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This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
cis(pi)/(10).cis(2pi)/(10).cis(3pi)/(10).cis(4pi)/(10)= |
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Answer» 1 |
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| 2. |
Removethe thirdtermfromtheequationx^4 +2x^3 -12 x^2 +2x -1=0 |
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Answer» 0 |
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| 3. |
Solve the differential equation (tan^(-1)y - x)dy = ( 1 + y^(2))dx. |
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Answer» |
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| 4. |
The solution of x (dy)/(dx) + y log y = xy e^(x) is |
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Answer» `x LOG y = (x+1)E^(x) + c` |
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| 5. |
If y ^(-2) =1+ 2 sqrt2 cos 2x,then : (d^(2) y)/(dx ^(2)) =y (py ^(2)+1) (qy ^(2) -1) then the vlaue of (p+q) equals to: |
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Answer» 7 |
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| 6. |
intdx/sqrt(4x^2-4x+5) |
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Answer» Solution :`INTDX/SQRT(4x^2-4x+5)=intdx/(sqrt((2x-1)^2) +2^2)` =`1/2log(2x-1+sqrt(4x^2-4x+5)+C` `[because intdt/sqrt(t^2+a^2)=log(t+sqrt(t^2+a^2)+C]` |
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| 7. |
Let X be the soultion set of the equation A^(x)=-I, where A = [[0 , 1, -1],[4, -3, 4],[3, -3, 4]] and I is the corresponding unit matrix and x subseteq N,the minimum value ofsum ( cos ^(x) theta + sin ^(x) theta )theta in R - { (npi)/2 , n in I }is |
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Answer» `therefore A^(2) = A cdot A =[[0,1,-1],[4,-3,4],[3,-3,4]][[0,1,-1],[4,-3,4],[3,-3,4]] = [[1,0,0],[0,1,0],[0,0,1]]= I ` `RARR A^(2) = I rArr A^(4) = A^(6) = A^(8) = ... = I` Now, `A^(X) = I` `rArr x = 2, 4, 6, 8...` `therefore sum (cos ^(x) theta + sin ^(x) theta ) = ( cos^(2) theta + sin ^(2) theta) + (cos ^(4) theta + sin^(4) theta ) + (cos^(6) theta + sin ^(6) theta) + ...` `=(cos^(2) theta + cos^(4) theta + cos ^(6) theta +...) ` `+ (sin^(2) theta + sin^(4) theta + sin^(6) theta + ...)` `= (cos^(2) theta)/(1- cos^(2) theta) + (sin ^(2) theta)/(1- sin ^(2) theta)` `= cot^(2) theta + tan ^(2) theta ge 2` hences, minimumvalue of `sum (cos^(x) theta+ sin ^(x) theta)` is 2. |
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| 8. |
Differentiate w.r.t x the function x^(x) + x^(a) + a^(x) + a^(a), for some fixed a gt 0 and x gt 0. |
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| 9. |
Let points A,B and C lie on lines y-x=0, 2x-y=0 and y-3x=0, respectively. Also, AB passes through fixed point P(1,0) and BC passes through fixed point Q(0,-1). Then prove that AC also passes through a fixed point and find that point. |
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Answer» Solution :Let the coordiantes of points A,B and C be `(alpha, alpha), (beta,2beta) " and " (gamma, 3gamma)`, respectively. Points A,B, P are COLLINEAR. `therefore |{:(1,0,1),(alpha,alpha, 1),(beta, 2beta,1):}| = 0` `rArr alpha-2beta+alphabeta=0 "" (1)` ALSO, points B,C,Q are collinear. `therefore |{:(0, -1,1),(beta,2beta,1),(gamma, 3gamma,1):}| = 0` `rArr beta-gamma +beta gamma = 0` `rArr beta = (gamma)/(1+gamma) "" (2)` PUTTING value of `beta` in equation (1), we get `alpha +2alpha gamma = 2gamma.` Let AC pass through fixed point R(h, K). Since C, A and R are collinear, `|{:(alpha,alpha,1),(gamma,3 gamma, 1),(h, k,1):}| = 0` `rArr h(alpha-3gamma) - k(alpha-gamma) +2alphagamma = 0` `rArr h(alpha-3gamma) - k(alpha-gamma) +2gamma-alpha = 0` `rArr alpha(h-k-1) + gamma(-3h+k+2)=0 " for all "alpha,gamma` `therefore h-k-1=0 " and "-3h+k+2 =0` `therefore h =(1)/(2), k =-(1)/(2)` `"Thus, AC passes through the point " ((1)/(2), -(1)/(2)).` |
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| 10. |
If s_(n) = sum_(r = 0)^(n) 1/(""^(n)C_(r)) and t_(n) = sum _(r = 0)^(n) r/(""^(n)C_(r)),then t_(n)/s_(n) is equal to |
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Answer» `N - 1` |
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| 11. |
Compute ((1+k)(1+k/2) ..... (1+k/n))/((1+n)(1+n/2) ..... (1+n/k)) |
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Answer» SOLUTION :((1+K)(1+k/2) .... (1+k/N))/((1+n)(1+n/2) .... (1+n/k)) ((1+k)(2+k) .... (n+k))/(1.2.3.......n)/((1+n)(2+n) .... (k+n))/(1.2.3.......k) (1.2.3...k(k+1)(k+2)....(k+n))/(1.2.3....n(n+1)(n+2)....(n+k)) `((k+n!))/((n+k!))` = 1 |
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| 12. |
