28278 + Interview Questions in Maths in Class 12 Page 545 InterviewSolution

27201.	If 1 , omega , omega^(2) are the cube roots of unity, then prove that (2 - omega)(2 -omega^(2)) (2 - omega^(10)) ( 2 - omega^(11)) = 49.
Answer»

Discussion

27202.	The odds against A solving a certain problem are 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. Find the probability that the problem will be solved.
Answer» ANSWER :`(3)/(8) = 0.38`

Discussion

27203.	Differentiate the following w.r.t. x : sin (log x), x gt 0
Answer» ANSWER :`=(COS (LOGX))/(X)`.

Discussion

27204.	If veca=2hati-3hatj+4hatk, veca.vecb=2 and veca xx vecb=hati+2hatj+hatk, then vecb is equal to
Answer» `15hati-8hatj+hatk` `(1)/(29)(15hati-8hatj+hatk)` `(1)/(5)(2hati+hatj+hatk)` `2hati+hatj+hatk` ANSWER :B

Discussion

27205.	I : The vector 6i + 2j + k, 2i - 9j + 6k are mutually perpendicular. II : The vectors I + 2j - , 2i + j + k are mutually perpendicular
Answer» only I is ture Only II is ture both I and II are true Neither I nor II are true Answer :A

Discussion

27206.	Evaluate: (1+cosalphacosx)/(cosalpha+cosx)dx
Answer» SOLUTION :`xcosalpha+sinalpha" LN "\|(cos1/2(alpha-x))/(cos1/2(alpha+x))\|+C`

Discussion

27207.	Prove the inequalities 1/2 le int _(0) ^(2) (dx)/(2+x ^(2)) le 5/6
Answer» ANSWER :`1/e`

Discussion

27208.	If (-2,1) is a limiting point of a coaxial system of circle of which x^(2)+y^(2)-4x-6y+7=0 is a member, then the other limiting point is
Answer» `((4)/(5),(-12)/(5))` `((-4)/(5),(12)/(5))` `((-4)/(5),(-12)/(5))` `((4)/(5),(12)/(5))` ANSWER :D

Discussion

27209.	Prove that the portion or the tangent intercepted,between the point of contact and the directrix of the parabola y^(2)= 4ax subtends a right angle at its focuc.
Answer» `30^(@)` `45^(@)` `60^(@)` `90^(@)` ANSWER :D

Discussion

27210.	The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?
Answer» ANSWER :`bsqrt(3) CM^(3)//s`

Discussion

27211.	If x + 1/x = 2 cos theta, then for any integer n, x^(n)+1/(x^(n))=
Answer» `2cosntheta` `2sinnthet a` `2icosntheta` `2isinntheta` ANSWER :A

Discussion

27212.	P is a point on the ellipse x^2/a^2+y^2/b^2=1(a gt b) and Q is the point corresponding to P on the auxiliary circle x^2+y^2=a^2.N is the foot of the perpendicular from P on the major axis of the ellipse. (PN)/(PQ) is equal to:
Answer» `B/(a-b)` `a/(a-b)` `b/(a+b)` `a/(a+b)` ANSWER :A

Discussion

27213.	Let A = R- {(3)/(5)} " and" B =R -{(7)/(5)} Let f: A to B: f (x) =(7x +4)/(5x-3) " and" g: B to A: g (y)= (3y+4)/(5y-7) Show that(g o f)I_(A) " and" ( fog) = I_(B)
Answer» Solution :Let `x in A. `Then (g o f)(x) = g [f (x)] `=g ((7x+4)/(5x-3))` `=g(y) " where " y =(7x+4)/(5x-3)` `=(3y+4)/(5y-7) =(3((7x+4)/(5x-3))+4)/(5((7x+4)/(5x-3))-7) "" "[using (i)]"` `=((21 x+12 +20x-12))/((5x-3)) XX ((5x-3))/((35 x+20 -35 x+21))` `=(41x)/(41) =x= I_(A) (x)` `:. (gof)=I_(A)` Againlet `y in B .` Then ( FOG)(y) = f[g (y)] `=f((3y+4)/(5y-7))` `=f(z)" where"z= (3y+4)/(5y-7)` `=(7z+4)/(5z-3) =(7((3y+4)/(5y-7))+4)/(5((3y+4)/(5y-7))-3)` `=((21 y+28 +20y-28))/((5y-7)) xx ((5y-7))/((15y+20 -15y+21))` `=(41y)/(41) =y= I_(B) (y)` `:. (f o g)=I_(B)` hence(f o g ) `=I_(A)" and" (f o g) =I_(B)`

