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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. |
Ifalpha , beta, gammaare therootsof x^3 +qx +r=0 thensum( beta + gamma )^(-1)= |
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Answer» `(Q)/(R )` |
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| 2. |
If A.M and G.M of the roots of a quadratic equation in x are p and q repectively then its equation is |
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Answer» `X^(2) - 2PX + Q^(2) = 0` |
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| 3. |
Let a_n(10^n)/(n!) for0 len le 1then theminimumvalueofn !(1-n)!is attainedwhena valueofn= |
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Answer» 11 |
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| 4. |
If alpha+beta, gamma epsilon R^(+) such that alpha gamma =1/(beta) then maximum value of alpha^(beta+gamma), .beta^(beta+gamma).gamma^(alpha+beta) is_____ |
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Answer» Assume `alpha GE beta ge gamma` (SAY) `impliesalpha ge 1` and `gamma LE1` `:.(alpha^(beta+gamma).gamma^(alpha+beta))/((alpha gamma)^(alpha+gamma))=(gamma^(beta-gamma))/(alpha^(alpha-beta)) le 1` |
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| 5. |
Find the area of region : {(x,y) : 0 le y le x^(2) + 1, 0 le y le x + 1,0 le x le 2}. |
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| 6. |
A die is thrown. If E is the event the number appearing is a multiple of 3 and F be the event the number appearing is even'. Then find wheather E and F are independent ? |
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| 7. |
Let n be a fixed positive integer and X take values 1,2,3,………..n. If P(X=k)=(1)/(n) for |
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Answer» `sqrt(N^(2)-1)/(2SQRT3)` |
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| 8. |
Find the middle term inn (i) ((2y^(2))/(3)+(3)/(2y^(2)))^(9),y ne0 (ii) (4x^(2)+9y^(2)+12xy)^(n) |
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Answer» Solution :(i) Number of terms in the expansion is (9+1)=10 (EVEN) `therefore` There are two middle terms given by `T_((9+1)/(2)) annd T_((9+3)/(2))` i.e., `T_(5) and T_(6)` We know that `T_(r+1)=.^(n)C_(r)x^(n-r)a^(r)` `thereforeT_(5)=T_(4+1)=.^(9)C_(4)((2y^(2))/(3))^(5)((3)/(2y^(2)))^(4)` `=(9xx8xx7xx6)/(4xx3xx2xx1)xx(2^(5))/(3^(5))y^(10).(3^(4))/(2^(4)y^(8))` `=(9xx8xx7xx6)/(4xx3xx2xx1)xx(2)/(3)xxy^(2)` `=84y^(2)`. and `T_(6)=T_(5+1)=.^(9)C_(5)((2y^(2))/(3))^(4)((3)/(2y^(2)))^(5)` `=(9xx8xx7xx6)/(4xx3xx2xx1)xx(2^(4))/(3^(4))xx(3^(5))/(2^(5))xx(1)/(y^(2))` `=(9xx8xx7xx6)/(4xx3xx2xx1)xx(3)/(2)xx(1)/(y^(2))` `=(189)/(y^(2))` (ii) `(4x^(2)+9y^(2)+12xy)^(n)` Since `4x^(2)+9y^(2)+12xy=(2x+3y)^(2)`, therefore `(4x^(2)+9y^(2)+12xy)^(n)=(2x+3y)^(2n)` Hence, there is only one middle TER namely `((2n)/(2)+1)^(th)` term i.e., `(n+1)^(th)` term. `therefore` The required middle term `=T_(n+1)=.^(2n)C_(n)(2x)^(2n-n)(3y)^(n)` . . . `=.^(2n)C_(n)(2x)^(n)(3y)^(n)` `=(lfloor(2n))/(lfloor(n)lfloor(n)).2^(n).3^(n)x^(n).y^(n)` hence, the middle term int he expansion of `(4x^(2)+9^(y)+12xy)^(n)` is `(lfloor(2n))/(floor(n)floor(n)).2^(n).3^(n)x^(n).y^(n)` |
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| 9. |
Minimise and maximise Z=3x+9y subject to the constraints. x+3y le 60 x+y ge 10 x le y x ge 0, y ge 0 |
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| 10. |
