28278 + Interview Questions in Maths in Class 12 Page 69 InterviewSolution

3401.	Evaluate int_(0)^(100)e^(x-[x])dx where [ ] denotes the greatest integer function.
Answer» ANSWER :`100(e-1)`

Discussion

3402.	State which of the following are positive ?sec 199^@
Answer» Solution :`SEC 199^@` is -ve as `199^@` lies in 3RD quadrant and sec is -ve there.

Discussion

3403.	A:int tan^(4)x sec^(2)x dx=(1)/(5)tan^(5)x+c R:int[f(x)]^(n) f'(x) dx=([f(x)]^(n+1))/(n+1)+c
Answer» Both A and R are TRUE and R is the CORRECT EXPLANATION of A Both A and R are true and R is not correct explanation of A A is true R is false A is false but R is true. Answer :1

Discussion

3404.	If the sides of a triangle are 3cm, 2cm and 4cm then the cosine of the greatest angle is equal to
Answer» `(7)/(8)` `(11)/(16)` `-(1)/(4)` `(1)/(4)` ANSWER :C

Discussion

3405.	If f is greatest integer function and g is a modulus functions the find . (gof) (-1/3)-(fog)(-1/3) .
Answer» SOLUTION :N/A

Discussion

3406.	If f(x)={{:(x^2+Ax+5, x in Q),(1+x , x in R ~ Q):} is continuous at exactly two points , then the possible values of A are in
Answer» `(1,oo)` `(-3,oo)` `(5,oo) UU (-oo,-3)` `(-oo,oo)` ANSWER :C

Discussion

3407.	Let a, b and c be three non-zero vectors such that no two of these are collinear. If the vector a+2b is collinear with c and b+3c is collinear with a (lambda being some non-zero scalar), then a+2b+6c equals to
Answer» `lambdaa` `LAMBDAB` `lambdac` 0 Solution :If `a + 2B` is collinearwith c, then `a + 2b = tc "…"(i)` Also, `b+3c` is collinearwith a, then `b + 3c = lambda` `RARR b= lambda a - 3c "…."(iii)` From Eqs. (i) and (ii), we get `a+ 2(lambdaa- 3c) = t c` `rArr (a-6c) = t c - 2lambda a` On comparingthe COEFFICIENTS of a and b,we get `1= -2lambda` `rArr lambda = - 1/2 ` and `- 6 = t` `rArr t = - 6` `rArrt = - 6` From Eq. (i) we get `a + 2b = -6c` `rArr a + 2b+6c = 0`

Discussion

3408.	The equation of asymptotes of the hyperbola 4x^(2)-9y^(2)=36 is
Answer» ANSWER :2X + 3Y = 0 and 2x - 3y = 0

Discussion

3409.	If a student is eligible for all the prizes is
Answer» 625 620 1024 1020 Answer :A

Discussion

3410.	All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?
Answer» mean median mode variance Answer :D

Discussion

3411.	Find the order and degree (if defined) of the following differential equations. (d^4y)/(dx^4) + sin (y''') = 0
Answer» Solution :The HIGHEST ORDER derivation in the DIFFERENTIAL equation `(d^4 y)/(dx^4)` and its order = 4 . The degree of the differential equation is not defined.

Discussion

3412.	Evalute the following integrals int (1)/(n sqrt((x - a)^(n- k) (x - b)^(n+k) ) )dx ( Hint Put (x- a)/(x - b ) = t )
Answer» Answer :`(1)/(a -b) (n)/(k) ((X - a)/(x - b))^(k//n)` + C

Discussion

3413.	Find intcosxlog(sinx)dx.
Answer» ANSWER :`intt.dt=t^(2)/2+c=([LOG(SINX)]^(2))/2+c`

Discussion

3414.	If f and g two functions are defined as : f= {(1,2),(3,6),(4,5)} and g = {(2,3),(6,7),(5,8)}, then find gof.
Answer» Solution :Domain of F = {1,3,4} Now, (gof)(1) = g[f(1)] = g (2)(`:'` f (1) = 2) = 3 (`:'` g(2) = 3) (gof) (3) = g [f (3)] = g (6) (`because` f (3) - 6) = 7 (`because` g (6) = 7 ) and(gof) (4) = g [f(4)] = g (5) (`because` f (4) = 5) = 8 (`because` f (5)= 8 ) `:.` (gof) = {(1,3),(3,7),(4,8)}. ANS.