Discussion

27214.	Let A = {a,b,c} and R_(1) = {(a,a), (c,b), (b,c)} R_(2) = {(b,b), (c,c)} Which of the following is true?
Answer» `R_(1)`and `R_2` are both transitive `R_(1)`and `R_2` are both SYMMETRIC `R_(1)`and `R_2` are both reflexive `R_2` is an equivalence RELATION but `R_(1)`is not Answer :B

Discussion

27215.	Find the area of the region bounded by y= sin x, y= 0, x= 0 and x= pi
Answer» ANSWER :2

Discussion

27216.	The probability of drawing 3 white and 4 green balls from a bag containing 5 white and 6 green balls if the seven balls are drawn at random simultaneously is
Answer» `(5)/(11)` `(6)/(11)` `(7)/(11)` `(8)/(11)` Answer :A

Discussion

27217.	Find the area of a triangle with angles alpha,beta " and "gammaknowing that the distance from an arbitrary point M taken inside the triangle to its sides are equal to m,n and k.
Answer» ANSWER :`=((K SIN GAMMA +n sin alpha +m sin beta)^2)/(2 sin alpha sin beta sin gamma)`.

Discussion

27218.	If \|(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))\|=Delta, then \|(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))\|=
Answer» 0 `DELTA` `Delta^(2)` `Delta^(3)` ANSWER :C

Discussion

27219.	If l_(n)=int_(pi//2)(oo)e^(-x)cos^(n)xdx=(720)/(3145)e^(-pi//2)AA n =2m in N, then find the value of n.
Answer» ANSWER :`N!=720 RARR n=6.`

Discussion

27220.	Find the least value of a such that the function given by f(x)=x^(2)+ax+1 is strictly increasing on (1, 2).
Answer» ANSWER :`-2`

Discussion

27221.	If a line makes alpha=45^(@),beta=60^(@) with positive direction of axis x and y, then the angle it makes with the z-axis (gamma) is
Answer» `60^(@)` `SIN^(2)alpha+sin^(2)beta+sin^(2)gamma=1` `cos^(2)alpha+cos^(2)beta+cos^(2)gamma=1` `sin^(2)alpha+sin^(2)beta+sin^(2)gamma=2` ANSWER :A::B

Discussion

27222.	Integrate the function in Exercise. x cos^(-1)x
Answer» Answer :`(COS^(-1)X)/(4)(2x^(2)-1)-(x)/(4) SQRT(1-x^(2))+c`

Discussion

27223.	Find the probability of getting 2 heads when 4 coins are tossed
Answer» ANSWER :`3//8`

Discussion

27224.	Construct 3xx3 matrix A=[a_(ij)] whose elements are given by a_(ij)=2i-3j.
Answer» Answer :`A=[{:(-1,-4,-7),(1,-2,-5),(3,0,-3):}]`

Discussion

27225.	The solution of cos^(2) x (dy)/(dx) + y = tan x is
Answer» `y e^(Tan x) = (TANX- 1)e^(Tan x) - Tanx + C` `y e^(Tan x) = (Tanx + 1)e^(Tan x) + Tanx + c` `y e^(Tan x) = (Tanx- 1)e^(Tan x)+ c` `y e^(Tan x) = (Tanx - 1)e^(Tan x) + Tanx + c` Answer :C

Discussion

27226.	If a+b+c=0 , one root of : \|(a-x,c,d),(c,b-x,a),(b,a,c-x)\|=0 is :
Answer» `X=1` `x=2` `x=a^2b^2+c^2` x = 0 Answer :D

