28278 + Interview Questions in Maths in Class 12 Page 74 InterviewSolution

3651.	Prove that the following equations has no solutions. (i) sqrt((2x+7))+sqrt((x+4))=0 (ii) sqrt((x-4))=-5 (iii) sqrt((6-x))-sqrt((x-8))=2 (iv) sqrt(-2-x)=root(5)((x-7)) (v) sqrt(x)+sqrt((x+16))=3 (vi) 7sqrt(x)+8sqrt(-x)+15/(x^(3))=98 (vii) sqrt((x-3))-sqrt(x+9)=sqrt((x-1))
Answer» Solution :(i) We have `sqrt((2x+7))+sqrt((x+4))=0` This equation is defined for `2x+7ge0` and `x+4ge0implies{(xge-7/2),(xge-4):"` `:.xge-7/2` For `xge-7/2` , the left hand SIDE of the original equation is positive, but right hand side is ZERO. Therefore, the equation has no roots. (ii) We have `sqrt(x-4))=-5` The equation is defined for `x-4ge0` `:.xge4` For `xge4`, the left hand side of the original equation is positive but right hand side is negative. Therefore, the equation has no roots. (iii) We have `sqrt((5-x))-sqrt(x-8)=2` The equation is defined for `6-xge0` and `x-8ge0` `:.{(xle6),(xge8):}` Consequently, there is no `x` for which both expressions WOULD have sense. Therefore, the equation has no roots. (iv) We have `sqrt((-2-x))=root(5)((x-7))` This equation is defined for `-2-xge0impliesxle-2` For `xle-2` the left hand side is positive, but right hand side is negative. Therefore, the equation has no roots. (v) We have `sqrt(x)+sqrt((x+16))=3` The equation isdefined for `xge0` and `x+16ge0implies{(xge0),(xge-16):}` Hence `xge0` For `xge0` the left hand side `GE4`, but right hand side is 3. Therefore, the equation has no roots. (vi) We have `7sqrt(x)+8sqrt(-x)+15/(x^(3))=98` For `xlt0` the expression `7sqrt(x)` is meaningless, For `xgt0`, the expressionn `8sqrt(-x)` is meaningles and for `x=0` the expression `15/(x^(3))` is meaningless. Consequently the left hand side of the original equation ils meaningless for any `x epsilonR`. Therefore the equation has no roots. (vii) We have `sqrt((x-3))-sqrt((x+9))=sqrt(x-1)` This equation is defined for `{(x-3ge0),(x+9ge0),(x-1ge0):}implies{(xge3),(xge-9),(xge1):}` Hence `xge3` For `xge3, sqrt(x-3)ltsqrt(x+9)` i.e. `sqrt((x-3))-sqrt((x+9))LT0` Hence for `xge3`,the left hand side of the original equation is negative and right hand side is positive. Therefore, the equation has no roots.

Discussion

3652.	Evaluate int" e"^(x)sinx dx
Answer» ANSWER :`1=1/2e^(X)(sinx-cosx)`

Discussion

3653.	Usingintegrationfind theareaof theregionboundedby thetrianglewhosevertices are (-1,1) , (0,5) and (2,3)
Answer» ANSWER :`(15)/(2)` SQ . UNITS

Discussion

3654.	If (1 + ax)^(n) = 1 + 6x + 27/2 x^(2) + ...... + a^(n)x^(n), then the values of a and n are respectively
Answer» 2,3 3,2 `3/2`,`4` 1,6 Answer :C

Discussion

3655.	Fill in the gaps with correct answer . sin 105^@.cos105^@ = _____.
Answer» SOLUTION :`-(1)/4`

Discussion

3656.	Let E and F be the events such that, P(E )=(1)/(3), P(F) = (1)/(4) and P(E cap F)=(1)/(5), find P((barF)/(barE)).
Answer» `(37)/(40)` `(3)/(4)` `(4)/(7)` NONE of these Solution :N/A

Discussion

3657.	Find the power of the point P w.r.t. the circle S =0 whenP(2,4)and S = x^(2) + y ^(2)- 4x - 6y -12
Answer» ANSWER :` -24`

