28278 + Interview Questions in Maths in Class 12 Page 56 InterviewSolution

2751.	If A=[{:(1,3),(3,4):}]andA^(2)=kA=5I=O,then k =....
Answer» 3 7 5 9 Answer :C

Discussion

2752.	Evaluate int e^(-2x) [ 2" tan x " - sec^(2) x ]dx
Answer» ANSWER :`-e^(-2x)` TAN X + C

Discussion

2753.	(x+1) sqrt(2x^(2)+3)
Answer» Answer :`:. I=(1)/(6)(2x^(2)+3X)^(2)+(x)/(2) sqrt(2x^(2)+3)+(3sqrt(2))/(4)log\|x+ sqrt(x^(2)+(3)/(2))\|+c`

Discussion

2754.	If the plane 3x - 2y - z - 18 = 0meets the coordinates axes in A,B,C then the centroid of triangle ABC is
Answer» `(2,3,-6)` `(2,-3,6)` `(-2,-3,6)` `(2,-3,-6)` ANSWER :D

Discussion

2755.	Let S = {a,b,c} and T = {1,2,3} . Find F^(-1) of the following functions F from S to T , if it exists . F = {(a,3),(b,2),(c,1)}
Answer» SOLUTION :N/A

Discussion

2756.	If A={:[(3,-2),(4,-2)], find k such that A^(2)=kA-2I_(2)
Answer» ANSWER :k=1

Discussion

2757.	Obtain the general solution of the following differential equations.tan y dx+cot x dy=0
Answer» Solution :TAN y dx+cot X DY=0 `rArr tan xcdotdx+cot y dy =0` `rArrint tan xdx+intcot y dy =0` `-In COS x +In sin y =In C` `In siny/cosx=InC` `rArr siny/cosx=C rArr sin y=C cosx`

Discussion

2758.	Differentiatex^3-5x
Answer» SOLUTION :LET `y= x^3-5x` Then `dy/dx=3x^2-5`

Discussion

2759.	A particle moves along the curve 6y= x^(3)+2 find the points on the curve at which the y-coorinate is changing 8 times as fast as the x- coordinate.
Answer» ANSWER :(4,11) and `(-4, (-31)/(3))`

Discussion

2760.	Let z_(1),z_(2),z_(3) be the three nonzero comple numbers such thatz_(2) ne 1, a= \|z_(1)\|, b = \|z_(2)\| and c= \|z_(3)\|. Let \|{:(,a,b,c),(,b,c,a),(,c,a,b):}\|= 0 Then
Answer» `arg((z_(3))/(z_(2))) = arg((z_(3)-z_(1))/(z_(2)-z_(1)))` ortho centre of triangleformed by `z_(1),z_(2),z_(3)` is `z_(1)+z_(2)+z_(3)` if trinagleformed by `z_(1),z_(2),z_(3) ` is `z_(1)+z_(2)+z_(3)` is `(3sqrt(3))/(2)\|z_(1)\|^(2)` if triangleformed by `z_(1),z_(2),z_(3)` is equlateral, then `z_(1)+z_(2)+z_(3)=0` Solution :Given `\|{:(,a,b,c,),(,b,c,a,),(,c,a,b,):}\|=0` `rArr a^(3) + b^(3) + c^(3) - 3abc = 0` `rArr (a+b+c)(a^(2) + b^(2) + c^(2) - ab -ab-ca)=0` `rArr (1)/(2)(a+b+c)[(a-b)^(2) + (c-a^(2)) = 0` `rArr (a-b)^(2) + (b-c)^(2) + (c-a)^(2) = 0` `rArr a=b=c [ therefore a+b + c ne 0, because z_(1) ne 0, therefore\|z_(1)\| = a ne 0 etc.]` Hence, `OA= OB = OC`, where O is the ORIGIN and A, B, C arethe points represeniting `z_(1),z_(2)` and `z_(3)` respectively . Therefore ,O is circumcenter of `Delta ABC `. Now `arg((z_(3))/(z_(2)))= /_BOC""(1)` `=2 /_BAC = 2 arg((z_(3)-z_(1))/(z_(2)-z_(1))) ""(2)` `arg ((z_(3)-z_(1))/(z_(2)-z_(1)))""[because /_BOC = 2/_ BAC]` Hence, `arg((z_(3))/(z_(2))) =arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2)` Also, centroid is `(z_(1) +z_(2) +z_(3))//3`. Since HG: GO -= 2:1 (where H isorthocenter and G is centroid),then orthocentre is `z_(1) + z_(2) +z_(3)` (by sectionformula). When triangle is equilateral centroid conicides with circumcenter, hence `z_(1) + z_(2) +z_(3)= 0` Also, the area for equilateral triangle is `(sqrt(3)//4L)`, where Lis lenghtof side. Since radius is `\|z_(1)\|,L = sqrt(3)\|z_(1)\|`, the area is `(3sqrt(3)//4)\|z_(1)\|`

