqrt x) = 1/{2(x + sqrt x)}`

Discussion

2844.	If \|a\|=\|b\|=1 and \|a+b\|=sqrt(3), then the value of (3a-4b).(2a+5b) is
Answer» `-21` `-21/2` `21` `21/2` Solution :`because\|a+B\|=sqrt3rArr\|a\|^2+\|b\|^2+2ab=3` `RARR""ab=(3-1-1)/2=1/2"...(i)"[because\|a\|=\|b\|=1, "GIVEN"]` `:.""(3a-4b)(2a+5b)=6\|a\|^2+15ab-20\|b\|^2` `=6+7a*b-20` `=6+7/2-20"[from Eq. (i)]"` `=(7-28)/2-21/2`

Discussion

2845.	Distance between the two planes : 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
Answer» 2 units 4 units 8 units `(2)/(SQRT(29))` units Answer :D

Discussion

2846.	Fundamental theorem of definite integral : (d)/(dx)(F(x))=(e^(sinx))/(x),ngt0. If int_(1)^(4)(2e^(sinx^(2)))/(x)dx=F(k)-F(1) then the possible value of k is……..
Answer» 4 8 16 32 Answer :C

Discussion

2847.	int(cos4x+1)/(cosx-tanx)dx=k cos4 x+c then.....
Answer» `k=-(1)/(2)` `k=-(1)/(8)` `k=-(1)/(4)` NONE of these Answer :B

Discussion

2848.	int tan^(-1) (sqrt((1-x)/(1+x)))dx is equal to
Answer» `1/2(x cos^(-1)x-sqrt(1-x^(2))+C)` `1/2(x cos^(-1)x+sqrt(1-x^(2))+c)` `1/2(xsin^(-1)x-sqrt(1-x^(2))+c)` `1/2(x SIN^(-1)x + sqrt(1-x^(2))+c)` ANSWER :A

Discussion

2849.	Let A(3, 7) and B(6, 5) are two points. C:x^(2)+y^(2)-4x-6y-3=0 is a circle. Q.Equation of the member of the family of circles S that bisects the circumference of C is :
Answer» `X^(2)+y^(2)-5x-1=0` `x^(2)+y^(2)-5x+6y-1=0` `x^(2)+y^(2)-5x-6y-1=0` `x^(2)+y^(2)+5x-6y-1=0` ANSWER :C

Discussion

2850.	Let be a binary operation on the set of natural numbers given by ab = L.C.M of a and b, find 5**7,
Answer» ANSWER :`5**7 =` LCMOF 5AND `7=35`.

Discussion

Explore topic-wise InterviewSolutions in .

Evaluate int_(0)^(oo) log ( x + 1/x) . (1)/(1+x^(2))dx

((costheta+isintheta)^5(cos3theta-isin3theta)^6)/((cos2theta+isin2theta)^3(cos4theta-isin4theta)^5)=

Show that if R is an equivalence relation on X then dom R=rngR =X.

Let phi(x)=(x)/(x^(2)+1)(x+1) if a, band c are the roots of the equation x^(3) -3x+lambda=0(lambda ne 0) Then phi(a)phi(b)phi(c ) =

Refers to question 14. How many sweaters of each type of should the company make in an day to get a maximum profit? What is the maximum profit?

intsqrt((a-x)/(a+x))dx=....+c

Find the area bounded by the curves y = log_(2) x and x = 2 y - y^(2) and the lines y = 0, y = 2

Consider the function f(x)cosx^2. Then

Rewrite the following sets by the set builder method. (i) {1,3,5,7,9}, (ii){(1)/(9),(1)/(25),(1)/(49),(1)/(81)}.

int(tan(x-theta)tan(x+theta)tan2xdx

One root of the equation (12x -1)(6x -1)(4x-1)(3x-1)=5 is

Find derivatives of the following functions(e^(sin x) - a^(cos x))

If (2,3,-3) is one end of a diameter of the sphere x^(2)+y^(2)+z^(2) - 6x - 12y - 2z +20 = 0then the other end of the diameteris

Given that the events E and F are such that P(E) = (1)/(2), P(E cup F) = (3)/(5) and P(F) = p. Find the p, if E and F are independent events.