Discussion

3658.	The value of sum_(r=1)^(n) (-1)^(r+1)(""^(n)C_(r))/(r+1) is equal to
Answer» `-1/(N+1)` `-(1)/(n)` `(1)/(n+1)` `(n)/(n+1)` SOLUTION :`UNDERSET(r=1)overset(n)SUM(-1)^(r+1)(.^(n)C_(r))/((r+1))= (1)/(n+1)underset(r=1)overset(n)sum(-1)^(r+1).^(n+1)C_(r+1)` `= 1/(n+1)(0-1+(n+1))= (n)/(n+1)`

Discussion

3659.	int(dx)/((x^(2)+a^(2))^(2)')(agt0)
Answer» ANSWER :`(1)/(2A^(3))[tan^(-1)((x)/(a))+(1)/(2)sin2(tan^(-1)((x)/(a)))]+C`

Discussion

3660.	Find the value of k if the point (4,2)and (k,-3)are conjugate points with respect to the circles x^(2)+ y ^(2)-5x +8y +6=0
Answer» ANSWER :` RARR K= (28)/(3)`

Discussion

3661.	Expand the following (-9a/2+b)^7
Answer» SOLUTION :`((-9A)/2+b)^7` = `((-9a)/2)^7+"^7C_1 ((-9a)/2)^(7-1) b^1+^7C_2 ((-9a)/2)^7-2b^2+ .... + b^7`

Discussion

3662.	An integrating factor of the differential equation (1-x^(2))(dy)/(dx)+xy=(x^(4))/((1+x^(5)))(sqrt(1-x^(2)))^(3) is
Answer» `SQRT(1-X^(2))` `(x)/(sqrt(1-x^(2)))` `(x^(2))/(sqrt(1-x^(2)))` `(1)/(sqrt(1-x^(2)))` ANSWER :D

Discussion

3663.	Five red marbles, four white marbles and three blue marbles of the same shape and size are placed in a row .Find the total number of possible arrangements.
Answer» Solution :5 red , 4 white and 3 blue marbles of the same SIZE and shape are PLACED in a row. `:.` The total number of marbles is 12 out of which 5 are of ONE kind, 4 are of 2nd kind and 3 are of 3rd kind. `:.`The total number of possible ARRANGEMENTS `=(12!)/(5!4!3!)=(1211109876)/(43232) `=12111037=27720`

Discussion

3664.	Find the equation of tangent to the curve given by x=a sin^(3)t, y=b cos^(3)t at a point where t=(pi)/(2).
Answer» ANSWER :y = 0

Discussion

3665.	The sum of first n terms of the series 1 + (1 + x) y+ (1 + x + x^(2)) y^(2) + (1 + x + x^(2) + x^(3))y^(3) + ……….. is
Answer» `(1/(1 - X))[(1-y^(N))/(1 - y) - y ((1 - x^(n)y^(n))/(1 - XY))]` `(1/(1 - x))[(1-y^(n))/(1 - y^(2)) - x ((1 - x^(n)y^(n))/(1 - xy))]` `(1/(1 - x))[(1-y^(n))/(1 - y) - x^(2) ((1 - x^(n)y^(n))/(1 - xy))]` `(1/(1 - x))[(1-y^(n))/(1 - y) - 2x ((1 - x^(n)y^(n))/(1 - xy))]` Answer :D

Discussion

3666.	If int(e^(x)-1)/(e^(x)+1)dx = f(x)+c, then f(x) is equal to
Answer» `2log(E^(X)+1)` `log(e^(2X) -1)` `2log(e^(x)+1)-x` `log(e^(2x)+1)` ANSWER :A

Discussion

3667.	Tibu and Babu are playing a game in which both the players roll a pair of dice. Tibu wins if the sum of the numbers appearing on the dice is a prime number while Babu wins if the product of the numbers appearing on the dice is prime. The chance that no one wins is (both players are allowed to win simultaneously)
Answer» `(19)/(36)` `(1)/(2)` `(17)/(36)` `(4)/(9)` ANSWER :C

Discussion

3668.	Which of the following fractions is larger? 9//10""10//1
Answer» ANSWER :`10//11`

Discussion

3669.	If a compound proposition obtained by combining two propositions p and q is false only when both p and q are false and true in all the other cases, then the compound proposition is
Answer» CONJUCTION disjunction implication equivalence. Answer :B

