28278 + Interview Questions in Maths in Class 12 Page 14 InterviewSolution

651.	Differentiate the following functions with respect to x: x+ sin x = sin y
Answer» ANSWER :`(1+COS X)/(cos y)`

Discussion

652.	If f(x)=int_(-1)^(x)abs(t)dt, x ge 1, then
Answer» F is CONTINUOUS at x=0 but `f^(')` is not continuous both f and `f^(')` are continuous for all `x gt -1` f is continuous for `x gt -1 " but " f^(')` is not continuous f and `f^(')` are differentiable at x=0 Answer :A

Discussion

653.	int _(0) ^(pi//2) sin phi cos phi sqrt((a^(2) sin ^(2) phi cos ^(2)))d phi a ne b a gt 0, b gt 0)
Answer» ANSWER :`1/3 (a^(2) -B ^(2))/(a ^(2) -b ^(2)) `

Discussion

654.	If f_(1)(x) = 2x + 3, f_(2)(x) = 3x^(2) + 5, f_(3)(x) = x + cos x are defined from R rarr R, then f_(1), f_(2) and f_(3) are
Answer» ONE-one onto Many one into One-one -into Many one onto Answer :A

Discussion

655.	sinh^(-1)2+sinh^(-1)3 = x rArr "coshx"is equal to
Answer» `1/2(3 SQRT(5)+2sqrt(10))` `1/2 (3sqrt(5) - 2 sqrt(10))` `1/2 (12 +2sqrt50)` `1/2 (12 - 2 sqrt(50))` Answer :C

Discussion

656.	If I=int_(3)^(4) (1)/(3sqrt(logx))dxthen
Answer» `0.92 LT I lt 1` `I GT 1` `I lt 0.92` NONE of these ANSWER :A

Discussion

657.	For a poisson variate X, P(X=2) = P(X = 3) find variance of X.
Answer» ANSWER :3

Discussion

658.	Find the equation of ellipse whose vertices are (5,0) and (-5,0) and whose focus lies on the line 3x-5y=9
Answer» ANSWER :1

Discussion

659.	The solution set of the inequation 5^((1//4)(log5 x)^(2)) ge 5x^((1//5)(log5 x)) is
Answer» `(0,5^(-2sqrt(5))]` `[5^(2sqrt(5)),OO)` `(0,5^(-2sqrt(5))] cup[5^(2sqrt(5)),oo)` `(0,oo)` ANSWER :C

Discussion

660.	If2^(2006) + 2006is divided by 7, the remainder is
Answer» ANSWER :8

Discussion

661.	If the solution of the differential equation (dy)/(dx)-y((x^(4)+3x^(2))/((x^(2)+1)^(2)))=(4x+3)e^((x^(3))/(e^(2)+1)) is in the form y =f(x) (where f(0)=1), then f(A)+f(-1) is
Answer» `2sqrt(E)` `3sqrt(e)` `4sqrt(e)` `6sqrt(e)` Solution :I.F.`=e^(-INT(x^(4)+3x^(2))/((x^(2)+1)^(2))dx=e^((-x^(3))/(x^(2)+1))` `impliesye^((-x^(5))/(x^(2)+1))=int(4x+3)dx` `impliesf(x)=y=(2x^(2)+3x+1)e^((x^(3))/(x^(2)+1)` `impliesf(1)=6sqrt(e)` & `f(-1)=0` `impliesf(-1)=0`

Discussion

662.	If X is a Poisson variate such that P(X=2)=9P(X=4)+90P(X=6), then the ordered pair (Mean (X), Variance (X) =
Answer» (1, 2) (1, 1) (2, 1) (2, 2) ANSWER :B

Discussion

663.	Equation of a hyperbola such that the distance between the foci is 16 and eccentricity is sqrt(2) is
Answer» `x^2 - y^2 = 16` `x^2 - y^2 = 32` `x^2 - 2y^2 = 16` `2x^2 - y^2 = 27` ANSWER :B

