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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 12751. |
Find the antiderivative (or integral) of the following functions by the method of inspection. (ax+b)^2 |
| Answer» SOLUTION :`int (ax+b)^2 dx = (ax+b)^3/(3a) +C | |
| 12752. |
Let I = int_(0)^(1) (sin x)/(sqrt(x)) dx and J= int_(0)^(1) (cos x)/(sqrt(x))dx which one of the following is true |
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Answer» `I lt 2/3 & J lt 2` |
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| 12753. |
Find the volume of the ellipsoid (x^(2))/(a^(2)) + (y^(2))/(b^(2)) + (z^(2))/(c^(2))=1 |
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| 12754. |
If lim_(x to 1) (1+alpha x +beta^2 x)^((gamma//(x-1))=e^3 then the value of 2betagamma+alpha gamma is equal to |
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| 12755. |
Find the shortest distance and the vector equation of the line of shortest distance between the lines given by : (i) vec(r) = (-4 hati + 4 hatj + hatk ) + lambda ( hati + hatj - hatk) and vec(r) = (-3 hati - 8 hatj - 3 hatk) + mu (2 hati + 3 hatj + 3 hatk) (ii) vec(r) = (- hati + 5 hatj ) + lambda ( - hati + hatj + hatk) and vec(r) = ( - hati - 3 hatj + 2 hatk) + mu ( 3 hati + 2 hatj + hatk ). |
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Answer» (ii) `sqrt(42) , vec(r) = (hati + 3 hatj - 2 hatk) + mu ( hati - 4 hatj + 5 hatk)`. |
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| 12756. |
{:(" " Lt),(n rarroo):}pi/n (sin. pi/n + sin. (2pi)/(n)+sin. (3pi)/(n)+......+sin. ((n-1)pi)/(n))= |
| Answer» Answer :C | |
| 12757. |
A man throws a pair of fair dice until he gets a doublet for the first time. Find the probability of getting sum 10 in last throw. |
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| 12758. |
If the polar of (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is always touching the ellipse (x^(2))/(b^(2))+(y^(2))/(a^(2))=1, then the locus of the polar is : |
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Answer» a CIRCLE |
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| 12759. |
Find the equation of the circle which cuts the following circles orthogonally. x^2 + y^2 + 2x + 17y + 4 =0, x^2 + y^2 + 7x + 6y + 11 = 0, x^2 + y^2 - x + 22y + 3 = 0 . |
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| 12760. |
f:[1, infty) rightarrow R : f(x) is a monotonic and differentiable function and f(1)=1 , then number of solutions of the equation f(f(x))= 1/x^(2)-2(x)+2 is /are |
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Answer» 2 |
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| 12761. |
Let nge3 be an integer. For a permutaion sigma=(a_(1),a_(2),.....,a_(n)) of (1,2,.....,n) we let f_(sigma)(x)=a_(n)X^(n-1)+a_(n-1)X^(x-2)+....+a_(2)x+a_(1). Let S_(sigma) be the sum of the roots of f_(sigma)(x)=0 and let S denote the sum over all permutations sigma of (1,2,.....,n) of the numbers S_(sigma). Then- |
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Answer» `Slt0n!` `AAlambda=a_(1)+a_(2)+....+a_(n)` `S=-[(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+.....+(1)/(a_(n)))-n]` `S=n-(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+.....+(1)/(a_(n)))` FOM `A.M.geH.M` `(a_(1)+a_(2)+....+a_(n))((1)/(a_(1))+(1)/(a_(2))+.....+(1)/(a_(n)))GEN^(2)` `Sle-n(n-1)` |
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| 12762. |
if Delta (x)=|{:(a_(1)+x,,b_(1)+x,,c_(1)+x),(a_(2)+x,,b_(2)+x,,c_(2)+x),(a_(3)+x,,b_(3)+x,,c_(3)+x):}| then show thatDelta (x)=0 andthat Delta (x)=Delta(0)+sx. where sdenotesthe sum of all thecofactors of allthe elements in Delta (0) |