Discussion

27227.	intsin^20xcos^3xdx
Answer» SOLUTION :`intsin^20xcos^3xdx` =`intsin^20x.(1-sin^2x)cosxdx` [PUT sinx=t Then cosxdx=DT] =`intt^20(1-t^2)dt=int(t^20-t^22)dt` =`1/21t^21-1/23t^23+C` =`1/21sin^21x-1/23sin^23x+C`

Discussion

27228.	Evaluate : int_(0)^(pi)(\|cos x\| + \|sin x\|)dx.
Answer» ANSWER :4

Discussion

27229.	All possible roots of polynomial with integral coefficients can be identified by "RATIONAL ROOT TEST"according to rational root test if a plynomial ltbr. a_(n)p^(n)+a_(n-1)p^(n-1)q+a_(n-2)p^(n-2)q^(2)+…..+a_(2)p^(2)q^(n-2)+a_(1)pq^(n-1)+a_(0)q^(n)=0 Every term in above equation except possible the last one is divisible by p hence p should also divide a_(0)q^(n), since q and p are relatively prime p must divide a_(0). Similarly q also divides a_(n). Now consider an equation 6x^(5)-19x^(4)-9x^(3)-16x^(2)+9x-1=0 and answer the following questions Q. if given equation and equation x^(3)+ax^(2)+ax+1=0 have two roots common then possible value(s) of is/are?
Answer» `0` `-2` `-3` `3` Solution :(13 to 14) `6x^(5)-19x^(4)-9x^(3)-16x^(2)+9x-1=0` rational roots of equation `x=+-1,+-(1)/(2),+-(1)/(2),+-(1)/(6)` are possible `x=(1)/(6)` satisfies the equation `6x^(5)-x^(4)-18X^(4)+3x^(3)-12x^(3)+2x^(2)-18x^(2)+3x+6x-1=0` `(6x-1)(x^(4)-3x^(3)-2x^(2)-3x+1)=0` `x^(4)-3x^(3)-2x^(2)-3x+1)=0` `x^(4)-3x^(3)-2x^(2)-3x+1=0` `x^(2)+(1)/(x^(2))-3(x+(1)/(x))-2=0` `(x+(1)/(x))^(2)-3(x+(1)/(x))-4=0` `t^(2)-3t-4=0` `(t-4)(t+1)=0` `x+(1)/(x)=4` or `x+(1)/(x)+1=0` `x^(2)-4x+1=0` `x^(2)+x+1=0` sum of REAL roots will be `4+(1)/(6)=(25)/(6)` for rational values of a roots of CUBIC `x^(2)+ax^(2)+ax+1=0` will be in conjugate pair so either `x^(2)-4x+1=0` or `x^(2)+x+1=0` is a factor of `x^(3)+ax^(2)+1=0` if `x^(2)-4x+1=0` is a factor of `x^(3)+ax^(2)+ax+1=0` then `a=-3` and `x^(2)+x+1=0` is a factor of `x^(3)+ax^(2)+ax+1=0` then `a=2`

Discussion

27230.	If f is continuous on its domain D, then \|f\| is also continuous on D
Answer» ANSWER :1

Discussion

27231.	Check the validity of If n is a real number with n > 3, thenn2 > 9by the method of contradiction.
Answer» SOLUTION :Let the given statement is false. i.e. for a natural NUMBER `n GE3 n^2 ge 9` `impliesn^2 le 3 implies n le 3` ( n is a natural number which contradicts the FACT that n>3) This contradiction is due to our false assumption. Thus for any natural number `n gt 3,n^2 gt 9`.