From a the point P(h, k) three normals are drawn to the parabola x^2=8y and m_1m_2" and "m_3 are the slopes of three normals, if the two normals from P are such that they make complementary angles with the axis then the directrix of the locus of point P (conic) is :

Verify that |P(phi)|=2^@

Find the angle between two vectors a and b with magnitudes sqrt(3) and 2 respectively, having a.b=sqrt(6).

Let P={(x,y)//x in R, y in R, x^(2)+y^(2)=1}, then P is

Sum to infininty of the series 1 + 4/5 + 7/5^(2) + 10/5^(3) + ...... is

Find the order and the degree of the following differential equation (d^2y)/(dx^2) + 5x ((dy)/(dx))^2 - 6y = log x

Find a vector in the direction of vector veca=hati-2hatj that has magnitude7 units.

Evaluate the limit . underset(n to 00)("lim")(1)/(n) [tan (pi)/(4n) + tan (2pi)/(4n) +………..tan (npi)/(4n)]

Determine P(E|F) A dice is thrown three times, E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses

Integrate thefunction in Exercise. (x^(3) sin(tan^(-1)x^(4)))/(1+x^(8))

If (2^(200)-2^(192).31+2^n) is the perfect square of a natural number , then find the sum of digits of 'n' .

If ""^(n)C_(r-1)=36 ,""^(n)C_(r)= 84 and ""^(n)C_(r+1)=126,then the value of ""^(n)C_(8) is

Let g(x)=e^f(x)), g(x+1)=xg(x), then (g'((2n+1)/(2))g((1)/(2))-g'((1)/(2))g((2n+1)/(2)))/(g((2n+1)/(2))g((1)/(2))), where n in N, is

Prove that : int_(9)^(a) (dx)/(x+sqrt(a^(2)-x^(2)))=(pi)/(4)

Evaluate the following definite integrals as limit of sums : int_(1)^(2)3^(x)dx

Construct Collection of all the nationalised political parties in the form of set and describe it with the help of proposition.

int_(2)^(e) ((1)/(ln x) - (1)/((ln x)^(2)))dx=

Prove that the circles x^(2) + y^(20 - 8x - 6y + 21 =0 and x^(2) + y^(2) - 2y - 15 = 0have exactly two common tangents. Also find the point of intersection of those tangents.

The pint of extremum of f(x)=int_0^x(t-2)^2(t-1)dtis a

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

Find derivatives of the following functions.sec^(-1)(2x+1)

If the line y = x cuts the curve x^(3) + 3y^(3) - 30xy + 72x - 55 = 0 in points A,B and C, then the value of(4sqrt(2))/(55) OA.OB.OC (where O is the origin), is

Evaluate the following integrals: int_2^3 xdx/(x^2+1)

thevalueof tan 40^@+ tan20^@+ sqrt(3) tan 20^@tan 40^@isequalto

If lim_(x to 0) (2^(sin x^(2)) - 2^(tan x^2) - 2x^(7))/(x^n) exists finitely and equal to a non-zero number then n is

If lim_(xto0)f(x)=l in R, then

Evaluate the following integral : int(x+sqrt(a^(2)+x^(2)))^(2007)dx

Find derivatives of the following functions. In (sqrtx + 1)

If |a|=|b|=1 and |a+b|=sqrt(3), then the value of (3a-4b).(2a+5b) is

Distance between the two planes : 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

Fundamental theorem of definite integral : (d)/(dx)(F(x))=(e^(sinx))/(x),ngt0. If int_(1)^(4)(2e^(sinx^(2)))/(x)dx=F(k)-F(1) then the possible value of k is……..

int(cos4x+1)/(cosx-tanx)dx=k cos4 x+c then.....

int tan^(-1) (sqrt((1-x)/(1+x)))dx is equal to

Let A(3, 7) and B(6, 5) are two points. C:x^(2)+y^(2)-4x-6y-3=0 is a circle. Q.Equation of the member of the family of circles S that bisects the circumference of C is :

Let ** be a binary operation on the set of natural numbers given by a**b = L.C.M of a and b, find 5**7,

Let be a binary operation on the set of natural numbers given by ab = L.C.M of a and b, find 5**7,