28278 + Interview Questions in Maths in Class 12 Page 29 InterviewSolution

1401.	Find the vector and cartesian equations of the plane which passes through the point (5, 2, - 4) and perpendicular to the line with direction ratios 2, 3, -1.
Answer» ANSWER :vector equations `[vecr - (5hati + 2hatj - 4HATK)] · (2hati + 3hatj - hatk) = 0,` cartesian equations 2x + 3Y -z= 20)

Discussion

1402.	int e^(x) (x -1)/((x + 2)^(4))dx =
Answer» `(E^(x))/((x + 2)^(3)) + C ` `(e^(x))/((x + 2)^(4)) +c ` `-(e^(x))/((x + 2)^(3)) +c ` `e^(x) (x - 1)/((x + 2)^(3)) +c ` ANSWER :A

Discussion

1403.	Find two positive numbers x and y such that their sum is 35 and the product x^(2)y^(5) is a maximum.
Answer» ANSWER :10 and 25

Discussion

1404.	int (x)/(sqrt(1- x^(2)) cos^(2) sqrt(1 - x^(2)))dx =
Answer» TAN `sqrt(1 - X^(2)) ` + C `- tan sqrt(1-x^(2)) + c ` `cot sqrt(1 -x^(2))` + c `- cot sqrt(1 - x^(2)) + c ` Answer :B

Discussion

1405.	Let A and B be the ends of the diameters 4x-y-15=0 of the circle x^(2)+y^(2)-6x-6y-16=0. A and B lie on tangents at the end point on the major axis of an ellipse such that line joining A and B is a tangent to the same ellipse at a point P. If the equation of major axis of the ellipse is y=x then the distance between the foci is
Answer» `2sqrt(2)` `4sqrt(2)` `8` `2sqrt(3)` SOLUTION :Solve `y=x` with circle Circle MUST passes through the FOCI of the ellipse

Discussion

1406.	Find a unit vector perpendicualr to the plane which passes through the point P(1,-1,2),Q(2,0,-1) and R(0,2,1).
Answer» ANSWER :`(1)/(SQRT(6))(2hati+hatj+hatk)`

Discussion

1407.	Find the co-ordinates of the point on the parabola y^(2)=8x whose focal distance is 10.
Answer» ANSWER :`(8, PM 8)`

Discussion

1408.	Given that E and F are events such that P(E) = 0.6,P(F)= 0.3 and P(EnnF)=0.2.Find P (E/F).
Answer» <P> SOLUTION :P(E/F)=(P`(ENNF))/(P(F))`=0.2/0.3=2/3

Discussion

1409.	A double decker bus has 15 seats in the lower deck and 13 seats in the upper deck. In how many ways can a marriage party of 28 persons be arranged if 4 old people refuse to go to the upper deck and 4 children wish to travel in the upper deck only.
Answer» ANSWER :`""^(20)C_9xxlfloor15xxlfloor13`

Discussion

1410.	Using elementary row transformations , find the inverse of [{:(3,10),(2,7):}]
Answer» ANSWER :`thereforeA^(-1)=[{:(7,-10),(-2,3):}]`

Discussion

1411.	Find the area of one of the curvilinear triangles bounded by y=sinx,y=cosx and X-axis.
Answer» D, C, B, A B, C, D, A C, D, A, B D, A, B, C Answer :A

Discussion

1412.	Minimise z = 13x - 15y subjectto the constraints : x+y le 7, 2x-3y+6 ge 0, x ge 0, y ge 0
Answer» Answer :Minimum value of objective function Z = 13X -15Y is -30 for point (0, 2)

Discussion

1413.	Evaluate the following integrals: int_2^3 1/x dx
Answer» SOLUTION :`int_2^3 1/x DX = [LOG\|x\|]_2^3` =`log\|3\|-log\|2\|` ` =log3-log2 ` `=log(3/2)`