Discussion

3415.	Let set S consists of all the points (x, y) satisfying 16x^2+25y^2 le 400. For points in S let maximum and minimum value of (y-4)/(x-9) be M and m respectively, then:
Answer» `M=1` `M=(65)/(7)` `m=1` `m=(7)/(65)` ANSWER :A::D

Discussion

3416.	Consider the equation sec theta +cosec theta=a, theta in (0, 2pi) -{pi//2, pi, 3pi//2} If the equation has two distinct real roots, then
Answer» `\|a\| ge 2sqrt(2)` `a LT 2sqrt(2)` `\|a\| lt 2sqrt(2)` none of these Solution :We have equation `sec x + cosec x =a` To analyze the roots of the equation, we draw the GRAPH of function `y= sec x+ cosec x` and CHECK how many times line `y=a` intersects this graph. Period of `y=sec x + cosec x` is `2pi`. So, we draw the graph of the function for `x in [0, 2pi]`. The GARPH of function can be easily drawn by drawing the graph of `y=sec x` and `y=cosec x` and then adding the values of `sec x` and `cosec x` by inspection. For EXAMPLE, in first quadrant, `sec x, cosec x gt 0`. Also, when x approaches to zero, `cosec x` approaches to infinity. So, `f(x)` approaches to infinity. Similarly, when x approaches to `pi//2 sec x` approaches to infinity. So, `f(x)` approaches to infinity. At `x=pi//4, f(x)` attains its least value which is `2sqrt(2)`. With similar arguments, we can draw the graph of `y=f(x)` in intervals `(pi//2, pi), (pi, 3pi//2)` and `(3pi//2, 2pi)` We have following graph of `y=f(x)`. From the figure, we can say that `f(x)=a` has two distinct solution if line `y=a` cuts the graph `y=f(x)` between `y=2sqrt(2)` and `y=-2sqrt(2)` i.e., `\|a\| lt 2sqrt(2)`. If line `y=a`, cuts the graph of `y=f(x)` above `y=2sqrt(2)` and below `y=-2sqrt(2)`, then `f(x)=a` has four distinct solutions. So, `\|a\| gt 2sqrt(2)`.

Discussion

3417.	If P(x_(1), y_(1)) is a point such that the length of the tangents from it to the circles x^(2) + y^(2) - 4x - 6y - 12 = 0 and x^(2) + y^(2) + 6x + 18y + 26 = 0 are in the ratio 2 : 3, then the locus of P is
Answer» `x^(2) + y^(2) + 24x - 36Y + 62 = 0` `x^(2) + y^(2) - 24x + 36y + 62 = 0` `x^(2) + y^(2) - 24x - 54y - 88 = 0` `x^(2) + y^(2) + 24x + 36y + 62 = 0`

Discussion

3418.	Find all 3-digit numbers which are the sums of the cubes of their digits.
Answer» ANSWER :153, 370, 371, 407

Discussion

3419.	Find the second order derivatives of the function. x^(2)+3x+2
Answer» ANSWER :`2`

Discussion

3420.	If veca is a non zero vector of magnitude 'a' and lambda a nonzero scalar, then lambda vec a is unit vector if
Answer» `lambda=1` `lambda=-1` `a=\|lambda\|` `a=1//\|lambda\|` ANSWER :D

Discussion

3421.	Shows that the following functions do not possess maximum or minimum. x^3
Answer» Solution :`y = x^3` `dy./dx = 3x^2, (d^2y/dx^2 = 6x` For extremum `dy/dx = 0 RARR 3x^2 = 0 rArr x = 0` `(d^2y)/dx^2]_(x=0) = 0` Thus the function does not POSSESS a MAXIMUM or a minimum.

