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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. |
If sin beta=sin(2alpha+beta) then find the value of (cotalpha+cot(alpha+beta))(cot beta-5cot(2alpha+beta))? |
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Answer» Solution :`((COS ALPHA)/(sinalpha)+(cos(alpha+BETA))/(sin (alpha+beta)))((cosbeta)/(sinbeta)-(5 cos(2alpha+beta))/(sin(2alpha+beta)))` `(sin(2alpha+beta))/(sinalpha(alpha+beta))((cosbeta)/(sinbeta)-(5 cos(2alpha+beta))/(5 sinbeta))` `(5 sin beta)/(sin alpha.sin(alpha+beta))((cosbeta-cos(2alpha+beta))/(sin beta))` `(10 (sin alpha.sin(alpha+beta)))/(sinalpha.sin(alpha+beta))=10 ` |
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| 2. |
Three non-zero real numbers form a AP and the squares of these numbers taken in same order form a GP. If the possible common ratios are (2pm sqrt(k)) where k in N, then the value of [k)/(8)-(8)/(k)) is (where [] denites the greatestinteger function). |
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Answer» SOLUTION :Let number of AP are `(a-d),a,(a+d)`. According to the question, `(a-d)^(2),a^(2),(a+d)^(2)` are in GP. `:.(a^(2))^(2)=(a-d)^(2)(a+d)^(2)` `implies a^(4)=(a^(2)-d^(2))^(2)` `implies a^(4)=a^(4)+d^(4)-2a^(2)d^(2)` `implies a^(2)(a^(2)-2D^(2))=0` `implies ane0," So "a^(2)=2d^(2)` `implies a=pm sqrt(2d)"" "......(i)"` Let common ratio of GP is r. `:.r^(2)=((a+d)^(2))/((a-d)^(2))` `implies r^(2)=(a^(2)+d^(2)+2ad)/(a^(2)+d^(2)-2ad)` `implies r^(2)=(2d^(2)+d^(2)+2sqrt(d^(2)))/(2d^(2)+d^(2)-2sqrt(d^(2)))` `"" [" from EQ. (i) for"a= sqrt(2)d]` `implies r^(2)=((3+2sqrt(2))d^(2))/((3-2sqrt(2))d^(2))` `implies r^(2)=((3+2sqrt(2))(3+2sqrt(2)))/(9-8)` `implies r^(2)=(3+2sqrt(2))^(2)` `implies r^(2)=(3+sqrt(8))^(2)` `:. r=pm(3+sqrt(8))` `impliesr=3+sqrt(8) "" [:." ris positive "]` Similarly, for `a=-sqrt(2)d,` we get `r=pm(3-sqrt(8))` `implies r=(3-sqrt(8)) "" [:. " r is positive "]` COMPARE r with `3pmsqrt(K)`, we get `k=8` `[(k)/(8)-(8)/(k)]=[(8)/(8)-(8)/(8)]` `= [1-1]=[0]=0`. |
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| 3. |
Find the volume of the parallelepiped with segments AB, AC and AD as concurrent edges, where the position vectors of A, B, C, D are hat(i)+hat(j)+hat(k),2hat(i)-hat(j)+3hat(k),3hat(i)-2hat(j)-2hat(k)and3hat(i)+3hat(j)+4hat(k) respectively. |
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Answer» 27 CU. units |
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| 4. |
Find least positive integer x, satisfying 276x+128=4 (mod 7). |
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Answer» Solution :Now 128 -= mod 7 Now `176 x+128 -=4 "mod" 7` `implies 176x -=(4-2)"mod" 7` `implies 176 x -=2 "mod" 7` `176 XX x=2 "mod" 7`, ` "But" 276 -= 3 "mod" 7,` `"THUS" x=3`. |
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| 5. |
If ui=axi+b and vi =cyi+d then |
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Answer» cov (u,v) |
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| 6. |
If the foot of perpendicular from origin to the plane is (1,2,3) then the equation of the plane is............ |
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Answer» `(x)/(1) + (y)/(2) + (Z)/(3) = 1` |
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| 7. |
int(1)/(x)[sqrt[x-1)/(sqrt(x+1])]dx= |
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Answer» `Cos H^(-1)x-Sec^(-1)x+c` |
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| 8. |
If the 4 points made by intersection of lines 2x-y+1=0, x-2y+3=0 with the coordinate axes are concylic then centre of circle is |
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Answer» (7/4,5/4) |
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| 9. |
Find the mean deviation of the first three odd natural numbersfrom their mean. |
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| 10. |
int (1)/((x^(2)+9)sqrt(x^(2)-9))dx= |
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| 11. |
y= f(mu), where f(mu) = (3)/(2mu^(2) + 5mu -3) and mu=(1)/(x+2). Find the points of discontinuity of y. |
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| 13. |
The area of the region bounded by the curves y=x^(2)andy=(2)/(1+x^(2)) is |
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| 14. |
Prove that : Find the sum of the infinite series 1+(2)/(3).(1)/(2)+(2.5)/(3.6)((1)/(2))^(2)+(2.5.8)/(3.6.9)((1)/(2))^(3)+......oo |
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Answer» `ROOT(3)(3)` |
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| 15. |
Find the sum of all 4 digited numbers that can be formed using the digits 0,2,4,7,8 without repetition. |
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| 17. |
An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. The probability that the second ball drawn is red will be |
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| 18. |
Compute P(X=k) for the binominal distribution, B(n,p) where n=10, p=1/5, k=4 |
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| 19. |
Evaluate the following integrals int(cosx)/(3cosx+4sinx)dx |
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| 20. |
Find the area of the region bounded by the curve y^(2) = 4x and the line x = 3. |
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| 21. |
If a_(1), a_(2), a_(3), ………, a_(n)….. are in G.P., then the determinant Delta =|(""loga_(n)" "loga_(n + 1)" "log_(n + 2)""),(""loga_(n + 3)" "loga_(n + 4)" "log_(n + 5)""),(""loga_(n + 6)" "loga_(n + 7)" "log_(n + 8)"")| is equal to |
