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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.
| 4701. |
A: Ifthereare8 pointsin a planenothreeofwhichare onthe samestraight lineexcept4 pointsarecollinearthen thenumberofstraightlinesformedbyjoiningthemis 23. R :IF therearen pointsin a planenothreeof whichare onthe samestraightlineexceptp points are collinearthen thenumberof straightlinesformedbyjoiningthemis""^(n) C_2- ""^(p) C_2 +!. |
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Answer» BothAand Raretrueand RIS thecorrectexplanationof A . |
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| 4702. |
If veca is a nonzero vector of mangitude 'a' and lambda a nonzero scalar , then lambdaveca is unit vector if |
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Answer» `lambda=1` |
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| 4703. |
Order of the differential equation of the family of all concentric circles centered at (h,k) is |
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Answer» 1 |
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| 4706. |
Find the point of intersection of the circle x^(2)+y^(2)+4x+6y-39=0 and the normal at (2,3). |
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| 4707. |
Prove that the point (-1, -2) lies on the circle x^(2) + y^(2) - x y - 8 = 0. Find the coordinates of the other extremity of the diameter through (-1, -2). |
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| 4708. |
Let a,b,c are positive real numbers such that p =a ^(2) b + ab ^(2) c - ac ^(2), q =b ^(2) c+bc^(2) -a ^(2)b-ab ^(2) and r =ac ^(2) +a^(2) c - cb^(2) - bc ^(2) and the quadratic equation px ^(2) +qx+r=0 has equal roots , then a,b,c are in : |
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Answer» A.P. |
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| 4709. |
If x,y,z are nonzero real numbers, then the inverse of matrix A=[{:(x,0,0),(0,y,0),(0,0,z):}] is ……… |
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Answer» `[{:(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1)):}]` |
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| 4710. |
Given that lim_(nrarroo) sum_(r=1)^(n) (log_(e)(n^(2)+r^(2))-2log_(e)n)/n = log_(e)2+pi/2-2, then evaluate :lim_(nrarroo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+2^(2))^(m) "......"(2n^(2))^(m)]^(1//n). |
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| 4711. |
Prove that the sum of all the vectors drawn from the centre of a regular octagon to its vertices is the null vector. |
Answer» Solution : Let O be the centre of a regular OCTAGON ABCDEFGH. Then `vec(OA)` = `-vec(OE)`, `vec(OB)` = `-vec(OF)` `vec(OC)` = `-vec(OG)`, `vec(OD)` = `-vec(OH)` Now `vec(OA)+vec(OB)+vec(OC)+vec(OD)+vec(OE)+vec(OF)+vec(OG)+vec(OH)` =`(vec(OA)+vec(OE)) + (vec(OB)+vec(OF)) + (vec(OC)+vec(OG)) + (vec(OD)+vec(OH))` = `vec0+vec0+vec0+vec0` = 0 |
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| 4712. |
Find the number of proper divisors of 2520. How many of them are divisible by 10. Find their sum. |
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| 4713. |
If AB=0 for the matrices |
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Answer» an ODD MULTIPLE of `(PI)/(2)` |
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| 4714. |
Find the value of b for which difference between maximum and minimum value of x^2 - 2bx - 1 in [0, 1] is 1. |
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| 4715. |
Consider a set of point Rin which is at a distance of 2 units from the line (x)/(1)= (y-1)/(-1)= (z+2)/(2) between the planes x-y+2z=3=0 and x-y+2z-2=0. |
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Answer» The volume of the bounded figure by POINTS R and the planes is `(10//3sqrt3)pi` cube units. ALSO the figure formed is a CYLINDER, whose radius is `r=2` units. Hence, the volume of the cylinder is `""pir^(2)h= pi(2)^(2)*(5)/(sqrt6)= (20pi)/(sqrt6)` cubic units. Also the curved surface area is `""2pirh= 2pi(2)*(5)/(sqrt(6))= (20pi)/(sqrt6)` |
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| 4716. |
If x=a cos ^(3) theta and y = a sin^(3) thetathen (dy)/( dx) = |
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| 4717. |
