28278 + Interview Questions in Maths in Class 12 Page 248 InterviewSolution

12351.	Find the number of quadruplets of positive integers (a,b,c,d) satisfying the following relations . 1 le a le b le c le d and ab + cd = a + b + c + d + 3
Answer» ANSWER :4

Discussion

12352.	f : R rarr R , f(x) = x^(2) +2x +3 is .......
Answer» ONE - one but not ONTO. onto but not one - one onto but not one one MANY - one and not onto . SOLUTION :N/A

Discussion

12353.	Evaluate the following integrals : int_(1)^(3)(logx)/((x+1)^(2))dx
Answer» ANSWER :`(3)/(4)log3-log2`

Discussion

12354.	The value of (i)^(i) is
Answer» `E^(-(PI)/(2))` `e^(i"" (pi)/(2))` `e^(- (pi)/(2))` `e^(-i "" (pi)/(2))` ANSWER :A

Discussion

12355.	f : R - {(-4)/3} rarr R be a function defined as f(x) = (4x)/(3x +4). The inverse of f is the map g : Range f rarr R - {(-4)/3} given by
Answer» `g(y) = (3Y)/(3 - 4Y)` `g(y) = (4y)/(4 - 3y)` `g(y) = (4y)/(3 - 4y)` `g(y) = (3y)/(4 - 3y)` ANSWER :B

Discussion

12356.	Differentiate the following w.r.t x sin^3x+cos^6x
Answer» SOLUTION :LET `y=sin^3x+cos^6x` Then`(DY)/dx=3sin^2x(COSX)+6cos^5x(-SINX)=3sin^2xcosx-6cos^5xsinx=3sinxcosx(sinx-2cos^4x)`

Discussion

12357.	If \|x\|lt 1, Coefficient of x^3 in 1/(e^x (1+x)) is
Answer» `17/6` `-8/3` `-11/6` `11/6` ANSWER :B

Discussion

12358.	IntriangleVWY below X lies on bar(WY) , Z lies on bar(VY) , and a,b,c and d are angle measures , in degrees. The measure of angleY is 45^@. What is a+b+c+d ?
Answer» 315 270 225 135 Answer :B

Discussion

12359.	The radius of a sphere is 4 cm with a possible error of 0.01 cm . Then absolute error in volume is
Answer» `64 cm^(3)` `PM 0 . 64 pi cm^(3)` ` 6 . 4 pi cm^(3)` `pm 0.0075 cm^(3)` Answer :B

Discussion

12360.	If \|(x^(2)+kx +1)/(x^(2) + x + 1)\| lt 3 for all real numbers x, then the range of the parameter k is
Answer» (0, 4) (-1, 5) (-4, 0) (-5, 1) Answer :B

Discussion

12361.	f(x)= {(-2",","if" x le -1),(2x",","if" -1 lt x le 1),(2",","if " x gt 1):}
Answer» ANSWER :F(1)

Discussion

12362.	Evaluate :intsqrt(cotx)+sqrt(tanx)dx.
Answer» ANSWER :`sqrt2sin^(-1)(sinx-cosx)+C`

Discussion

12363.	Which of the followingstatement is true?
Answer» Which of the followingstatement is TRUE? Which of the followingstatement is true? The real PART of a COMPLEX number cannot exceed its modulus. If the sum of TWO complex quantities `z_(1)and z_(2)` is real then `z_(1)and z_(2)` are conjugate of each other. Answer :C

Discussion

12364.	int (2x+3)/(sqrt(4x+3))dx=
Answer» Answer :`(1)/(12)(4x+3)^(3//2)+(3)/(4)SQRT(4x+3)+c`

Discussion

12365.	Evaluate the following define integrals as limit of sums : lim_(n rarroo) sum_(i=1)^(n) (i^(3))/(i^(4)+n^(4))
Answer» ANSWER :`1/4 LOG 2`

Discussion

12366.	Find the area of the region bounded by y = f(x), y = \|g(x)\| and the lines x = 0, x = 2 , where 'f' . 'g' are continuous functions satisfying f(x+y)=f(x)+f(y)-8xyAA x, y inR and g(x+y) = g(x) + g(y) + 3xy(x+y) x, y in R also f'(0) = 8 and g'(0) = - 4.
Answer» SOLUTION :N//A

