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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.
| 9701. |
Let A,B,C be the three points on the parabola y ^(2)=4ax. If the orthocentre of the triangle ABC is at the focus then show that the circuimcircle of Delta ABC touches the y-axis. |
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| 9702. |
(i) Find maximum value of f(x)=|{:(1+sin^(2)x,cos^(2)x,4sin2x),(sin^(2)x,1+cos^(2)x,4sin2x),(sin^(2)x,cos^(2)x,1+4sin2x):}|. (ii) Let A,B and C be theangles of triangle such that Agt=Bgt=C. Find the minimum value of Delta where Delta=|{:(sin^(2)A,sinAcosA,cos^(2)A),(sin^(2) B,sinBcosB,cos^(2)B),(sin^(2)C,sinCcosC,cos^(2)C):}|. |
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| 9703. |
Find the number of ways of arranging 15 students A_(1),A_(2),…….,A_(15) in a row such that neither A_(2) nor A_(3) be seated before A_(1) |
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| 9704. |
Find the values of k so that the function f is continuous at the indicated point f(x)= {(kx^(2)",","if" x le 2),(3",","if" x gt 2):} at x=2 |
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| 9706. |
If the arithmetic mean of first n natural numbers is ( 38)/( 75)n,then n is equal to ________ |
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| 9707. |
If the line of regression are parallel to the coordinate axis, then the coefficient of correlation is |
| Answer» ANSWER :B | |
| 9709. |
iff(x) ={x^(2)}, where{x} denotesthe fractional partof x , then |
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Answer» F(X)is continuousat x=-2 but not at x=2 |
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| 9710. |
Find the distance of a point (2, 4, -1) from the line (x+5)/(1)=(y+3)/(4)=(z-6)/(-9). |
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| 9711. |
intcosxdx |
| Answer» SOLUTION :`intcosxdx=sinx+C` | |
| 9712. |
The orthocentre of the triangle formed by (8,0) and (4, 6) with the origin is |
| Answer» Answer :A | |
| 9713. |
After having celebrated his brother’s birthday, Max makes his way to Fuchsia city, to take a boat to Cinnabar Island. The sailor, Mr. Hendricks, is however in a dilemma. There is a castle in the middle of the sea on the way to Cinnabar Is land. Owing to issues, he needs to bribe each and every soldier with 10 PokéDollars to pass through. He is not sure of the number of soldiers in the main towers due to the new rules that have been imposed. He seeks Max’s help. In the castle, there are passways as straight row and column (passways are the dark grey strips) at any point when you see left or right or up or down . Also there are some magical tunnels joining one stop (stops are the places where troops of soldiers stand) to the other in a non-straight manner. The magic of the tunnel is clon- ing the soldiers. If you place any number of soldiers in one stop, then all stops joined to it through the tunnel will have same number of cloned soldiers. Cloned soldiers can also be cloned if there is one more. Number of soldiers at any stop in any row or column are more than one and sum of all soldiers at all stops in any row or column is same. Every stop is marked a number from 1 to 6 and stops in any row or column have different numbers. Soldier troop with ‘x’ number of soldier will stand on the stop marked as ‘x’. There are some soldier having all time duty and they are depicted in the figure with their positions. But fearing for safety, the king increased the security and ordered to place a soldier on every stop in the above way of his convenience. The circle in the middle represents the main tower and Max has to find the number of extra soldiers. The answer is ________ |
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Answer» 62  The question asked is, how many soldiers are outside the central 4 blocks without the CLONED EXTRA soldiers. CLEARLY, from the given INFORMATION, REST of the squares can be filled. |
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| 9715. |
ABCD is a square with side 16 units and A is the origin. If the equation of the circle circumscribing the square ABCD is x^(2) + y^(2) = 4k(x + y), then k= |
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Answer» 2 |
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| 9716. |
Prove that ""^(n)C_(3)+""^(n)C_(7) + ""^(n)C_(11) + ...= 1/2{2^(n-1) - 2^(n//2 )sin"" (npi)/(4)} |