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find 5 * 7. |
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| 13. |
Letf'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2)) and g(x) =f(sinx)+f(cosx) If x = 3 is the only point of minima in its neighborhood and x=4 is neither a point of maxima nor a pointminima, then which of the following can be true? |
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Answer» `a GT 0, b lt 0` or `-9+12+a=3a+b or 2a+b=3` Also `f(4^(+)) or 4a+b=-b+6 r 2a+b=3` Thus f(x) is contnous for infinite values of a and b also `f(x)={{:(-2x+4,xlt3),(a,3ltxlt4),((-b)/(4),xgt4):}` For f(x) to be diffentiable `f(3^(-))=f(3^(+))` or `a=-2 and -(bb)/(4) =a=-2 or b=8` But these values do not SATISFY equation (1) HENCE f(x) cannot be differentiable 
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| 14. |
Integrate the following functions x sec^2x |
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Answer» Solution :`INT x sec^2 dx = x TANX-int 1 XX tanx dx` ` =x tanx-log|secx|+c` |
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| 15. |
Coordinates of the vertices B and C of the base of a triangle ABC are (-a,0) and (a, 0) respectively. If C-B=pi//3, the vertex A lies on the curve |
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Answer» `X^(2)-y^(2)+2sqrt(3)xy-a^(2)=0` |
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| 16. |
A young couple has two children. The probability that both children are boys, if it is known that atleast one of the children is a boy is |
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Answer» `(1)/(4)` |
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| 17. |
If a complex number z satisfies log_(1//sqrt2) ((|z|^2 + 2 |z|+6)/(2 |z|^2 + 2|z|+1)) lt 0, then locus/ region of the point represented by z is |
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Answer» `|z| = 5` |
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| 18. |
a, b, c, d are integers such that ad + bc divides each of a, b, c and d.Prove that ad + bc = pm1 |
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| 19. |
int(2cosx)/(1-cos^2x)dx |
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Answer» SOLUTION :`INT(2cosx)/(1-cos^2x)dx=2intcosx/(sin^2x)dx` `2int(1)/sinx xxcosx/sin^2xdx` =`2intcosecx XX cotxdx` =`-2cosecx+C` |
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| 22. |
Determine the differentials in each of the following cases. r = 4/(1 +sin theta) |
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Answer» Solution :`r = 4/(1 +SIN THETA)` `DR =(- 4 COS theta)/((1 + sin theta)^2` =`d theta` = -(4 cos theta)/((1 + sin theta)^2` `dtheta` |
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| 23. |
If P(A)= (7)/(13), P(B)= (9)/(13)and P( Acap B) = (4)/(13) evaluate P(A | B) |
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| 24. |
Resolve (3x^(2)+x-2)/((x-2)^(2)(1-2x)) into partial fractions. |
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| 25. |
Find the values of x,y,z if the matrix A=[{:(0,2y,z),(x,y,-z),(x,-y,z):}]satisfy the equation A'A=I. |
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| 26. |
Assertion(A ) :theequationwhoserootsareexceedby2thenthoseof2x^3 +3x^2 -4x +5=0 is2x^3 -9x^2 +8x+9=0 Reason (R ): theequationwhoserootsareexceedby hthanthoseoff(x)=0 isf(x-h) =0 |
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Answer» BOTHA and RaretrueR isthe correctexplanationof A |
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| 27. |
sqrt(4ab - 2i (a^(2) - b^(2) ) = |
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Answer» `PM {((a-B) + i(a-b))/(2)}` |
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| 28. |
The order degree of the D.E. corres ponding to the family of curve y = a(x+a)^(2) where a is an arbitrary constant is |
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Answer» 1,2 |
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| 29. |
If z_(1)" and "z_(2) are two complex numbers such that Im(z_(1)+z_(2))=0, Im(z_(1)z_(2))=0 then |
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Answer» `z_(1)= -z_(2)` |
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| 30. |
Four positive integers are taken at random and are multiplied together. Then theprobabilitythat the product ends in an odd digit otherthan 5 is |
| Answer» Answer :B | |
| 31. |
If n(A)=3 and n(B)=4 , then no. of of one-one function from A to B is : |
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Answer» 12 |
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| 32. |