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Answer» 0 |
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| 22. |
If a=(2, 1, -1), b=(1, -1, 0), c=(5, -1, 1), then unit vector parallel to a + b - c but in opposite direction is |
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Answer» `(1)/(3)(2hati-hatj+2hatk)` Then, `a+b-c=(2+1-5)hati+(1-1+1)hati+(-1+0-1)hatk` `=-(2hati-hatj+2hatk)` `because` Unit VECTOR of `(a+b-c)=-((2hati-hatj+2hatk))/(3)` `therefore` Required unit vector of `(a+b-c)=((2hati-hatj+2hatk))/(3)` |
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| 23. |
If points A(bara) = 10i + 3j . B (barb) = 12i - 5j and C (barc) = ai + 11j are collinear, then a = |
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Answer» `-8` |
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| 25. |
Findthe quotientand theremainderwhenx^4 -6x^3 +3x^2 + 26- 24isdividedbyx-4 |
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| 26. |
int(sec^2x.cosec^2x)dx |
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Answer» SOLUTION :`INT(sec^2x.cosec^2x)DX` =`INT1/(sin^2xcos^2x)dx` =`int(sin^2x+cos^2x)/(sin^2xcos^2x)` =`int(sec^2+cosec^2)dx` =tanx-cotx+C |
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| 28. |
Find the matrix X such that , X[{:(5,-7),(-2,3):}]=[{:(-16,-6),(7,2):}]. |
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| 29. |
Acceleration -position graph for a particle moving along x-axis is given below. At origin velocity of particle is 5 m/s then velocity of particle at x=6 m is : |
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Answer» 5 m/s |
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| 30. |
Let A=[{:(2,4),(3,2):}],B=[{:(1,3),(-2,5):}],C=[{:(-2,5),(3,4):}] Find each of the following : (i) A+B ,(i) A-B, (iii) 3A-C, (iv) AB, (v)BA |
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Answer» (ii) `=[{:(1,1),(5,-3):}]` (iii) `=[{:(8,7),(6,2):}]` (iv) `=[{:(-6,26),(-1,19):}]` (V) `=[{:(11,10),(11,2):}]` |
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| 31. |
If A=[{:(1,0,0),(0,1,0),(0,0,1):}]then A^(2)+2A= ………. |
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Answer» 4A |
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| 32. |
Find all points of discontinuityof f, where f is defined by f(x)={{:(x^(10)-1," if "x le1),(x^(2)," if "x gt 1):} |
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| 33. |
Using elementary row transformations , find the inverse of [{:(2,0,-1),(5,1,0),(0,1,3):}] |
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| 34. |
Differentiate the functions with respect to x in Exerecises 1 to 8. sin (x^(2)+5) |
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| 35. |
If y = sin ^(2) alpha + cos ^(2) (alpha + beta) + 2 sin alpha sin betacos(alpha+beta) then (d^(3)y)/(da^(3))=? |
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Answer» `sin^(3) (ALPHA +beta)/(COSALPHA)` |
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| 36. |
Integrate the following functions : int(dx)/(sqrt(1-(4x+5)^(2))) |
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| 38. |
The internal centre of similitude of the two circles x^(2)+y^(2)+6x-2y+1=0, x^(2)+y^(2)-2x-6y+9=0 is |
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| 39. |
If the normals at the points t_(1) and t_(2) on y^(2) = 4ax at the point t_(3) on the parabola, the t_(1)t_(2)= |
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Answer» 4 |
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| 40. |
Find the values of each of the expression following : sin^(-1)("sin"(pi)/3) |
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| 41. |
How many ordered pair of integers ( a,b) satisfy all the following inequalities a^(2) +b^(2) lt 16 , a^(2) +b^(2) lt 8 a,a^(2) +b^(2) lt 8b? |
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| 42. |
IFalpha, betabe theroots of6x^2-6x +1=0 then (1)/(2) (a+ ba+ calpha^2+ dalpha ^3)+(1)/(2)(a+ b beta + cbeta^2 +d beta ^3)= |
| Answer» Answer :C | |
| 43. |
{:(" "Lt),(n rarr oo):}1/n sum_(r=1)^(2n)(r)/(sqrt(n^(2)+r^(2)))= |
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Answer» `1+sqrt(5)` |
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| 45. |
Solve that |{:(1+x,1-x,1-x),(1-x,1+x,1-x),(1-x,1-x,1+x):}|=0 |
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Answer» o OR 1 |
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| 46. |
Find the shortest distance between the lines(x-1)/(2)=(y-2)/(3)=(z-3)/(4) and (x-2)/(3)=(y-4)/(4)=(z-5)/(5) |
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| 47. |
Evalute the following integrals int (1)/(4 sin^(2) x + 3 sin" x cos x " +2 cos^(2)x)dx |
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| 48. |
The sum upto (2n+1) terms of the series a^(2)-(a+d)^(2)+(a+2d)^(2)-(a+3d)^(2)+… is |
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Answer» `a^(2)+3ND^(2)` |
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| 49. |
In a bulb factory, three machines, A, B, C, manufacture 90%, 25% and 15% of the total production respectively. Of their respective outputs, 1 %, 2% and 1 % are defective. A bulb is drawn at random from the total product and it is found to be defective. Find the probability that it was manufactured by machine C. |
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