28278 + Interview Questions in Maths in Class 12 Page 194 InterviewSolution

9651.	Let ABC be triangle whose equation of sides are AB=x+2y=3, AC=2x+y=3 and BC=x+y=4 . S_1,S_2,S_3 are three circles drawn considering AB,AC and BC as diameter respectively. The radical axis of S_1 of S_2,S_1 and S_3 and S_2and S_3 are L_12 =0 , L_13 =0 ,L_23=0 .These radical axes meet sides BC,AC and AB at D,E,Frespectively and triangle DEF is formed. Equation of L_12=0 is
Answer» x-y=2 2x+3y=5 x-y=0 none of these ANSWER :C

Discussion

9652.	Let ABC be triangle whose equation of sides are AB=x+2y=3, AC=2x+y=3 and BC=x+y=4 . S_1,S_2,S_3 are three circles drawn considering AB,AC and BC as diameter respectively. The radical axis of S_1 of S_2,S_1 and S_3 and S_2and S_3 are L_12 =0 , L_13 =0 ,L_23=0 .These radical axes meet sides BC,AC and AB at D,E,Frespectively and triangle DEF is formed. Radical centre of circles S_1,S_2,S_3 is
Answer» (3,-3) (-3,-3) (-7,-7) NONE of these ANSWER :C

Discussion

9653.	Let ABC be triangle whose equation of sides are AB=x+2y=3, AC=2x+y=3 and BC=x+y=4 . S_1,S_2,S_3 are three circles drawn considering AB,AC and BC as diameter respectively. The radical axis of S_1 of S_2,S_1 and S_3 and S_2and S_3 are L_12 =0 , L_13 =0 ,L_23=0 .These radical axes meet sides BC,AC and AB at D,E,Frespectively and triangle DEF is formed. Equation of circumcircleof triangle ADC is a
Answer» `x^2+y^2-6x+4=0` `x^2+y^2-6y+4=0` `x^2+y^2+6x+4=0` NONE of these ANSWER :B

Discussion

9654.	Evaluate int(1)/((x-1)sqrt(x+2))dx
Answer» ANSWER :`1/(SQRT(3))log\|(sqrt(x+2)-sqrt(3))/(sqrt(x+2)+sqrt(3))\|+C`

Discussion

9655.	f and g are real valued function f(x) =x^(2) + x + 7 , x in R and g(x) = 5x -3 , x in R. Find fog and gof. Also find (fog) (2) and (gof) (1).
Answer» SOLUTION :N/A

Discussion

9656.	For\|x\|lt(1)/(5)thecoefficientofx ^ 3intheexpansion of(1)/((1 - 5x ) ( 1 - 4 x))is
Answer» 369 370 371 372 Solution : ` (1)/(( 1 - 5 x)( 1 - 4x )) = (1 - 5x ) ^(-1) (1 - 4x ) ^( -1 ) ` `= [ 1 + 5x+(5x ) ^ 2+ (5x ) ^3 + …. ][ 1+ 4x+ (4x) ^ 2+(4x ) ^3 + … ]` `therefore`Co- efficientof` x ^ 3` intheaboveexpansion `= (1)(4) ^3+ 5 (4)^2 + (5 ^ 2 ) (4)+1 (5) ^ 3 ` `= 64+80+ 100 +125 ` ` = 369 `

Discussion

9657.	Find the approximate value of each of the following :cos 29^(@)
Answer» ANSWER :`(SQRT(3))/(2)+(PI)/(360)`

Discussion

9658.	Evaluate the definite integrals int_(0)^(pi/2)cosxdx
Answer»

Discussion

9659.	If the distance between two points A and B is d, and the lengths of the projections of AB on the coordinate planes are d_(1), d_(2), d_(3), then
Answer» `2d^(2) = d_(1)^(2) + d_(2)^(2) + d_(3)^(2)` `d_(1) + d_(2) + d_(3) = 0` `d_(1)^(2) + d_(2)^(2) + d_(3)^(2) = d^(2)` `d_(1) + d_(2) + d_(3) = d` ANSWER :A

Discussion

9660.	Statement I : The points 4i + 5j + k, -j -k, 3i + 9j + 4k and -4i + 4j + 4k are coplanar. Statement II : The given points from the vertices of a parallelogram. Which of the following is true ?
Answer» Both statements are true and statement II is CORRECT EXPLANATION of staement I Both statements are true and statement II is not a correct explanation of statement I Statement I is true and Statement II is false Statement I is false and STATEMETN II is true Answer :C

