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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. |
Integrate the functions (e^(x))/((1+e^(x))(2+e^(x))) |
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| 3. |
Show that the normal at any point theta to the curve x = a cos theta + a theta sin theta, y = a sin theta - a theta cos theta is at the constant distance from origin. |
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| 4. |
intx.tan^(-1)xdx=......+c |
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Answer» `(x^(2)+1)/(2)tan^(-1)x+(x)/(2)` |
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| 5. |
If A={:[( 3,1),( -1,2) ]:}Show that A^(2) -5A +7I=O.Hence find A^(-1) |
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| 6. |
If f (x) = (sin 4pi [ pi ^(2) x ])/( 7 + [x ] ^(2)), [.] denotes ghe greatest integer function, then f (x) is |
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Answer» continuous `AAX, ` but f '(X) does not EXIST. |
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| 7. |
A number n is chosen at random from S = {1,2,3…….50}. Then correct order of their probabilities is |
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Answer» <P>`P(A) LT P(B) lt P("C")` |
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| 8. |
Iff (2) =4 and f (2)=1 , " then "lim_(x to 2)(x f(2) - 2f(x))/(x-2)is equal to |
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Answer» `-2` |
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| 9. |
Using the properties of determinants, prove the following |{:(a^2,b^2,c^2),((a+1)^2,(b+1)^2,(c+1)^2),((a-1)^2,(a-1)^2,(c-1)^2):}|=4|{:(a^2,b^2,c^2),(a,b,c),(1,1,1):}| |
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| 10. |
Find minors and cofactors of the elements a_(11),a_(21) in the determinant Delta=|{:(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33)):}| |
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Answer» `a_(21)=M_(21)=a_(12)a_(33)-a_(13)a_(32)` COFACTOR : `a_(11)=a_(22)a_(33)-a_(23)a_(32)` `a_(21)=-a_(12)a_(33)+a_(13)a_(32)` |
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| 11. |
If two circles (x+4)^(2)+y^(2)=1 and (x-4)^(2)+y^(2)=9 are touched extermally by a circle, then locus of centre of variable circle is |
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Answer» `(x^(2))/(15)-(y^(2))/(1)=1`  `CS=r+1` `CS'=r+3` `CS'-CS=2` Locus of C is hyperbola with `(-4,0)` and `(4,0)` as foci, Here, 2a=2, so a = 1. Also, 2ae = 8, so ae = 4 `therefore""b^(2)=a^(2)(e^(2)-1)=15` |
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| 12. |
Resolve (x^(2)+4x+7)/((x^(2)+x+1)^(2)) into partial fractions. |
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| 13. |
int (cos x + x sin x )/(x (x + cos x ))dx = |
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Answer» `log |("x + COS x")/(x )| + c ` |
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| 14. |
Find the latus rectum of the parabola 2x^2+3y=0 |
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| 15. |
If f: R to R defined as f(x) = (2x-7)/( 4) is an invertible function, write f^(-1) (x). |
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| 16. |
int [ log(log x) + (log x )^(-2) ]dx = |
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Answer» `X { LOG (log x) - (1)/(log x ) } + C` |
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| 17. |
Metal (M) crystallizes in a cubic unit cell with density 3.2 g/cc. Edge-length of the unit cell is 4.37Å picometre (pm) The nearestneighbour will be |
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Answer» On the corners |
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| 18. |
Metal (M) crystallizes in a cubic unit cell with density 3.2 g/cc. Edge-length of the unit cell is 4.37Å picometre (pm) The number of nearest neighbours of a Ca atom are |
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Answer» 4 |
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| 19. |
If the first and the n^("th")term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P^2 = (ab)^n. |
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| 20. |
If f : R to R is defined by f (x) =x ^(2) - 3x + 2, find f (f (x)). |
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| 21. |
If P (A) = 0.4,P (B | A)= 0.3 and P (B^c | A^c)=0.2. find P ( B | A^c) |
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Answer» <P> SOLUTION :`P(B|A^c)=(P(BcapA^c))/(P(A^C))=(P(B-A))/(1-P(A))``=(P(B)-P(A capB))/(1-P(A))=(0.6-0.12)/(1-0.4)=0.48/0.6=0.8=4/5` |
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| 22. |
Find longest wavelength in Lyman series of hydrogen atom spectrum. |
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Answer» 1216Å |
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| 25. |
(p rArr q) rarr [(r vv p) rArr (r vv q)] is |
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Answer» a TAUTOLOGY |
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| 26. |
If (0, 1//2), (1//2, 1//2), (1//2, 0) are the midpoints of the sides of a triangle, then incentre of the triangle is |
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Answer» `((1)/(SQRT(2)), (1)/(sqrt(2)))` |
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| 27. |
Molarity and Molality of a solution of an liquid (mol. Wt. = 50) in aqueous solution is 9 and 10 respectively. What is the density of solution ? |
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Answer» 1 g/cc |
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| 28. |
The saving of an employee equals Income minus Expenditure. If their Incomes ratio is 1 : 2 : 3 and their Expenses rato is 3 : 2 : 1, then what is the order of the employees, A , B and C in the increasing order of the size of their saving? |
