28278 + Interview Questions in Maths in Class 12 Page 237 InterviewSolution

11801.	Integrate the following functions : intsin2x.e^(cos^(2)x)dx
Answer» ANSWER :`-E^(COS^(2))+C`

Discussion

11802.	A man, 2m tall, walks at the rate of 1(2)/(3) m/s towards a street light which is 5(1)/(3) m above the ground. At what rate is the tip of his shadow moving and at what rate is the length of the shadow changing when he is 3(1)/(3)m from the base of the light ?
Answer» ANSWER :`(DX)/(DT)=-1`

Discussion

11803.	A has 3 shares in a lottery containing 3 prizes and 6 blanks . B has two shares in a lottery containing 2 prizes and 6 blanks . The ratio of their chances of success is
Answer» `952 : 715` `274 : 659` `113 : 907` `64 : 39` Answer :D

Discussion

11804.	An equilateral triangle is inscribed in the circle x ^(2) + y ^(2) =1 which one vertex at the point (1,0). Length of each side of the tringle is
Answer» 1 `SQRT2` `SQRT3//2` `sqrt3` ANSWER :D

Discussion

11805.	A company has two plants to manufacturebicycles. The first plant manufacture 60% of the bicycles and the second plant, 40%. Also, 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant.A bicycle is picked up at random and found to be of standard quality. Find the probability that it comes from the second plant.
Answer» Solution :Let `E_1 and E_2` be the events of CHOOSING a BICYCLE from the first plant and the second plant respectively. Then, `P(E_1)=60/100=3/5,and P(E_2)=40/100=2/5`. Let E be the event of choosing a biacycle of standard quality. Then, `P(E//E_1)`= PROBABILITY of choosing a bicycle of standard quality, given that it is PRODUCED by the first plant `=80/100=4/5`. `P(E//E_2)`=probability of choosing a bicycle of standard quality, given that it is produced by the second plant `=90/100=9/10`. The required probability `P(E//E_2)` =probability of choosing a biacycle from the second plant, given that it is of standard quality `=(P(E_2).P(E//E_2))/(P(E_1).P(E//E_1)+P(E_2).P(E//E_2))`[by Bayes's theorem] `((2/5xx9/10))/((3/5xx4/5)+(2/5xx9/10))=3/47`.

Discussion

11806.	If the comples cube roots of (-i) are alpha, beta ,gamma the alpha^2+beta^2+gamma^2=
Answer» 1 `-1` `-i` 0 Answer :D

Discussion

11807.	(1+(2^2)/(2!)+(2^4)/(3!)+(2^6)/(4!)+.....oo)/(1+(1)/(2!)+(2)/(3!)+(2^2)/(4!)+.....oo) =
Answer» `E^2` `e^(2)+1` `(e-1)/(e+1)` `(e+1)(e-1)` ANSWER :D

Discussion

11808.	intsqrt(1+2 tan x (tan x + sec x ) ) dx
Answer» ANSWER :`LOG \|SEC x+tan x\|+log\|sec x\|+c`

Discussion

11809.	Let f (x) be invertible function and let f ^(-1) (x) be is inverse. Let equation f (f ^(-1) (x)) =f ^(-1)(x) has two real roots alpha and beta (with in domain of f(x)), then :
Answer» `f (x) =x ` also have same two rreal roots `f ^(-1) (x)=x` also have same two REAL roots `f (x)=f^(-1)(x)` also have same two real roots Area of triangle formed by `(0,0), (alpha, f (alpha)), and (BETA, f (beta))` is 1 unit ANSWER :A::B::C

Discussion

11810.	A person is to walk from A to B. However, he is restricted to walk only to the right of A or upwards of A. but not necessarily in the order shown in the figure. Then find the number of paths from A to B.
Answer» ANSWER :126

Discussion

11811.	A threedigitnumbern issuchthat the lasttwodigitsof itareequaland differentfromthefirst, thenumberof suchn'sis
Answer» 64 72 81 900 Answer :C