Discussion

3670.	Find all positive real numbers x, y, z such that: 2x-2y + 1/z = 1/(2014), 2y-2z + 1/x = 1/(2014), 2z-2x + 1/y = 1/(2014)
Answer» ANSWER :`x=y=z=2014`

Discussion

3671.	Find the value of the integral underset(0)overset(pi)int sin^(3)dx cos^(6) x dx
Answer» ANSWER :`(4)/(63)`

Discussion

3672.	After somehow escaping from the previous castle, pangs of hunger grip him. Mario visits a chocolate store nearby where the shopkeeper has 9 jars filled with 4,2,6,7,3,4,5,8,3 jellyfishes respectively. Each second he is allowed to either double the content of one jar, or eat 9 jellyfishes (one from each jar). Can Mario always empty all the jars using these moves?If yes, then give the minimum time required to empty the jars ? (Answer 0 if no)
Answer» Solution : First note that a jar shall not become zero with other jars being nonzero.(Think !). Yes mario can always empty the jars. Basic idea is to reduce one from every jar until a jar contains only one, then double that jar and CONTINUE. This will ensure emptying, though notin minimum steps. For minimum steps we shall reduce the unnecessary steps. The method is to first double each jar until it reaches a number less than or equal to the highest number, then deduct all jars by one until a jar BECOMES one OR if a jar can again be doubled under above condition (note that this doubling will not add any extra work & infact reduce some.). Then check above rule after every deduction & double the necessaryjars. For eg, CONSIDER only 2 jars with 1 4 as initial counts. Then he shall follow these steps 1. Double the jar 1 (GIVES 2 4 ) 2. Double the jar 1 (gives 4 4 ) 3. Reduce both jars one by one `(3 3 gt 2 2 gt 1 1 gt 0 0 )` ( 4 times )Which gives 6as the minimum steps to empty the jars.eg 2: in the case 2 6 8, the number shall drop as follows (11 steps)2 6 8 gtgt 4 6 8 gtgt 8 6 8 (double each jar close to highest number)`8 6 8 gt 7 5 7 gt6 4 6 gt 5 3 5 gt4 2 4` (reduce all ONLY until there is a double possibleagain!)`4 2 4 gtgt 4 4 4 gt3 3 3gtgt2 2 2gtgt1 1 1gtgt0 0 0`(then follow same procedure)Now that you have an idea, the original question will be easy to answer. Which gives that minimum 19 steps are required for emptying the jars.

Discussion

3673.	Find a vector in the opposite direction of a vector -3hati+2sqrt(3)hatj-2hatk which has magnitude 20 units.
Answer» ANSWER :`12hati-8sqrt(3)hatj+8hatk`

Discussion

3674.	Let A (0,0) and B(8,2) be two fixed points on the curve y^(3) =\|x\| A point C (abscissa is less than 0) starts moving from origin along the curve such that rate of change in the ordinate is 2 cm/sec. After t_(0) seconds, triangle ABC becomes a right triangle. After t_(0) secods, tangent is drawn to teh curve at point C to intersect it again at (alpha,beta). Then the value of 4alpha+3 beta is
Answer» `4/3` `3/4` 2 1 Answer :D

Discussion

3675.	Find the number of non negative integral solutions of x_1+x_2+x_3+4x_4=20
Answer» ANSWER :536

Discussion

3676.	Let A (0,0) and B(8,2) be two fixed points on the curve y^(3) =\|x\| A point C (abscissa is less than 0) starts moving from origin along the curve such that rate of change in the ordinate is 2 cm/sec. After t_(0) seconds, triangle ABC becomes a right triangle. The value of t_(0) is
Answer» 1 SEC 2 sec `1/4` sec `1/2` sec ANSWER :C

Discussion

3677.	If veca = (2,-2,1), vecb = (2,3,6) and vecc = (-1,0,2), Find the magnitude and direction of veca-vecb+2vecc.
Answer» Solution :`veca-vecb+2vecc = (2,-2,1) - (2,3,6) + 2(-1,0,2) = (2,-2,1) + (-2,-3,-6) + (-2,0,4) = (2-2,-2,-2-3+0, 1-6+4) = (-2, -5, -1) MAGNITUDE of `veca-vecb+2vecc = sqrt(4+25+1) = sqrt(30)` D. CS of `veca-vecb+2vecc` are `-2/sqrt(30), -5/sqrt(30), -1/sqrt(30)`