Discussion

12367.	int_(0)^(1) sin (2 tan^(-1) sqrt((1+x)/(1-x)))dx =
Answer» `PI/6` `pi/4` `pi/2` `pi` ANSWER :B

Discussion

12368.	The value of x for which the angle between the vectors veca = xhati + 3hatj -hatk and vecb = 2xhati + xhatj -hatk is acute and the angle between vecb and Y-axis lies between pi/2 and pi are
Answer» `-1` `X gt 0` 1 `x LT 0` ANSWER :D

Discussion

12369.	Let d in R, and A[{:(,-2,4+d,(sin theta-2)),(,1,(sin theta)+2,d),(,5,(2sin theta)d,(-sin theta)+2+2d):}]=theta in [0,2pi] If the minimum value of det(A) is B. Then the value of d is:
Answer» `-5` `2(sqrt2+2)` `2(sqrt2+1)` `-7` ANSWER :A

Discussion

12370.	Evaluate int(x^(2))/(x^(4)+x^(2)+1)dx
Answer» Answer :`I_(9)=(1)/(2)(I_(3)+I_(6))`

Discussion

12371.	Integrate the following functions. intsin^(-1)sqrt((x)/(x+a))dx
Answer» Answer :`(x+a)TAN^(-1)SQRT((x)/(a))-sqrtax+atan^(-1)sqrt((x)/(a))+C`

Discussion

12372.	Integrate :int tan^(-1)x dx,hence, find the value of int cot^(-1)x dx.
Answer» Answer :`=X COT^(-1)x+(1)/(2)log(1+x^(2))+K`.

Discussion

12373.	Inverse of (0,0) w.r.t to circle x^(2)+y^(2)-4x-6y+3=0 is
Answer» `(6/13,9/13)` `((-5)/13,9/13)` `(6/13,(-9)/13)` `((-6)/13,(-9)/13)` ANSWER :A

Discussion

12374.	Let int(10lnx)/(x^(2))dx=f(x), for all positive x. If f(e )=(1)/(e ), then f(2)+f(4) is equal to
Answer»

Discussion

12375.	Evaluate int(2x+5)/( sqrt(x^(2)-2x+10))) dx .
Answer» ANSWER :`2sqrt(X^(2)-2x+10)+7sinh^(-1)((x-1)/(3))+c`

Discussion

12376.	Find lambda and mu if : (2hati+6hatj+27hatk)xx(hati+lambdahatj+muhatk)=vec0.
Answer» p=6,Q=27 p=3,q=(27/2) p=6,q=(27/2) p=3,q=27 Answer :B

Discussion

12377.	Two circular path of radii a and b intersect at a point O and AB is a line through O meeting the circles at A and B respectively. Chords OA and OB subtend equal angles of 60^(@) at their respective centres. A vertical pole at O subtends angles alpha and beta respectively at A and B then height of the pole is
Answer» `a COT alpha` `B cot BETA` `(a + b)/(cot alpha + cot beta)` NONE of these Answer :C

Discussion

12378.	Foot of perpendicular drawn from the origin to the plane 2x-3y + 4z=29 is_________
Answer» `(5, -2,3)` `(7, -1, 3)` `(2, -3, 4)` `(5, -1, 4)` Answer :C

Discussion

12379.	If 2,5,7,-4are the rootsofax^4+ bx^3 + cx^2 + dx+e=0thentherootsofax^4 - bx^3+ cx^2 -dx +e=0are
Answer» `2,5,7,-4` `-2,-5,-7,4` `2,5,7,4` `2,-5,7,-4` ANSWER :B

Discussion

12380.	Evaluate the following integrals int(1)/((1-sqrtx)sqrt(x-x^(2)))dx
Answer» ANSWER :`(2sqrt(x-1))/(SQRT(1-x))+C`

Discussion

12381.	What is the area of the triangle?
Answer» 6 7.5 8 11 Answer :A

Discussion

12382.	d/dx(e^(2sin^-1x)) =
Answer» `2E^(2sin^-1x)` `(2e^(2sin^-1x))/SQRT(1-x^2)` `E^(2sin^-1x)/sqrt(1-x^2)` `e^(2sin^-1x)` ANSWER :B