Discussion

11812.	Using Lagrange's Mean Value theorem , find the co-ordinates of a point on the curve y = x^(2) at which the tangent drawn is parallel to the line joining the points (1,1) and (3,9).
Answer» Solution :LET f(x) = `x^(2)` . Here given INTERVAL is [1,3]. (i) The function f(x) has definite and unique value in [1,3] `therefore` f(x) is continuous in [1,3] . (ii) f'(x) = 2x Which is defined in ]1,3[. `therefore` f(x) is differentiable in ]1,3[ . Thus the conditions of lagrange's Mean Value THEOREM satisfies . Now there exists at least one value of `c in ]1,3[` such that `f'(c) = (f(3) - f(1))/(3-1)` `implies "" 2C = (9-1)/(2) = 4` `implies "" c = 2 in ]1 , 3[` Hence Lagrange's Mean Value theorem verified . `therefore "" f(c) = c^(2)` `implies "" f(2) = 2^(2) = 4` `therefore "" ` The required point = (2,4). `""` Ans.

Discussion

11813.	Consider a plane pi:vecr*vecn=d (where vecn is not a unti vector). There are two points A(veca) and B(vecb) lyingon the same sideof the plane. Q.Reflection of A(veca) in the plane pi has the position vector :
Answer» `VECA+(2)/((VECN)^(2))(d-vecavecn)vecn` `veca-(1)/((vecn)^(2))(d-vecavecn)vecn` `veca+(2)/((vecn)^(2))(d+veca*vecn)vecn` `veca+(2)/((vecn)^(2))vecn` ANSWER :A

Discussion

11814.	Consider a plane pi:vecr*vecn=d (where vecn is not a unti vector). There are two points A(veca) and B(vecb) lyingon the same sideof the plane. Q.If a plane pi_(1) is drawn from the pointA(veca) and anotherplane pi_(2)is drawn point B(vecb)parallelto pi, then the distancebetweenthe planes pi_(1) and pi_(2) is :
Answer» `(\|(veca-vecb)VECN\|)/(\|vecn\|)` `\|(veca-vecb)vecn\|` `\|(veca-vecb)xxvecn\|` `(\|(veca-vecb)xxvecn\|)/(\|vecn\|)` ANSWER :A

Discussion

11815.	What is the derivative of f(In x) with respect to x where f(x) = In x ?
Answer» SOLUTION :`F(X)=In x RARRF(INX)=In (Inx)rArrd/dxf(Inx)=d/dxIn(In)=-/(xInx)`

Discussion

11816.	The system of equations x+2y+3z=4 2x+3y+4z=5 3x+4y+5z=6 has
Answer» INFINITELY MANY solutions no solution a UNIQUE solution none Answer :A

Discussion

11817.	int(dx)/(cosa+cosx)=
Answer» `(" cosec "a)LOG\|(cot((a)/(2))+TAN((x)/(2)))/(cot((a)/(2))-tan((x)/(2)))\|+C` `(" cosec "a)log\|(cot((a)/(2))-tan((x)/(2)))/(cot((a)/(2))+tan((x)/(2)))\|+c` `(" cosec "a)log\|(tan((x)/(2))+tan((a)/(2)))/(tan((x)/(2))-tan((a)/(2)))\|+c` `(" cosec "a)log\|(tan((x)/(2))-tan((a)/(2)))/(tan((x)/(2))+tan((a)/(2)))\|+c` Answer :A

Discussion

11818.	A man starts repaying a loan as first instalment of Rs. 100. Ifthe increases the instalment by Rs. 5 every month, then the amount he will pay in the 30th instalment is
Answer» 245 345 445 550 Answer :A

Discussion

11819.	Write solution ofdy/dx=2y,y(0)=2
Answer» Solution :`dy/dx=2yrArrintdy/y=2intdxrArr1ny=2x+C` Using the condition y(0)=2 we GET 1N 2 = C `THEREFORE` The required solution is In `y=2x+In2`

Discussion

11820.	The value of ""^(40) C_(31) + sum _(r = 0)^(10) ""^(40 + r) C_(10 +r)is equal to
Answer» `""^(51)C_(20)` `2""^(50)C_(20)` `2""^(45)C_(15)` `""^(51)C_(21)` ANSWER :A