Discussion

3678.	The sum of the coefficients of odd powers of x in the expansion of (1+x-x^2+x^3)^5 is
Answer» 510 512 521 522 Answer :B

Discussion

3679.	If for a particle position x prop t^(2) then :-
Answer» velocity is CONSTANT ACCELERATION is constant acceleration is variable None of these Answer :A

Discussion

3680.	If A , B , C , D are four points in a straight line such that distance from A to B is 10 , B to C is 5 , C to D is 4 and A to D is 1, then their correct sequence is
Answer» `A - B - C - D ` `C - D - A - B` `A - D - C - B ` `C- B - A - D ` ANSWER :C

Discussion

3681.	If xdy=y(dx+y dy), y gt 0 and y(1)=1, then y(-3)=
Answer» 1 3 5 `-1` ANSWER :2

Discussion

3682.	intsqrt((x+1)/(x+2))dx=
Answer» `SQRT(x^2+3x+2)-(1)/(4)log\|x+(3)/(2)+sqrt(x^2+3x+2)\|+c` `sqrt(x^2+3x+2)+(1)/(4)log\|x+(3)/(2)+sqrt(x^2+3x+2)\|+c` `sqrt(x^2+3x+2)-(1)/(2)log\|x+(3)/(2)+sqrt(x^2+3x+2)\|+c` `sqrt(x^2+3x+2)+(1)/(2)log\|x+(3)/(2)+sqrt(x^2+3x+2)\|+c` Answer :C

Discussion

3683.	Box B_(1) contains 4 white and 6 black balls and box B_(2) contains 6 white and 4 black balls. Now one balanced coin is tossed. If two head comes up then 2 balls are drawn from box B_(1) other wise 2 white balls are drawn from Box B_(2). Then find probability of an event that selected both balls are of white colour.
Answer» `(19)/(60)` `(21)/(60)` `(17)/(60)` `(23)/(60)` Answer :C

Discussion

3684.	findthe area ofthe regionenclosedby theparabola x^2 =yand the lineliney=x +2.
Answer» ANSWER :`9/2` SEQ. UNIT

Discussion

3685.	if ab = 2a^2 + ab then 25 =
Answer» 19 24 15 18 Answer :D

Discussion

3686.	From first 100 natural numbers five are selected at random. Find the probability that no two of them are consecutive
Answer» ANSWER :`1-(96C_(5))/(100C_(5))`

Discussion

3687.	If int(dx)/(x+x^(2011))=f(x)+C_(1) and int(x^(2009))/(1+x^(2010))dx=g(x)+C_(2)(where C_(1) and C_(2) are constants of integration). Let h(x)=f(x)+g(x). If h(1)=0 then h(e) is equal to :
Answer» ANSWER :1

Discussion

3688.	If A is invertible matrix of order 3xx3 then \|A^(-1)\|="........."
Answer» ANSWER :`THEREFORE(A^(-1))=1/(\|A\|)`

Discussion

3689.	If the tangents at and t_(1) and t_(2)= on y^(2)=4ax meets on its axis then
Answer» `t_(1)=t_(2)` `t_(1)=-t_(2)` `t_(1)t_(2)=2` `t_(1)t_(2)=-1` ANSWER :B

Discussion

3690.	Match the equation of the plane in column I and columa II
Answer» ANSWER :A::B::C::D

Discussion

3691.	From 1980 to 1990, the population of Mitannia increased by 6%. From 1990 to 2000, it decreased by 3%. What was the overall percentage change in the population of Mitannia from 1980 to 2000?
Answer» ANSWER :`2.82%` INCREASE

Discussion

3692.	int(dx)/((x+1)sqrt(x^2-2))=
Answer» `-SIN^(-1)((x+2)/(sqrt(2)(x+1)))+C` `-COS^(-1)((x+2)/(sqrt(2)(x+1)))+c` `-sin^(-1)((sqrt(2)(x+1))/(x+2))+c` `- cos^(-1)((sqrt(2)(x+1))/(x+2))+c` ANSWER :A

