InterviewSolution
Saved Bookmarks
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.
| 3752. |
If (1+x+x^(2))^(20) = a_(0) + a_(1)x^(2) "……" + a_(40)x^(40), then following questions. The value of a_(0) + 3a_(1) + 5a_(2) + "……" + 81a_(40) is |
|
Answer» `161 xx 3^(20)` Replacing x by `1//x`, we get `(1+x1/x+1/(x^(2)))^(20) = underset(r=0)overset(40)suma_(r)(1/x)^(r )` or `(1+x+x^(2))^(20) = underset(r=0)overset(40)suma_(r)x^(40-r) ""(2)` SINCE (1) and (2) are same series, COEFFICIENT of `x^(r )` is (1) `=` coefficient of `x^(r)` in (2). `rArr a_(r) = a_(40-r)` In (1) Putting `x = 1`, we get `3^(20) = a_(0)+a_(1)+a_(2)+"...."+a_(40)` `= (a_(0)+a_(1)+a_(2)+"...."+a_(19))+a_(20)+(a_(21)+a_(n+2)+"..."+a_(40))` `= 2(a_(0)+a_(1)+a_(2)+"...."+a_(19))+a_(20)""( :' a_(r) = a_(40-r))` or `a_(0) + a_(1) + a_(2) + "......."+ a_(19) = 1/2 (3^(20)-a_(20)) = 1/2(9^(10) - a_(20))` Also, `a_(0)+3a_(1)+5a_(2)+81a_(40)` `= (a_(0)+81a_(40))+(3a_(1)+79a_(39))+"...."+(39a_(19)+43a_(21))+41a_(20)` `= 82(a_(0) + a_(1) + a_(2) + "......" + a_(19)) + 41a_(20)` `= 41 xx 3^(20)` `a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - a_(3)^(2) + "....."` suggests that we have to MULTIPLY the TWO expansions. Replacing x by `-1//x` in (1), we get `(1-1/x+1/(x^(2)))^(20) = a_(0) - (a_(1))/(x)+(a_(2))/(x^(2))-"...."+(a_(40))/(a_(40))` `rArr (1-x+x^(2))^(20) = a_(0)x^(40) - a_(1)x^(39) + a_(2)x^(38) - "....."a_(40)""(3)` Clearly, `a_(0)^(3) - a_(1)^(2) + a_(2)^(2) + "....."+ a_(0)^(2)` is the coefficeint of `x^(40)` in `(1+x+x^(2)) (1-x+x^(2))^(20)` = Coefficient of `x^(40)` in `(1+x^(2)+x^(4))^(20)` In `(1+x^(2)+x^(4))^(20)` replace `x^(2)`, by y, then the coefficientof `y^(20)` in `(1+y+y^(2))^(20)` is `a_(20)`. Hence `a_(0)^(2) - a_(1)^(2) -"......"+a_(40)^(2) = a_(20)` or `(a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - "....." - a_(19)^(2)) + a_(20)^(2) + (-a_(21)^(2) + "....." + a_(40)^(2)) = a_(20)` or `2(a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - "....." - a_(19)^(2)) + a_(20)^(2) = a_(20)` or `a_(0)^(2) - a_(1)^(2) -"......" - a_(19)^(2) = (a_(20))/(2)[1-a_(20)]` |
|
| 3753. |
If z = x + iy and if the point P in the argand plane represents z then find the locus of P satisfying the following equations Re ((z-4)/(z-2i)) = 0 (z ne 2i) |
|
Answer» |
|
| 3754. |
(iii) If (x^(2)+x+2)/(x^(2)+2x+1)=A+(B)/(x+1)+(C)/((x+1)^(2)), then find the value of A+B+C. |
|
Answer» |
|
| 3755. |
Consider a silver target in coolidge tube to produce x-rays. The acceleratingpotential is 31 kV. E_(K)=25.51 KeV, E_(L)=3.5 1 KeV. "If" lambda_(K alpha)-lambda_("min") is approximately8N pm(in pm), where N is an integer find N. Round off to nearest integer. (Take : hc = 1240 e Vnm) |
|
Answer» `E_(K)-E_(L)=(hc)/(lambda_(k alpha))=25.51-3.5` `lambda_(k alpha)=(1240)/(22)xx10^(-3)=56.36` pm `=16.36 ~=16`pm |
|
| 3756. |
(1+x+x^2 + ……+x^p)^n = a_0 + a_1x + a_2 x^2 + ….+a_(np) x^(np) rArr a_1 + 2a_2 + 3a_3 + ……+np .a_(np) = |
|
Answer» `(np(p+1)^n)/(2)` |
|
| 3757. |
If the tangents at t_(1) and t_(2) on y^(2)=4ax meet on the directrix then |
|
Answer» `t_(1)=t_(2)` |
|
| 3758. |
A car hire firm, fires 2 cars everyday. The number of demands for car per day on an average is 1.5. Find the expected number of days it can reject a demand in 100 working days. |
|
Answer» |
|
| 3759. |
If A=[((2)/(3), 1,(5)/(3)),((1)/(3), (2)/(3), (4)/(3)),((7)/(3), 2, (2)/(3))] and B=[((2)/(5), (3)/(5), 1),((1)/(5), (2)/(5), (4)/(5)),((7)/(5),(6)/(5),(2)/(5))], then compute 3A-5B. |
|
Answer» |
|
| 3760. |
