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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. |
Match the items in column I with those in column II. Here omegain1 is a cube root of unity . {:("Column I","Column II"),("(A)The value of "1/3(1-omega)(1-omega^2)(1-omega^4)(1-omega^8)"is ",(p)-128),((B) omega(1+omega-omega^2)^7=,(q)6),((C)"The least positive integer n such that "(1+omega^2)^n=(1+omega^4)^n " is " ,(r)0),((D) (1)/(1+2omega)+(1)/(2+omega)-(1)/(1+omega)" is equal to ",(t)3):} Now match for Column I from Column II |
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Answer» <P>`{:("(A)",(B),(C),(D)),(q,s,t,p):}` |
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| 2. |
3 dice are rolled and given that one or more dice shows 6. Find the probability that atleast one die shows 5. |
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| 3. |
If the harmonic mean of the roots ofsqrt(2)x^(2)-bx+(8-2sqrt(5))=0 is 4, then the value of b= |
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Answer» 2 |
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| 4. |
Find both the maximum value and the minimum value 3x^(4) - 8x^(3) + 12x^(2) - 48x + 25 on the interval [0,3]. |
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| 5. |
If the letters of the word 'STREAM' are arranged in all possible ways and the words thus formed are arranged as in a dictionary. Find the word whose rank is 257. |
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| 6. |
Three cards are drawn from an ordinary deck of 52 cards. Each card is replaced in the deck before the next card is drawn. What is the pobability that at least one of the cards will be a spade? |
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Answer» `3/12` was a spade = `39/52cdot39/52cdot39/52cdot=3/4cdot3/4cdot3/4=27/64`. Probability that I was a spade=`1-27/64=37/64`. |
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| 8. |
If three unit vectors a,b,c satisfy a + b + c = 0 then the angle between a and b is : |
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Answer» `(2 pi)/(3)` |
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| 9. |
Find the area bounded by xy=a^2,y=0,x=alpha,x=beta(beta gt alpha gt0) |
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Answer» SOLUTION :AREA = `int_alpha^betaydx=int_alpha^betaa^2/xdx` `=a^2[INX]_alpha^beta=a^2 In(beta//alpha)` |
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| 10. |
What is the value of sec^2(tan^(-1)2)+cosec^2(cot^(-1)3)? |
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| 11. |
Differentiate w.r.t.x the function in Exercises 1 to 11. (cos^(-1)""(1)/(2))/(sqrt(2x+7)), -7 lt x lt2. |
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| 13. |
** is a binary operation o Z. If x"*" y = x^(2)+y^(2)+xythen find [(1"*"2)+(0"*"3)]^(2). |
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| 15. |
If A+B+C=270^(@) then cos 2A+cos2B=cos2C+4sinA sinB sinC= |
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Answer» 0 |
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| 16. |
Find the projection of the vector veca=hati+3hatj+7hatk on the vector vecb=7hati-hatj+8hatk. |
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| 17. |
A wire of length 28 m, is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum ? |
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| 18. |
Construct truth tables for the following and indicate which of these are tautologies p rarr( ~ q^^r) |
Answer» SOLUTION :
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| 19. |
The volume of sphere is increasing at the rateof 1200 cu cm/s. The rate of increase in its surface area when the radius is 10 cm is |
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Answer» 120 SQ cm/s |
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| 20. |
If [x] and {x} represents integral and fractional parts of a real number x, and f(x) = (a^(2[x] +{x})-1)/(2[x} + {x}), x ne 0, f(0) = log_(e) a, where a gt 0, a ne 1, then |
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Answer» `f(X)` is continous at x=0 |
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| 21. |
The value of cos^(2)10^(@)-cos10^(@)cos 50^(@)is k/2 . The value of k is ________. |
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Answer» `=1/2 [ 2 cos^(2) 10^(@)-2cos10^(@)cos 50^(@)+2cos^(2)50^(@)]=1/2[1+cos20^(@)-cos40^(@)-cos60^(@)+1+cos 100^(@)]` `=1/2[3/2 + (cos 20^(@)-cos40^(@))+cos(90^(@)+10^(@))]=1/2[3/2 +2*sin30^(@)SIN 10^(@)-sin 10^(@)]=3/4` |