Discussion

2761.	Show that the height of the cylinder of maximum volumethat can be inscribed in a sphere of radius Ris(2R)/(sqrt(3)). Also find the maximumvolume.
Answer» ANSWER :`(4PI R^(3))/(3sqrt(3))`

Discussion

2762.	If (1)/(n+1)+(1)/(2(n+1)^(2))+(1)/(3(n+1)^(3))+….= lambda((1)/(n)-(1)/(2n^(2))+(1)/(3n^(3))-……) then lambda=
Answer» 2 `n+1` 1 n Answer :C

Discussion

2763.	underset0overset(pi)intcos^2xdx=_______
Answer» SOLUTION :`I=underset0overset(pi//2)intcos^5xdx=4/5 2/3=8/15`

Discussion

2764.	Find by integration the area of the trapezoid bounded gy y= 2x+ 1,y= 0, x =2 andx=4 Verify your result by finding the area of a trapezoid as the product of half the sum of the two parallel sides and the distance between them.
Answer» ANSWER :` = 14 `SQUARE UNITS

Discussion

2765.	Find derivatives of the following functions.e^(sin t)
Answer» SOLUTION :`y + E^(SIN t) dy/dx = d/dt (e^sin t).d/dt(sin t) =e^(sin t). COS t`

Discussion

2766.	5 dice are thrown then the number of out comes in which there is atleast one three is
Answer» `6^(5)-1` `6^(5)-5^(5)` `6^(5)-6.5^(4)` `5^(5)-1` Answer :B

Discussion

2767.	LetP(n): 2^(n +2) lt 3 ^n , is truefor
Answer» `N in N` ` n gt n , AAnin N` `n gt2, AA n in N` Noneof these Answer :B

Discussion

2768.	The total revenue received from the sale of x units of a product is given by R(x) = 20 x - 0.5x^(2). Find Difference of average revenue and marginal revenue when x = 10
Answer» ANSWER :₹ 5

Discussion

2769.	The set of all values of lambda for which the system of linear equation \|(2x_(1)-2x_(2)+x_(3),=lambdax_(1)),(2x_(1)-3x_(2)+2x_(3),=lambdax_(2)),(-x(1)+2x_(2),=lambdax_(3))\| hasa non trivial solution
Answer» is an empty set is a SINGLETON CONTAINS TWO ELEMENTS contains more than two elements . Answer :C

Discussion

2770.	S = n xx 10^(6m) + n xx 10^(6m + 1) + n xx 10^(6m-2)+ "…." + nxx 10 +n - 10 m in I, n = single digit +ve integer. Then the remainder if (1) S is divided by 13 is a , (2) ItS is divided by 11 is b , (3)It S is divided by 7 is c a+b+c+ =
Answer» 5 6 7 8 Answer :A

Discussion

2771.	A rod AB of length 3 units moves vertically with its bottom B always an the circle x^(2)+y^(2)=25 then the equation of the locus of A is
Answer» `X^(2)+(y+3)^(2)=25` `(x-3)^(2)+y^(2)=25` `(x+3)^(2)+y^(2)=25` `x^(2)+(y-3)^(2)=25` ANSWER :D

Discussion

2772.	cos^(-1)((1-x^(2))/(1+x^(2)))= k tan^(-1)xthen k is :
Answer» `-1` 2 `-2` 1 Answer :B

Discussion

2773.	int(1)/(x^(2)+(a+b)x+ab)dx=
Answer» `(1)/(B-a)LOG[(x+b)/(x+a)]+c` `(1)/(b-a)log[(x+a)/(x+b)]+c` `(1)/(AB)tan^(-1)[(AX)/(b)]+c` `(1)/(ab)tan^(-1)[(bx)/(a)]+c` ANSWER :B

Discussion

2774.	Let P be a variable point on the ellipsewith foci S_(1) and S_(2) . If A be the area of trianglePS_(1)S_(2)then findthe maximumvalueof A
Answer» ANSWER :ABE