Consider two curves y^2 = 4a(x-lambda) and x^(2)=4a(y-lambda), where agt0 and lambda is a parameter. Show that (i) there is a single positive value of lambda for which the two curves have exactly one point of intersection in the 1st quadrant find it. (ii) there are infinitely many nagetive values of lambda for which the two curves have exactly one points of intersection in the 1st quadrant. (iii) if lambda=-a , then find the area of the bounded by the two curves and the axes in the 1st quadrant. |
|
Answer» |
|
| 3761. |
Translate "A year consists of twelve months while a month does not consist of more than thirty one days" propositions into symbolic form, stating the prime components |
|
Answer» SOLUTION :LET p: A year consists of twelve months. Q :A MONTH consists of more than 31 DAYS. `:.` Answer is `p^^~~q`. |
|
| 3762. |
If oversetrarra"and"oversetrarrbare two collinear vectors, then which of the following are incorrect:a)oversetrarrb =oversetrarra lambdascalar"lambdab) oversetrarra = +-oversetrarrbc)The respective components of oversetrarra "and "oversetrarrb are proportionald)Both oversetrarra"and" oversetrarrb have same direction, but different magnitude. |
|
Answer» `OVERSETRARRB =LAMBDA oversetrarra`"for some scalar"lambda` |
|
| 3763. |
Two coins are tossed once,where E: no tail appears, F: no head appears.Find P (E/F) in each case above. |
|
Answer» Solution :Here S = {HH,HT,TH,TT} E={HH}, F={TT} `RARR` `EnnF`=`phirArr P(EnnF)=0` therefore `P(E/F)=(P(EnnF))/(P(F))`=0 |
|
| 3764. |
Let alpha, betabe two distinct roots of a cos theta+b sin theta =c,where a, b and c are three real constants and theta in [0, 2pi].Then alpha+betais also a root of the same equation, if |
|
Answer» `a+b=c` |
|
| 3765. |
Let P be the point of intersection of the lines represented by r=(i+2j-k)+lambda(2i+3j+4k) (1) and r=(-i-3j+7k)+mu(i+2j-k) (2) If the position vector of P is ai+bj+ck, then |a^(2)-b^(2)+c^(2)|=________ |
|
Answer» |
|
| 3766. |
If 2^(2020)+2021 is divided by 9, then the remainder obtained is |
|
Answer» 0 |
|
| 3767. |
Let f: R to R be defined as f(x) = x^(4) . Choose the correct answer. |
|
Answer» `f` is one-one onto Let `x, y in R and f(x) = f(y)` `rArr "" x^(4) = y^(4) rArr x = pm y` `thereforef` is not one-one. `rArr f` is many one. Again let `f(x) =y ` where `y in R` `rArr "" x^(4) = y ` `rArr "" x= (y)^(1//4)notin R if y = -1 ` `therefore f ` is not onto. Therefore, `f` is neither one-one nor onto. |
|
| 3768. |
The third vertex of an equilateral triangle whose vertices are (2, 4), (2, 6) |
|
Answer» (`2+SQRT3`, 5) |
|
| 3770. |
Integrate the function in Exercise. (sin^(-1)x)^(2) |
|
Answer» |
|
| 3771. |
A cylinder manufacturer makes small and large cylinders from a large piece of cardboard. The large cylinder requires 4 sq. m and small cylinder requires 3 sq. m of cardboard. The manufacturer is required to make at least 3 large cylinder and at least twice as many small cylinder as large cylinders. If 60 sq. m of cardboard is in the stock and profit on the small and large cylinders are Rs. 25 and Rs 35 respectively. How many of each type of cylinders to made to maximize the profit. Also, find the maximum profit. |
|
Answer» |
|
| 3773. |
If bar(x)=(1,-1,0),bar(y)=(0,1,3) and bar(z)=(2,1,1) then bar(x)xx (bar(y)xx bar(z)) = …………. |
|
Answer» `(2,4,2)` |
|
| 3774. |
If the volume of the parallelopiped with a, b and c as coterminous edges is 40 cu units, then the volume of the parallelopiped having b+c, c+a and a+b as coterminous edges in cubic units is |
|
Answer» 80 `therefore` Volume of parallelopiped `=[B+c c+a a +b]=2[abc]` `2xx40=80` CU units. . |
|
| 3776. |
If the straight lines y = 2x, y = 2x + 1, y= -7x, y= -7x+ 1 form a parallelogram, then the area of the parallelogram (in square units) is |