Discussion

3422.	If A(i=1,2,3….n) are n independent events with P(A) = (1)/(1+i) for each i , then the probability that none of A, occur is :
Answer» `(N-1)/(n+1)` `(n)/(n+1)` `(1)/(n+2)` `(1)/(n+1)` ANSWER :D

Discussion

3423.	A set of parallel chords of the parabola y^(2) =4ax have their mid-points on
Answer» any straight LINE through the VERTEX any straight line throught the FOCUS a straight line parallel to the AXIS a straight line parallel to the latusrectum Answer :C

Discussion

3424.	If (1+2x)/(1-x-x^(2)) =a_(0)+a_(1)x +a_(2)x^(2) +a_(3)x^(3)+… then find (a_(0), a_(1), a_(2), a_(3)).
Answer» SOLUTION :N/A

Discussion

3425.	which of the following is divisible byx^(2) - y^(2) AAxne y ?
Answer» `x^(N) - y^(n) ,AAn in N` `x^(n) +y^(n),AAn in N` `(x^(n) - y^(n))(x^(2n+1)+y^(2n+1)),AAnin N` `(x^(n)-y^(n))(x^(m)+y^(m)),AAm,nin N` ANSWER :C

Discussion

3426.	Find the area of the region bounded by y== cos x and y=1 -(2x)/(pi)
Answer» ANSWER :`(1- (PI)/(4))`

Discussion

3427.	If A = (1, 8, 4) , B= (2, -3, 1) , then the direction cosines of a normal to the plane AOB is
Answer» A`2/sqrt(78) , 5 /sqrt(78), (-7)/sqrt(78)` B`(2sqrt(10))/9, (7sqrt(10))/90, (-19sqrt(10))/90` C`4/sqrt(218) , 9 /sqrt(218), (-11)/sqrt(218)` D`2/11, 6/11, (-9)/11` ANSWER :B

Discussion

3428.	(-3, lemda, 1) bot(1,0,-3) implieslemda = ____
Answer» 0 1 impossible to find any REAL number Answer :C

Discussion

3429.	Objective function of a linear programming problem is a-
Answer» FUNCTION to be optimized constraint linear function of the VARIABLES to be optimized relation AMONG the variables Answer :C

Discussion

3430.	If a, b, c are unit vectors, then the maximum value of abs(a+2b)^(2)+abs(b+3c)^(2)+abs(c+4a)^(2) is
Answer» 28 21 48 58 Answer :B

Discussion

3431.	The integrating factor of the differential equation : (dy)/(dx)(xlogx)+y=2 logxis :
Answer» Answer :`y LOG x = (-2)/(x)(1 + log \|x\|) + C`

Discussion

3432.	Integration of some particular functions : int(e^(x))/(e^(2x)+e^(x)+1)dx=.....+c
Answer» `(1)/(SQRT(3))sec^(-1)((2E^(x)+1)/(sqrt(3)))` `TAN^(-1)(1+e^(x))` `(2)/(sqrt(3))tan^(-2)((2e^(x)+1)/(sqrt(3)))` `(1)/(sqrt(3))tan^(-2)((e^(x)+1)/(sqrt(3)))` Answer :C

Discussion

3433.	If [x] denotes the greatest integer le x, and , b are two odd integers, then number of solution of [x]^(2) + a[x] + b =0 is
Answer» 1 0 2 infinite Answer :B

Discussion

3434.	A(3,2,0),B(5,3,2) and C(-9,6,-3) are the vertices of triangle ABC. If the bisec-tor of angleBAC meets BC at D then D is
Answer» `((19)/(8),(57)/(16),(17)/(16))` `((-19)/(8),(57)/(16),(17)/(16))` `((19)/(8),(-57)/(16),(17)/(16))` `((19)/(8),(57)/(16),(-17)/(16))` ANSWER :A

