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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. |
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle alpha is one - third that of the cone and the greatest volume of cylinder is (4pi)/(27)h^(3) tan^(2)alpha. |
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| 2. |
The value of the integral int_(-pi//4)^(pi//4) log (sec theta - tan theta)d theta is |
| Answer» ANSWER :A | |
| 3. |
Evaluation of definite integrals by subsitiution and properties of its : F(x)=int_(x^(2))^(x^(3))logt.dt(xgt0) then F'(x)= …………. |
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Answer» `(9X^(2)-4x)logx` |
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| 6. |
Consider A={1,2,3,......,10} Find the sum of all products of numbers by taking one or more from A. |
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| 7. |
Form the differential equation by eliminating the arbitrary constant from the equation y = ax^(2) + be^(-x) |
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Answer» `x (x + 2) (d^(2)y)/(DX^(2)) + (x^(2) - 2) (dy)/(dx) + 2 (x + 1) y = 0` |
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| 9. |
If alpha, beta are the roots of x^(2) + bx + c = 0 and, alpha +h, beta+h are the roots of x^(2) + qx +r = 0 then h = |
| Answer» ANSWER :D | |
| 10. |
If int e^(2x) .x^(3) dx = (e^(2x))/(16) [ ax^(3) + bx^(2) + cx + d] + K then (a, b, c,d ) = |
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Answer» (8, -12, 12, -6) |
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| 11. |
Why was Franz surprised? |
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Answer» Because of VILLAGE elders |
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| 13. |
Find the maximum and minimum values, if any, of thefunctions given by h(x) = x + 1, x in(– 1, 1) |
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| 14. |
The tangent to the ellipse 3x^(2) +16y^(2) =12,at the point (1,3//4), interects the curve y^(2) +x= 0at |
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Answer» no point |
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| 15. |
Each question in this section has four choices A , B, C , and D out of which only one is coR Rect. Mark your choices as follows . Let R be a relation on the set N N of the natural numbers defined by nRm ltimplies n is a factor of m.LetR be the relation over the set of integers such that mRn if and only if m is a multiple of n Then R is reflexive because |
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Answer» MRN as m is MULTIPLE of n |
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| 16. |
Satement-1: if 1/2lexle1then cos^(-1)x-sin^(-1){x/2+sqrt(3-3x^(2))/(2)} is equal to (pi)/(5) Statement-2: sin^(-1)(2xsqrt(1-x^(2))=2sin^(-1)x if x in -(1)sqrt(2),(1)sqrt(2)) |
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Answer» Statement-1 is is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1. `1/2 LE x le 1 rarr 1/2 le cos theta le 1 rarr 0 le theta le (PI)/(3)` `therefore -theta-sin^(-1){1/2cos theta +sqrt(2)/(2)sin theta}` `=theta -sin^(-1){sin(pi)/(6)cos theta +cos (pi)/(6) sin theta}` `=theta -sin^(-1){sin(theta+(pi)/(6))}` `=theta -(theta +(pi)/(6))=-(pi)/(6)` so statement 1 true let x =sinx `theta` then `(1)/sqrt(2) le x le (1)sqrt(2) rarr -(1)/sqrt(2) le sin thetale (1)/sqrt(2) rarr =(pi)/(4) le theta le (pi)/(4)` `there sin^(-1)2xsqrt(1-x^(2))=sin^(-1)(sin 2 theta) =2 sin^(-1)x` so statement 2 true we have `1/2 le x le 1 rarr x^(2)+3/4gt 1` using : `sin^(-1) x + sin^(-1)y =pi -sin^(-1)xsqrt(1-y^(2))+ysqrt(1-x^(2))` when `0 lt x,y le 1 and x^(2) a+ y^(2) gt1` we have `thereforesin^(-1)((x)/(2)+(sqrt(3-3x^(2))/(2))=pi-sin^(-1)x-(pi)/(3)=2pi)/(3)-=sin^(-1)x` `=cot^(-1)x=(2x)/(3)+sin^(-1)x=(pi)/(2)-(2pi)/(3)=-(pi)/(6)` so statement-1 is true statement -2 true (see theory ) |
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| 17. |
The radius of the circle passing through the foci of the ellipse x^(2)/16+y^(2)/9=1,and having its centre at (0, 3) is |
| Answer» ANSWER :A | |
| 18. |
What are the value of theta, between 0 and 2 pi, when tan theta = -1? |
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Answer» `PI/4 and (3 pi)/4` only |
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| 19. |