Discussion

1414.	If x, y, z are in A.P then 1/(sqrt(x) + sqrt(y)) , 1/(sqrt(z) + sqrt(x)) , 1/(sqrt(y) + sqrt(z)) are in
Answer» A.P G.P H.P A.P and H.P Answer :A

Discussion

1415.	int (x^(4)+1)/(1+x^(6))dx=
Answer» `Tan^(-1) (X)-Tan^(1) x^(3))+C` `Tan^(-1) (x)-(1)/(3)Tan^(-1) (x^(3))+c` `Tan^(-1) (x)+Tan^(-1) (x^(3))+c` `Tan^(-1) (x)+(1)/(3)Tan^(-1) (x^(3))+c` Answer :D

Discussion

1416.	A : Ifx^(2) + y^(2) - 2x + 3y + k = 0 , x^(2) + y^(2) + 8x 0 6y - 7 = 0cut each other orthogonally then k = 10 . R : The circlesx^(2) + y^(2) + 2gx + 2fy + c = 0 , x^(2) + y^(2) + 2g'x + 2f'y + c' = 0cut each other orthogonally iff2gg' + 2ff'= c + c'
Answer» Both A and R are true and R is the correct EXPLANATION of A Both A and R are true but R is not CORRCET explanation of A A is true but R is FALSE A is false but R is true . Answer :A::D

Discussion

1417.	Prove that the orthocenter of the triangle formed by any three tangents to a parabola lies on the directrix of the parabola.
Answer» ANSWER :`x=a(t_(1)-t_(3))` IE,x=-a

Discussion

1418.	An ellipse hase semi-major of length 2 and semi-minor axis of length 1. It slides between the coordinates axes in the first quadrant while mantaining contact with both x-axis and y-axis. The locus of the centre of the ellipse is
Answer» `X^(2)+y^(2)=8` `(x^(2))/(9)+(y^(2))/(4)=1` `y^(2)=4X` `x^(2)+y^(2)=5` Solution :Let the centre of the ellipse be C(x,y) Clearly, axeas are tangents to the ellipse to the ellipse which are perpendicular and intesect at origin O. So, origin lies on the director CIRCLE Thus, origin lies on the director circle. OC is radius of he director circle . So, `OC^(2)=a^(2)+b^(2)=22+12` or `x^(2)+y^(2)=5`

Discussion

1419.	If the normals at four points (x_(1),y_(1)),(x_(2),y_(2)), (x_(3),y_(3)) and (x_(4),y_(4)) on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 are concurrent then sumcosalpha.sumsecalpha is (where alpha, beta, gamma, delta are the eccentric angles of the points):
Answer» 1 2 3 4 Solution :d `(a^(2)-b^(2))x^(4)-2ha^(2)(a^(2)-b^(2))x^(3)-x^(2)+2a^(4)H(a^(2)-b^(2))-a^(6)h^(2)=0implies(sumx_(1))(sum1/(x_(1)))=4`

Discussion

1420.	Two dice A and B are rolled. If it is known that the number on B is 5, then the probability that the sum of the numbers on the two dice will be greater than 9 is
Answer» `1/3` `1/4` `1/5` `1/2` ANSWER :A

Discussion

1421.	The system of equations {:(kx+(k+1)y+(k-1)z=0),((k+1)x+ky+(k+2)z=0),((k-1)x + (k+2)y+kz=0):} has a nontrivial solution for :
Answer» Exactly three REAL value of k Exactly two real VALUES of k Exactly one real value of k Infinite NUMBER of values of k ANSWER :C

Discussion

1422.	If the system [(2,8),(3,7)][(a),(b)]= k [(a),(b)] has non-trivial, solution then the positive value of k and a solution of the system for that value of k are
Answer» `9, [(3),(-8)]` `10, [(-8),(3)]` `6, [(-1),(-1)]` `10, [(1),(1)]` Answer :D