Discussion

2775.	xdy-ydx=(x^(2)-3xy+2y^(2))dx=0
Answer» ANSWER :`y=xtan(x+c)`

Discussion

2776.	Find int(2x^(2)+e^(x))dx.
Answer» ANSWER :`(2X^(3))/3+e^(X)+C`

Discussion

2777.	If A and B are two square matrices of the same order then which of the following is true.
Answer» (AB)'= A'B' (AB)'=B'A' \|AB\|=0 `IMPLIES \|A\|=0` and \|B\|=0 none of these ANSWER :B

Discussion

2778.	Let A=[(1,1,0),(0,1,0),(0,0,1)],and let l denote the 3 xx 3 identity matrix. Then 2A^(2)-A^(3)=……....
Answer» `A+1` `A-1` `l-A` A Solution :`underset(Lrarr0)(Lt)[(X)/(SINX)]=1" and "underset(xrarr0)(Lt)[(sin^(-1)(x))/(x)]=1 " and "underset(xrarr0)(Lt)[(TAN^(-1)x)/(x)]=0`

Discussion

2779.	If x = cisA, y=cisB then find the value of cos(A+B) interms of x and y.
Answer» ANSWER :`1/2(XY+(1)/(xy))`

Discussion

2780.	Refer to question 11. How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximise his profit? Determine the maximum profit.
Answer» ANSWER :Maximum value is 480 COMPANY has to produce 6 and 3 CIRCUITS of the TYPE A and B

Discussion

2781.	If the equationax^(4)+bx^(3)+cx^(2)+kx=0 has a rootalpha gt 0, then the equation4ax^(3)+3bx^(2)+2cx+k=0 has -
Answer» ONE negative and two POSITIVE roots at LEAST one ROOT in `(0, alpha)` no root in `(0, alpha)` none of these ANSWER :B

Discussion

2782.	int_0^(pi/2)sin^10thetad theta
Answer» SOLUTION :`int_0^(pi/2)sin^10thetad theta=9/10 CDOT 7/8 cdot 5/6 cdot 3/4 cdot 1/2 cdot pi/2=(405pi)/7680`

Discussion

2783.	The normal to a curve at P(x,y) meets the x-axis at G. If the distance of G from the origin is twice the abscisa of P, then the curve is a
Answer» Ellipse Parabola Circle Hyperbola Answer :D

Discussion

2784.	A body is projected from ground with speed 20 m/s making an angle of 45^(@) with horizontal. The equation of path is h = Ax-Bx^(2), where h is height, x is horizontal distance, A and B are constant. The ratio A : B is :- (g = 10 m//s^(2))
Answer» 1 : 5 5 : 1 1 : 40 40 : 1 Answer :A

Discussion

2785.	For what value of lambda the equation of conic 2xy+4x-6y+ lambda=0 represents two intersecting straight lines ? If lambda= 17 then does this equation represents ?
Answer» Answer :the given equation REPRESENTS a HYPERBOLA for `LAMBDA = 17`

Discussion

2786.	Find the number of rectangles from a rectangles of size 10xx9. Also find the number of square with size 4xx4. Also find the total number of rectangles excluding squares.
Answer» ANSWER :2145

Discussion

2787.	IFxin Rthen( x^2+ 2x+ a) /(x^2+ 4x + 3a )cantakeall realvaluesif
Answer» `a in(0,2)` `a in[0,1]` `a in[-1,1]` none Answer :B

Discussion

2788.	Water is being poured into a vessel, in the form of an inverted right circular cone of semi-vertical angle 45^(@), in sucha way that the rate of change of volume at any time is proportional to the area of the curved surface which is wet at that time. Intially, the vessel is full to a height to 2 cm and after 2 seconds, the height is 10 cm. After 3.5 seconds, the height is
Answer» 18 cm 16 cm 14 cm 12 cm Answer :B

Discussion

2789.	Solve the differential equations: (dy)/(dx) + y secx = tanx0 le x le pi/2
Answer» ANSWER :`SECX + TANX -X + C`

Discussion

2790.	If the position vectors of three points A, B and Care respectivelyveci+vecj+3veck,4veci+vecj+5veckand7(veci+veck). Findvec(AB)xxvec(AC).. Interpret the result geometrically.
Answer» ANSWER : ` VEC0 `

Discussion

2791.	If A=[[1,2],[-2,3]]B=[[3,2],[1,4]],C=[[2,2],[1,3]]Calculate AB.
Answer» SOLUTION :`AB=[[1,2],[3,4]][[3,2],[1,4]]` `=[[1.3 +2.1""1.1+2.1],[2.3+1.1""2.1+1.1]]=[[5,3],[7,3]]`