| Answer» Answer :D | |
| 29. |
Find the equation of the parabola whose axis is parallel to X-axis and which passes through these points. (-2,1),(1,2), and (-1,3) |
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| 30. |
What is the sum of the squares of the roots of the equation x^(2)-7[x] + 5 = 0 ? (Here [x] denotes the greatest integer less than or equal to x. For example [3.4] = 3 and [-2.3] =-3). |
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| 31. |
The point on the curve x^2=2y which is nearest to the point (0,5) is -- |
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Answer» `(2sqrt2,4)` |
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| 32. |
Determine the sine of the angle between the vectors 5hati-3hatj, 3hati-2hatk |
Answer» Solution : `|vecaxxvecb| = sqrt(36+100+81) = sqrt(217)` `|veca| = sqrt(25+9) = sqrt(34)` `|vecb| = sqrt(9+4) = sqrt(13)` If `theta` is the ANGLE between `veca` and `vecb` then `sintheta = |vecaxxvecb|/(|veca||vecb| = sqrt(217)/(sqrt(34) sqrt(13)) = sqrt((217)/442). |
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| 33. |
Estimate the integral int_(0)^(1) sqrt(1 + x^(4)) dx using (a) themean-valuetheoremfor a definite integral, (b) the result of the preceding problem, (c) the inequalitysqrt(1 + x^(4)) lt 1 + (x^(4))/(2) (d)the Schwarz-Bunyakovaskyinequality(see Problem 6.3.6) |
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Answer» (B) 1.207 (C)1 . 1 (d)1 . 095 |
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| 34. |
Integrate the function (6x+7)/(sqrt((x-5)(x-4))) |
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| 35. |
Solve as directed: 2x + 3 gt 15 in integers, in natural numbers. |
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Answer» SOLUTION :2x + 3 `GT` 15 `rArr 2x + 3-3 `gt` 15-3 `rArr 2x gt 12` `(2x)/2 gt (12)/2` `x gt 6` If x `in` Z then the solution set isS = {x:x `in` Z and x `gt` 6} = {7,8,9………} If x `in` then the solution set is S = {x:x `in` N and x `gt` 6} {7,8,9.......} |
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| 36. |
Find the value of 1/x for the following values of x : (a)[-5 , -1](b)(3,6)(c )(-2 , 3)(d )(-oo , -2]( e)[-3,4] |
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Answer» ` rArr1/2 gt 1/xgt1/5` `RARR 1/(x)in(1/5,1/2)` (b) `-5 le x lt -1` `rArr-1/5ge 1/x gt-1` `rArr1/(x)in(-1,-1/5]` (c) `x gt3 or 3 lt x lt oo` `x le -2 or - oo lt x le-2` `rArr0 gt1/xge-1/2` `rArr1/x""in[-1/2,0)` (e ) `x in [-3 4]or -3 le x le 4` Now for `1/x `to get defined ,we must have `-3 le x lt 0 ` or `0 lt x le 4` `rArr -1/3 ge 1/xgt -ooor 1/4 le 1/x""lt oo` `rArr1/x""in(-oo,1/3]CUP[1/4,oo)` |
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| 37. |
(2x^(4)+3x^(2)+1)/((x^(2)+1)^(4))=(A)/((x^(2)+1))+(B)/((x^(2)+1^(2)))+(C)/((x^(2)+1)^(3))+(D)/((x^(2)+1)^(4)) then Match the following. {:("List - I","List - II"),("1) A","(a) 2"),("2) B","(b) 1"),("3) C","(c) -1"),("4) D","(d) 0"), (,"(e) 1/2"):} |
| Answer» Answer :A | |
| 38. |
If the variance of first n natural numbers is 2 and the variance of first m odd natural numbers is 40, then m+n = |
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Answer» variance of first m ODD natural numbers is `2^2 ((m^2-1))/12=40 rArr (m^2-1)/12=10` `rArr m^2-1 =120 rArr m^2=121 rArr` m=11 |
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| 39. |
If A = [(2,-1,3),(-4,5,1)]and B = [(2,3),(4,-2),(1,5)]then : |
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Answer» only AB is DEFINED |
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| 40. |
The sum of the series 1+(1^2+2^2)/(2!)+(1^(2)+2^(2)+3^(2))/(3!)+(1^(2)+2^(2)+3^(2)+4^2)/(4!)+.. Is |
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Answer» 3e |
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| 42. |
If hat(i)+hat(j)-hat(k), 2hat(i)+3hat(j), 3hat(i)+5hat(j)-2hat(k) and -hat(j)+hat(k) are the position vectors of vertices of a quadrilateral, then it is a |
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Answer» parallelogram |
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| 43. |
Find the locus of the point whose polars with respect to the circles x^(2) + y^(2) -4x -4y -8=0 and x^(2) + y^(2) -2x + 6y -2= 0are mutuallyperpendicular. |
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| 44. |
If the circles of same radii and with centres (1, 3), (9, 11) cut orthogonally then radius is |
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Answer» 2 |
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| 45. |
The value ofint_(-(1)/(2))^((1)/(2)){[x]+log((1+x)/(1-x))}dx is equal to - |
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Answer» 0 |
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| 46. |
If abs(z-2)lesqrt2, then the maximum value of abs(3+i(z-1)) is |
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Answer» `SQRT2` |
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| 47. |
If f (x) = sin (2x ^(2) -2 [x]) for 0 lt x lt 1, then f ' ((sqrtpi)/(2))= |
| Answer» Answer :C | |
| 48. |
If the line joining the points hati+hatjand3hati+hatj-hatk meets the plane that passes through the point 2hati+4hatj and parallel to the vectors 3hati+5hatkand3hati-hatk at, P, then the position vector of the point P is |
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Answer» `-27hati+hatj+14hatk` |
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| 49. |
For all real value of k, the polar of the point (2k,k-4) with respect to x^2+y^2-4x-6y+1=0 passes through the point |
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Answer» (1,1) |
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