Discussion

27232.	All possible roots of polynomial with integral coefficients can be identified by "RATIONAL ROOT TEST"according to rational root test if a plynomial ltbr. a_(n)p^(n)+a_(n-1)p^(n-1)q+a_(n-2)p^(n-2)q^(2)+…..+a_(2)p^(2)q^(n-2)+a_(1)pq^(n-1)+a_(0)q^(n)=0 Every term in above equation except possible the last one is divisible by p hence p should also divide a_(0)q^(n), since q and p are relatively prime p must divide a_(0). Similarly q also divides a_(n). Now consider an equation 6x^(5)-19x^(4)-9x^(3)-16x^(2)+9x-1=0 and answer the following questions Q. Sum of real roots of equation will be
Answer» 4 `(25)/(6)` `(1)/(6)` can not determine Answer :C

Discussion

27233.	If a unit vector veca makes angles (pi)/(3) with hati,(pi)/(4) with hatj and an acute angle theta with hatk then find theta and hence , the components of veca.
Answer» ANSWER :`(pi)/(3);(1)/(2),(1)/(sqrt(2)),(1)/(2)`

Discussion

27234.	The quadrilateral formed by the lines x+y-3=0, x-y+3=0, x+y+1=0, x-y-1=0 is
Answer» PARALLELOGRAM RECTANGLE RHOMBUS SQUARE ANSWER :D

Discussion

27235.	If y = ax ^(n+1)+ bx ^(-n) then x ^(2) y _(2) =
Answer» `N(n +1) y` `n ` `n ( n+1)` `1` ANSWER :A

Discussion

27236.	Consider two vectors oversetrarra "and" oversetrarrbso that\|oversetrarra\|=2,\|oversetrarrb\|=5 "and"\|oversetrarraxxoversetrarrb\|=8. Suppose the angle between a and b is acute.Prove that oversetrarra.oversetrarrb=6
Answer» Solution :`veca.vecb =\|veca\|\|vecb\| cos THETA` `2xx5xx SQRT(1-sin^2 theta)` `=10 sqrt(1-(16)/(25))=10xx3/5=6`

Discussion

27237.	If A = [(a,b),(b,a)] and A^(2) = [(alpha, beta ),(beta,alpha)], then :
Answer» `ALPHA =2 AB , BETA =a^(2) +b^(2)` `alpha =a^(2) +b^(2) , beta = ab ` `alpha =a^(2) +b^(2),beta=2ab` `alpha = a^(2)+b^(2),beta=a^(2) -b^(2)` Answer :C

Discussion

27238.	Consider two vectors oversetrarra" and " oversetrarrbso that\|oversetrarra\|=2,\|oversetrarrb\|=5 and \|oversetrarraxxoversetrarrb\|=8 .Suppose the angle between a and b is acute. findsin theta, where theta is the angle between oversetrarra"and"oversetrarrb
Answer» SOLUTION :`\|vecaxxvecb\|=\|veca\|\|vecb\| SIN THETA`

Discussion

27239.	An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver ?
Answer» ANSWER :`(1)/(52)`

Discussion

27240.	Evalute the following integrals int (sec^(2) "x tan x")/(sec^(2) x + tan^(2) x)dx
Answer» ANSWER :`(1)/(4)log(1+sin^(2)X)-(1)/(2)log\|cosx\|+c`

Discussion

27241.	Let A be a 2xx2 invertible matrix . For which of the following functions det (f(A)) = f (det (A)) is not true ?
Answer» f(X) `= x^(3)` f(x)`=x^(-1), x ne 0` f(x) = 1 +x f(x) `=x^(-3) , x ne 0` ANSWER :C

Discussion

27242.	If the vectors AB=3hati+4hatk and AC=(5hati-2hatj+4hatk) are the sides of a DeltaABC, then the length of the median through A is
Answer» `sqrt(18)` `sqrt(72)` `sqrt(33)` `sqrt(45)` Solution :We know that, the sum of three vectors of a triangle is zero. `therefore AB+BC=CA=0` `rArr BC=AC-AB` `rArr BM=(AC-AB)/(2) `[SINCE, M is mid-point of BC] ALSO, `AB+BM+MA=0 ` [ property of triangle] `rArr AB+(AC-AB)/(2) =AM` `rArr AM=(AB+AC)/(2) = AM` `rArr AM=(AB+AC)/(2) = (3 hati+4hatk+5hati-2hatj+4hatk)/(2)` `=4hati - hatj +4 hatk` `rArr \|AM\| = sqrt(4^(2)+1^(2)+4^(2)) = sqrt(33)`