Discussion

12383.	Ifalpha, beta,1are rootsofx^3 -2x^2 -5x +6=0 ( alphagt 1)then3 alpha+ beta=
Answer» 7 5 14 10 Answer :A

Discussion

12384.	If (sqrt(8)+i)^(50)=3^(49)(a+ib)" then "a^(2)+b^(2) is
Answer» `3` `8` `9` 4` ANSWER :A

Discussion

12385.	Evaluate int_(1)^(2)((x+1)^(3))/(x^(2))dx
Answer» ANSWER :`5+3 LOG 2`

Discussion

12386.	If x+y+z = 0 prove that \|{:(xa,yb,zc),(yc,za,xb),(zb,xc,ya):}\|=xyz\|{:(a,b,c),(c,a,b),(b,c,a):}\|
Answer» ANSWER :`XYZ\|{:(a,B,C),(c,a,b),(b,c,a):}\|`

Discussion

12387.	If an object is moving at an average rate of speed of 18(km)/("min"), how many meters does it travel is 5 seconds?
Answer» ANSWER :`1,500`m

Discussion

12388.	Evaluate overset(1)underset(-1)int 5x^(4) sqrt(x^(5)+1)dx
Answer» ANSWER :`(4 SQRT(2))/(3)`

Discussion

12389.	A speaks truth in 75% of the cases and B in 80% of the cases. What is the prbability that (a) both speak truth (b) both speak lie (c) their statements about an incident do not match.
Answer» Answer :(a) `(3)/(5)` (b) `(1)/(20)` (c) `(7)/(20)`

Discussion

12390.	Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2)A. If Rolle's theorem is applicable to phi(x) on, [a,b], answer following questions. If there exists some unmber c(a lt c lt b) such that phi'(c)=0 and f(b)=f(a)+(b-a)f'(a)+lambda(b-a)^(2)f''(c), then lambda is
Answer» 1 0 `(1)/(2)` `-(1)/(2)` Solution :`phi(x)=F(b)-f(x)-(b-x)f'(x)-(b-x)^(2)A` `phi(b)=0` `phi(a)=f(b)-f(a)-(b-a)f'(a)-(b-a)6(2)A` Since Rolle'stheorem is APPLICABLE `therefore""A=(f(b)-f(a)-(b-a)f'(a))/((b-a)^(2))"(i)"` `phi'(x)=-f'(x)+f'(x)-(b-x)f''(x)+2(b-x)A` `therefore"There EXISTS some number " cin (a,b)` such that `0=phi'(c)=-(b-c)f''(c)+2(b-c)A` `"i.e.A=(1)/(2)f''(c)"(ii)"` From (i) and (ii), `f(b)-f(a)-(b-a)f'(a)=(1)/(2)(b-a)^(2)f''(c)` `"i.e."f(b)=f(a)+(b-a)f'(a)+(1)/(2)(b-a)^(2)f''(c)` `therefore""LAMBDA=(1)/(2)`

Discussion

12391.	Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2)A. If Rolle's theorem is applicable to phi(x) on, [a,b], answer following questions. Let f(x)=sin x, a = alpha and b=alpha+h. If have exists a real number t such that 0lt t lt 1, phi'(alpha+th)=0 and (sin(alpha+h)-sinalpha-h cosalpha)/(h^(2))=lambdasin(alpha+th), then lambda=
Answer» `(1)/(2)` `-(1)/(2)` `(1)/(4)` `(1)/(3)` Solution :`sin(ALPHA+h)=sin alpha+h COS alpha+(1)/(2)h^(2)(-sin (alpha+th))` `THEREFORE""(sin (alpha+h)-sin alpha-h cos alpha)/(h^(2))=-(1)/(2)sin(alpha+th)` `therefore""alpha=-(1)/(2)`