Discussion

11821.	If a is perpendicular to b and c, \|a\|=2, \|b\|=3, \|c\|=4 and the angle between b and c is (2pi)/(3), then [a,b,c] is equal to
Answer» `4sqrt3` `6sqrt3` `12sqrt3` `18sqrt3` SOLUTION :`:. [(a,b,c)]=a. (bxxc)=a. (\|b\|\|c\| "SIN" (2PI)/3 hat(N))` `=\|b\|\|c\| "sin" (2pi)/3 (a.hat(n))` `=\|a\|\|b\|\|c\| ("sin" (2pi)/3)` `[ :' a. hat(n)=\|a\|\|hat(n)\| cos 0^(@)=\|a\|]` `=2xx3xx4xxsqrt(3)/2=12 SQRT(3)`

Discussion

11822.	The numbers of ways of arranging the word 'ARRANGE' so that neither 2A's nor 2R's occur together are
Answer» 900 240 660 71 Answer :C

Discussion

11823.	Value of F(3)=
Answer» 1 -3 5 13 Answer :C

Discussion

11824.	A line makes angles alpha,beta,gammawith the coordinate axes. If alpha + beta = pi//2, then (cos alpha+cosbeta+cosgamma)^(2) is equal to
Answer» `1+SIN 2 ALPHA` `1+sin 2ALPHA` `1-sin 2 alpha` `sin 2 alpha` ANSWER :A

Discussion

11825.	If z_k=cos((kpi)/(10))+isin((kpi)/(10)) , then z_1z_2z_3z_4 is equal to
Answer» -1 1 -2 2 Answer :A

Discussion

11826.	Evaluate the following integrals. int(1)/(3cosx+4sinx+6)dx
Answer» ANSWER :`(2)/(sqrt(11))TAN^(-1)[(3TAN""(X)/(2)+4)/(sqrt(11))]+C`

Discussion

11827.	int(e^(2x)+1)/e^xdx
Answer» SOLUTION :`INT(E^(2x)+1)/e^xdx=int(e^x+e^-x)dx` =`e^x-e^-x+C`

Discussion

11828.	If barr=hati+hatj+lambda(2hati+hatj+4hatk)" and "barr*(hati+2hatj-hatk)=3 are the equations of a line and plane respectively, then which of the following is truwe?
Answer» The line is PERPENDICULAR to the plane The line LIES in the plane The line is PARALLEL to the plane but doesn't lie in the plane. The line cuts the plane obliquely. Answer :B

Discussion

11829.	Method of integration by parts : int[(1)/(logx)-(1)/((logx)^(2))]dx=.......
Answer» `(1)/(LOGX)+c` `(X)/(logx)+c` `(x)/((logx)^(2))` NONE of these Answer :B

Discussion

11830.	The solution of the differential equation (d ^(3) y)/( dx ^(3)) - 8 (d ^(2)y )/( dx ^(2))=0 safisfing y (0) =1/8, y'(0) and y" (0) =1 is equal to 1/p [ (e ^(8x))/( 8) -+ + (7)/(8)], then find the value of p.
Answer» ANSWER :8

Discussion

11831.	The solution of (dy)/(dx) = (x + y +1)/(x + y -1) is
Answer» `E^(y-x) = C(x+y)` `e^(y-x) = c(x-y)` `e^(y+x) = c(x+y)` `e^(y-x) = c(2x+y)` ANSWER :A

Discussion

11832.	The value of the integral int_(-1)^(1){(x^(2015))/(e^(\|x\|)(x^(2)+cosx))+(1)/(e^(\|x\|))}dx is equal to
Answer» 0 `1-e^(-1)` `2E^(-1)` `2(1-e^(-1))` ANSWER :D

Discussion

11833.	R(f)=
Answer» `[-(1)/(2),(1)/(2)]-{0}` `[-2,(1)/(2)]` `[-2,0]` `[-1,0]` ANSWER :a

Discussion

11834.	Consider a function f: [0, (pi)/(2)] to R given by f (x) = sin x and g : [0, (pi)/(2)] to R given by g (x) = cos x, Show that f and g are one-one but f + g is not one-one.
Answer» ANSWER :`F + G`