Discussion

1423.	There are 10 greetings cards, each of a different colour and 10 envelopes of the saem ten colours. The number of ways in which the cards can go to the envelopes such that exactly 6 cards go into the envelopes of the corresponding colour is
Answer» 2520 151200 1890 None of these Answer :C

Discussion

1424.	ABC is a triangle with vertices A(-1,4), B(6,-2),and C(-2, 4). D, E and F are the points which divide each AB, BC, and CA respectively in the ratio 3:1 internally. Then the centroid of the triangle DEF is
Answer» (3, 6) (1, 2) (4, 8) (-3, 6) Answer :B

Discussion

1425.	If the events A and B are independent, then P(A cap B) is equal to ……..
Answer» <P>`P(A) + P(B)` `P(A) - P(B)` `P(A) * P(B)` `P(A) // P(B)` ANSWER :C

Discussion

1426.	bar(x)=(2,3) and bar(y)=(5,-2) are ………….. Vectors.
Answer» COLLINEAR NON collinear co DIRECTION OPPOSITE direction ANSWER :B

Discussion

1427.	Which of the following is minimize the objective function P=2x/5+3x/7 is
Answer» (5,0) (0,5) (5/2,7/3) (2/5,3/7) ANSWER :D

Discussion

1428.	If a and b arc rcul numbers between O and 1 such that z_(1) = a + i, z_(2) = 1 + bi, z_(3) = 0 form an equilateral triangle, then
Answer» `a=sqrt2-1` `b=sqrt-1` `a=2-sqrt3` `b=2-sqrt3` ANSWER :C::D

Discussion

1429.	Evaluate:int sin ^(-1) sqrt((x)/(x+a)) dx
Answer» ANSWER :` TAN ^(-1) sqrt((X)/(a)),(a+ x)- sqrt(ax)+c`

Discussion

1430.	If y= e^(4x) + e^(-3x) satisfies the relation (d^(3)y)/(dx^(3)) + A (dy)/(dx) + By=0 then A and B respectively are
Answer» 12,13 `-12,13` `12,-13` `-13,-12` ANSWER :D

Discussion

1431.	The standard deviations of two sets containing 10 and 20 members are 2 and 3 respectively measured from their common mean 5. Find the S.D. for the whole set of 30 members.
Answer» ANSWER :`SQRT(((22)/(3)))`

Discussion

1432.	Evaluate the integrals by using substitution int_(1)^(2)(1/x-1/(2x^(2)))e^(2x)dx
Answer» ANSWER :`(E^(2)(e^(2)-2))/4`

Discussion

1433.	If 1,omega,omega^(2) are three cube roots of unity, then (1-omega+omega^(2))(1+omega-omega^(2)) is
Answer» 2 4 1 3 Answer :B

Discussion

1434.	Given pair of lines 2x^(2) +5xy +2y^(2) +4x +5y +a = 0 and the line L: bx +y +5 = 0. Then
Answer» `a = 2` `a =- 2` There exists no circle which touches the pair of lines and the line L if `b = 5`. There exists no circle which touches the pair of lines and the line L if `b =- 5` Solution :`2x^(2) +5xy +2Y^(2) +4x 5Y +a =0` represents the pair of lines if `\|{:(2,5//2,2),(5//2,2,5//2),(2,5//2,a):}\| =0` `rArr a = 2` So pair of lines is `2x^(2)+5xy +2y^(2) +4x +5y +2 =0` or `x +2y +1 = 0, 2x +y +2 =0` These lines are concurrent with `BX +y +5 =0` If `\|{:(1,2,1),(2,1,2),(b,1,5):}\| =0` `rArr b =5` Which lines are concurrent, no circle can be drawn touching all THREE lines.

Discussion

1435.	A metal box with a square base and vertical sides is to contain 1024 cm^(3). The materical for the top and bottom costs Rs. 5//cm^(2) and the material for the sides costs Rs. 2.50//cm^(2). Find the least cost of the box.
Answer» ANSWER :1920 RS.