Discussion

3693.	If the points P(a + 2b + c), Q(2a + 3b), R (b + tc) are collinear, where a, b and c are three non-coplanar vectors, the value of t is
Answer» `-2` `-1//2` `1//2` 2 Solution :It is GIVEN that the POINTS P(a + 2b + c), Q (2a + 3b) and R(B + t c) are collinear. `therefore PQ = lambda QR" for some SCALAR "lambda` `implies a+b-c=lambda(-2a-2b+tc)` `implies (2lambda+1)a+(1+2lambda)b-(t lambda+1)c=0` `implies 2lambda+1=0, 2lambda+1=0, tlambda+1=0` [`because` a, b and c are non-coplanar] implies t = 2

Discussion

3694.	Using Cofactors of elements of second row, evaluate Delta ={:\|( 5,3,8),( 2,0,1),( 1,2,3) \|:}
Answer» ANSWER :`7`

Discussion

3695.	If A = ((cos alpha , -sin alpha ),(sin alpha,cos alpha)) then :
Answer» A is symmetric MATRIX A is a skew - symmetric matrix A is an aothogonal matrix NONE of these Answer :C

Discussion

3696.	The value of lambda for which the vectors 3bar(i)-6bar(j)+bar(k) and 2bar(i)-4bar(j)+lambda bar(k) are parallel is …..
Answer» `(2)/(3)` `(3)/(2)` `(5)/(2)` `(2)/(5)` ANSWER :A

Discussion

3697.	Using the property of determinants andd without expanding in following exercises 1 to 7 prove that \|{:(1+a^2-b^2,2ab,-2b),(2ab,1-a^2+b^2,2a),(2b,-2a,1-a^2-b^2):}\|=(1+a^2+b^2)^3
Answer» ANSWER :`(1+a^2+b^2)^3`

Discussion

3698.	The numberof waysthat theChief ministerand the14ministersof ourstatecansitaroundtablefora conferenceso thatthechiefministercanoccupy thefixedseatis
Answer» `13!` `14!` `(14!)/(2)` `15!` Answer :B

Discussion

3699.	Find which of the following statements convey the same meanings? i . If it is the bride's dress then it has to be red. ii. If it is not bride's dress then it cannot be red. iii. If it is a red dress then it must be the bride's dress. iv. If it is not a red dress then it can't be the bride's dress
Answer» (i, IV) and (II, III) (i, ii) and (iii, iv) (i), (ii), (iii) (i, iii) and (ii, iv) ANSWER :A

Discussion

3700.	Find the odd one out in the following :
Answer» `VECA+vecb` `veca XX vecb` `veca xx (vecb xx VEC c)` `[veca, vecb, vecc]` SOLUTION :N/A

Discussion

Explore topic-wise InterviewSolutions in .

Prove that the following equations has no solutions. (i) sqrt((2x+7))+sqrt((x+4))=0 (ii) sqrt((x-4))=-5 (iii) sqrt((6-x))-sqrt((x-8))=2 (iv) sqrt(-2-x)=root(5)((x-7)) (v) sqrt(x)+sqrt((x+16))=3 (vi) 7sqrt(x)+8sqrt(-x)+15/(x^(3))=98 (vii) sqrt((x-3))-sqrt(x+9)=sqrt((x-1))

Evaluate int" e"^(x)sinx dx

Usingintegrationfind theareaof theregionboundedby thetrianglewhosevertices are (-1,1) , (0,5) and (2,3)

If (1 + ax)^(n) = 1 + 6x + 27/2 x^(2) + ...... + a^(n)x^(n), then the values of a and n are respectively

Fill in the gaps with correct answer . sin 105^@.cos105^@ = _____.

Let E and F be the events such that, P(E )=(1)/(3), P(F) = (1)/(4) and P(E cap F)=(1)/(5), find P((barF)/(barE)).

Find the power of the point P w.r.t. the circle S =0 whenP(2,4)and S = x^(2) + y ^(2)- 4x - 6y -12

The value of sum_(r=1)^(n) (-1)^(r+1)(""^(n)C_(r))/(r+1) is equal to

int(dx)/((x^(2)+a^(2))^(2)')(agt0)

Find the value of k if the point (4,2)and (k,-3)are conjugate points with respect to the circles x^(2)+ y ^(2)-5x +8y +6=0

Expand the following (-9a/2+b)^7

An integrating factor of the differential equation (1-x^(2))(dy)/(dx)+xy=(x^(4))/((1+x^(5)))(sqrt(1-x^(2)))^(3) is

Five red marbles, four white marbles and three blue marbles of the same shape and size are placed in a row .Find the total number of possible arrangements.