Discussion

27243.	Consider the following statements : (a) If any two rows or columns of a determinant are identical, then the value of the determinant is zero (b) If the corresponding rows and columns of a determinant are interchanged, then the value of the determinant does not change. (c) If any two rows or columns of a determinant are interchanged, then the value of the determinant changes in sign. Which of these are correct ?
Answer» (a) and (C) (a) and (B) (a), (b) and (c) (b) and (c) ANSWER :C

Discussion

27244.	Find the differential equation of family of all straight lines passing throughthe origin .
Answer» ANSWER :`(DY)/(DX) = (y)/(X)`

Discussion

27245.	A : Area of triangle formed by tangent to the curve xy=c^2 with coordinates axes is 2c^2 sq . Units. R: Area of triangle formed by the line x/a+y/b=1 with coordinate axes is 1/2 \|ab\| sq. units
Answer» A and R are TRUE and R is the CORRECT EXPLANATION of A A and R are true and R is not correct explanation of A A is true but R is FALSE A is false but R is true Answer :A

Discussion

27246.	int_(0)^((pi)/(2n))(dx)/(1+Cot^(n)nx)=
Answer» `(PI)/(2N)` `(pi)/(4N)` `(pi)/(8N)` `pi/n` ANSWER :B

Discussion

27247.	Evaluate the following:""^(10)C_(3)=
Answer»

Discussion

27248.	If f(x)=(p-x^(n))^(1//n), p gt 0 and n is a positive integer, then f[f(x)] is equal to
Answer» <P>X `x^(N)` `p^(1//n)` `p-x^(n)` ANSWER :A

Discussion

27249.	If int_(ln 2)^(x) (e^(x)-1)^(-1)dx= ln (3/2) then x=
Answer» `LN 4` 1 `e^(2)` `1/e` ANSWER :A

Discussion

27250.	For all natural number of n, 2^(2n).3^(2n)-1-35n is divisible by
Answer» `(35)^(3)` `(35)^(2)` `(35)^(4)` `(35)^(5)` ANSWER :2

Discussion

Explore topic-wise InterviewSolutions in .

If 1 , omega , omega^(2) are the cube roots of unity, then prove that (2 - omega)(2 -omega^(2)) (2 - omega^(10)) ( 2 - omega^(11)) = 49.

The odds against A solving a certain problem are 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. Find the probability that the problem will be solved.

Differentiate the following w.r.t. x : sin (log x), x gt 0

If veca=2hati-3hatj+4hatk, veca.vecb=2 and veca xx vecb=hati+2hatj+hatk, then vecb is equal to

I : The vector 6i + 2j + k, 2i - 9j + 6k are mutually perpendicular. II : The vectors I + 2j - , 2i + j + k are mutually perpendicular

Evaluate: (1+cosalphacosx)/(cosalpha+cosx)dx

Prove the inequalities 1/2 le int _(0) ^(2) (dx)/(2+x ^(2)) le 5/6

If (-2,1) is a limiting point of a coaxial system of circle of which x^(2)+y^(2)-4x-6y+7=0 is a member, then the other limiting point is

Prove that the portion or the tangent intercepted,between the point of contact and the directrix of the parabola y^(2)= 4ax subtends a right angle at its focuc.

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?

If x + 1/x = 2 cos theta, then for any integer n, x^(n)+1/(x^(n))=

P is a point on the ellipse x^2/a^2+y^2/b^2=1(a gt b) and Q is the point corresponding to P on the auxiliary circle x^2+y^2=a^2.N is the foot of the perpendicular from P on the major axis of the ellipse. (PN)/(PQ) is equal to:

Let A = R- {(3)/(5)} " and" B =R -{(7)/(5)} Let f: A to B: f (x) =(7x +4)/(5x-3) " and" g: B to A: g (y)= (3y+4)/(5y-7) Show that(g o f)I_(A) " and" ( fog) = I_(B)

Let A = {a,b,c} and R_(1) = {(a,a), (c,b), (b,c)} R_(2) = {(b,b), (c,c)} Which of the following is true?