Discussion

1436.	If C(x)=3x ((x+7)/(x+5))+5 is the total cost of production of x units a certain product, then then marginal cost
Answer» A) FALLS continuously as the OUTPUT X increases B) Increases continuously as the output x increase C) Neither increases nor decreases as the output x increases D) None of these Answer :A

Discussion

1437.	lim_(x to pi//2) (cos x)/(root(3)((1-sinx)) is equal to
Answer» 1 2 `21//3` NONE of these ANSWER :D

Discussion

1438.	Find the equations of the tangent and normal to the curve y = 1/(1 + x^2) at (0,1).
Answer» ANSWER :1,0

Discussion

1439.	Evaluate the following lim_(xto2)(1/x^2-1/4)/(x-2)
Answer» SOLUTION :`lim_(xto2)(1/x^2-1/4)/(x-2)` `=lim_(xto2)(4-x^2)/((x-2)4x^2)` `=lim_(xto2)(-(x+2))/(4x^2)=-4/16=-1/4`

Discussion

1440.	Differentiate the following w.r.t. x : log((x+3)+sqrt(x^(2)+6x+3))
Answer» ANSWER :`1/sqrt(X^(2)+6x+3)`

Discussion

1441.	Three dice are rolled and told that exactly two of them are showing the same number. The probability of getting sum 16 is
Answer» `(1)/(14)` `(1)/(15)` `(1)/(17)` `(1)/(16)` Answer :B

Discussion

1442.	Find approximate values of the following : (1.99)^7
Answer» Solution :Let `y = X^7` Then `dy = 7x^6 dx rArr deltay = 7x^6 DELTAX` `rArr (x + deltax)^7 - x^7 = 7x^6 deltax` `rArr (x + deltax)^7 = x^7 + 7x^6 CDOT deltax` Put x = 2 and `deltax = -0.01` Then `(1.99)^7 = 2^7 - 7 XX 2^6 xx 0.01` ` = 128 - 7 xx 64 xx 0.01` = `128 - 4.48 = 123 cdot 52`

Discussion

1443.	Match the following
Answer» `(-infty , COT 3)` `[CO 3, COT2]` `[cot 3, infty]` `none of these ANSWER :(A) (q) (B)(s)(C )(R )(D)(p)

Discussion

1444.	Match the following
Answer» `(cos1,1)` `(-1,cos 1)` `SIN 1,1)` `(cos 1, sin 1)` ANSWER :(A)(r )(B)(s)(C )(s)(D)(Q)

Discussion

1445.	If A=[{:(3,1),(-1,2):}], show that A^(2)-5A+7I=O.
Answer» ANSWER :`=[{:(0,0),(0,0):}]=O=R.H.S`.

Discussion

1446.	Let f(x)={a^2+a+1}+b "cosec"^(-1)x where a in l and b > 0.If range of f(x) is [p, r) uu(r, q], then thevalue of p^2/(q^2+r^2) equals :-
Answer» 1 2 3 4 Solution :RANGE is `[-(BPI)/2,0) UU (0,(bpi)/2]`

Discussion

1447.	intdx/(1+tanx)
Answer» SOLUTION :`I=intdx/(1+tanx)` =`INT(cosxdx)/(sinx+cosx)` =`1/2int((cosx-sinx)+(cosx+sinx))/(sinx+cosx)DX` =`1/2int(cosx-sinx)/(sinx+cosx)dx+1/2intdx` =`1/2Inabs(cosx-sinx)+1/2x+c` `because d/dx(sinx+cosx)=cosx-sinx`