Discussion

12392.	Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2)A. If Rolle's theorem is applicable to phi(x) on, [a,b], answer following questions. Let f(x)=x^(3)-3x+3, a=1 and b=1+h. If there exists c in (1,1+h) such that phi'(c)=0 and (f(1+h)-f(1))/(h^(2))=lambdac, then lambda=
Answer» `1//2` 2 3 does not exist Solution :`f(1+h)=f(1)+hf'(1)+(1)/(2)h^(2)f''(c)` `THEREFORE""f(1+h)=1+3h^(2)c` `therefore""(f(1+h)-f(1))/(h^(2))=3e` `therefore""lambda=3`

Discussion

12393.	Let A(2hati+3hatj+5hatk),B(-hati+3hatj+2hatk) and C(lamdahati+5hatj+muhatk) be the vertices of DeltaABC and its median through A be equally inclined to the positivedirections of the coordinate axds. Then, the value of 2lamda-mu is
Answer» 0 1 4 3 Solution :NA

Discussion

12394.	Integrate the rational functions (2x)/((x^(2)+1)(x^(2)+3))
Answer» Answer :`INT(1)/(x^(2))+a^(2)=(1)/(a)tan^(-1)((x)/(a))+C`

Discussion

12395.	f(x)= [sin x], x in [0, 2pi] At which points, f(x) is discontinuous?
Answer» ANSWER :`X= (PI)/(2), pi , 2PI`

Discussion

12396.	If mu the mean of distribution (y_(i),f_(i)), then sum f_(i)(y_(i)-mu)=
Answer» M.D. S.D. 0 Relative frequency Solution :We have, `sumf_(i)(y_(i)-MU)=SUM f_(i)y_(i)-mu sum f_(i)=mu sum f_(i)-mu sum f_(i)=0 "" [because mu=(sum f_(i)y_(i))/(sum f_(i))]`

Discussion

12397.	Area bounded by y=f^(-1)(x) and tangent and normal drawn to it at points with abscissae pi and 2pi, where f(x)=sin x-x is
Answer» `(pi^2)/(2)-1` `(pi^2)/(2)-2` `(pi^2)/(2)-4` `(pi^2)/2` ANSWER :B

Discussion

12398.	Show that the right circular cone of least curved surface and given volume has an altitude equal to sqrt(2) time the radius of the base.
Answer» ANSWER :`h=sqrt(2)R`

Discussion

12399.	The probabilities of a problem being solved by three students are 1/3 , 1/4 and 1/6 . The probability of the problem being solved is
Answer» ANSWER :`(7)/(12)`

Discussion

12400.	Find the equation of circles determined by the following conditions. The ends of diameter are (-5, 3) and (7, 5).
Answer» SOLUTION :The ENDPOINTS of diameter of the circle are (-5, 3) and (7, 5). `THEREFORE` EQU. Of the circle is `(x-h)^2 + (y-k)^2 = a^2` `(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0` or, `(x+5)(x-7)+(y-3)(y-5)=0` or, `x^2-7x+5x-35+y^2-5y-3y+15=0` or, `x^2+y^2-2x-8y-20=0`

Discussion

Explore topic-wise InterviewSolutions in .

Find the number of quadruplets of positive integers (a,b,c,d) satisfying the following relations . 1 le a le b le c le d and ab + cd = a + b + c + d + 3

f : R rarr R , f(x) = x^(2) +2x +3 is .......

Evaluate the following integrals : int_(1)^(3)(logx)/((x+1)^(2))dx

The value of (i)^(i) is

f : R - {(-4)/3} rarr R be a function defined as f(x) = (4x)/(3x +4). The inverse of f is the map g : Range f rarr R - {(-4)/3} given by

Differentiate the following w.r.t x sin^3x+cos^6x

If |x|lt 1, Coefficient of x^3 in 1/(e^x (1+x)) is

IntriangleVWY below X lies on bar(WY) , Z lies on bar(VY) , and a,b,c and d are angle measures , in degrees. The measure of angleY is 45^@. What is a+b+c+d ?

The radius of a sphere is 4 cm with a possible error of 0.01 cm . Then absolute error in volume is

If |(x^(2)+kx +1)/(x^(2) + x + 1)| lt 3 for all real numbers x, then the range of the parameter k is

f(x)= {(-2",","if" x le -1),(2x",","if" -1 lt x le 1),(2",","if " x gt 1):}

Evaluate :intsqrt(cotx)+sqrt(tanx)dx.