Find the area of the region bounded by y= sin x, y= 0, x= 0 and x= pi

The probability of drawing 3 white and 4 green balls from a bag containing 5 white and 6 green balls if the seven balls are drawn at random simultaneously is

Find the area of a triangle with angles alpha,beta " and "gammaknowing that the distance from an arbitrary point M taken inside the triangle to its sides are equal to m,n and k.

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))|=Delta, then |(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))|=

If l_(n)=int_(pi//2)(oo)e^(-x)cos^(n)xdx=(720)/(3145)e^(-pi//2)AA n =2m in N, then find the value of n.

Find the least value of a such that the function given by f(x)=x^(2)+ax+1 is strictly increasing on (1, 2).

If a line makes alpha=45^(@),beta=60^(@) with positive direction of axis x and y, then the angle it makes with the z-axis (gamma) is

Integrate the function in Exercise. x cos^(-1)x

Find the probability of getting 2 heads when 4 coins are tossed

Construct 3xx3 matrix A=[a_(ij)] whose elements are given by a_(ij)=2i-3j.

The solution of cos^(2) x (dy)/(dx) + y = tan x is

If a+b+c=0 , one root of : |(a-x,c,d),(c,b-x,a),(b,a,c-x)|=0 is :

intsin^20xcos^3xdx

Evaluate : int_(0)^(pi)(|cos x| + |sin x|)dx.

If f is continuous on its domain D, then |f| is also continuous on D

Check the validity of If n is a real number with n > 3, thenn2 > 9by the method of contradiction.

If a unit vector veca makes angles (pi)/(3) with hati,(pi)/(4) with hatj and an acute angle theta with hatk then find theta and hence , the components of veca.

The quadrilateral formed by the lines x+y-3=0, x-y+3=0, x+y+1=0, x-y-1=0 is

If y = ax ^(n+1)+ bx ^(-n) then x ^(2) y _(2) =

Consider two vectors oversetrarra "and" oversetrarrbso that|oversetrarra|=2,|oversetrarrb|=5 "and"|oversetrarraxxoversetrarrb|=8. Suppose the angle between a and b is acute.Prove that oversetrarra.oversetrarrb=6

If A = [(a,b),(b,a)] and A^(2) = [(alpha, beta ),(beta,alpha)], then :

Consider two vectors oversetrarra" and " oversetrarrbso that|oversetrarra|=2,|oversetrarrb|=5 and |oversetrarraxxoversetrarrb|=8 .Suppose the angle between a and b is acute. findsin theta, where theta is the angle between oversetrarra"and"oversetrarrb

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident are 0.01, 0.03 and 0.15 respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver ?

Evalute the following integrals int (sec^(2) "x tan x")/(sec^(2) x + tan^(2) x)dx

Let A be a 2xx2 invertible matrix . For which of the following functions det (f(A)) = f (det (A)) is not true ?

If the vectors AB=3hati+4hatk and AC=(5hati-2hatj+4hatk) are the sides of a DeltaABC, then the length of the median through A is

Find the differential equation of family of all straight lines passing throughthe origin .

A : Area of triangle formed by tangent to the curve xy=c^2 with coordinates axes is 2c^2 sq . Units. R: Area of triangle formed by the line x/a+y/b=1 with coordinate axes is 1/2 |ab| sq. units

int_(0)^((pi)/(2n))(dx)/(1+Cot^(n)nx)=

Evaluate the following:""^(10)C_(3)=

If f(x)=(p-x^(n))^(1//n), p gt 0 and n is a positive integer, then f[f(x)] is equal to

If int_(ln 2)^(x) (e^(x)-1)^(-1)dx= ln (3/2) then x=

For all natural number of n, 2^(2n).3^(2n)-1-35n is divisible by