Discussion

11835.	Let F_1(x_1,0)" and "F_2(x_2,0), for x_1 lt 0 " and" x_2 gt 0, be the foci of the ellipse (x^2)/(9)+(y^2)/(8)=1. Suppose a parabola having vertex at the origin and focus at F_2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of the area of the triangle MQR to area of the quadrilateral MF_1NF_2 is :
Answer» `3 : 4` `4 : 5` `5 : 8` `2 : 3` ANSWER :C

Discussion

11836.	IF theequationx^2-x-p=0andx^2 + 2px -12 =0havea commonrootthenthatroot is
Answer» 1 p+2 2 can not be determined Answer :C

Discussion

11837.	Find the equation of normal to the ellipse (x^(2))/(16)+(y^(2))/(9) = 1 at the point whose eccentric angle theta=(pi)/(6)
Answer» ANSWER :`7sqrt(3)`

Discussion

11838.	The value of int_(-pi//4)^(pi//4) sin^(103) x*cos^(101) xdx is
Answer» `((PI//4))^(103)` `((pi)/(4))^(101)` 2 0 Answer :D

Discussion

11839.	C_0 - [C_1 -2.C_2+ 3.C_3-……..+(-1)^(n-1).n.C_n] =
Answer» 0 1 `-1` 2 Answer :B

Discussion

11840.	Discuss the applicability of the Rolle's theorem for the following functions in the given intervals. (i) f(x) = (x-2) (2x-1) in the interval [1,2]. (ii) f(x) = tan x in the interval [0, pi] . (iii) f(x) = sin (1)/(x) in the interval [-2,2] . (iv) f(x) = \|x\| in the interval [-2,2] . (v) f(x) = x^(1//3) in the interval [-2,2].
Answer» ANSWER :not APPLICABLE

Discussion

11841.	By using the properties of definite integrals, evaluate the integrals int_(0)^(pi/2)(2log sin x log sin 2x)dx
Answer» ANSWER :`pi/2log1/2`

Discussion

11842.	Integration using rigonometric identities : int sec x tan^(3)x dx=....
Answer» `(1)/(3)SEC^(3)x-secx+c` `sec^(3)x-secx+c` `(1)/(3)sec^(3)x+secx+c` NONE of these Answer :A

Discussion

11843.	If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+………+C_(n)x^(n) AA n in N and (C_(0)^(2))/(1)+(C_(1)^(2))/(2)+(C_(2)^(2))/(3)+……..+(C_(n)^(2))/(n+1)=(lambda(2n+1)!)/((n+1)!)^(2), then the vlaue of lambda is equal to
Answer» ANSWER :1

Discussion

11844.	If the curve y=a sqrt(x)+bxpasses through P(1,2) and lies above the x-axis for x in [0,9] and the area bounded by the curve, the x-axis and x=4 is 8 sq. units the 2a-3b=
Answer» 6 9 0 10 Solution :`a+b=2 & underset(0)OVERSET(4)(INT)(asqrt(X)+bx)dx=8implies(2a)/(3)xx8+(b)/(2)xx16=8implies(2a)/(3)+b=1`

Discussion

11845.	Find the order and the degree of the following differential equation (d^4y)/(dx^4) - sin ((d^3y)/(dx^3)) = 0
Answer» SOLUTION :ORDER = 4

Discussion

11846.	Three positive numbers form an increasing G.P. if the middle term in this G.P is doubled, the new numbers are in A.P then the common ratio of the G.P. is
Answer» 2 - `SQRT(3)` `2 + sqrt(3)` `sqrt(2) + sqrt(3)` `3 + sqrt(2)` ANSWER :B