Discussion

2792.	Find the solution of the equation [x] + {-x} = 2x, (where [] and {} represents greatest integer function and fractional part function respectively.
Answer»

Discussion

2793.	int_(0)^(pi//2)(1)/(4 cos^(2)x+9sin^(2)x)dx=
Answer» `pi/4` `pi/8` `pi/12` `pi/16` ANSWER :C

Discussion

2794.	Consider a parabola P touches coordinate axes at (4,0) and (0,3). Equation of directrix of parabola P is
Answer» 4x+3y=0 3x+4y=12 3x+4y=0 4x+3y=12 Answer :C

Discussion

2795.	If int (1)/(x sqrt(1 - x^(3))) " dx = a log" \| (sqrt(1- x^(3)) - 1)/(sqrt(1 -x^(3) )+ 1) \| + bthen a =
Answer» `(2)/(3)` `(1)/(3)` `- (2)/(3)` `- (1)/(3)` ANSWER :B

Discussion

2796.	Form a differential equation representing the given family of curves by eliminating arbitrary constants a and by = ae^(3x) + be^(-2x)
Answer» ANSWER :`y " - y' - 6Y = 0`

Discussion

2797.	int(sin2x/(acos^(2)x)+bsin^(2)x)dx =
Answer» `1/B-a log\|acos^(2)X - BSIN^(2)x\| + c` `1/b-a log\|acos^(2)x + bsin^(2)x\| + c` `1/b+a log\|acos^(2)x + bsin^(2)x\| + c` `1/b+a log\|acos^(2)x - bsin^(2)x\| + c` ANSWER :B

Discussion

2798.	If f(x)={(1,-,kx, x,le3),(2x,+,3,x,>3):} is a continuous function, then the value of k is
Answer» `8/3` `-8/3` `3/8` NONE of these Answer :B

Discussion

2799.	Evaluate the following determinants: [[a-b,b-c,c-a],[x-y,y-z,z-x],[p-q,q-r,r-p]]
Answer» SOLUTION :`[[a-b,b-c,c-a],[x-y,y-z,z-x],[p-q,q-r,r-p]]` =`[[0,b-c,c-a],[0,y-z,z-x],[0,q-r,r-p]](C_1=C_1+C_2+C_3)` =0`(THEREFORE C_1=0)`

Discussion

2800.	The position vectors of A, B and C are 2hati-hatj+hatk,hati-3hatj-5hatk and x hati-3hatj+hatk respectively in DeltaABC. If angleC=(pi)/(2) then the value of x is ……………..
Answer» `-2` and `-1` `-2` and 1 2 and `-1` 2 and 1 Answer :D

Discussion

Explore topic-wise InterviewSolutions in .

If A=[{:(1,3),(3,4):}]andA^(2)=kA=5I=O,then k =....

Evaluate int e^(-2x) [ 2" tan x " - sec^(2) x ]dx

(x+1) sqrt(2x^(2)+3)

If the plane 3x - 2y - z - 18 = 0meets the coordinates axes in A,B,C then the centroid of triangle ABC is

Let S = {a,b,c} and T = {1,2,3} . Find F^(-1) of the following functions F from S to T , if it exists . F = {(a,3),(b,2),(c,1)}

If A={:[(3,-2),(4,-2)], find k such that A^(2)=kA-2I_(2)

Obtain the general solution of the following differential equations.tan y dx+cot x dy=0

Differentiatex^3-5x

A particle moves along the curve 6y= x^(3)+2 find the points on the curve at which the y-coorinate is changing 8 times as fast as the x- coordinate.

Let z_(1),z_(2),z_(3) be the three nonzero comple numbers such thatz_(2) ne 1, a= |z_(1)|, b = |z_(2)| and c= |z_(3)|. Let |{:(,a,b,c),(,b,c,a),(,c,a,b):}|= 0 Then

Show that the height of the cylinder of maximum volumethat can be inscribed in a sphere of radius Ris(2R)/(sqrt(3)). Also find the maximumvolume.

If (1)/(n+1)+(1)/(2(n+1)^(2))+(1)/(3(n+1)^(3))+….= lambda((1)/(n)-(1)/(2n^(2))+(1)/(3n^(3))-……) then lambda=

underset0overset(pi)intcos^2xdx=_______

Find by integration the area of the trapezoid bounded gy y= 2x+ 1,y= 0, x =2 andx=4 Verify your result by finding the area of a trapezoid as the product of half the sum of the two parallel sides and the distance between them.