Discussion

1448.	squareABCD is a parallelogram vec(AB)=bar(q),vec(AD)=bar(p),angleBAC is on acute angle. From the point B, the perpendicular is drawn on side vec(AD). The vector along with it is vec( r ). Then vec( r ) = …………. .
Answer» `VEC( r )=3bar(Q)-(3(bar(p).bar(q)))/((bar(p).bar(p)))bar(p)` `vec( r )=-bar(q)+((bar(p).bar(q)))/((bar(p).bar(p)))bar(p)` `vec( r )=bar(q)-((bar(p).bar(q)))/((bar(p).bar(p)))bar(p)` `vec( r )=-3bar(q)+(3(bar(p).bar(q)))/((bar(p).bar(p)))bar(p)` ANSWER :B

Discussion

1449.	intsec^(3)xdx
Answer» ANSWER :`(1)/(2)[secxtanx+log\|secx+tanx\|]+C`

Discussion

1450.	If sin(A+iB)=x+iy then ((x^2)/(sin^2A)-(y^2)/(cos^2A))+((x^2)/(cosh^2B)+(y^2)/(sinh^2B))=
Answer» 1 2 3 4 Answer :B

Discussion

Explore topic-wise InterviewSolutions in .

Find the vector and cartesian equations of the plane which passes through the point (5, 2, - 4) and perpendicular to the line with direction ratios 2, 3, -1.

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

Find two positive numbers x and y such that their sum is 35 and the product x^(2)y^(5) is a maximum.

int (x)/(sqrt(1- x^(2)) cos^(2) sqrt(1 - x^(2)))dx =

Find a unit vector perpendicualr to the plane which passes through the point P(1,-1,2),Q(2,0,-1) and R(0,2,1).

Find the co-ordinates of the point on the parabola y^(2)=8x whose focal distance is 10.

Given that E and F are events such that P(E) = 0.6,P(F)= 0.3 and P(EnnF)=0.2.Find P (E/F).

A double decker bus has 15 seats in the lower deck and 13 seats in the upper deck. In how many ways can a marriage party of 28 persons be arranged if 4 old people refuse to go to the upper deck and 4 children wish to travel in the upper deck only.

Using elementary row transformations , find the inverse of [{:(3,10),(2,7):}]

Find the area of one of the curvilinear triangles bounded by y=sinx,y=cosx and X-axis.

Minimise z = 13x - 15y subjectto the constraints : x+y le 7, 2x-3y+6 ge 0, x ge 0, y ge 0

Evaluate the following integrals: int_2^3 1/x dx

If x, y, z are in A.P then 1/(sqrt(x) + sqrt(y)) , 1/(sqrt(z) + sqrt(x)) , 1/(sqrt(y) + sqrt(z)) are in

int (x^(4)+1)/(1+x^(6))dx=

A : Ifx^(2) + y^(2) - 2x + 3y + k = 0 , x^(2) + y^(2) + 8x 0 6y - 7 = 0cut each other orthogonally then k = 10 . R : The circlesx^(2) + y^(2) + 2gx + 2fy + c = 0 , x^(2) + y^(2) + 2g'x + 2f'y + c' = 0cut each other orthogonally iff2gg' + 2ff'= c + c'

Prove that the orthocenter of the triangle formed by any three tangents to a parabola lies on the directrix of the parabola.

An ellipse hase semi-major of length 2 and semi-minor axis of length 1. It slides between the coordinates axes in the first quadrant while mantaining contact with both x-axis and y-axis. The locus of the centre of the ellipse is

If the normals at four points (x_(1),y_(1)),(x_(2),y_(2)), (x_(3),y_(3)) and (x_(4),y_(4)) on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 are concurrent then sumcosalpha.sumsecalpha is (where alpha, beta, gamma, delta are the eccentric angles of the points):

Two dice A and B are rolled. If it is known that the number on B is 5, then the probability that the sum of the numbers on the two dice will be greater than 9 is

The system of equations {:(kx+(k+1)y+(k-1)z=0),((k+1)x+ky+(k+2)z=0),((k-1)x + (k+2)y+kz=0):} has a nontrivial solution for :