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| 22. |
S is a sample sapce. S={x in N : 1 lt x le 100} and E={x : (x+1) (x-1) in S}. Then P(E )= |
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Answer» `(1)/(10)` |
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| 23. |
Ify= (sin x )/( 1+(cos x )/(( sin x)/(1+ ....infty ))),then (dy)/(dx) = |
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Answer» ` ((1+y)COS X+ y SIN x)/( 1+2y +cos x -sin x )` |
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| 25. |
If the circle x^2 + y^2 + 2x - 2y + 4 = 0 cuts the circle x^2 + y^2 + 4x - 2fy +2 = 0 orthogonally, then f = |
| Answer» ANSWER :C | |
| 26. |
cos (sin x) |
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| 27. |
Compute the surface area of the torus generated by revolving the circle x^(2) + [y -b]^(2)= r^(2) (0 lt r lt b) about the x-axis |
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| 28. |
Number of way in which a composite number N can be resolved into two factors which are co-prime to each other if N is of the form 2^(2)3^(2)5^(2)7^(2) ,is |
| Answer» ANSWER :D | |
| 29. |
A car starts from a point P at time t = 0 seconds and stops at point Q. The distance x, in metres, covered by it, in t seconds is given byx=t^(2)(2-(t)/(3))Find the time taken by it to reach Q and also find distance between P and Q. |
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| 30. |
Find the mean number of heads in three tosses of a fair coin. |
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| 31. |
Corner points of the feasible region determined by the system of linear constraints are (0, 3), (1, 1) and (3, 0). Let Z = px + qy, where p, q gt 0. Condition on p and q, so that the maximum of Z occurs at (3, 0) and (1, 1) is ………. |
| Answer» Answer :B | |
| 32. |
Thevectors vec(a) = - 4hat(i) + 3hat(k), vec(b) |
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Answer» `HAT(i) + hat(J) + 2hat(k)` |
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| 33. |
Let ABC be a triangle in which the line joining the circumecentre and incentre is parallel to base BC of the triangle. Then answer the following questions : If angleA=60^(@), then Delta ABC is |
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Answer» isoceles `because A = 60^(@)` but `(r )/(R )LE (1)/(2)` in any `Delta ABC` Hence, `Delta` is equilateral |
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| 34. |
Let ABC be a triangle in which the line joining the circumecentre and incentre is parallel to base BC of the triangle. Then answer the following questions : Then range of angle A is |
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Answer» `[(pi)/(6),(pi)/(3)]`  `because` The line joining O and I is parallel to BC `therefore OI = DE` and OD = IE OD = R COS A, IE = r `therefore cos A = (r )/(R ) le (1)/(2)` `rArr 0 lt cos A le (1)/(2)rArr (pi)/(3)le A lt (pi)/(2)` |
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| 35. |
Let ABC be a triangle in which the line joining the circumecentre and incentre is parallel to base BC of the triangle. Then answer the following questions : If ODEI is a square where O and I stands for circumcentre and incentre, respectively and D and E are the point of perpendicular from O and I on the base BC, then |
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Answer» `(r )/(R )=(3)/(8)` Now, `OI = sqrt(R^(2)-2Rr)` `therefore sqrt(R^(2)-2Rr) = R cos A` `RARR R^(2)-2Rr = R^(2)cos^(2)A` or `1-cos^(2)A=(2R)/(R )` Also `cos A = (r )/(R )` `rArr 1-((r )/(R ))^(2)=(2r)/(R )` `rArr ((r )/(R ))^(2)+(2r)/(R )-1=0` `rArr (r )/(R )= sqrt(2)-1` |
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| 36. |
Find the domain of definitions of the functions (Read the symbols [*] and {*} as greatestintegers and fractional part functions respectively.) y = log_(10) sin (x-3) + sqrt(16 -x^2) |
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| 37. |
Find the domain of definitions of the functions (Read the symbols [*] and {*} as greatestintegers and fractional part functions respectively.) f(x) = log_7log_3 log_2(2x^3 + 5x^2 - 14 x) |
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| 38. |
If veca=hati+2hatk,vecb=hati+hatk,vecc=7hati-3hatj+4hatk, then the vector vecd such that vecd"xxvecb=veccxxvecb,veca.vecd=0 is |
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Answer» `2hati-5hatj+2hatk` |
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| 39. |
