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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. |
Examine the continuity of the function f(x) = {(|x-a| sin ((1)/(x-a))",","if " x ne a),(0",","if' x = a):} at x=a |
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| 2. |
What are the coordinates of a point which is common to all the coordinate planes. |
| Answer» SOLUTION :ORIGIN O(0,0,0) is COMMON to all COORDINATE PLANES. | |
| 3. |
Let p, q be real numbers. If alphais the root ofx^(2) +3p^(2)x+5q^(2)=0, betais a root of x^(2) + 9p^(2)x + 15q^(2) = 0 and 0 lt alpha lt beta, then the equationx^(2) +6p^(2)x +10q^(2)=0has a root gammathat always satisfies |
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Answer» `GAMMA =ALPHA//4 +BETA` |
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| 4. |
If f(x)=((a^(x)-1)^(3))/(sin (x log a)log (1+x^(2) log a^(2)) is continuous at x=0 , then f(0)= |
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Answer» LOG a |
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| 5. |
intdx/(sqrtx-root(3)(x))(x=t^6) |
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Answer» SOLUTION :`INTDX/(sqrtx-root(3)(x))` [Put`x=t^6 `dx=6t^5`] =`(6t^2dt)/(t^3-t^2)=6int(t^3dt)/(t-1)=6int((t^3-1)+1)/(t-1)DT` =`6int{t^2+t+1+1/(t-1)}dt` =`6{1/3 t^3+1/2 t^2+t+Inabs(t-1)}+C` =`2sqrtx+3root(3)(x)+6x^(1/6)+6Inabs(x^(1/6)-1)+C` |
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| 6. |
Findthecoordinatesofthepointofintersection for the following lines :( x-3 ) /(1) = (y-5 ) /( 2 )=(z- 1 ) /(- 1)and( x-4 ) /(2) =(y- 2 ) /( - 1)=(z -4 ) /(2) |
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| 7. |
If ((1-3x)^(1//2)+(1-x)^(5//3))/(sqrt(4-x)) is approximately equal to a+bx for small values of x, then (a,b)= |
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Answer» `(1,(35)/(24))` `=([1+(1)/(2)(-3x)]+[1+(5)/(3)(-x)])/(2[1+(1)/(2)(-(x)/(4))])` (Neglecting HIGHER POWERS of `x`) `=([1-(19)/(12)x])/([1-(x)/(8)])` `=[1-(19)/(12)x][1-(x)/(8)]^(-1)` `=[1-(19)/(12)x][1+(x)/(8)]=1-(35)/(24)x` then `a+bx=1-(35)/(24)ximpliesa=1`, `b=-(35)/(24)` |
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| 9. |
The solution of (dy)/(dx)+ (xy)/(1 + x^(2)) = (1)/(x (1 + x^(2))) is |
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Answer» `ysqrt(1+x^(2)) = sec^(-1)x+c` |
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| 10. |
If f has a local extremum at a and if f'(a) exists then |
| Answer» Answer :C | |
| 11. |
If p is the point of contact of thecircles x^2 + y^2 + 4x + 4y - 10 = 0and x^2 + y^2 - 6x - 6y + 10 = 0and Q is theirexternal centre of similitude, then the equation of the circle with P and Q as the extremities of its diameter is |
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Answer» `x^2 + y^2 + 14X + 14y - 26 = 0` |
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| 12. |
Minimize z=sum_(j=1)^(n)""sum_(i=1)^(m)c_("ij ")x_("ij") Subject to : sum_(j=1)^(n)x_("ij")=a_(i),i=1,..........,m sum_(i=1)^(n)x_("ij")=b_(i),j=1,..........,n is a LPP with number of constraints |
| Answer» ANSWER :A | |
| 13. |
A common tangent tox^(2)-2y^(2)=18and x^(2)+ y^(2) =9 is |
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Answer» ` y =2x + 3SQRT5` |
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| 14. |
A realvalued functionf (x)satisfiesthe functionalequationf (x-y)= f(x) f(y) - f(a-x)f(a +y)wherea isa givenconstantand f (0)=1,f(2a -x)isequalto |
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Answer» `F(X)` |
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| 15. |
The probability that an event A occurs in a single trial of an experiment is 0.6. Three independent trials of the experiment are performed. Find the probability that the event at least once occurs. |
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Answer» SOLUTION :P (event at LEAST once) `=1-[1-0.6]^3` =1-0.064 =0.936 |
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| 16. |
If A is an invertible matrix of order 2, then det(A^(-1)) is equal to |
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Answer» DET (A) |
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| 17. |
Find the least and the greatest value of 2sin x + sin 2x over [0, 2pi]. |
| Answer» Answer :C | |