Discussion

3435.	Evaluate Lt_(x to 0)(e^(sin x) - 1)/(x - 3)
Answer» ANSWER :A

Discussion

3436.	Three points A,B and C taken on rectangular hyperbola xy = 4 where B(-2,-2) and C(6,2//3). The normal at A is parallel to BC, then
Answer» circumcentre of `DeltaABC` is `(2,-2//3)` equation of circumcircle of `DeltaABC` is `3X^(2)+3y^(2) -12x + 4y - 40 =0` orthocenter of `DeltaABC` is `((2)/(sqrt(3)),2sqrt(3))` none of these Solution :Given `B(-2,-2)` and `C(6,2//3)` on hyperbola Let `A(2t,2//t)` Differentiating curve `xy' +y =0` or `y' =- y//x =- 1//t^(2)` Slope of normal at A is `t^(2)` Normal is PARALLEL to BC, which has slope `((2)/(3)+2)/(6+2)=(1)/(3)=t^(2)` `:.t= (1)/(sqrt(3))` `:. A ((2)/(sqrt(3)),2sqrt(3))` ALSO (slope of AB) `xx` (Slope of `AC) =-1` `:. DeltaABC` is a right angled `Delta`. Circum-center of `DeltaABC` is mid-point `BC,(2,-2//3)` Circum-circle of `DeltaABC` is circle on BC as diameter. Orthocenter of `DeltaABC` is at A.

Discussion

3437.	Different words are being formed by arranging the letters of the word "SUCCESS". The probability of word in which no two C and no two S are together is
Answer» `2/35` `8/35` `24/105` 1 Answer :B

Discussion

3438.	If I_(n) = int Sec^(n) xdx thenI_(8) - (6)/(7) I_(6) =
Answer» `(Sec^(6) " X tanx")/(7)` `(-Sec^(6) " x tanx")/(7)` `(Sec^(6) " x tanx")/(8)` `(-Sec^(6) " x tanx")/(9)` ANSWER :A

Discussion

3439.	The solution of the differential equation (2x-3y+5)dx+(y-6x-7)dy=0, is
Answer» `3x-3Y+8 LOG \|6x-9y-1\|=c` `3x-9y+8 log \|6x-9y-1\|=c` `3x-9y+8 log \|2X-3y-1\|=c` `3x-9y+4 log \|2x -3y -1\|=c` ANSWER :B

Discussion

3440.	Show that the given differential equation is homogeneous and solve each of them. ydx+xlog((y)/(x))dy-2xdy=0
Answer» ANSWER :`CY = LOG \|(y)/(X)\|-1`

Discussion

3441.	If n(A) = 4, then number of symmetric relations that can be defined on A is
Answer» `2^(10)` `2^(6)` `2^(8)` `2^(16)` ANSWER :A

Discussion

3442.	IF ais a unitvector, then\| a xxhati\|^2 +\| a xx hatj \|^2 + \|a xx hatk \|^2 =
Answer» 2 4 1 0 Answer :A

Discussion

3443.	I. The geometric mean of 2,4,16 and 32 is a II. The strength of 7 colleges in a city are 385, 1748, 1343, 1935, 786, 2874 , 2108. Then the median strength is b. II. The algebric sum of the deviations of 20 observations measured from 30 is 2. The mean of these observations is c.
Answer» `a LT B lt C` `b lt c lt a` `c lt a lt b` `a lt c lt b` ANSWER :D

Discussion

3444.	If 'alpha' be the non-real n^(th) roots of unity, then 1+3alpha+5alpha^2+…...(2n-1)alpha^(n-1) is equal to
Answer» `(2N)/(1-alpha)` `(N)/(1-alpha)` `(n)/(2(1-alpha))` `(2n)/(alpha-1)` ANSWER :D