Discussion

9661.	Solve : x " sin"(y)/(x).(dy)/(dx)=y " sin"(y)/(x)-x
Answer» `E^((y)/(x)) = CY` `e^(cosy//x) = cx` `e^((x)/(y)) = cx` `e^((2Y)/(x)) = cx` Answer :B

Discussion

9662.	If the projection of the line x/2=(y-1)/2=(z-1)/1 on a plane P is x/1=(y-1)/1=(z-1)/-1. Then the distance of plane P from origin is
Answer» `SQRT(3)` `sqrt(3/2)` `sqrt(6)` `2/sqrt(3)` SOLUTION :Projection of line `x/2=(y-1)/2=(z-1)/1` on a plane P is `x/1=(y-1)/1=(z-1)/-1`. `therefore` Plane through these lines is PERPENDICULAR to the plane P. Normal to the plane determined by the given lines is. `\|{:(hati,HATJ,hatk),(2,2,1),(1,1,-1):}\|=-3hati+3hatj`

Discussion

9663.	int _(0) ^(1//2)[sin 4 pix] dx=
Answer» `pi-1` `2/pi` `1/pi` `0` ANSWER :C

Discussion

9664.	Method of integration by parts : int e^(x)((x^(2)+1))/((x+1)^(2))dx=....
Answer» `((X-1)/(x+1))E^(x)+C` `e^(x)((x+1)/(x-1))+c` `e^(x)(x+1)(x-1)+c` NONE of these Answer :A

Discussion

9665.	Consider aDelta ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C, respectively. Suppose a=2, b = 3, c = 4and H be the orthocenter. Find 15(HA)^(2).
Answer» ANSWER :11

Discussion

9666.	Multiply(x-(1+sqrt(-3))/2)(x-(1-sqrt(-3))/2)
Answer» SOLUTION :`(X-(1+sqrt(-3)/2)(x-(1-sqrt3)/2)` `=(X+(-1-isqrt3)/2)(X+(-1+isqrt3)/2)` `=(X+omega)(X+omegea^2)=X^2+omega^2X+omegaX +omega^3` `=X^2+X (omega^2+omega) +1=X^2-X+1`

Discussion

9667.	Fill in the gaps with correct answer .sin^2(24)^@-sin^2(6)^@ = _____.
Answer» SOLUTION :`(sqrt5-1)/8`

Discussion

9668.	Let a,lambda,mu in R, Consider the system of linear equations ax+2y=lambda 3x-2y=mu Which of the flollowing statement (s) is (are) correct?
Answer» If `a=-3`, then the SYSTEM has infinitely MANY solutions for all values of `lambda and mu` If `a ne-3`, then the system has a UNIQUE solution for all values of `lambda and mu` If `lambda+mu=0`, then the system has infinitely many solution for `a=-3` IF `lambda+mune0`, then the system has no solution for `a=-3`

Discussion

9669.	Evaluate (sqrt(3) + sqrt(2))^(6) - (sqrt(3) - sqrt(2))^(6)
Answer» ANSWER :`396sqrt(6)`

Discussion

9670.	Find the equation of the circle which touches X-axis at a distance of 3 units from the origin and making an intercept of length 6 on Y- axis.
Answer» Answer :` X ^(2) + y ^(2)- 6x + 6 SQRT2 y + 9=0 and x^(2) + y ^(2)- 6x -6 sqrt2 y + 9 =0`

Discussion

9671.	In the three element group {e,a,b} where e is the identity ,a^(5)b^(4) is equal to
Answer» a e ab b Answer :A

Discussion

9672.	Find the sum of all 4 digit numbers formed with the digits 1, 2, 4 and 6.
Answer» 36664 26664 24 46632 Answer :B

Discussion

9673.	Evaluate the following integrals (iii) int_(0)^(pi) x sin^(7) x cos^(6) x dx
Answer» ANSWER :`(16pi)/(3003)`

Discussion

9674.	The product of (32)(32)^(1//6)(32)^(1//36) …….. To infty is
Answer» 16 32 64 0 Answer :C

Discussion

9675.	Which of the following when simplified, reduces to unity?
Answer» `log_(10)5. log_(10) 20+ log_(10)^(2)2` `(2 log 2+ log 3)/(log 48 - log4)` `-log_(5),log_(3) = sqrt(ROOT5(9))` `(1)/(6) log_((sqrt(3))/(2))((64)/(27))` ANSWER :A::B::C::D