| Answer» Answer :C | |
| 3778. |
A bag contains 6 black balls and unknown number (le 6) of white balls. Three balls are successively drawn and not replaced and are all found to be white. Prove that the chance that a black ball will be drawn in the next draw is (677)/(909). |
|
Answer» |
|
| 3779. |
A tower of height h stands at a point O on the ground. Two poles of height a and b stand at the points A and B respectively such that O lies on the line joining A and B. If the angle of elevation of the top of the tower at the foot of one pole is ame as at the top of the other pole, then h is equal to |
|
Answer» `(a+B)/(AB)` |
|
| 3781. |
For therectangleshownin thestandard( x, y)Coordinateplanebelow, whatare thecoordinateof theunlabeled vertex ? |
|
Answer» `(4,5)` |
|
| 3782. |
Integrate the following functions sqrt(1+x^2/9) |
|
Answer» SOLUTION :`INT sqrt(1+x^2/9) dx = int sqrt(1+(x/3)^2) dx` 1/(1/3)[(x/3)/2 sqrt(1+(x/3)^2) + 1/2 log|x/3 + sqrt(1+(x/2)^2)|]+C` `3[x/6 sqrt(1+x^2/9) + 1/2 log|x/3 + sqrt(1+x^2/9)|]+c` |
|
| 3783. |
There are 20 points in a plane out of which 7 points are collinear and no three of the points are collinear unless all the three are from these 7 points. Find the number of different straight lines. |
|
Answer» |
|
| 3784. |
The vertices of the hyperbola((x-2)^(2))/( 9) -((y-3)^(2))/( 4) =1 are |
|
Answer» (2,3), (-1,3) |
|
| 3785. |
Biological and chemical factory of body is :- |
|
Answer» SPLEEN |
|
| 3786. |
int(1)/([(x-1)^(3)(x+2)^(5)]^((1)/(4)))dx=... |
|
Answer» `(4)/(3)((x-1)/(x+2))^((1)/(2))+C` |
|
| 3787. |
If veca, vecb and vecc are three vectors such that 3veca+4vecb+6vecc=vec0, |veca|=3, |vecb|=3 and |vecc|=4, then the value of -864((veca.vecb+vecb.vecc+vecc.veca)/(6)) is equal to |
|
Answer» |
|
| 3789. |
intx^4e^(2x)dx= |
|
Answer» `(e^(2X))/(4)(2x^4 -4x^3+6x^2 -6x +3 )+C` |
|
| 3790. |
The vectors bar(a) and bar(b) are perpendicular then bar(a)xx{bar(a)xx{bar(a)xx(bar(a)xx bar(b))}} = …………. |
|
Answer» `|bar(a)|^(2)bar(B)` |
|
| 3791. |
The value of sin^(-1) {( sin. pi/3) x/sqrt((x^(2) + k^(2) - kx))} - cos^(_1) {( cos. pi/6) x/sqrt((x^(2) + k^(2) - kx))}" , where" (k/2 lt x lt 2k, k gt 0) is |
|
Answer» `TAN^(-1) ((2x^(2) + SK - k^(2))/(X^(2) - 2xk + k^(2)))` |
|
| 3792. |
The position vectors of four points A, B, C and D in the plane are vec(a),vec(b),vec( c ) and vec(d). If (vec(a)-vec(d)).(vec(b)-vec( c ))=(vec(b)-vec(d)).(vec( c )-vec(a))=0 then D is a ………………is DeltaABC. |
|
Answer» In CENTRE |
|
| 3793. |
Match the items of List-I with the items of List - II andchoose the correct option. |
|
Answer» `{:(A,B,C,D),(II,IV,III,I):}` |
|
| 3796. |
Find the angles which the vector veca = hati-hatj+sqrt2 hatk makes with the coordinates axes. |
|
Answer» Solution :`veca` = `hati-hatj+sqrt2 K` `|veca|` = `sqrt(1+1+2)` = 2 `HATA` = `veca/|veca|` = `1/2hat-1/2hatj+1/sqrt2 k` If `ALPHA,beta` and `gamma` are the angles made by `veca`with co-ordinate axes then `COSALPHA` = `1/2,cosbeta` = -1/2 and `cosgamma` = `1/sqrt2` therefore `alpha` = `pi/3,beta` = `2pi/3` and `gamma` = `pi/4` |
|
| 3797. |
If the equation x^(2)+ax+b=0 and x^(2)+bx+a=0 have a common root, then their other roots satisfy the equation |
|
Answer» `X^(2) + x + AB = 0` |
|
| 3798. |
If direction cosines of a vector of magnitude 3 are (2)/(3), -(a)/(3), (2)/(3) and a gt 0, then vector is ________ |
|
Answer» `i+2j+ 2K` |
|
| 3799. |
let a be unit vector, b = 2i + j - k and c = I + 3k. The maximum value of [a b c] is |
| Answer» | |
| 3800. |
For p, q in R, the roots of (p^(2) + 2)x^(2) + 2x(p +q) - 2 = 0 are |
|
Answer» REAL and equal |
|