| 18. |
If 0 lt y lt 2^(1//3) and x(y^(3)-1)=1 then (2)/(x)+(2)/(3x^(3))+(2)/(5x^(5))+…….oo |
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Answer» `LOG((y^(3))/(2-y^(3)))` |
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| 20. |
the point lies in half plane x-2y |
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Answer» (5,2) |
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| 21. |
How many 10 letter words can be formed using 26 English letters such that each word must include A, B but separated exactly with 3 letters. |
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| 22. |
Which among the following is/are true ? I. The values of cosec x repeat after an interval of 2pi. II. The values of sec x repeat after an interval of 2pi. III. The values of cot x repeat after an interval of pi. |
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Answer» I is TRUE |
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| 23. |
Compute the magnitude of the following vectors : a = hat(i)+hat(j)+hat(k),b=2hat(i)-7hat(j)-3hat(k),c=(1)/(sqrt(3))hat(i)+(1)/(sqrt(3))hat(j)-(1)/(sqrt(3))hat(k). |
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| 24. |
Integration by partial fraction : int(x^(2)+x-1)/(x^(2)+x-6)dx=.... |
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Answer» `x+LOG(x+3)+log(x-2)+C` |
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| 25. |
Find the centre and radius of each of the circles whose equations are given below. 2x^(2) + 2y^(2) - 3x + 2y - 1 =0 |
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| 26. |
cos^(-1)((-1)/2)-2 sin^(-1)(1/2)+3 cos^(-1) ((-1)/sqrt2) -4 tan^(-1) (-1) equals |
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Answer» `(19pi)/12` |
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| 27. |
The value of determinant |(a-b ,b+c,a),(b-a,c+a,b),(c-a,a+b,c)| is |
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Answer» 1.`a^(3)+B^(3)+C^(3)` |
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| 28. |
Find the rate at which the volume of a spherical balloon will increase when its radius is 2 meters if the rate of increase of its redius is 0.3 m/min. |
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Answer» Solution :Let r be the radius of a spherical ballon. Give that dr/dt = 0.3 m/min. If V be the volume of the ballon then `dV/dt = 3/4 PI r _3`. `rArr Dv/dt = 3/4 pi cdot 3r^2 dr/dt = 4 pi r^2 dr/dt` `therefore Dv/dt](r=2) = 4pi xx4 XX cdot 3 = 4 cdot 8 pi m^3/min` `therefore` The volume increase at the rate of `4 cdot 8 pi m^3/min`. |
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| 29. |
Form the differential equation of all circles touching the x-axis at the origin and center on y-axis. |
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| 30. |
int_(0)^(b) (dx)/(1+x^2)= int_(b)^(oo) (dx)/(1+x^2), then b is equal to |
| Answer» Answer :D | |
| 31. |
If y = (cos x - sin x)/( cos x + sin x ) , "then " (dy)/(dx) |
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Answer» `X^(2)` |
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| 32. |
Ifx^4 - 12x ^3 +52 x^2 - 96x+64=0havepairsof equalrootsthen therootsare |
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Answer» `1,1,5,5` |
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| 33. |
If a number x is selected from natural numbers 1 to 100 find the prbability for x + (100)/(x) gt 29. |
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| 34. |
The position vector of the point P is (4, 5, -3). The distance of the point P from the plane XY, YZ and XZ is P_(1),P_(2) and P_(3) respectively then underset(i=1)overset(3)sumP_(i)=……….. |
| Answer» ANSWER :B | |
| 35. |
If the equations of two circles whose radii are a and a' are S=0 and S'=0 , then show that the circles S/a+(S')/(a')=0 and S/a-(S')/(a')=0 intersect orthogonally. |
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Answer» `(pi )/(3) ` |
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| 36. |
Arrange the following angles in ascending order (A) Angle between the vectors I + 3j + 4k and I - 3j + 2k (B) Angle between the planes r. (2i - j + k) = 7 and r. (I + j + 2k) = 11 (C ) Angle between the lines r = x(I + 2j + 2k) and r = t(3i + 2j + 6k) |
| Answer» Answer :B | |