Discussion

664.	If a,b,c are in A.P.and f(x)=\|(x+a,x^2+1,1),(x+b, 2x^2-1,1),(x+c,3x^2-2 , 1)\|, thenf'(x)is :
Answer» 0 1 a+bc `(ABC)/(a+b+c)` ANSWER :A

Discussion

665.	If n is positive integer then greatest value of x(a-x)^non (0,a) is
Answer» 0 `(2A)^(N+1)` `a^n` `(a(an)^n)/((n+1)^(n+1))` ANSWER :D

Discussion

666.	The values of (sqrt3+i)^(1//4) are
Answer» `2^(1//4)CIS[(2kpi+pi//6)/(4)],k=0,1,2,3` `2^(1//4)cis[(2kpi+pi//3)/(4)],k=0,1,2,3` `2^(1//4)cis[(2kpi-pi//6)/(4)],k=0,1,2,3` `2^(1//4)cis[(2kpi-pi//3)/(4)],k=0,1,2,3` ANSWER :A

Discussion

667.	S is a sample spaceS= (x in N : 1 x le 100) and E=(x : x(+1)(x-1) in S) . Then P(E)=
Answer» `1//10` `2//25` `99//100` `1//11` ANSWER :A

Discussion

668.	Let f be the real valued differentiable function on R such that e^(-x)f(x)=3/(e^(2))+4e^(-x) int_(2)^(x) sqrt(2t^(2)+6t+5)dt AA x in R and let g(x)=f^(-1) (x) then [g^(')(3)]+[\|g^('')(3)\|] is equal to _____ (where [.] denote the greatest integer function)
Answer» Solution :`g(f(X))=x` So, `g^(')(f(x)).f^(')(x)=1` `g^('')=(f(x))=-(f^('')(x))/((f^(')(x))^(3))` So, `g^(')(3)=1/23` and `g^('')(3)=(-43)/(5(23)^(3))`

Discussion

669.	If O is the circumcentre and O' the orthocenter of DeltaABC prove that (i) SA+SB+SC=3SG, where S is any point in the plane of DeltaABC. (ii) OA+OB+OC=OO' Where, AP is diameter of the circumcircle.
Answer» Solution :Let G be the centroid of `DeltaABC`, first we shall show that CIRCUMCENTRE O, ORTHOCENTER O' and centroid G are collinear and O'G=2OG. Let AL and BM be perpendiculars on the sides BC and CA, respectively. Let AD be the median and OD be the perpendicular from O on side BC. iff R is the circumradius of circumcircle of `DeltaABC`, then OB=OC=R. In `DeltaOBD,` we have OD=R cosA. . . (i) In `DeltaABM,AM=AB cos A=c cosA`. . . (ii) FORM `DeltaAO'M,AO'=AMsec(90^(@)-C)` =c cos A cosec C `=(c)/(SINC)cosA=2RcosA ""(because(a)/(sinA)=(b)/(sinB)=(c)/(sinC)=2R)` `AO'=2(OD)`. . . (iii) Now, `DeltaAGO' and DeltaOGD` are similar. `therefore(OG)/(OG)=(GD)/(GA)=(OD)/(AO')=(1)/(2)`[using Eq. (iii)] `implies 2OG=O'G` (i) We have, `SA+SB+SC=SA+(SB+SC)` `=SA+2SD`(`because` D is the mid-point of BC) `=(1+2)+SG=8SG`. (ii) On REPLACING S by O in Eq. (i), we get `OA+OB+OC=3OG` `=2OG+OG=GO'+OG` `=OG+GO'+O O'`.