Discussion

9676.	Evaluation of definite integrals by substitution and properties of its : If n is an integer int_(0)^(pi)e^(cos^(2)x)cos^(3)(2n+1)xdx=............
Answer» 1 `-1` 0 `PI` ANSWER :C

Discussion

9677.	If A = [(alpha ,0),(1,1)] and B = [(1,0),(5,1)]whenever A^(2) = Bthen value of alphais :
Answer» 5 -1 11 no REAL VALUE of `ALPHA ` ANSWER :D

Discussion

9678.	{:(" "Lt),(n rarr oo):}{ (1)/(2n+1)+(1)/(2n+2)+......+(1)/(3n)}=
Answer» ANSWER :`LOG((3)/(2))`

Discussion

9679.	Equation of the tangent to the circle x^(2)+y^(2)-2x+4y-4=0 which is parallel to the line 3x+4y-1=0 is
Answer» `3x+4y=5` `3x+4y=15` `3x+4y=10` `3x-4y=10` ANSWER :C

Discussion

9680.	Determine with proof all triples (a, b, c) of positive integers satisfying 1/a+2/b+3/c=1where a is a prime number and a le b lec
Answer» ANSWER :(2,5,30)(2,6,18)(2,7,14)(2,8,12)(2,10,10)(3,4,6)(3,6,9)

Discussion

9681.	Differentiate the following w.r.t.x e^(mtan^-4x)
Answer» SOLUTION :`d/dx(E^(mtan^-1x))=e^(mtan^-1x)d/dx(mtan^-1x)e^(mtan^-1xm/(1+x^2)=(me^(mtan^-1x))/(1+x^2)`

Discussion

9682.	A particle is moving along a curve 6y=x^(3)+2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.
Answer» ANSWER :(4, 11), `(-4, -31/5)`

Discussion

9683.	Solve the following differential equations.dy/dz=secy
Answer» Solution :`DY/DZ=secyrArr cos y CDOT dy =dz` `RARR sin y =z+C`

Discussion

9684.	A: In a Delta ABC (a+b+c) (b+c-a) =Abc if 0 lt lamda lt 4 R: In a Delta ABC,-1 le cos A le 1
Answer» A is TRUE, R is true and R is CORRECT explanation of A A is true, R is true and R is not correct explanation of A A is true, R is FALSE A is false, R is true Answer :C

Discussion

9685.	LetD and E be the midpoints of the sides AC and BC of a triangle ABC respectively . If O is an interior point of the triangle ABC such that OA + 2OB+ 3OC= 0, then the area (in sq units) of the triangle ODE is
Answer» 6 5 `(3)/(4)` 0 Answer :D

Discussion

9686.	Evaluate int(cos4x+cos2x)/(sin4x+sin2x)dx.
Answer» SOLUTION :`INT(cos4x+cos2x)/(sin4x+sin2x)==int("2cos"(6x)/2"cos"(2X)/2)/(2"sin"(6x)/2"cos"(2x)/2)dx =intcot3xdx=1/3In\|sin3x\|+C`

Discussion

9687.	If 2[(3,4),(5,x)]+[(1,y),(0,1)]=[(7,0),(10,5)], find (x-y).
Answer» ANSWER :10

Discussion

9688.	A: In a Delta ABC, if the sides a,b,c are in A.P. then a cos^2""C/2+c cos^2""A/2=3b R: In a Delta ABC, if A,B,C are in A.P. then B=pi//3
Answer» A is true, R is true and R is CORRECT EXPLANATION of A A is true, R is true and R is not correct explanation of A A is true, R is false A is false, R is true Answer :D

Discussion

9689.	Number of diagonals in a decagon is
Answer» 45 36 46 35 Answer :D

Discussion

9690.	Length of the perpendicular form the point (-6,2,3) on the plane 3x-6y+2z+10=0 is
Answer» 2 `(13)/(7)` `(10)/(7)` `(8)/(7)` ANSWER :A

Discussion

9691.	If [(a-b,2a+c),(2a-b,3c+d)]=[(-1,5),(0,13)], then a = __________.
Answer» ANSWER :`a=1`

Discussion

9692.	If int((sqrt(x))^(5))/((sqrt(x))^(7)+x^(6))dx=lamdaln(1+2cosx)/((2+cosx)^(2))+C, then a+lamda is
Answer» `=2` `gt2` `LT2` `=1` Answer :B