Let D is a point on the line l_(1):x+y-2=0, S(3, 3) is a fixed point and line l_(2) is the perpendicular to DS and passingthrough S. If MK is another point on the line l_(1) (other than D), then the locus of the point of intersection of l_(2) and angle bisector of the angle MDS is a conic whose length of latus rectus rectum is equal to |
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Answer» `4sqrt2` |
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| 20. |
The point from which the tangents to the circles x^2+y^2-8x+40=0, 5x^2+5y^2-25x+80=0 and x^2+y^2-8x+16y+160=0 are equal in length is |
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Answer» `(8,(-15)/2)` |
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| 21. |
If a is the matrix [(1,-3),(-1,1)], then A-(1)/(3)A^(2)+(1)/(9)A^(3)……….. +(-(1)/(3))^(n)A^(n+1)+ ……… oo=(3)/(13)[(1,a), (b,1)]. Find |(a)/(b)|. |
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| 22. |
Two vehicles C_(1) and C_(2) start from a point P and travel east of P at the speeds 20 km/hr and 60 km/hr respectively. If an observer, one kilometre north of P, is able to see both the vehicles at the same time, then the maximum angle of sight between the observer's view of C_(1) and C_(2), is : |
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Answer» `(PI)/(3)` |
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| 23. |
If the 6^(th) term in the expansion of ((1)/(x^(6//3))+x^(2)log_(10)+x)^(8)is 5600, then the value of x is |
| Answer» Solution :`a^(sqrt(X))rarr0 " "&" "a^(1//sqrt(x))rarr+oo" as "x rarr0^(+)` | |
| 24. |
Evaluate the following definite integrals : int_(0)^(9)(dx)/(1+sqrtx) |
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| 25. |
If alpha, beta are roots of the quadratic equationx^(2)-x-1=0,then the quadratic equation whose roots are (1+alpha)/( 2-alpha), (1+beta)/(2-beta) is |
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Answer» `Z^(2)+z+1=0` |
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| 26. |
(a) Let '**' be a binary operation defined on Q, the set of rational numbers, as follows : (i) a**b=a-b, for a,binQ (ii) a**b=a^(2)+b^(2), for a,binQ (iii) a**b=a+ab, for a,binQ (iv) a**b=(a-b)^(2), for a,binQ (v) a**b=(ab)/(4), for a,binQ (vi) a**b=ab^(2), for a,binQ. Find which of the binary operations are commutative and which are associative. |
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| 27. |
Show that the circles x^(2) +y^(2) -4x-6y-12=0 and 5(x^(2)+y^(2))-8x - 14y - 32 = 0 touch each other and find their point of contact. |
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Answer» (iv) `(1/13,(-21)/13),5x+12y+19=0` (V) `(5,1),4x-7y-13=0` (iv) `(5,5),4x+3y-35=0` |
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| 28. |
Assertion A: In DeltaABC, sum(cos A)/(sin B sin C)=2 Reasin(R):In DeltaABC, sin A +sinB+sin C= 4"cos"A/2"cos"B/2"cos"C/2 |
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Answer» A is true, R is true and R is correct explanation of A |
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| 29. |
Differentiate w.r.to x : x^(x) + a^(x) + x^(a) + a^(a) |
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| 30. |
Let a, b, c and m in R^(+). The possible value of m (independent of a, b and c) for which atleast one of the following equations have real roots is {:(ax^(2)+bx+cm=0),(bx^(2)+cx+am=0),(cx^(2)+ax+bm=0):}} |
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Answer» `(1)/(2)` `D_(1)+D_(2)+D_(3) ge 0` `(b^(2)-5acm)+(c^(2)-4bam)+a^(2)-4cbm ge 0` `a^(2)+b^(2)+c^(2) ge 4(ab+bc+ca)m` `4m ge (a^(2)+b^(2)+c^(2))/(ab+bc+ca)`………`(1)` `AA a,b,c in R+` but `a^(2)+b^(2) ge 2ab` ETC. `:. a^(2)+b^(2)+c^(2) ge ab+bc+ca` `(a^(2)+b^(2)+c^(2))/(ab+bc+ca) ge 1` `:.(a^(2)+b^(2)+c^(2))/(ab+bc+ca)|_(min)=1` , Hence `4m` must be less than or equal to theminimum VALUE. `:. 4m le 1` gtbrgt `implies m le (1)/(4)` `implies m in (0,(1)/(4)]` |
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| 32. |
Find the value of K if the lines x+y-5=0 and 2x+ky-8=0 are conjugate with respect to the circle x^(2)+y^(2)-2x-2y-1=0 |
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| 33. |
Find the order and the degree of the following differential equation ((dy)/(dx))^3 - 4((dy)/(dx))^2 + 7y = sin x |