Discussion

670.	ABCDE is a pentagon. If the sum of the vectors bar(AB),bar(AE), bar(BC), bar(DC), bar(ED), bar(AC) is lambda bar(AC) then find the value of lambda.
Answer» ANSWER :3

Discussion

671.	Find int "cosec"x ( "cosec" x + cot x ) dx
Answer» ANSWER :`=- COT X - COSECX + C`

Discussion

672.	Let{:{(\|x-1\|+a, x lt 1), (2x+3, x ge 1):} if f(x)has a local minima at x=1 , then
Answer» `a ge 5` `a GT 5` `a gt 0` NONE of these Answer :A

Discussion

673.	p: Two circles are congruent if and only if their radii are equal. Check the validity of statement P
Answer» Solution : In ORDER to prove the statement "p if and only if q", we need to show that (i) If p is TRUE, then q is true and (ii) If q is true, then p is true or show that `~q rArr ~p`. Now, Let p : Two circles are congruent. q : Radius of two circles are equal. Assume that pis true i.e., circles are congruent, we KNOW that two figures are congruent thentheir areas are also equal. Let radius of two circles be `r_1 and r_2` Now, Area of circles are equal So,`pir_1^2=pir_2^2` `rArr r_1^2=r_2^2` `rArr r_1=r_2` So, their radii are equal , so q is also true Now, soppose q is true, i.e., radius of both these circles are equal. When radius of both these circles are equal then they will of same size. So, two circles are of same SHAPE and size which i is the required criteria of congruency. Hence, both circles are congruent. So p is also true.

Discussion

674.	Differentiate the following with respect to x: x cos x + e^(x)
Answer» ANSWER :`COS X- x SIN x + E^(x)`

Discussion

675.	If x^(2) + 6x-27 gt 0, -x^(2) + 3x + 4 lt 0, then x lies in the interval
Answer» `(-OO, -9) CUP (4, oo)` [3, 4] `(-oo, 3) cup (4, oo)` (-9, 4) ANSWER :A

Discussion

676.	Which of the following in not true is a-=bx (modm)and x in Z?
Answer» `a + X -=b + x` (MOD m) `ax -= BX` (mod m) `a - x -= b -c` (mod m) `a -:x -=b -:x` (MODM) ANSWER :D

Discussion

677.	int(dx)/((x+1)sqrt(2x^(2)+3x+4)) =K logf(x) + C them
Answer» `K = - (1)/(SQRT(3)), f(x)` `=(5-x)/(sqrt(23)(x+1))+sqrt(((5-x)/(sqrt(23)(x+1)))^(2)+(23)/(16))` `K=(1)/(sqrt(3))f(x)=(4-x)/(x+1)+sqrt(((4-x)/((x_+1)))^(2)+(23)/(16))` `K=2,f(x)=y+sqrt(y^(2)+1),y(5-x)/(x+1)` none of these Answer :D

Discussion

678.	Calculate magnitude of DeltaH in calorie for 1 mol of an ideal gas undergoingadiabatic reversible process from 8 atm, 300 K to 2 atm.(Given : gamma=2, R=2Cal//K//mol)
Answer» <P> SOLUTION :[0600] `(T_(2))/(T_(1))=((P_(1))/(P_(2)))^((1-GAMMA)/(gamma))=((8)/(2))^((1-2)/(2))=((4)/(1))^((-1)/(2))=((4)/(1))^((1)/(2))=(1)/(2)` `T_(2)=150K` `DeltaH=1xx(gammaR)/(gamma-1)(-150)=(2xx2)/(1)xx(-150)=-600`

Discussion

679.	STATEMENT-1 A,B,C,D are four sets such that n(A cap C)=3 and n(P(A cap C)) ne n(P(B cap D)). STATEMENT-2: If A cup B= A cap B, then it is NOT necessary that A=B.
Answer» ANSWER :B

Discussion

680.	Equation of the tangent at theta=pi//4 to the curve x=a cos2theta,y=2sqrt(2)asintheta is
Answer» `x+y=a` `x+y=2a` `y=x+2a` none Answer :B