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Answer» Solution :In given series difference in lower suffices is 4. i.e., ` 7 -3 =11 = …= 4 ` Now , ` (1)^(1//4) = (cos 0+ I sin 0 )^(1//4)` `= (cos 2r pi + i sin2r pi)^(1//4)` `= cos ""(rpi)/(2) + i sin"" (rpi)/(4) "where r" = 0,1,2,3 ` FOUR roots of unity` = 1,i,-1, 1,alpha, alpha^(2) , alpha^(3)` [SAY] and `(1 + x)^(n) = sum_(r=0)^(n) ""^(n)C_(r) x^(r)` Putting ` x = 1, alpha, alpha^(2) , alpha^(3) , " we GET " 2^(n) = sum_(r=0)^(n) ""^(n)C_(r)` ...(i) ` (1 + alpha)^(n) = sum_(r=0)^(n) ""^(n)C_(r) alpha^(r)` ...(ii) ` (1 + alpha ^(2))^(n) = sum_(r=0)^(n) ""^(n)C_(r) alpha^(2r) `...(iii) `(1 + alpha^(2))^(n) - sum_(r=0)^(n) ""^(n)C_(r) alpha^(3r)` ...(iv) On multiplying Eq.(i) by 1, Eq.(ii) By ` alpha ` , Eq.(iii) by ` alpha^(2)` and Eq.(iv) by ` alpha^(3)` and adding , we get ` rArr 2^(n) + alpha (1 + alpha)^(n) + alpha^(2) (1 + alpha^(2))^(n) + alpha^(3) (1 + alpha^(3))^(n)` ` =sum_(r=0)^(n) ""^(n)C_(r) (1 + alpha ^(r+1) + alpha^(2r +2)+ alpha^(3r +3))` ...(v) for r = 3,7,11,... RHS of Eq.(v) ` =""^(n)C_(r) (1 + alpha ^(4) + alpha^(8)+ alpha^(12))+""^(n)C_(7) (1 + alpha ^(4) + alpha^(16)+ alpha^(24))+""^(n)C_(11) (+ alpha ^(12) + alpha^(24)+ alpha^(36))+ ... ` ` = 4(""^(n)C_(3) + ""^(n)C_(7) + ""^(n)C_(11) +...) "" [ because alpha^(4) = 1]` and LHS of Eq.(v) `= 2^(n) + ii (1 + i)^(n) + i^(2) (1 + i^(2))^(n) + i^(3) (1 + i^(3))^(n)` ` = 2^(n) + i (1+ i)^(n) + 0 - i (1 - i)^(n)` ` = 2^(n) + i {(1 + i)^(n) - (1 - i)^(n)}` Since , `[(1 + i)^(n)= [sqrt(2)((1)/(sqrt(2))+ (i)/(sqrt(2)))]^(n)]` ` = 2^(n) + i2^(n//2)* sin"" (npi)/(4) = 2^(n//2) {cos""(pi)/(4) + i sin"" (pi)/(4)}^(n)` ` = 2^(n) - i2^(n//2)* sin"" (npi)/(4) = 2^(n//2) {cos""(pi)/(4) + i sin"" (pi)/(4)}` HENCE , ` 4(""^(n)C_(3) + ""^(n)C_(7) + ""^(n)C_(11)+ ...) = 2 (2^(n-1) - 2^(n//2) sin" " (npi)/(4))` ` rArr ""^(n)C_(3) + ""^(n)C_(7) + ""^(n)C_(11) + ... = 1/2 (2^(n-1) - 2^(n//2) sin"" (npi)/(4))`. |
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| 9717. |
Find (dy)/(dx) in the following : x^(3)+x^(2)y+xy^(2)+y^(3)= 81. |
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| 9718. |
If the 3^(rd),8^(th) and 13^(th) terms of a G.P are x,y,z respectively, then prove that x,y,z are in G.p.. |
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| 9719. |
If x is so small such that its square and higher powers may be neglected, the find the value of ((1-2x)^(1//3)+(1+5x)^(-3//2))/((9+x)^(1//2)) |
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| 9720. |
If V(x) is larger of e^(x) -1 and (1 +x) log (1 + x) for x in (0, oo) then log (V (8) + 1) is equal to |
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| 9721. |
STATEMENT-1 :For three positive unantities a , b,care in H.P., we must havea^(2008) + c^(2008) gt 2b^(2008)and STATEMENT-2 :A.M.ge G.M. ge H.M.for positive numbers |
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Answer» Statemant-1 is True , STATEMENT-2 is True, Statement -2 is a correct EXPLANATION for Statement-1 |
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| 9722. |
If[veca, vecb, vecc]=1then the value of(veca.(vecbxxvecc))/((veccxxveca).vecb)+(vecb.(veccxxveca))/((vecaxxvecb).vecc)+(vecc.(vecaxxvecb))/((veccxxvecb).veca)is________ |
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Answer» 1 |
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| 9723. |
If the equation cx^(2)+bx-2a=0 has no real roots and a lt (b+c)/2 then |
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Answer» `aclt0` |
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| 9724. |
If the weights of 10 persons (in kgs) are observed as : 45, 49, 55, 50, 41, 60, 58, 53, 55, then the variance of their weights is |
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Answer» A51 |
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| 9725. |
Differentiate w.r.t x the function (pi)/(4) lt x lt (3pi)/(4), (sin x - cos x)^((sin x-cos x)) |
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| 9726. |
If int sin^(-1)(2x)/(1+x^(2))dx = f(x) - log(1+x^(2))+c, then f(x) is equal to |
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Answer» `2X TAN^(-1)X` |
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| 9727. |
If one of the lines given by 6x^(2) - xy + 4cy^(2) = 0 is 3x + 4y = 0 then c equals |
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Answer» 3 |
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| 9728. |
Evalute the following integrals int sin^(3) 2x dx |
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| 9729. |
Given the function f(x) = x^(3) on the interval [-2,3] , find the lower (s_(n))and the upper (S_(n)) integral sums for thegiven interval by subdividing it into n equalparts. |
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| 9730. |
Let a, b and c be distinct and none of them is equal to 1. If the lines x + ay + a= 0, bx + y + b = 0 and cx + cy + 1 = 0 are concurrent, then the value of (a)/(a-1) + (b)/(b-1) + (c )/(c-1) is |
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Answer» 1 |