Discussion

9693.	Evaluate the following integrals: int_0^(pi/4) tanx dx
Answer» SOLUTION :`int_^(pi/4) tanx dx = [log \|SECX\|]_0^(pi/4)` =`log\|SEC pi/4 \|-log \|sec 0` =`log \|sqrt2\| -log\|1\|` =`log\|sqrt2\| = log^(1/2) = 1/2 log2`

Discussion

9694.	If the lines x^(2) + 2xy- 35 y^(2) - 4x +44y - 12 = 0 and 5x + lambda y - 8 = 0 are concurrent , then the value of lambda is ,
Answer» 0 1 `-1` 2 Answer :D

Discussion

9695.	The solution for x of the equation int_(sqrt(2))^(x) (dt)/(\|t\|sqrt(t^(2)-1))=pi/2 is
Answer» `2sqrt(2)` 2 `PI` `-SQRT(2)` ANSWER :D

Discussion

9696.	What is the value of lambda if the triangle whose vertices are hat(i), hat(j) and hat(i)+hat(j)+lambdahat(k) will be right angled?
Answer» 2 0 -1 1 Solution :Let us consider TRIANGLE ABC. Suppose `hat(i), hat(j) and hat(i)+hat(j)+lambda hat(K)` are the position vector of A, B and C. Then `vec(AB)=hat(j)-hat(i), vec(AC)=hat(j)+lambda hat(k), vec(BC)=hat(i)+lambda hat(k)` `\|vec(AB)\|=sqrt((-1)^(2)+(1)^(2))=sqrt(2)` `\|vec(BC)\|=sqrt((1)^(2)+(lambda))^(2))=sqrt(1+lambda^(2)` `\|vec(AC)\|=sqrt((1)^(2)+(lambda))^(2)=sqrt(1+lambda^(2))`

Discussion

9697.	Find the equation of the circle passing through the three points (4, 7), (5, 6) and (1, 8). Also find the coordinates of the point of intersection of the tangents to the circle at the points where it is cut by thestraight line 5x + y + 17 = 0.
Answer» ANSWER :`(-4, 2), x^(2) + y^(2) - 2X - 6y - 15 = 0`

Discussion

9698.	A monopolist's demand function is p= 300 - 5x. At what price is the marginal revenue zero.
Answer» ANSWER :RS 150

Discussion

9699.	Sum of n terms of the series (1^(4))/(1.3)+(2^(4))/(3.5)+(3^(4))/(5.7)+…. is equal to
Answer» `(N(n+1)(2N^(2)+n+1))/(6(2n+1))` `((n+1)(n^(2)+1))/(6(2n+1))` `((n+1)((2n+1)^(2)+1))/(8(2n+1))` `(n(n+1)((2n+1)^(2)+1))/(16(2n+1))` ANSWER :B

Discussion

9700.	A merchant plans to sell two types of personal computers -a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop modelis Rs. 4500 and on portable model is Rs. 5000.
Answer» Solution :Let `X` and `y` be the number of desktop and protable COMPUTERS respectively. According to the problem, MAXIMUM profit `Z=4500x+5000y` and constraints `xge0, yge0` `x+yle250` `25000x+40000yle7000000` `implies25x+40yle7000` `implies25x+40y le7000` First, draw the graph of the lines `x+y=250` and `25x+40y=7000`. Now, find the feasible region from the constraints`xge0,yge0,x+yle250, 25x+40yle7000` and shade it. The vertices of this shaded region are `A(250,0),B(200,50)` and `C(0,175)`. We find the value of `Z` at these vertices. For the the maximum profit 200 desktop and 50 portable computers should be produced.

Discussion

Explore topic-wise InterviewSolutions in .

Evaluate int(1)/((x-1)sqrt(x+2))dx

f and g are real valued function f(x) =x^(2) + x + 7 , x in R and g(x) = 5x -3 , x in R. Find fog and gof. Also find (fog) (2) and (gof) (1).

For|x|lt(1)/(5)thecoefficientofx ^ 3intheexpansion of(1)/((1 - 5x ) ( 1 - 4 x))is

Find the approximate value of each of the following :cos 29^(@)

Evaluate the definite integrals int_(0)^(pi/2)cosxdx

If the distance between two points A and B is d, and the lengths of the projections of AB on the coordinate planes are d_(1), d_(2), d_(3), then

Statement I : The points 4i + 5j + k, -j -k, 3i + 9j + 4k and -4i + 4j + 4k are coplanar. Statement II : The given points from the vertices of a parallelogram. Which of the following is true ?