Discussion

681.	Show that "tan"^(-1)1/2+"tan"^(-1)2/11="tan"^(-1)3/4
Answer» ANSWER :R.H.S

Discussion

682.	Evaluate the following integrals intcos(logx)dx
Answer» Answer :`(x)/(2)COS(LOGX)+sin(logx)+C`

Discussion

683.	Findthe areaboundedby thecurvey= sqrt(x), x= 2 y+3 in thefirstquadrantandY- axis
Answer» SOLUTION :WRONG QUESTION

Discussion

684.	If the eqution of the circle cutting orthogonally the circlesx^(2)+ y^(2) - 6x = 0 , x^(2) + y^(2) + 4x + 3y + 1 = 0and which has its its centre on the line x + 2y = 5 isx^(2) + y^(2) - 2ax - 2by + c = 0then the dessending order of a,b,c is
Answer» a,B,C b,c,a b,a,c a,c,b Answer :A::B

Discussion

685.	Which is the following is a homogeneous differential equation ?a)(4x + 6y + 5)dy - (3y + 2x + 4) dx = 0b) (xy) dx - (x^3 + y^3)dy = 0c)(x^3 + 2y^2)dx + 2xy dy = 0d)y^2 dx + (x^2 - xy - y^2) dy = 0
Answer» (4x + 6y + 5)DY - (3y + 2x + 4) dx = 0 (XY)dx - `(x^3 + y^3)`dy = 0 `(x^3 + 2y^2)`dx + 2xy dy = 0 `y^2 dx + (x^2 - xy - y^2) dy = 0` ANSWER :D

Discussion

686.	The vertex of the parabola y^2 = 4x + 4y is
Answer» `(1,-2)` `(-1,2)` `(2,1)` `(-2, 1)` ANSWER :B

Discussion

687.	Let veca = 2hati+hatj,vecb = -hati+3hatj+hatk and vecc = hati+2hatj+5hatk be three vectors. Find veccxxveca
Answer» SOLUTION :

Discussion

688.	Let us consider one vertex and one side through the vertex along x -axis of a triangle . Now the coordinates of the vertices B,C and A of any triangle ABC (0,0) ,(a,0) and (h,k) respectively should be taken .If the altitude (AE)of the triangle in question (ii) greater than sqrt(10)and the lenght of AB and AC are of integral value , then the lenght of AC is -
Answer» 3 6 4 or 5 none of these Answer :C

Discussion

689.	If z_1=1+2i,z_2=2+3i,z_3=3+4i, " then " z_1,z_2and z_3 represents the vertices of
Answer» EQUILATERAL TRIANGLE RIGHT angled triangle ISOSCELES triangle collinear Answer :D

Discussion

690.	int_(-3pi//2)^(-pi//2)[(x+pi)^(3)+cos^(2)(x+3pi)]dx=
Answer» `PI/2` `pi/4 -1` `(pi^(4))/(32)` `(pi^(4))/(32) + (pi)/(2)` Answer :A

Discussion

691.	A coin is tossed three times. Find the probability of getting at most 2 heads
Answer» Answer :A coin is tossed THREE times. `S={HHH,HT T, HTH,THH,T TH, THT,HHT,T T T}` `\|S\|=8` Let B be the event of getting almost 2 heads `thereforeB={HT T,HTH,THH,T TH,THT, HHT, T T T}` `implies \|B\|=7` `therefore P(B)=\|B\|/\|S\|=7/8`

Discussion

692.	Evalute the following integrals int (1)/((x + 1)sqrt(x - 1)) dx
Answer» Answer :`SQRT(2)tan^(-1)(sqrt((x-1)/(2)))+C`

Discussion

693.	If y=x^2+3x+6 then find triangley and dy when x=10, trianglex=0.01.
Answer» ANSWER :`0.0026`

Discussion

694.	Fill in the blanks in each of the following, using the answers given against each of them : The slope and x-intercept of the line 3x-y+ k = 0 are equal if k= _____ .
Answer» 0 `-1` 3 `-9` ANSWER :D

Discussion

695.	(x)/(x+1)+(1)/(2)((x)/(x+1))^(2)+(1)/(3)((x)/(x+1))^(3)+....=
Answer» `LOG(1+X)` `log(1-x)` `log x` `log((1)/(x))` ANSWER :A