the ( relative) minimumvalue of( x^2- 3x +2) /(x^2+ 3x+2)is |
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Answer» `-1//11` |
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| 4718. |
Prove that the sum of two functions increasing on a certain open interval is a function monotonically increasing on this interval. Will the difference of increasing functions be a monotomic function?s |
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| 4719. |
Show that the equaton of any circle in the complex plane is of the form z barz+b bar z+ b bar z+c=0,( b in C, c in R) |
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Answer» Solution :Assume the genral FRORM of the equation of a circle in cartesion co-ordinates as `x^(2)+y^(2)2ag+2fy+c=0` `(g , f in R)""..(1)` To write this equation in the complex VARIABLE formlet `(x,y)=Z` Then `(z+ BAR z)/(2)= x, (z- bar z)/(2 i)` `=y=(-i (z bar z))/(2)` `:.x^(2)+y^(2)=|z|^(2)=z barz` Substituting these,RESULTING in equation (1), we obtain `z bar z+g(z+barz)+f(z- barz)(-i)+c=0` i.e., `z bar z+(g- if)z+ (g+ if) bar z+c=0 "" (2)` If `(g+if)=b` , then equation (2) can be writenas `z bar z+ bar b z+b bar z+c=0` |
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| 4720. |
"^(30)C_(0)*^(20)C_(10)+^(31)C_(1)*^(19)C_(10)+^(32)C_(2)*18C_(10)+....^(40)C_(10)*^(10)C_(10) is equal to |
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Answer» `"^(51)C_(41)` `="coefficient of " x^(10) " in" (1-x)^(-11)(1-x)^(-31)` `="coefficient of " x^(10) " in" (1-x)^(-42)` `='^(42+10-1)C_(42-1)=^(51)C_(41)=^(51)C_(10)` |
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| 4721. |
If (y)/(3)=6x, then in terms of y, which of the following is equivalent to x? |
| Answer» ANSWER :D | |
| 4722. |
If P(A)=0.7, P(B)=0.4 then the interval in which P(AnnB) lies is |
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Answer» `[0.1,0.4]` |
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| 4723. |
Explain why the experiment of tossing a coin three times is said to have binomial distribution. |
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| 4724. |
If A is a 2xx2 matrix such that |
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Answer» `A^(2008+I)` |
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| 4726. |
Find the transformed equation of x^(2)+2sqrt3 xy-y^(2) = 2a^(2) when the axes are rotated through an angle 30^(0). |
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| 4727. |
Evalute the following integrals int ((x +1)(x + "log x")^(2))/(x )dx |
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| 4728. |
If |{:(a,b,ax+by),(b,c,bx+cy),(ax+by,bx+cy,0):}|=0 and ax^2+2abxy+cy^2ne0," then "...... |
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Answer» a,B,C are in A.P |
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| 4729. |
Find all polynomials whose coefficients are equal to 1 or - 1 and whose all roots are real. |
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| 4730. |
If the coefficients of r^(th), (r+1)^(th) " and "(r+2)^(th) terms in the expansion of (1+x)^(14) are in an arithmetic progression, then r = |
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Answer» 4 or 10 |
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| 4731. |
A bag contains 6 white balls and 4 black balls. A ball is drawn and is put back in the bag with 5 balls of the same colour as that of the ball drawn. A ball is drawn again at random. What is the probability that the ball drawn now is white. |
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| 4732. |
If x^3 -3x^2 + 4=0 hasa multiplerootthenthatmultipleroot is |
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Answer» 0 |
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| 4733. |
f: R rarr R , f(x) ={{:(-1,xlt0,),(0,x=0 , g:R rarr R ","g(x)=),(1,xgt0,):} 1 + x - [x] then for all x, f (g(x)) = ........ |
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Answer» 1 |
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| 4734. |
Using Rolle's theroem, find the point on the curve y = x (x-4), xin [0,4], where the tangent is parallelto X-axis. |