A pair of dice rolled 24 times. A person wins by not getting a pair of 6's on any of the 24 rolls. What is the probability of his winning? |
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| 40. |
If x in(1, 30) and (1+tan x^(@))(1+tan(x+1)^(@))...(1+tan(x+44)^(@)) = 2^(23) then value of x is |
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Answer» `0` `x = 1` `(1+tan x^(@))(1+tan(x+1)^(@))…. (1+tan(x+44)^(@))ge(1+tan 1^(@))` `(1+tan 2^(@)) ……….. (1+tan 45^(@))` Now `(1+tan1^(@))(1+tan 44^(@))(1+tan2^(@))(tan 43^(@)+1)`……….. using `(1+tan x^(@))(1+tan(45-x^(@)))=2` `(1+tan 1^(@))(1+tan 2^(@))............ (1+tan 45^(@)) = 2^(23)` |
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| 41. |
If a complex number z satisfies |z| = 1 and arg(z-1) = (2pi)/(3), then (omega is complex imaginarynumber) |
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Answer» `z^(2) + z` is PURELY imaginary number `|z|=1` Thismeans that z lies on a circle whose centre is at ORIGIN and radius is 1. Also ,`arg(z-1)= (2pi)/(3)` Thisimpliesthatz lies ona ray emanating (1,0) making an angle of `(2pi)/(3)` with positive real axis. `RARR /_PQX =(2pi)/(3)` ` rArr /_PQO = (pi)/(3)` and OP=OQThus, triangle OPQ is equilaterl.  `rArr OP = OQ = PQ` `rArr |z|=|z-1| =1` Also `arg(z) = 60^(@)` `therefore z = COS60^(@) + i sin 60^(@) = (1+ isqrt(3))/(2) = -omega^(2)` `z^(2)= omega^(4) = omega` `rArr z^(2) + z = omga - omega^(2) = ((-1+isqrt(3))/(2)) -((-1-isqrt(3))/(2)) = isqrt(3)` Therefore , `z^(2) +z` is purely imaginary number . |
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| 42. |
The equation of the directrix of the parabola y^(2)+4y+4x+2=0 is |
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Answer» ` X=- 1 ` |
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| 43. |
In the redox reaction . MnO_(4)^(-)+C^(2)O_(4)^(2-)+H^(+) rarr Mn^(2+)+CO_(2)+H_(2)O (Unbalance equation) 20 mL of 0.1 M KMnO_(4) react quantitively with |
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Answer» 20 ML of 0.1 M oxalate |
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| 44. |
The slope of a chord of the parabolay^(2) = 4xis 2. Show that the locus of the point which divides the chord internally in the ration 1 : 2is a parabola whose equation is (y - (8)/(9))^(2) = (4)/(9) (x - (2)/(9)). |
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| 45. |
Following is the graph ofy = f' (x) , given thatf(c) = 0. Analyse the graph and answer the following questions. (a) How many times the graph ofy = f(x) will intersect thex - axis? (b) Discuss the type of roots of the equation f (x) = 0,a le x le b. (c) How many points of inflection the graph ofy = f(x), a le x le b, has? (d) Find the points of local maxima/minima of y = f(x), a lt x b. (e) How many roots equationf''(x) = 0 has? |
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Answer» Solution :(a) ` f'(x) le 0, forall x in [a, b], ` so f(x)is a decreasing function and `f(c) = 0 RARR f(x)`is a decreasing function and` x (c) = 0 rArrf(x) ` cutsx - axis once when x = c. (b) We NOTE thatf(c) = 0, f'(c) = 0. Also tangent tof'(x)atx = cis y = 0. So f''(c) = 0Therefore, x = c is repeated root of third order. So theequation f(x) = 0 has at least three repeated roots. (c) We have f''(c) = 0 . So thegraph of y = f (x)has ONE point of inflection at x = c. (d) Asf (x) is a decreasing function for all` x in (a, b) ,f(x)` has no local maxima or minima. (E) ` f''(c) = 0 rArrc `ISA root off''(x) = 0. |
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| 46. |
If f(x)=1-x+x^(2)-x^(3)+x^(4)+….oo then int_(0)^(x)f(x)dx= |
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Answer» LOG X |
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| 47. |
If alpha" and "beta arethe roots of the equation x^(2)+3x-4=0, then (1)/(alpha)+(1)/(beta) is equal to |
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Answer» `(-3)/(4)` |
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| 49. |
If n is a positive interger, then the coefficient of x^(6) in the expansion of (1-2x + 3x^(2) -4x^(3) + …)^(-n) is |
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Answer» `""^((2n))C_(4)` |
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| 50. |
A plane P_(1) is making intercepts 2, 3, 4 on X, Y and Z-axes respectively. Another plane P_(2) is passing through (-1,6,2) and is perpendicular to the line joining the points (1,2,3) and (-2,3,4). Let theta be an angle between P_(1) and P_(2), then 61cos^(2)theta=______ |
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