Discussion

696.	Write down the first three terms is the following expansions(1 + x/2)^(-5)
Answer» ANSWER :`1, (-5X)/(2), 10/4 x^2`

Discussion

697.	Classify 20m//sec^2 measures as scalar and vector.
Answer» SOLUTION :The GIVEN MEASURE is a SCALAR.

Discussion

698.	An ellipse with major and minor axis 6sqrt3 and 6 respectively, slides along the coordinates axes and always remains confined in the first quardrant. If the length if arc decribed by center of ellipse is (pilambda)/6 then the value of lambda is
Answer» ANSWER :6

Discussion

699.	Show that the circumcircle of the triangle formed by the lines ax+by+c=0,bx+cy+a=0 and cx+ay+b=0 passes through the origin if (b^(2)+c^(2))(c^(2)+a^(2))(a^(2)+b^(2))=abc(b+c)(c+a)(a+b).
Answer» ANSWER :`=ABC(a+b)(b+c)(c+a)`

Discussion

700.	Evaluate the following integrals. int(secx)/((secx+tanx)^(n))dx
Answer» ANSWER :`(-1)/(N(secx+tanx)^(n))+C`

Discussion

Explore topic-wise InterviewSolutions in .

Differentiate the following functions with respect to x: x+ sin x = sin y

If f(x)=int_(-1)^(x)abs(t)dt, x ge 1, then

int _(0) ^(pi//2) sin phi cos phi sqrt((a^(2) sin ^(2) phi cos ^(2)))d phi a ne b a gt 0, b gt 0)

If f_(1)(x) = 2x + 3, f_(2)(x) = 3x^(2) + 5, f_(3)(x) = x + cos x are defined from R rarr R, then f_(1), f_(2) and f_(3) are

sinh^(-1)2+sinh^(-1)3 = x rArr "coshx"is equal to

If I=int_(3)^(4) (1)/(3sqrt(logx))dxthen

For a poisson variate X, P(X=2) = P(X = 3) find variance of X.

Find the equation of ellipse whose vertices are (5,0) and (-5,0) and whose focus lies on the line 3x-5y=9

The solution set of the inequation 5^((1//4)(log5 x)^(2)) ge 5x^((1//5)(log5 x)) is

If2^(2006) + 2006is divided by 7, the remainder is

If the solution of the differential equation (dy)/(dx)-y((x^(4)+3x^(2))/((x^(2)+1)^(2)))=(4x+3)e^((x^(3))/(e^(2)+1)) is in the form y =f(x) (where f(0)=1), then f(A)+f(-1) is

If X is a Poisson variate such that P(X=2)=9P(X=4)+90P(X=6), then the ordered pair (Mean (X), Variance (X) =

Equation of a hyperbola such that the distance between the foci is 16 and eccentricity is sqrt(2) is

If a,b,c are in A.P.and f(x)=|(x+a,x^2+1,1),(x+b, 2x^2-1,1),(x+c,3x^2-2 , 1)|, thenf'(x)is :

If n is positive integer then greatest value of x(a-x)^non (0,a) is

The values of (sqrt3+i)^(1//4) are

S is a sample spaceS= (x in N : 1 x le 100) and E=(x : x(+1)(x-1) in S) . Then P(E)=

Let f be the real valued differentiable function on R such that e^(-x)f(x)=3/(e^(2))+4e^(-x) int_(2)^(x) sqrt(2t^(2)+6t+5)dt AA x in R and let g(x)=f^(-1) (x) then [g^(')(3)]+[|g^('')(3)|] is equal to _____ (where [.] denote the greatest integer function)

If O is the circumcentre and O' the orthocenter of DeltaABC prove that (i) SA+SB+SC=3SG, where S is any point in the plane of DeltaABC. (ii) OA+OB+OC=OO' Where, AP is diameter of the circumcircle.