| 37. |
find the general solution of sec^(2)xtan y dx + sec^(2) y tan x dy = 0 |
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| 38. |
Find the slope of the tangent to the curve y=x^(3)-3x+2 at the point whose x - coordinate is 3. |
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| 39. |
If veca and vecb are unit vectors such that [veca, vecb, veca xx vecb]=(1)/(4), then the angle between veca and vecb is : |
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Answer» `(PI)/(6)` |
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| 40. |
Internal bisectors of DeltaABC meet the circumcircle at point D, E, and F Length of side eF is |
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Answer» `2R cos.(A)/(2)` `angleADE = angleABE = (B)/(2)` SIMILARLY, `angleFDA = angleFCA = (C)/(2)` `rArr angleFDE = (B+C)/(2)` and `angleDEF = (A+C)/(2) and angleDFE = (A +B)/(2)`  In `DeltaDEF`, by SINE rule, `(EF)/(sin (angleFDE)) = 2R` `rArr EF = 2R cos ((A)/(2))` Then, area of `DeltaDEF` `= 2R^(2) sin ((A+B)/(2)) sin ((B+C)/(2)) sin ((A+C)/(2))` `=2R^(2) cos ((A)/(2)) cos ((B)/(2)) cos ((C)/(2))` Now `(Delta_(ABC))/(Delta_(DEF)) = (2R^(2) sin A sin B sin C)/(2R^(2) cos ((A)/(2)) cos((B)/(2)) cos((C)/(2)))` `= 8sin ((A)/(2)) sin ((B)/(2)) sin ((C)/(2)) le 1` `rArr Delta_(ABC) le Delta_(DEF)` |
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| 41. |
Evaluate underset(-pi//2)overset(pi/2)int sin|x| dx |
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Answer» 0 |
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| 42. |
(i) If 'a' is a complex number such that |a| = 1. Find the values of a, so that equation az^2 + z+1=0 has one purely imaginary root. (ii) Show that ((pi+1)/(pi-1))^(m) e^(2mi cot^(-1) (p))=1. |
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| 43. |
If( 1+ x ) ^n= C _ 0+ C _ 1x+C_ 2x ^ 2+ … + C_n x ^ n, thenC _0+2 C_1+ 3C_2+ … + (n + 1 ) C_ nisequal to |
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Answer» `2^N+n.2^(n -1) ` ` RARRX ( 1 + x ) ^n= C _0x+ C _ 1x ^ 2+C _ 2x ^3+… +C_n x ^ ( n+ 1 ) ` Differentiatingonbothsides `XN (1 +x) ^(n - 1 )+(1 + x ) ^n= C_0+2C_1 x+3 C_2x ^ 2+... ( n + 1 ) C _ n x ^ n ` Substituting`x= 1`inaboveequation, `thereforen ( 2 )^(n - 1 )+2^n=C_0+2C_1+ 3 C_2+... +(n + 1 )C_ n ` |
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| 44. |
The straight line y=2x meets y=f(x) at P, where f(x) is a solution of the differential equation (dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2)) such that f(1)=3, then f'(x) at point P is |
| Answer» Answer :A | |
| 45. |
If A,B,C,D are the length of normals to the curves 1) y=4x^2 at (-1,4) 2) y=x^3+1 at (1,2)3) y=(x^3)/(2-x) at (1,1) 4) 2x^2+3xy-2y^2=8 at (2,3)then the ascending order of A,B,C,D is |
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Answer» A,B,C,D |
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| 46. |
a circles S of radius 'a' is the director circle of another circleS_(1).S_(2) is the director circle of S_(2) and so on. If the sum of radius of S, S_(1), S_(2), S_(3) …. circlesis '2' and a=(k-sqrt(k)), then the value of k is …… |
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| 47. |
Let E = 1^(2017) + 2^(2017) + 3^(2017) + ... + 2016^(2017), then E is divisible by |
| Answer» ANSWER :B | |
| 48. |
Find lambda if the vectors hati-hatj+hatk, 3hati-hatj+2hatk and hati+lambda hatj-3hatk are coplannar. |
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| 49. |
For a biased die the probability for different faces to turn up are given below. {:("Face",1,2,3,4,5,6),("Probability",0.1,0.32,0.21,0.15,0.05,0.17):} The die is tossed and you are told either face 1 or 2 has turned up. Then the probability that it is face 1 is |
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Answer» `5//21` |
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| 50. |
Express A= [{:(3,5),(1,-1):}] as sun of symmetric and skew symmetric matrix. |
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Answer» (ii) `A=[(6,-2,2),(-2,3,-1),(2,-1,3)]+[(0,0,0),(0,0,0),(0,0,0)]` (iii) `A=[(3,(1)/(2),(-5)/(2)),((1)/(2),-2,-2),((-5)/(2),-2,2)]+[(0,(5)/(2),(3)/(2)),((-5)/(2),0,3),((-3)/(2),-3,0)] (IV)A=[(1,2),(2,2)]+[(0,3),(-3,0)]` |
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