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Answer» Solution :`Delta (x)=|{:(a_(1)+x,,b_(1)+x,,c_(1)+x),(a_(2)+x,,b_(2)+x,,c_(2)+x),(a_(3)+x,,b_(3)+x,,c_(3)+x):}|` ` :. Delta (x) =|{:(a_(1)+x,,b_(1)+x,,c_(1)+x),(a_(2)+x,,b_(2)+x,,c_(2)+x),(a_(3)+x,,b_(3)+x,,c_(3)+x):}|+|{:(a_(1)+x,,1,,c_(1)+x),(a_(2)+x,,1,,c_(2)+x),(a_(3)+x,,1,,c_(3)+x):}|` `+|{:(a_(1)+x,,b_(1)+x,,1),(a_(2)+x,,b_(2)+x,,1),(a_(3)+x,,b_(3)+x,,1):}|` APPLYING`C_(2) toC_(2)-xC_(1),C_(3) to C_(3) xC_(1)` in thefirstdet. `C_(1) to C_(1) -xC_(2),C_(3) to C_(3) -xC_(3)-xC_(2)` in theseconddet. `" and " C_(1) to C_(1)-xC_(3),C_(2)to C_(2)-xC_(3)` in thethirddet. we get `Delta (x)=|{:(1,,b_(1),,C_(1)),(1,,b_(2),,c_(2)),(1,,b_(3),,c_(3)):}|+|{:(a_(1),,1,,c_(1)),(a_(2),,1,,c_(2)),(a_(3),,1,,c_(3)):}|+|{:(a_(1),,b_(1),,1),(a_(2),,b_(2),,1),(a_(3),,b_(3),,1):}|` `Delta (0)=|{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|` whichare `b_(2) C_(3) =b_(3)C_(2),C_(2)a_(3)-C_(3)a_(2),a_(2)b_(3)-b_(2)a_(3)` etc Clearly `|{:(1,,b_(1),,c_(1)),(1,,b_(2),,c_(2)),(1,,b_(3),,c_(3)):}| = (b_(2)b_(3) -b_(3)c_(2))+(c_(1)b_(3)-c_(3)b_(1))+(b_(1)c_(2)-b_(2)c_(1))` Whichis the sum ofcofactors of the firstrow ELEMENTS of `Delta (0)` `" similarly "|{:(a_(1),,1,,c_(1)),(a_(2),,1,,c_(2)),(a_(3),,1,,c_(3)):}|" and "|{:(a_(1),,b_(1),,1),(a_(2),,b_(2),,1),(a_(3),,b_(3),,1):}|` are thesum of cofactorsof 2ndrow and3rdelementsrespectively of `Delta (0)` . Hence`Delta(x)=s` where S denotesthe sumof allcofactors of elementsof `Delta(0)` `:. Delta ''(x) =0` SINCE `Delta (x) = s Delta (x) sx+k` So `Delta (0) =k` Hence `Delta(x)= x S+Delta (0)` |
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| 12763. |
Consider the vectors oversetrarra=overset^^i-overset^^j+overset^^k"and"oversetrarrb=2overset^^i-3overset^^j-5overset^^k:If oversetrarra "and" oversetrarrb' are two adjactent sides of a parallelogram, find the area of the parallelogram. |
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Answer» Solution :Area of the PARALLELOGRAM`=|VECAXXVECB|` `SQRT(114)`sq.units. |
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| 12764. |
If A=[{:(cos alpha,sin alpha),(-sin alpha, cos alpha):}] and A^(-1)=A' then find the value of alpha. |
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Answer» SOLUTION :We have, `A=[{:(cos a,sina),(-SIN a,cosa):}]` and `A'=[{:(cos a,-sina),(sina,cosa):}]` Also, `A^(-1)=A`' `rArr A A^(-1)=A A`' `rArr I=[{:(cos alpha,sin alpha),(-sin alpha,cos alpha):}]` `rArr [{:(1,0),(0,1):}]=[{:(cos^(2)alpha+sin^(2)alpha, 0),(0,sin^(2)alpha+cos^(2)alpha):}]` By using EQUALITY of matrices, we get `cos^(2)alpha+sin^(2)alpha=1` which is true for all REAL values of `alpha`. |
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| 12765. |
If (1)/(1!(n-1)!) + (1)/(3!(n-3)!) + (1)/(5!(n-5)!) +….= |
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Answer» `(2^(N-1))/(n!) AA n in N` |
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| 12766. |
Let L denote the set of all straight lines in a plane. Let a Relation R be defined on L by xRy iff x||y, for x,y in L then R is |
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Answer» only REFLEXIVE |
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| 12767. |
Find the second order derivatives of the function x^(3) log x. |
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| 12768. |
Let f:(11,oo)rarr(0,oo) be given by f(x)=underset(l=l)overset(10)prod(1)/((x-1)) |
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| 12769. |
Let X={1,2,3,.......,10} and A={1,2,3,4,5}. Then the number of subsets B of X such that A-B={4} is |
| Answer» Answer :A | |