The point on the ellipse 16x^(2)+9y^(2)=400, where the ordinate decreases at the same rate at which the abscissa increases is (a, b), then a+3b can be |
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Answer» 16 |
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| 33. |
If p is any statement, then which of the following is a contradiction? |
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Answer» <P>`p ^^ p ` |
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| 34. |
The mean and standard deviation of a random variable X are given by E(X) = 5 and sigma_(x)=3 respectively, then (1) E(X^(2))=……. (2) E[(3X-2)^(2)]= ………. (3) V(3-2X)= ………… |
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Answer» (1) 34, (2) 250, (3) 36 |
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| 35. |
Consider a LPP given by minimise Z = 6x + 10y. Subject to x ge 6, y ge 2, 2x+y ge 10, x ge 0, y ge 0. Redundant constraints in this LPP are ……….. |
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Answer» `X ge 6, y ge 2` |
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| 36. |
If a,b,and c are sides of Delta ABC such that |{:(c,bcosB+alphabeta,acosA+balpha+cgamma),(a,c cosB+a beta,b cosA+c alpha+agamma),(b,acosB+b beta,c cosA+aalpha+bgamma):}| =0 (where alpha ,beta,gamma,in R ^(+)"and" angleA,angleB,angleCne(pi)/(2)), Delta ABC is |
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Answer» an isosceles |
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| 37. |
Find c so that f'(c)=(f(b)-f(a))/(b-a) " where " f(x)=e^(x), a=0, b=1 |
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Answer» |
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| 38. |
Three bags contain a number of red and white balls as follows: If a white ball is selected, what is the probability that it came from (i) Bag II (ii) Bag III. |
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| 39. |
If n is a positive integer and (1 + x + x^(2))^(n) = a_(0) + a_(1) x + …. + a_(2 n) x^(2 n). Then, show that, a_(0)^(2) - a_(1)^(2) + ….a_(2n)^(2) = a_(n). |
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Answer» Solution :`(1 + x + x^(2))^(N) = a_(0) + a_(1)x + ...+ a_(2N)x^(2n)…(i)` REPLACING x by `-1//x` , we get `(1- (1)/( x )+(1)/ x^(2))^(n) = a_(0) - (a_(1))/(x)+(a_(2))/(x^(2))+(a_(3))/(x^(3)) + ...+ (a_(2n))/(x^(2n))….(ii)` Now , `a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - a_(3)^(2) +...a_(2n)^(2) = ` , COEFFICIENT of the term independent of x in ` [ a_(0) + a_(1)x + a_(2) x^(2) +...+ a_(2n)x^(2n)] xx[a_(0) - (a_(1))/(x) + (a_(2))/(x^(2))- ... + (a_(2n))/(x^(2n))]` = Coefficient of the term independent of x in `(1 + x + x^(2))^(n) (1 - (1)/(x) + (1)/(x^(2)))^(n)` Now , RHS = ` (1 + x + x^(2))^(n) (1 - (1)/(x) + (1)/(x^(2)))^(n)` `((1 + x + x^(2))^(n) (x^(2) - x + 1 )^(n))/(x^(2n))=([(x^(2) + 1)^(2)- x ^(2)]^(n))/(x^(2n))` `((1 + 2x^(2) + x^(4) - x ^(2))^(n))/(x^(2n)) = (1 + x^(2) +x^(4))^(n)/(x^(2n))` Thus , ` a_(0)^(2) - a_(1)^(2) + a_(2)^(2) - a_(3)^(2) +...+ a_(2n)^(2)` = Coefficient of the term independent of x in ` (1)/(x^(2n)) (1 + x^(2) + x^(4))^(n)` = Coefficient of `x^(2n) ` "in" `(1 + x ^(2) + x^(4))^(n)` = COFFICIENT of ` t^(n)` "in" ` (1 + t + t^(2))^(n) = a_(n)`. |
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| 40. |
Total number of 10 – digit numbers in which only and all the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 appear, is: |
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Answer» `5/2(10!)` |
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| 41. |
Find the valur of |a| for which the area of triangle included between the coordinate axes and any tangent to the curve x ^(4) y = lamda ^(a) is constant (where lamda is constnat.), |
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| 43. |
Find underset(n to oo)lim x_(n), if (a) x_(n)=((3n^(2)+n-2)/(4n^(2)+2n+7))^(3), (b) x_(n)=((2n^(3)+2n^(2)+1)/(4n^(3)+7n^(2)+3n+4))^(4), (c) x_(n)=rootn(5n), (d) x_n=rootn(n^(8)), (e) x_(n)=rootn(n^(5)), (f) x_(n)=rootn(6n+3). |
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| 44. |
Find area of the triangle withh vertices at the point given in each of the following : (7,9),(10,8),(12,10) |
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| 45. |
Show that the area bounded by the lines x = 0, y = 1, y = 2 and the hyperbola xy = 1 is log 2. |
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| 47. |
If cosy=xcos(a+y)" with "cosa!= +-1, then dy/dx is equal to |
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Answer» 1.`(sina)/cos^(2)(a+y)` |
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