Let baru and barv be two non-collinear vectors. If the lengths a, b, c of the sides of DeltaABC are such that (a-b)baru+(b-c)barv+(c-a)(baruxxbarv)=bar0 then DeltaABC is |
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Answer» RIGHT angled |
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| 11. |
If A, B, C, in (-(pi)/(2), (pi)/(2)). Then prove that cos A+ cos B + cos C le 3/2 . |
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| 12. |
If f(x)=8x^2 +x-3, then find f(-1) |
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| 13. |
The p.m.f.ofa random variable X is (# TRG_MAT_MCQ_XII_P2_C08_E02_009_Q 01.png" width="80%"> If b= 2a, then |
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Answer» `a =1/2, b=1/3` |
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| 14. |
Solve the differential equation ydx = (x + y^(2))dy ( y ne 0). |
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| 15. |
int_(0)^((pi)/(2))(cosxdx)/(sqrt(1+cosxsinx))= |
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Answer» `sqrt(2)COS^(-1)((1)/(sqrt(3)))` |
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| 16. |
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs. 5 each for type A and Rs. 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit ? |
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| 19. |
If theanglesof a trianglesare inthe ratio3 : 4: 5then the ratioof thelargestsidetothesmallestsideof thetriangleis |
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Answer» A. `(sqrt(3))/(2)` |
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| 20. |
(vec(a).(vec(b)xxvec(c)))/(vec(b).(vec(c)xxvec(a)))+(vec(b).(vec(a)xxvec(b)))/(vec(a).(vec(b)xxvec(c))) is equal to |
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Answer» 1 |
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| 21. |
Equatinof linepassingthroughthe point(2,3,1)and parallelto the line ofintersection of the planes x−2y−z+6=0 and x+y+3z=5 is |
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Answer» `(x-2)/(4)=(y-3)/(3)=(z-1)/(2)` |
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| 22. |
Showthat|{:(a-x,,c,,b),(c ,,b-x,,a),( b,, a,,c-x):}|=0 where a+b+c ne0 |
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Answer» `Delta=|{:(a+B+c-X,,c,,b),(a+b+c-x,,b-x,,a),(a+b+c-x,,a,,c-x):}|` `=(a+b+c-x) |{:(1,,c,,b),(1,,b-x,,a),(1,,a,,c-x):}|` Applying `R_(1) to R_(1)-R_(2),R_(2)toR_(2)-R_(3),` we get `Delta =(a+b+c-x) |{:(0,,c-b+x,,b-a),(0,,b-a-x,,a-c+x),(1,,a,,c-x):}|` `=(a+b+c -x){(x^(2)+x(a-b)+c-b)(a-c)` `+(x+a-b)(b-a)` `=(a+b+c-x){(x^(2)+x(a-b)+(c-b)(a-c)` `+x(b-a)-(a-b)^(2)}` `=(a+b+c-x)(x^(2)+ac-c^(2)-ab+bc-a^(2)-b^(2)+2ab)` `=(a+b+c-x)(x^(2)-a^(2)-b^(2)-c^(2)+ab+bc+ca)` `=(a+b+c-x)(x^(2)-(1)/(2){(a-b)^(2)+(b-c)^(2)+(c-a)^(2))})` SINCE `Delta =0` we have `x=a +b +c, +- sqrt((1)/(2)((a-b)^(2)+(b-c)^(2)+(c-a)^(2)))` |
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| 23. |
The solution of the differential equaiton |
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Answer» `(dy)/(dx)+y TAN X =sec x,` is |
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| 24. |
One morning, each member of manjul's family drank an 8-ounce mixture of coffee and milk. The amount of coffee and milk varied from cup to cup but never zero. Manjul drank (1/7)^(th) of the total amount of milk and (2/7)^(th)of the total amount of coffee. How many people are there in manjul's family? |