Discussion

11847.	Which term of the geometric sequence5,2,4/5 , 8/(25) …is (128)/(15625) ?
Answer» 11 10 9 4 Answer :A

Discussion

11848.	Let 0^(@)ltthetalt45^(@).Which one of the following is correct ?
Answer» `sin^(2)theta+cos^(6)theta=sin^(6)theta+cos^(2)theta` `"cosec"^(2)theta+cot^(6)theta="cosec"^(6)thetacot^(2)theta` `sin^(2)theta-cos^(4)theta=sin^(4)theta+cos^(2)theta` `"cosec"^(2)theta+cot^(4)theta="cosec"^(4)theta+cot^(2)theta` SOLUTION :In such a problem , we have to check option , one- by - one `sin^(2)theta+cos^(6)theta=sin^(6)theta+cos^(2)theta` `rArrsin^(6)theta-cos^(6)theta=(sin^(2)theta)^(2)-(cos^(2)theta)^(2)` Taking L . H . S, `=(sin^(2)theta-cos^(2)theta)(sin^(4)theta+sin^(2)thetacos^(2)theta+cos^(4)theta)` `(because a^(3)-b^(3)=(a-b)(a^(2)+ab+b^(2)))` `=(sin^(2)theta-cos^(2)theta)(sin^(4)theta+cos^(4)theta+sin^(2)thetacos^(2)theta)` `=(sin^(2)theta-cos^(2)theta)(sin^(2)theta+cos^(2)theta)-2sin^(2)thetacos^(2)theta+sin^(2)thetacos^(2)theta` `=sin^(2)theta-cos^(2)theta-sin^(2)thetacos^(2)theta` Which is not EQUAL to R .H . S.,`sin^(2)theta-cos^(2)theta` Option (a )is not CORRECT . ( b )`therefore"cosec"^(6)theta-cot^(6)theta` `("cosec"^(2)theta-cot^(2)theta)[("cosec"^(2)theta-cot^(2)theta)^(2)+("cosec "thetacottheta)]` `therefore`Option (b ) is also not correct. (c ) `sin^(4)theta+cos^(4)theta=(sin^(2)theta+cos^(2)theta)^(2)-2sin^(2)thetacos^(2)theta` `=1-2sin^(2)thetacos^(2)theta`. Which is not equal to `sin^(2)theta-cos^(2)theta`. Hence , option ( c) is also not correct . (d)`("cosec"^(2)theta+cot^(4)theta)` `="cosec"^(2)theta+("cosec"^(2)theta-1)^(2)` `="cosec"^(2)theta+"cosec"^(4)theta+1-2" cosec"^(2)theta` `="cosec"^(4)theta+1-"cosec"^(2)theta` `="cosec"^(4)theta+cot^(2)theta` Thus option (d ) is correct .

Discussion

11849.	If A={x inR:0ltxlt3}andB={x inR:1lexle5} then ADeltaB is
Answer» `{xinR:0ltxlt1}` `{X inR:3lexle5}` `{x inR:0ltxlt1or3lexle5}` `PHI` ANSWER :C

Discussion

11850.	If the roots of the equation x^(2) + p^(2) = 8x + 6p are real, then p belongs to be interval
Answer» [2,8] [-8,2] [-2,8] [-8,-2] ANSWER :C

Discussion

Explore topic-wise InterviewSolutions in .

Integrate the following functions : intsin2x.e^(cos^(2)x)dx

A man, 2m tall, walks at the rate of 1(2)/(3) m/s towards a street light which is 5(1)/(3) m above the ground. At what rate is the tip of his shadow moving and at what rate is the length of the shadow changing when he is 3(1)/(3)m from the base of the light ?

A has 3 shares in a lottery containing 3 prizes and 6 blanks . B has two shares in a lottery containing 2 prizes and 6 blanks . The ratio of their chances of success is

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If the comples cube roots of (-i) are alpha, beta ,gamma the alpha^2+beta^2+gamma^2=

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intsqrt(1+2 tan x (tan x + sec x ) ) dx

Let f (x) be invertible function and let f ^(-1) (x) be is inverse. Let equation f (f ^(-1) (x)) =f ^(-1)(x) has two real roots alpha and beta (with in domain of f(x)), then :

A person is to walk from A to B. However, he is restricted to walk only to the right of A or upwards of A. but not necessarily in the order shown in the figure. Then find the number of paths from A to B.