ABCDE is a pentagon. If the sum of the vectors bar(AB),bar(AE), bar(BC), bar(DC), bar(ED), bar(AC) is lambda bar(AC) then find the value of lambda.

Find int "cosec"x ( "cosec" x + cot x ) dx

Let{:{(|x-1|+a, x lt 1), (2x+3, x ge 1):} if f(x)has a local minima at x=1 , then

p: Two circles are congruent if and only if their radii are equal. Check the validity of statement P

Differentiate the following with respect to x: x cos x + e^(x)

If x^(2) + 6x-27 gt 0, -x^(2) + 3x + 4 lt 0, then x lies in the interval

Which of the following in not true is a-=bx (modm)and x in Z?

int(dx)/((x+1)sqrt(2x^(2)+3x+4)) =K logf(x) + C them

Calculate magnitude of DeltaH in calorie for 1 mol of an ideal gas undergoingadiabatic reversible process from 8 atm, 300 K to 2 atm.(Given : gamma=2, R=2Cal//K//mol)

STATEMENT-1 A,B,C,D are four sets such that n(A cap C)=3 and n(P(A cap C)) ne n(P(B cap D)). STATEMENT-2: If A cup B= A cap B, then it is NOT necessary that A=B.

Equation of the tangent at theta=pi//4 to the curve x=a cos2theta,y=2sqrt(2)asintheta is

Show that "tan"^(-1)1/2+"tan"^(-1)2/11="tan"^(-1)3/4

Evaluate the following integrals intcos(logx)dx

Findthe areaboundedby thecurvey= sqrt(x), x= 2 y+3 in thefirstquadrantandY- axis

If the eqution of the circle cutting orthogonally the circlesx^(2)+ y^(2) - 6x = 0 , x^(2) + y^(2) + 4x + 3y + 1 = 0and which has its its centre on the line x + 2y = 5 isx^(2) + y^(2) - 2ax - 2by + c = 0then the dessending order of a,b,c is

Which is the following is a homogeneous differential equation ?a)(4x + 6y + 5)dy - (3y + 2x + 4) dx = 0b) (xy) dx - (x^3 + y^3)dy = 0c)(x^3 + 2y^2)dx + 2xy dy = 0d)y^2 dx + (x^2 - xy - y^2) dy = 0

The vertex of the parabola y^2 = 4x + 4y is

Let veca = 2hati+hatj,vecb = -hati+3hatj+hatk and vecc = hati+2hatj+5hatk be three vectors. Find veccxxveca

If z_1=1+2i,z_2=2+3i,z_3=3+4i, " then " z_1,z_2and z_3 represents the vertices of

int_(-3pi//2)^(-pi//2)[(x+pi)^(3)+cos^(2)(x+3pi)]dx=

A coin is tossed three times. Find the probability of getting at most 2 heads

Evalute the following integrals int (1)/((x + 1)sqrt(x - 1)) dx

If y=x^2+3x+6 then find triangley and dy when x=10, trianglex=0.01.

Fill in the blanks in each of the following, using the answers given against each of them : The slope and x-intercept of the line 3x-y+ k = 0 are equal if k= _____ .

(x)/(x+1)+(1)/(2)((x)/(x+1))^(2)+(1)/(3)((x)/(x+1))^(3)+....=

Write down the first three terms is the following expansions(1 + x/2)^(-5)

Classify 20m//sec^2 measures as scalar and vector.

An ellipse with major and minor axis 6sqrt3 and 6 respectively, slides along the coordinates axes and always remains confined in the first quardrant. If the length if arc decribed by center of ellipse is (pilambda)/6 then the value of lambda is

Show that the circumcircle of the triangle formed by the lines ax+by+c=0,bx+cy+a=0 and cx+ay+b=0 passes through the origin if (b^(2)+c^(2))(c^(2)+a^(2))(a^(2)+b^(2))=abc(b+c)(c+a)(a+b).

Evaluate the following integrals. int(secx)/((secx+tanx)^(n))dx