Find derivatives of the following functions.e^(sin t)

5 dice are thrown then the number of out comes in which there is atleast one three is

LetP(n): 2^(n +2) lt 3 ^n , is truefor

The total revenue received from the sale of x units of a product is given by R(x) = 20 x - 0.5x^(2). Find Difference of average revenue and marginal revenue when x = 10

The set of all values of lambda for which the system of linear equation |(2x_(1)-2x_(2)+x_(3),=lambdax_(1)),(2x_(1)-3x_(2)+2x_(3),=lambdax_(2)),(-x(1)+2x_(2),=lambdax_(3))| hasa non trivial solution

S = n xx 10^(6m) + n xx 10^(6m + 1) + n xx 10^(6m-2)+ "…." + nxx 10 +n - 10 m in I, n = single digit +ve integer. Then the remainder if (1) S is divided by 13 is a , (2) ItS is divided by 11 is b , (3)It S is divided by 7 is c a+b+c+ =

A rod AB of length 3 units moves vertically with its bottom B always an the circle x^(2)+y^(2)=25 then the equation of the locus of A is

cos^(-1)((1-x^(2))/(1+x^(2)))= k tan^(-1)xthen k is :

int(1)/(x^(2)+(a+b)x+ab)dx=

Let P be a variable point on the ellipsewith foci S_(1) and S_(2) . If A be the area of trianglePS_(1)S_(2)then findthe maximumvalueof A

xdy-ydx=(x^(2)-3xy+2y^(2))dx=0

Find int(2x^(2)+e^(x))dx.

If A and B are two square matrices of the same order then which of the following is true.

Let A=[(1,1,0),(0,1,0),(0,0,1)],and let l denote the 3 xx 3 identity matrix. Then 2A^(2)-A^(3)=……....

If x = cisA, y=cisB then find the value of cos(A+B) interms of x and y.

Refer to question 11. How many of circuits of Type A and of Type B, should be produced by the manufacturer so as to maximise his profit? Determine the maximum profit.

If the equationax^(4)+bx^(3)+cx^(2)+kx=0 has a rootalpha gt 0, then the equation4ax^(3)+3bx^(2)+2cx+k=0 has -

int_0^(pi/2)sin^10thetad theta

The normal to a curve at P(x,y) meets the x-axis at G. If the distance of G from the origin is twice the abscisa of P, then the curve is a

A body is projected from ground with speed 20 m/s making an angle of 45^(@) with horizontal. The equation of path is h = Ax-Bx^(2), where h is height, x is horizontal distance, A and B are constant. The ratio A : B is :- (g = 10 m//s^(2))

For what value of lambda the equation of conic 2xy+4x-6y+ lambda=0 represents two intersecting straight lines ? If lambda= 17 then does this equation represents ?

Find the number of rectangles from a rectangles of size 10xx9. Also find the number of square with size 4xx4. Also find the total number of rectangles excluding squares.

IFxin Rthen( x^2+ 2x+ a) /(x^2+ 4x + 3a )cantakeall realvaluesif

Solve the differential equations: (dy)/(dx) + y secx = tanx0 le x le pi/2

If the position vectors of three points A, B and Care respectivelyveci+vecj+3veck,4veci+vecj+5veckand7(veci+veck). Findvec(AB)xxvec(AC).. Interpret the result geometrically.

If A=[[1,2],[-2,3]]B=[[3,2],[1,4]],C=[[2,2],[1,3]]Calculate AB.

Find the solution of the equation [x] + {-x} = 2x, (where [*] and {*} represents greatest integer function and fractional part function respectively.

int_(0)^(pi//2)(1)/(4 cos^(2)x+9sin^(2)x)dx=

Consider a parabola P touches coordinate axes at (4,0) and (0,3). Equation of directrix of parabola P is

If int (1)/(x sqrt(1 - x^(3))) " dx = a log" | (sqrt(1- x^(3)) - 1)/(sqrt(1 -x^(3) )+ 1) | + bthen a =

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and by = ae^(3x) + be^(-2x)

int(sin2x/(acos^(2)x)+bsin^(2)x)dx =

If f(x)={(1,-,kx, x,le3),(2x,+,3,x,>3):} is a continuous function, then the value of k is

Evaluate the following determinants: [[a-b,b-c,c-a],[x-y,y-z,z-x],[p-q,q-r,r-p]]

The position vectors of A, B and C are 2hati-hatj+hatk,hati-3hatj-5hatk and x hati-3hatj+hatk respectively in DeltaABC. If angleC=(pi)/(2) then the value of x is ……………..

Find the solution of the equation [x] + {-x} = 2x, (where [] and {} represents greatest integer function and fractional part function respectively.