If the system [(2,8),(3,7)][(a),(b)]= k [(a),(b)] has non-trivial, solution then the positive value of k and a solution of the system for that value of k are

There are 10 greetings cards, each of a different colour and 10 envelopes of the saem ten colours. The number of ways in which the cards can go to the envelopes such that exactly 6 cards go into the envelopes of the corresponding colour is

ABC is a triangle with vertices A(-1,4),﻿ B(6,-2),and C(-2, 4). D, E and﻿ F are the points which divide each AB, BC,﻿ and CA respectively in the ratio 3:1﻿ internally. Then the centroid of the triangle﻿ DEF is﻿

If the events A and B are independent, then P(A cap B) is equal to ……..

bar(x)=(2,3) and bar(y)=(5,-2) are ………….. Vectors.

Which of the following is minimize the objective function P=2x/5+3x/7 is

If a and b arc rcul numbers between O and 1 such that z_(1) = a + i, z_(2) = 1 + bi, z_(3) = 0 form an equilateral triangle, then

Evaluate:int sin ^(-1) sqrt((x)/(x+a)) dx

If y= e^(4x) + e^(-3x) satisfies the relation (d^(3)y)/(dx^(3)) + A (dy)/(dx) + By=0 then A and B respectively are

The standard deviations of two sets containing 10 and 20 members are 2 and 3 respectively measured from their common mean 5. Find the S.D. for the whole set of 30 members.

Evaluate the integrals by using substitution int_(1)^(2)(1/x-1/(2x^(2)))e^(2x)dx

If 1,omega,omega^(2) are three cube roots of unity, then (1-omega+omega^(2))(1+omega-omega^(2)) is

Given pair of lines 2x^(2) +5xy +2y^(2) +4x +5y +a = 0 and the line L: bx +y +5 = 0. Then

A metal box with a square base and vertical sides is to contain 1024 cm^(3). The materical for the top and bottom costs Rs. 5//cm^(2) and the material for the sides costs Rs. 2.50//cm^(2). Find the least cost of the box.

If C(x)=3x ((x+7)/(x+5))+5 is the total cost of production of x units a certain product, then then marginal cost

lim_(x to pi//2) (cos x)/(root(3)((1-sinx)) is equal to

Find the equations of the tangent and normal to the curve y = 1/(1 + x^2) at (0,1).

Evaluate the following lim_(xto2)(1/x^2-1/4)/(x-2)

Differentiate the following w.r.t. x : log((x+3)+sqrt(x^(2)+6x+3))

Three dice are rolled and told that exactly two of them are showing the same number. The probability of getting sum 16 is

Find approximate values of the following : (1.99)^7

Match the following

Match the following

If A=[{:(3,1),(-1,2):}], show that A^(2)-5A+7I=O.

Let f(x)={a^2+a+1}+b "cosec"^(-1)x where a in l and b > 0.If range of f(x) is [p, r) uu(r, q], then thevalue of p^2/(q^2+r^2)​ equals :-

intdx/(1+tanx)

squareABCD is a parallelogram vec(AB)=bar(q),vec(AD)=bar(p),angleBAC is on acute angle. From the point B, the perpendicular is drawn on side vec(AD). The vector along with it is vec( r ). Then vec( r ) = …………. .

intsec^(3)xdx

If sin(A+iB)=x+iy then ((x^2)/(sin^2A)-(y^2)/(cos^2A))+((x^2)/(cosh^2B)+(y^2)/(sinh^2B))=

ABC is a triangle with vertices A(-1,4), B(6,-2),and C(-2, 4). D, E and F are the points which divide each AB, BC, and CA respectively in the ratio 3:1 internally. Then the centroid of the triangle DEF is

Let f(x)={a^2+a+1}+b "cosec"^(-1)x where a in l and b > 0.If range of f(x) is [p, r) uu(r, q], then thevalue of p^2/(q^2+r^2) equals :-