| 12770. |
If P is a point on the line passing through the point A with position vector 2i + j - 3k and parallel to i + 2j + k such thatAP = 2 sqrt(6) then the position vector of P is |
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Answer» 4I + 5J + k |
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| 12771. |
A coin is tossed for some times. Probability to get head 7 times is the same as probability to get head 9 times then ……….. is the probability to get head two times. |
| Answer» Answer :C | |
| 12772. |
Sketch the graph of the functionf(x)=x^(2//3)(6-x)^(1//3). |
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| 12773. |
A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time. |
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| 12774. |
The rate of change of surface area of a sphere w.r.t. radius is ……….. |
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Answer» `8pi` (DIAMETER) |
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| 12775. |
I: The equation of the plane passing through the point (1, 2, – 3) and parallel to the plane 2x – 3y +z + 5 = 0 is 2x – 3y +z + 7 = 0 II : The equation of the plane passing through the point (1, 2, 3) and parallel to xy-plane is x + 2 = 0 |
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Answer» only I is TRUE |
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| 12776. |
Given that all three faces are different in a throw of three dice, find the probability that the sum is 9. |
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Answer» <P> Solution :Let A be the EVENT thet the sum is 9.`THEREFORE A cap B`={(1,3,5),(1,5,3),(3,5,1),(3,1,5),(5,1,3),(5,3,1),(1,2,6),(1,6,2),(2,1,6),(2,6,1),(6,2,1),(6,1,2),(2,3,4),(2,4,3),(3,2,4),(2,3,4),(3,4,2),(4,3,2) } `therefore absA capB=18` `therefore P(A | B )=(P(A cap B))/(P(B))=(18//216)/(20//216)=9/10` |
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| 12777. |
A sofa costs $50 less than three times the cost of a shair. If the sofa and chair together cost $650, how much more does the sofa cost than the chair ? |
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Answer» `$175` `{{:(s = 3c -50),(s+c=650):}` The TOP EQUATION is already solved for s, so substitute `3c-50` into the bottom equation for s and solve for c: `3c-50+c =650` ` 4c-50=650` `4c =700` `c = 175` Remember to check if you solved for the right thing! The chair costs `$175,` so the sofa costs `3 (175)-50=525-50=$475.` This means the sofa costs `$475-$175=$330` more than the chair. Therefore, (C ) is correct. |
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| 12778. |
In three dimensions there may be more than one point, which are equidistant from three given non-collinear points A, B, C. One of these points will be circumcentre of the triangle ABC. The y-coordinate of orthocenter of the triangle ABC. |
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Answer» `(3A^(2)c^(2)-a^(2)b^(2)-b^(2)c^(2))/(a^(2)b^(2)+b^(2)c^(2)+c^(2)a^(2)).b` |
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| 12779. |
If A+B+C=90^(@) then sin^(2)A+sin^(2)B+sin^(2)C= |
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Answer» `1-2sin A sin B sin C` |
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| 12780. |
In three dimensions there may be more than one point, which are equidistant from three given non-collinear points A, B, C. One of these points will be circumcentre of the triangle ABC. The y-coordinate of the circumcentre of triangle ABC must be |
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Answer» `(ac)/(a+b+C)` |
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| 12781. |
Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6 - digit numbers that are divisible by 7. |
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| 12782. |
If the normal at t_(1)on the parabola y^(2)=4ax meet it again at t_(2) on the curve then t_(1)(t_(1)+t_(2))+2 = |
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Answer» t |
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| 12783. |
Coefficient of x^(11) in the expaJ}sion of (1 + 3x + 2x^(2))^(6) is ________ |
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| 12784. |
If a != 0, then the equation (x-a-1)/(x-a)=(a+1)-(1)/(x-a) has |
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Answer» TWO ROOTS |
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| 12785. |
Let the position vectors of vertices A,B,C of DeltaABC be respectively veca,vecb and vecc. If vecr is the position vector of the mid point of the line segment joining its orthocentre and centroid then (veca-vecr)+(vecb-vecr)+(vecc-vecr)= |
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Answer» 1 |
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| 12787. |
A triangle with vertices (4, 0), (-1, -1), (3, 5) is |
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Answer» ISOSCELES and RIGHT ANGLED |
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| 12789. |
Let P be an interior point of a triangle ABC and AP, BP, CP meet the sides BC, CA, AB in D, E, F, respectively. Show that (AP)/(PD)=(AF)/(FB)+(AE)/(EC). |
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Answer» SOLUTION :Since A, B, C, P are co-planar, there exists four scalars `X,y,z,w` not all zero simultaneously such that `""xveca+yvecb+zvecc+wvecp=0` where `"" x+y+z+w=0` Also, `""(xveca+wvecp)/(x+w)=(yvecb+zvecc)/(y+z)` HENCE, `""(AP)/(PD)=-(w)/(x)-1` Also `""(xveca +yvecb)/(x+y)=(zvecc+wvecp)/(z+w)` `RARR""(AF)/(FB)=(y)/(x)` Similarly, `""(AE)/(EC)=(z)/(x)` THUS, to show that `-(w)/(x)-1=(y)/(x)+(z)/(x)` `rArr""x+y+z+w=0` which is true. Hence proved. 
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| 12790. |
Evaluate the definite integrals int_(0)^(pi)(xtanx)/(secx+tanx)dx |
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| 12791. |
How many 4 letter words can be formed using the letters of the word 'ARTICLE' such that each word must contain atleast one vowel. |
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| 12792. |
-((4)/(5)+(4^(2))/(5^(2).2)+(4^(3))/(5^(3).3)+(4^(4))/(5^(4).4)+……oo)= |
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Answer» `"LOG"(9)/(5)` |
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| 12793. |
The solution set of the constraints 2x+3y le 6, 5x+3y le 15" and "x ge 0, y ge 0 does not include ……. point. |
| Answer» ANSWER :D | |
| 12796. |
Expand tan x maclaurin 's series in asending powers of x upto 5^(th) power for -(pi)/(2) lt x lt (pi)/(2). |
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| 12797. |
Converse of the stetement" if x^(2) is odd then x is odd" is |
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Answer» if `X^(2)` is EVEN then x is even |
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| 12798. |
Find all 6-digit natural numbers a_(1) a_(2)a_(3)a_(4)a_(5)a_(6) formed by using the digits 1,2,3,4,5,6 once each such that number a_(1) a_(2) a_(3)…a_(k) is divisible by k for 1 le k le 6 |
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| 12800. |
Let A_(n) = {x in C : |z|^(2) le 1/n} for each n in N. Then underset(n =1)overset(infty)cap A_(n) is |
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Answer» a singleton set |
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