Solve : x " sin"(y)/(x).(dy)/(dx)=y " sin"(y)/(x)-x

If the projection of the line x/2=(y-1)/2=(z-1)/1 on a plane P is x/1=(y-1)/1=(z-1)/-1. Then the distance of plane P from origin is

int _(0) ^(1//2)[sin 4 pix] dx=

Method of integration by parts : int e^(x)((x^(2)+1))/((x+1)^(2))dx=....

Consider aDelta ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C, respectively. Suppose a=2, b = 3, c = 4and H be the orthocenter. Find 15(HA)^(2).

Multiply(x-(1+sqrt(-3))/2)(x-(1-sqrt(-3))/2)

Fill in the gaps with correct answer .sin^2(24)^@-sin^2(6)^@ = _____.

Let a,lambda,mu in R, Consider the system of linear equations ax+2y=lambda 3x-2y=mu Which of the flollowing statement (s) is (are) correct?

Evaluate (sqrt(3) + sqrt(2))^(6) - (sqrt(3) - sqrt(2))^(6)

Find the equation of the circle which touches X-axis at a distance of 3 units from the origin and making an intercept of length 6 on Y- axis.

In the three element group {e,a,b} where e is the identity ,a^(5)b^(4) is equal to

Find the sum of all 4 digit numbers formed with the digits 1, 2, 4 and 6.

Evaluate the following integrals (iii) int_(0)^(pi) x sin^(7) x cos^(6) x dx

The product of (32)(32)^(1//6)(32)^(1//36) …….. To infty is

Which of the following when simplified, reduces to unity?

Evaluation of definite integrals by substitution and properties of its : If n is an integer int_(0)^(pi)e^(cos^(2)x)cos^(3)(2n+1)xdx=............

If A = [(alpha ,0),(1,1)] and B = [(1,0),(5,1)]whenever A^(2) = Bthen value of alphais :

{:(" "Lt),(n rarr oo):}{ (1)/(2n+1)+(1)/(2n+2)+......+(1)/(3n)}=

Equation of the tangent to the circle x^(2)+y^(2)-2x+4y-4=0 which is parallel to the line 3x+4y-1=0 is

Determine with proof all triples (a, b, c) of positive integers satisfying 1/a+2/b+3/c=1where a is a prime number and a le b lec

Differentiate the following w.r.t.x e^(mtan^-4x)

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

Solve the following differential equations.dy/dz=secy

A: In a Delta ABC (a+b+c) (b+c-a) =Abc if 0 lt lamda lt 4 R: In a Delta ABC,-1 le cos A le 1

LetD and E be the midpoints of the sides AC and BC of a triangle ABC respectively . If O is an interior point of the triangle ABC such that OA + 2OB+ 3OC= 0, then the area (in sq units) of the triangle ODE is

Evaluate int(cos4x+cos2x)/(sin4x+sin2x)dx.

If 2[(3,4),(5,x)]+[(1,y),(0,1)]=[(7,0),(10,5)], find (x-y).

A: In a Delta ABC, if the sides a,b,c are in A.P. then a cos^2""C/2+c cos^2""A/2=3b R: In a Delta ABC, if A,B,C are in A.P. then B=pi//3

Number of diagonals in a decagon is

Length of the perpendicular form the point (-6,2,3) on the plane 3x-6y+2z+10=0 is

If [(a-b,2a+c),(2a-b,3c+d)]=[(-1,5),(0,13)], then a = __________.

If int((sqrt(x))^(5))/((sqrt(x))^(7)+x^(6))dx=lamdaln(1+2cosx)/((2+cosx)^(2))+C, then a+lamda is

Evaluate the following integrals: int_0^(pi/4) tanx dx

If the lines x^(2) + 2xy- 35 y^(2) - 4x +44y - 12 = 0 and 5x + lambda y - 8 = 0 are concurrent , then the value of lambda is ,

The solution for x of the equation int_(sqrt(2))^(x) (dt)/(|t|sqrt(t^(2)-1))=pi/2 is

What is the value of lambda if the triangle whose vertices are hat(i), hat(j) and hat(i)+hat(j)+lambdahat(k) will be right angled?

Find the equation of the circle passing through the three points (4, 7), (5, 6) and (1, 8). Also find the coordinates of the point of intersection of the tangents to the circle at the points where it is cut by thestraight line 5x + y + 17 = 0.

A monopolist's demand function is p= 300 - 5x. At what price is the marginal revenue zero.

Sum of n terms of the series (1^(4))/(1.3)+(2^(4))/(3.5)+(3^(4))/(5.7)+…. is equal to