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| 9732. |
If P(0,7,10), Q(-1,6,6) and R (-4,9,6) are three points in the space , then PQR is |
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Answer» RIGHT ANGLED ISOSCELES triangle |
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| 9733. |
The vectors (3, 6, -9) and ………….have same direction ratio. |
| Answer» Answer :C | |
| 9734. |
Evaluate: int (sin 2x)/( a cos^2 x+b sin^2 x) dx. |
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| 9735. |
The adjacent sides of a parallelogram are hat(i) + 2 hat(j) + 3 hat(k) and 2 hat (i) - hat(j) + hat(k) . Find the unit vectors parallel to diagonals. |
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| 9736. |
Quadrilateral ABCD is composed of four 30 - 60 - 90 triangles. If B=10sqrt3, what is the perimeter of ABCD? |
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| 9737. |
Find all the roots (2-2i)^((1)/(3)) and also find the product of its roots. |
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| 9739. |
It is given that at x = 1, the function x^(4) - 62x^(2) + ax + 9 attains its maximum value, on the interval [0,2]. Find the value of a. |
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| 9740. |
The orbital angular momentum of 3p electron is :- |
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Answer» `SQRT3 h` |
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| 9741. |
If a in R and the equation |
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Answer» <P>`(-1,0)UU(0,1)` Give, `a in R ` and EQUATION is `-3{x-[x]}^(2)+2{x-[x]}+a^(2)=0` Let `=x-[x],`then equation is `-3t^(2)+2t+a^(2)=0` `impliest=(a+-sqrt(1+3a^(2)))/(3)` `because t=x-[x]={X}""["fractional part"]` `because 0letle1` `0le(1+-sqrt(1+3a^(2)))/(3)le1` Taking positive SIGN, we GET `0le (1+sqrt(1+3a^(2)))/(3)lt1""p[because{x}gt0]` `impliessqrt(1+3a^(2))lt2impliesa+3a^(2)lt4` `impliesa^(2)-1lt0implies(a+1)(a-1)lt0` `therefore a in (-1,1)` for no integer solution of a, we consider `(-1,0)ii(0,1)` |
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| 9742. |
If f(x)=lim_(n to oo)sin^(4)x+1/4sin^(4)2x+..=1/(4^(n)).sin^(4)(2^(n)x) and g(x) is a differentiable function satisfying g(x)+f(x)=1, then the maximum value of (sqrt(f(x))+sqrt(g(x)))^(4) is ………. |
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| 9744. |
Find the number of integral solutions of x_(1)+x_(2)+x_(3)=24 subjected to the condition that 1 le x_(1) le 5, 12 le x_(2) le 18 " and" -1 le x_(3). |
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Answer» =coefficient of `p^(24) " in" (p+p^(2)+p^(3)+p^(4)+p^(5))` `(p^(12)+p^(13)+..+p^(18))(p^(-1)+p^(0)+p^(1)+..+p^(11))` =coefficient of `p^(24) " in" (p(1-p^(5)))/(1-p).(p^(12)(1-p^(7)))/(a-p).(p^(-1)(1-p^(13)))/(1-p)` coefficient of `p^(12) " in" (1-p^(5))(1-p^(7))(1-p^(13))(1-p)^(-3)` =coefficient of `p^(12) "in" (1-p^(5)-p^(7)+p^(12)(1-p)^(-3)` `= .^(14)C_(12)- .^(9)C_(7)- .^(7)C_(5)+1` =91-36-21+1=35 |
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| 9745. |
Evaluate : (i) int_(-1)^(2){2x}dx (where function{*} denotes fractional part function) (ii) int_(0)^(10x)(|sinx|+|cosx|) dx (iii) (int_(0)^(n)[x]dx)/(int_(0)^(n)[x]dx) where [x] and {x} are integral and fractional parts of the x and n in N (iv) int_(0)^(210)(|sinx|-[|(sinx)/(2)|])dx (where [] denotes the greatest integer function and nin 1) |
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| 9746. |
The parametric angle alpha where-piltalphalepi of the point on the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 at which the tangent drawn cuts the intercept of minimum length on the coordinates axes, is/are |
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Answer» `tan-SQRT(B/a)` |
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| 9747. |
Find the global maximum and minimum values of the function f given by f(x) = 2x^(3) - 15x^(2) + 36x + 1, x in [1,5]. |
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| 9748. |
Translate the following statements. R is 45 more than B. |
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| 9749. |
Find the distance between the following pairs of points. (3,4) , (2,1). |
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Answer» SOLUTION :DISTANCE between the POINTS (3,4) and (2,1) is: `SQRT(3+2)^2+(4-1)^2=`sqrt925+9)`=`sqrt34` |
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| 9750. |
IF thepairofstraightlinesxy- x -y +1=0 and thelinex +ay -3=0areconcurrentthen theacuteanglebetweenthe pairof linesax ^2- 13xy- 7y^2+ x+ 23y-6=0 is |
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Answer» `cos ^(-1)((5)/(SQRT(218)))` |
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