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Answer» (i) y is a continuous function since`x(x-4)` is a polynomialfunction. Hence, ` y = x (x-4)` iscontinuousin `[0,4]`. (ii) `y'= (x-4).1+x.1=2x-4` which existsin `(0,4)`. Hence, `y` is differentiablein `(0,4)`. (iii) `y(0) = 0(0-4) = 0` and `y(4) = 4(4-4) = 0` ` rArr y(0) =y(4)` Since,conditions of Rolle's theorem are satisfied. Hence, there exists a POINT such that `f(c) = 0` in `(0,4), [:' f(x) = y']` `rArr 2c - 4 = 0` `rArr c = 2` `rArr x =2, y = 2(2-4) = - 4` THUS, `(2,-4)`is the pointon the curve at which thetangent drawnis parallel to X-axis. |
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| 4735. |
Solve for x and y : x[{:(2),(1):}]+y[{:(3),(5):}]+[{:(-8),(-11):}]=[{:(0),(0):}] |
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| 4736. |
int(1)/(root(n)((x-a)^(n-k)(x-b)^(n+k))) ("Hint put"(x-a)/(a-b)=t) |
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| 4737. |
int(dx)/(xsqrt(ax+b)) |
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| 4738. |
int(cos^(3)x)/(sinx)dx |
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| 4739. |
If z is a complex number satisfying |z|^(2)+2(z+2)+3i(z-barz)+4 =0, then complex number z+3+2i lies on |
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Answer» circle witih center 1-5i and radius 4 `|z|^(2)+2(z+barz)+3i(z-barz)+4=0` `RARR zbarz+(2+3i)z+(2-3i)barz+4=0` CLEARLY, it REPRESENTS a circle with centers as `(-2,3)` and radius `=sqrt(4+9-4)=3`. The above equation can be written as `|z-(2+3i)|=4` or , `|z+2-3i|=4 rArr |omega-3-2i+2-3i|=4 rArr |omega-1-5i|=4` `rArr omega` lies on the circle with center 1+5i and radius 4. |
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| 4740. |
If bara,barb barc and bard are non-zero, non-collinear vectors such that bard is perpendicular to bara,barb and barc and (bara.barc)bara=barc then bara.(barb times bard) is equal to |
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Answer» `(barc.bard)/|barc|^2` |
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| 4741. |
If x_(1), x_(2)=3, x_(3)+x_(4)=12 and x_(1), x_(2), x_(3), x_(4) are in increasing G.P then the equation having x_(1), x_(2) as roots is |
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Answer» `X^(2)-3x+2=0` |
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| 4742. |
overset(pi)underset(0)int (x)/(1+sinx)dx |
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| 4744. |
A shopkeeper sells three types of flower seeds A_(1), A_(2), and A_(3). They are sold as a mixture, where the proportions are 4:4:2 respectively. The germination rates of the three type ofseeds are 45%, 60% and 35%. Calculate the probability (i) Of a randomly chosen seed to germinate (ii) That it will not germinate given that the seed is of type A_(3). (iii) That it is of the type A_(2) given that a randomly chosen seed does not germinate. |
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| 4745. |
Provethe followinginequalities. (i) x gt tan^(-1) (x) """for """x in (0,oo) (ii) e^(x) gt x+1 """ for""" x in (0,oo) (iii) (x)/(1+x) le en (1+x) le x"""for """ x in (0,oo) |
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| 4746. |
A vector of magnitude sqrt2 coplanar with hati+hatj+2hati and hati+2hatj+hatk and perpendicular to hati+hatj+hatk is |
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Answer» `-hatj+hatk` `hat(i)+hat(j)+2hat(k)+lambda(hat(i)+2hat(j)+hat(k))=(1+lambda)hat(i)+(1+2 lambda)hat(j)+(2+lambda)hat(k)` it is perpendicular to `hat(i)+hat(j)+hat(k)` `:. 1+lambda+1+2 lambda+2+lambda =0""[ :' a.b=0]` `implies lambda=-1` `:.` Required vector is `-hat(j)-hat(k)`. |
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| 4747. |
Integrate the following inta^(2x)dx |
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Answer» SOLUTION :`inta^(2X)dx [put 2x=t then 2sx=dt or dx=(1/2)dt] `inta^tcdot(1/2)dt=(1/2)inta^tdt` `a^(2x)/(2Ina)+C` |
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| 4748. |
Find the number of all onto functions from the set {1,2,3,.......,n} to itself. |
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| 4749. |
If x=ct and y= (c )/(t), find (dy)/(dx) at t=2 |
| Answer» Answer :B | |
| 4750. |
If y = (log)^(cos x) find (dy)/(dx). |
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