| Answer» SOLUTION :ORDER = 1, DEGREE = 3 | |
| 34. |
A is a set containing 'n' elements . A subset P of A is chosen at random . The set A is reconstructed by replacing the elements of the subset of P, a subset Q of A is again chosen at random. Find the probability that (i) PnnQ=phi (ii) PuuQ=A (iii) PuuQ=A and PnnQ=phi (iv) Q is subset of P |
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Answer» (ii) `((3)/(4))^(n)` (iii) `((1)/(2))^(n)` |
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| 35. |
intsec^30xtanxdx |
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Answer» SOLUTION :`intsec^30xtanxdx` =`intsec^29x.secx.tanxdx` [PUT secx=t Then secx.tanxdx=dt] =`intt^29dt` =`1/30t^30+C=1/30sec^30x+C` |
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| 37. |
Evaluate the integrals in exercise. overset(2)underset(0)int sqrt(x+2)dx |
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| 38. |
Which of the following statements can be inferred from the information given? For all the time periods shown, borate production, in millions of 2005 dollars, was the same. Of the time periods shown, 1981 - 1985 was the one in which the mining industries product the greatest value of gold and silver, measured in 2005 dollars. Of the time periods, shown, 2001 - 2005 had the highest average annual GDP, measured in 2005 dollars. |
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| 39. |
int_(-pi//2)^(pi//2) sin^4 xcos^6 xdx is equal to |
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Answer» `(3pi)/(128)` |
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| 40. |
The value of 99^(50)-(99)/(1)*98^(50)+(99.98)/(1.2)97^(50)-……-(99.98)/(1.2)*2^(50)+(99)/(1)*1^(50) is |
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Answer» `0` `=^(99)C_(0)99^(50)-^(99)C_(1)(99-1)^(50)+^(99)C_(2)(99-2)^(50)-....+^(99)C_(98)(99-98)^(50)+^(99)C_(99)(99-99)^(50)` `=99^(50)('^(99)C_(0)-^(99)C_(1)+^(99)C_(2)-^(99)C_(3)+...)+^(50)C_(1)*99^(49)('^(99)C_(1)-2*^(99)C_(2)+3*^(99)C_(3)-....)+....` `=0+0+0+....+0=0` |
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| 41. |
Two players A and B each toss 5 coins. Find the probability that A and B get the same number of heads. |
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| 42. |
Solve the equationx^4 -6x^3 +18x^2 - 30x+ 25=0given that2+Iis a root |
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| 43. |
If the imaginary part of (2z + 1)/(iz+ 1) is -2 , then the locus of the point representing z in the complex plane |
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Answer» a straight LINE |
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| 44. |
Statement: If N = ((1)/(20))^(20) then N contains 7 digitsbefore decimal. |
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Answer» STATEMENT -1 is ture, statement -2 is ture and statement-2 is CORRECT explaination for statement -1 |
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| 45. |
Of the three independent eventsE_(1) , E_(2) and E_(3) the probability that only E_(1)occurs is beta and only E_(3) occurs is gamma. Let the probability pthat none of the eventsE_(1),E_(2), "or" E_(3) occurs satisfy the equations (alpha - 2 beta)p = alpha beta and (beta - 3 gamma) p = 2 beta gamma. All the given probabilities are assumed to lie in the interval (0,1).Then("Probability of occurrence of" E_(1))/(" Probability of occurrence of" E_(3)) |
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Answer» <P> |
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| 46. |
(sin 5 alpha-sin3alpha)/(cos 5 alpha+2 cos 4 alpha+cos 3alpha)= |
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Answer» `COS ALPHA//2` |
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| 47. |
The order, degree of the differentialequation satisfying the relation sqrt(1+x^(2))+sqrt(1+y^(2))+lambda(x sqrt(1+y^(2))-y sqrt(1+x^(2))) is |
| Answer» ANSWER :1 | |
| 48. |
Choose the correct answer int(dx)/(sqrt(9x-4x^(2))) equals |
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Answer» `1/9sin^(-1)((9x-8)/8)+C` |
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| 49. |
Find 'c' of the mean value theorem for the function f(x)=2x^(2)-10x+29 in [2,9]. |
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