Find the equation of tangent to the curve given by x=a sin^(3)t, y=b cos^(3)t at a point where t=(pi)/(2).

The sum of first n terms of the series 1 + (1 + x) y+ (1 + x + x^(2)) y^(2) + (1 + x + x^(2) + x^(3))y^(3) + ……….. is

If int(e^(x)-1)/(e^(x)+1)dx = f(x)+c, then f(x) is equal to

Which of the following fractions is larger? 9//10""10//1

If a compound proposition obtained by combining two propositions p and q is false only when both p and q are false and true in all the other cases, then the compound proposition is

Find all positive real numbers x, y, z such that: 2x-2y + 1/z = 1/(2014), 2y-2z + 1/x = 1/(2014), 2z-2x + 1/y = 1/(2014)

Find the value of the integral underset(0)overset(pi)int sin^(3)dx cos^(6) x dx

Find a vector in the opposite direction of a vector -3hati+2sqrt(3)hatj-2hatk which has magnitude 20 units.

Find the number of non negative integral solutions of x_1+x_2+x_3+4x_4=20

Let A (0,0) and B(8,2) be two fixed points on the curve y^(3) =|x| A point C (abscissa is less than 0) starts moving from origin along the curve such that rate of change in the ordinate is 2 cm/sec. After t_(0) seconds, triangle ABC becomes a right triangle. The value of t_(0) is

If veca = (2,-2,1), vecb = (2,3,6) and vecc = (-1,0,2), Find the magnitude and direction of veca-vecb+2vecc.

The sum of the coefficients of odd powers of x in the expansion of (1+x-x^2+x^3)^5 is

If for a particle position x prop t^(2) then :-

If A , B , C , D are four points in a straight line such that distance from A to B is 10 , B to C is 5 , C to D is 4 and A to D is 1, then their correct sequence is

If xdy=y(dx+y dy), y gt 0 and y(1)=1, then y(-3)=

intsqrt((x+1)/(x+2))dx=

findthe area ofthe regionenclosedby theparabola x^2 =yand the lineliney=x +2.

if a*b = 2a^2 + ab then 2*5 =

From first 100 natural numbers five are selected at random. Find the probability that no two of them are consecutive

If int(dx)/(x+x^(2011))=f(x)+C_(1) and int(x^(2009))/(1+x^(2010))dx=g(x)+C_(2)(where C_(1) and C_(2) are constants of integration). Let h(x)=f(x)+g(x). If h(1)=0 then h(e) is equal to :

If A is invertible matrix of order 3xx3 then |A^(-1)|="........."

If the tangents at and t_(1) and t_(2)= on y^(2)=4ax meets on its axis then

Match the equation of the plane in column I and columa II

From 1980 to 1990, the population of Mitannia increased by 6%. From 1990 to 2000, it decreased by 3%. What was the overall percentage change in the population of Mitannia from 1980 to 2000?

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

If the points P(a + 2b + c), Q(2a + 3b), R (b + tc) are collinear, where a, b and c are three non-coplanar vectors, the value of t is

Using Cofactors of elements of second row, evaluate Delta ={:|( 5,3,8),( 2,0,1),( 1,2,3) |:}

If A = ((cos alpha , -sin alpha ),(sin alpha,cos alpha)) then :

The value of lambda for which the vectors 3bar(i)-6bar(j)+bar(k) and 2bar(i)-4bar(j)+lambda bar(k) are parallel is …..

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1+a^2-b^2,2ab,-2b),(2ab,1-a^2+b^2,2a),(2b,-2a,1-a^2-b^2):}|=(1+a^2+b^2)^3

The numberof waysthat theChief ministerand the14ministersof ourstatecansitaroundtablefora conferenceso thatthechiefministercanoccupy thefixedseatis

Find which of the following statements convey the same meanings? i . If it is the bride's dress then it has to be red. ii. If it is not bride's dress then it cannot be red. iii. If it is a red dress then it must be the bride's dress. iv. If it is not a red dress then it can't be the bride's dress

Find the odd one out in the following :

if ab = 2a^2 + ab then 25 =