Discussion

3445.	Refer to question 41 above. If a white ball is selected, what is the probability that it come from ltbr?. (i) Bag II?(ii) Bag III?
Answer» <P> SOLUTION :REFERRING to the PREVIOUS solution, using Bay's theorem, we have (i) `P(E_(2)//F)=(P(E_(2))cdotP(F//E_(2)))/(P(E_(1)cdotP(F/E_(1))+P(E_(2))cdotP(F//E_(2))+P(E_(3))cdotP(F//E_(3)))` `=(2/6cdot1/3)/(1/6cdot0+2/6cdot1/3+3/6cdot1)=(2/18)/(2/18+3/6)` `(2//8)/((2+9)/(18))=2/11` (i) `P(E_(3)//F)=(P(E_(3))cdotP(F//E_(3)))/(P(E_(1)cdotP(F/E_(1))+P(E_(2))cdotP(F//E_(2))+P(E_(3))cdotP(F//E_(3)))` `=(3/6cdot1)/(1/6cdot0+2/6cdot1/3+3/6cdot1)` `=(3/6)/(2/18+3/6)=(3//6)/(2/18+9/18)=9/11`

Discussion

3446.	int2xsin2xcosxdx
Answer» SOLUTION :`int2xsin2xcosxdx` =`INTX(sin3x+sinx)dx` =`intx.sin3xdx+intxsinxdx` =`x.(-(COS3X)/3)-int1.(-(cos3x)/3)dx+x.(-COSX)-int1.(-cosx)dx` =-1/3xcos3x+1/9sin3x-xcosx+sinx+C

Discussion

3447.	Find derivatives of the following functions.cot^-1frac{sqrt(1-x^2)}{x}
Answer» Solution :`y=COT^(-1)frac{SQRT(1-x^2)}{x}` `[Put x=SIN THETA` `cot^(-1)fracsqrt(1-sin^2theta)sintheta=cot^(-1)fraccosthetasintheta` `cot^(-1)cottheta=theta =sin^(-1)x` `dy/dx=1/sqrt(1-x^2)`

Discussion

3448.	A cubical die is thrown. Find the mean and variance of X, giving the number on the face shows up.
Answer» ANSWER :`(35)/(12)`

Discussion

3449.	Let 3x^(2) + 8xy - 3y^(2) = 0 represent the lines L_(1), L_(2) and 3x^(2) + 8xy - 3y^(2) + 2x -4y -1 =0 represent the lines, L_(3), L_(4). Let L be the line joining the points of intersection of L_(1), L_(3) and L_(2) and L_(4). Then , the area (in sq units) of the triangle formed by L with the coordinate axes is
Answer» `(1)/(2)` `(1)/(4)` `(1)/(8)` `(1)/(16)` Answer :D

Discussion

3450.	If alpha, beta ,gamma, delta are the roots of the equation x^(4) - x^(3) - 7x^(2) + x + 6 = 0 then sum alpha^(6) =
Answer» 123 36 149 795 Answer :1

Discussion

Explore topic-wise InterviewSolutions in .

Evaluate int_(0)^(100)e^(x-[x])dx where [ ] denotes the greatest integer function.

State which of the following are positive ?sec 199^@

A:int tan^(4)x sec^(2)x dx=(1)/(5)tan^(5)x+c R:int[f(x)]^(n) f'(x) dx=([f(x)]^(n+1))/(n+1)+c

If the sides of a triangle are 3cm, 2cm and 4cm then the cosine of the greatest angle is equal to

If f is greatest integer function and g is a modulus functions the find . (gof) (-1/3)-(fog)(-1/3) .

If f(x)={{:(x^2+Ax+5, x in Q),(1+x , x in R ~ Q):} is continuous at exactly two points , then the possible values of A are in

Let a, b and c be three non-zero vectors such that no two of these are collinear. If the vector a+2b is collinear with c and b+3c is collinear with a (lambda being some non-zero scalar), then a+2b+6c equals to

The equation of asymptotes of the hyperbola 4x^(2)-9y^(2)=36 is

If a student is eligible for all the prizes is

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

Find the order and degree (if defined) of the following differential equations. (d^4y)/(dx^4) + sin (y''') = 0

Evalute the following integrals int (1)/(n sqrt((x - a)^(n- k) (x - b)^(n+k) ) )dx ( Hint Put (x- a)/(x - b ) = t )

Find intcosxlog(sinx)dx.

If f and g two functions are defined as : f= {(1,2),(3,6),(4,5)} and g = {(2,3),(6,7),(5,8)}, then find gof.