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Consider a plane pi:vecr*vecn=d (where vecn is not a unti vector). There are two points A(veca) and B(vecb) lyingon the same sideof the plane. Q.Reflection of A(veca) in the plane pi has the position vector :

What is the derivative of f(In x) with respect to x where f(x) = In x ?

The system of equations x+2y+3z=4 2x+3y+4z=5 3x+4y+5z=6 has

int(dx)/(cosa+cosx)=

A man starts repaying a loan as first instalment of Rs. 100. Ifthe increases the instalment by Rs. 5 every month, then the amount he will pay in the 30th instalment is

Write solution ofdy/dx=2y,y(0)=2

The value of ""^(40) C_(31) + sum _(r = 0)^(10) ""^(40 + r) C_(10 +r)is equal to

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The numbers of ways of arranging the word 'ARRANGE' so that neither 2A's nor 2R's occur together are

Value of F(3)=

A line makes angles alpha,beta,gammawith the coordinate axes. If alpha + beta = pi//2, then (cos alpha+cosbeta+cosgamma)^(2) is equal to

If z_k=cos((kpi)/(10))+isin((kpi)/(10)) , then z_1z_2z_3z_4 is equal to

Evaluate the following integrals. int(1)/(3cosx+4sinx+6)dx

int(e^(2x)+1)/e^xdx

If barr=hati+hatj+lambda(2hati+hatj+4hatk)" and "barr*(hati+2hatj-hatk)=3 are the equations of a line and plane respectively, then which of the following is truwe?

Method of integration by parts : int[(1)/(logx)-(1)/((logx)^(2))]dx=.......

The solution of the differential equation (d ^(3) y)/( dx ^(3)) - 8 (d ^(2)y )/( dx ^(2))=0 safisfing y (0) =1/8, y'(0) and y" (0) =1 is equal to 1/p [ (e ^(8x))/( 8) -+ + (7)/(8)], then find the value of p.

The solution of (dy)/(dx) = (x + y +1)/(x + y -1) is

The value of the integral int_(-1)^(1){(x^(2015))/(e^(|x|)(x^(2)+cosx))+(1)/(e^(|x|))}dx is equal to

R(f)=

Consider a function f: [0, (pi)/(2)] to R given by f (x) = sin x and g : [0, (pi)/(2)] to R given by g (x) = cos x, Show that f and g are one-one but f + g is not one-one.

IF theequationx^2-x-p=0andx^2 + 2px -12 =0havea commonrootthenthatroot is

Find the equation of normal to the ellipse (x^(2))/(16)+(y^(2))/(9) = 1 at the point whose eccentric angle theta=(pi)/(6)

The value of int_(-pi//4)^(pi//4) sin^(103) x*cos^(101) xdx is

C_0 - [C_1 -2.C_2+ 3.C_3-……..+(-1)^(n-1).n.C_n] =

By using the properties of definite integrals, evaluate the integrals int_(0)^(pi/2)(2log sin x log sin 2x)dx

Integration using rigonometric identities : int sec x tan^(3)x dx=....

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+………+C_(n)x^(n) AA n in N and (C_(0)^(2))/(1)+(C_(1)^(2))/(2)+(C_(2)^(2))/(3)+……..+(C_(n)^(2))/(n+1)=(lambda(2n+1)!)/((n+1)!)^(2), then the vlaue of lambda is equal to

If the curve y=a sqrt(x)+bxpasses through P(1,2) and lies above the x-axis for x in [0,9] and the area bounded by the curve, the x-axis and x=4 is 8 sq. units the 2a-3b=

Find the order and the degree of the following differential equation (d^4y)/(dx^4) - sin ((d^3y)/(dx^3)) = 0

Three positive numbers form an increasing G.P. if the middle term in this G.P is doubled, the new numbers are in A.P then the common ratio of the G.P. is

Which term of the geometric sequence5,2,4/5 , 8/(25) …is (128)/(15625) ?

Let 0^(@)ltthetalt45^(@).Which one of the following is correct ?

If A={x inR:0ltxlt3}andB={x inR:1lexle5} then ADeltaB is

If the roots of the equation x^(2) + p^(2) = 8x + 6p are real, then p belongs to be interval