Which of the followingstatement is true?

int (2x+3)/(sqrt(4x+3))dx=

Evaluate the following define integrals as limit of sums : lim_(n rarroo) sum_(i=1)^(n) (i^(3))/(i^(4)+n^(4))

Find the area of the region bounded by y = f(x), y = |g(x)| and the lines x = 0, x = 2 , where 'f' . 'g' are continuous functions satisfying f(x+y)=f(x)+f(y)-8xyAA x, y inR and g(x+y) = g(x) + g(y) + 3xy(x+y) x, y in R also f'(0) = 8 and g'(0) = - 4.

int_(0)^(1) sin (2 tan^(-1) sqrt((1+x)/(1-x)))dx =

The value of x for which the angle between the vectors veca = xhati + 3hatj -hatk and vecb = 2xhati + xhatj -hatk is acute and the angle between vecb and Y-axis lies between pi/2 and pi are

Let d in R, and A[{:(,-2,4+d,(sin theta-2)),(,1,(sin theta)+2,d),(,5,(2sin theta)d,(-sin theta)+2+2d):}]=theta in [0,2pi] If the minimum value of det(A) is B. Then the value of d is:

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

Integrate the following functions. intsin^(-1)sqrt((x)/(x+a))dx

Integrate :int tan^(-1)x dx,hence, find the value of int cot^(-1)x dx.

Inverse of (0,0) w.r.t to circle x^(2)+y^(2)-4x-6y+3=0 is

Let int(10lnx)/(x^(2))dx=f(x), for all positive x. If f(e )=(1)/(e ), then f(2)+f(4) is equal to

Evaluate int(2x+5)/( sqrt(x^(2)-2x+10))) dx .

Find lambda and mu if : (2hati+6hatj+27hatk)xx(hati+lambdahatj+muhatk)=vec0.

Foot of perpendicular drawn from the origin to the plane 2x-3y + 4z=29 is_________

If 2,5,7,-4are the rootsofax^4+ bx^3 + cx^2 + dx+e=0thentherootsofax^4 - bx^3+ cx^2 -dx +e=0are

Evaluate the following integrals int(1)/((1-sqrtx)sqrt(x-x^(2)))dx

What is the area of the triangle?

d/dx(e^(2sin^-1x)) =

Ifalpha, beta,1are rootsofx^3 -2x^2 -5x +6=0 ( alphagt 1)then3 alpha+ beta=

If (sqrt(8)+i)^(50)=3^(49)(a+ib)" then "a^(2)+b^(2) is

Evaluate int_(1)^(2)((x+1)^(3))/(x^(2))dx

If x+y+z = 0 prove that |{:(xa,yb,zc),(yc,za,xb),(zb,xc,ya):}|=xyz|{:(a,b,c),(c,a,b),(b,c,a):}|

If an object is moving at an average rate of speed of 18(km)/("min"), how many meters does it travel is 5 seconds?

Evaluate overset(1)underset(-1)int 5x^(4) sqrt(x^(5)+1)dx

A speaks truth in 75% of the cases and B in 80% of the cases. What is the prbability that (a) both speak truth (b) both speak lie (c) their statements about an incident do not match.

Let A(2hati+3hatj+5hatk),B(-hati+3hatj+2hatk) and C(lamdahati+5hatj+muhatk) be the vertices of DeltaABC and its median through A be equally inclined to the positivedirections of the coordinate axds. Then, the value of 2lamda-mu is

Integrate the rational functions (2x)/((x^(2)+1)(x^(2)+3))

f(x)= [sin x], x in [0, 2pi] At which points, f(x) is discontinuous?

If mu the mean of distribution (y_(i),f_(i)), then sum f_(i)(y_(i)-mu)=

Area bounded by y=f^(-1)(x) and tangent and normal drawn to it at points with abscissae pi and 2pi, where f(x)=sin x-x is

Show that the right circular cone of least curved surface and given volume has an altitude equal to sqrt(2) time the radius of the base.

The probabilities of a problem being solved by three students are 1/3 , 1/4 and 1/6 . The probability of the problem being solved is

Find the equation of circles determined by the following conditions. The ends of diameter are (-5, 3) and (7, 5).