Let set S consists of all the points (x, y) satisfying 16x^2+25y^2 le 400. For points in S let maximum and minimum value of (y-4)/(x-9) be M and m respectively, then:

Consider the equation sec theta +cosec theta=a, theta in (0, 2pi) -{pi//2, pi, 3pi//2} If the equation has two distinct real roots, then

If P(x_(1), y_(1)) is a point such that the length of the tangents from it to the circles x^(2) + y^(2) - 4x - 6y - 12 = 0 and x^(2) + y^(2) + 6x + 18y + 26 = 0 are in the ratio 2 : 3, then the locus of P is

Find all 3-digit numbers which are the sums of the cubes of their digits.

Find the second order derivatives of the function. x^(2)+3x+2

If veca is a non zero vector of magnitude 'a' and lambda a nonzero scalar, then lambda vec a is unit vector if

Shows that the following functions do not possess maximum or minimum. x^3

If A(i=1,2,3….n) are n independent events with P(A) = (1)/(1+i) for each i , then the probability that none of A, occur is :

A set of parallel chords of the parabola y^(2) =4ax have their mid-points on

If (1+2x)/(1-x-x^(2)) =a_(0)+a_(1)x +a_(2)x^(2) +a_(3)x^(3)+… then find (a_(0), a_(1), a_(2), a_(3)).

which of the following is divisible byx^(2) - y^(2) AAxne y ?

Find the area of the region bounded by y== cos x and y=1 -(2x)/(pi)

If A = (1, 8, 4) , B= (2, -3, 1) , then the direction cosines of a normal to the plane AOB is

(-3, lemda, 1) bot(1,0,-3) implieslemda = ____

Objective function of a linear programming problem is a-

If a, b, c are unit vectors, then the maximum value of abs(a+2b)^(2)+abs(b+3c)^(2)+abs(c+4a)^(2) is

The integrating factor of the differential equation : (dy)/(dx)(xlogx)+y=2 logxis :

Integration of some particular functions : int(e^(x))/(e^(2x)+e^(x)+1)dx=.....+c

If [x] denotes the greatest integer le x, and , b are two odd integers, then number of solution of [x]^(2) + a[x] + b =0 is

A(3,2,0),B(5,3,2) and C(-9,6,-3) are the vertices of triangle ABC. If the bisec-tor of angleBAC meets BC at D then D is

Evaluate Lt_(x to 0)(e^(sin x) - 1)/(x - 3)

Three points A,B and C taken on rectangular hyperbola xy = 4 where B(-2,-2) and C(6,2//3). The normal at A is parallel to BC, then

Different words are being formed by arranging the letters of the word "SUCCESS". The probability of word in which no two C and no two S are together is

If I_(n) = int Sec^(n) xdx thenI_(8) - (6)/(7) I_(6) =

The solution of the differential equation (2x-3y+5)dx+(y-6x-7)dy=0, is

Show that the given differential equation is homogeneous and solve each of them. ydx+xlog((y)/(x))dy-2xdy=0

If n(A) = 4, then number of symmetric relations that can be defined on A is

IF ais a unitvector, then| a xxhati|^2 +| a xx hatj |^2 + |a xx hatk |^2 =

I. The geometric mean of 2,4,16 and 32 is a II. The strength of 7 colleges in a city are 385, 1748, 1343, 1935, 786, 2874 , 2108. Then the median strength is b. II. The algebric sum of the deviations of 20 observations measured from 30 is 2. The mean of these observations is c.

If 'alpha' be the non-real n^(th) roots of unity, then 1+3alpha+5alpha^2+…...(2n-1)alpha^(n-1) is equal to

Refer to question 41 above. If a white ball is selected, what is the probability that it come from ltbr?. (i) Bag II?(ii) Bag III?

int2xsin2xcosxdx

Find derivatives of the following functions.cot^-1frac{sqrt(1-x^2)}{x}

A cubical die is thrown. Find the mean and variance of X, giving the number on the face shows up.

If alpha, beta ,gamma, delta are the roots of the equation x^(4) - x^(3) - 7x^(2) + x + 6 = 0 then sum alpha^(6) =