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| 25. |
If two tangents to the parabola y^(2)=8x meet the tangent at its vertex in M and N such that MN = 4, then the locus of the point of intersection of those two tangents is |
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Answer» `y^(2)=8(x+3)` |
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| 26. |
If the first degree terms of x^(2) + 4xy+y^2 - 2x + 2y-6=0 are eliminated by translation of axes then the transformed equation is |
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Answer» `X^(2) + 4xy + y^(2)=8` |
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| 27. |
A: underset(x to pi//2)"Lt" ((cot x)/(pi//2-x))=1 R: If underset(x to a)"Lt" "f(x) exists, then " underset(x to a)"Lt" f(x)=underset(x to 0)"Lt" f(a+x)=underset(x to a)"Lt" f(a-x) |
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Answer» Both A and R are true and R is the correct EXPLANATION of A |
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| 28. |
Differentiate the following functions with respect to x: sqrtx + sqrty = sqrta |
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| 29. |
The plane 3x+4y+6z+7=0 is rotated about the line r=(i+2j-3k)+t(2i-3j+k) (1) until the plane passes through the origin. If the equation of this plane is x+y+lambdaz=0, then lambda=_______ |
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| 30. |
int_(0)^(pi)e^(x)cos2xdx=............. |
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Answer» `(1)/(5)(e^(PI/2)-1)` |
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| 31. |
lim_(x to pi/6) { (3 sin x - sqrt3 cos x)/( 6x - pi) }is equal to |
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Answer» ` SQRT3` |
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| 32. |
If ((2n+1),(0))+ ((2n+1),(3)) +((2n+1),(6))+ ...= 170,then n equals |
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Answer» 2 ` + ""^(2n+1)C_(4)x^(4)+""^(2n+1)C_(5)x^(5)+""^(2n+1)C_(6)x^(6)+...` Putting `x=1, omega, omega^(2)` (where `omega` is cube root of unity and adding, we GET `2^(2n+1)+(1+omega) ""^(2n+1)+(1+omega^(2))^(2n+1)=3(""^(2n+1)C_(0) +""^(2n+1)C_(3)+""^(2n+1)C_(6)+...)` `rArr 2""^(2n+1)-omega ^(2) " "^(2n+1)- omega^(2n+1)=3(""^(2n+1)C_(0) +""^(2n+1)C_(3)+""^(2n+1)C_(6)+...)[because 1 + omega + omega^(2) = 0]` `rArr ""^(2n+1)C_(0) +^(2n+1)C_(3)+^(2n+1)C_(6)+...=1/3 ` `(2^(2n+1) -omega^(2) " "^(2n+1) -omega^(2n+1))` `((2n+1),(0))+ ((2n+1),(3)) +((2n+1),(6))+...=1/3(2^(2n+1) -omega^(2) " "^(2n+1) -omega^(2n+1))` For N `= 4,170 = 1/3 (512-1-1)=170 [because omega^(2)=1]` Hence,`n=4` |
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| 33. |
Let P(alpha,beta) and Q(gamma,delta) be two points that lie on the curve tan^(2)(x+y)+cos^(2)(x+y)+y^(2)+2y=0in the XY - plane . If the distance between P and Q is d then cos d= |
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Answer» 0 |
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| 34. |
If veca=hati+hatj+hatk, overset(-)(b)=veci+vecj, vecc=veci and (veca xx vecb) xx vecc=lambda veca+mu vecb," then "lambda +mu is equal to |
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Answer» 0 |
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| 35. |
From the set {1,2,3,4,5} two numbers a and b (a != b) are chosen at random . The probability that a/b is an integar is |
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Answer» `1/3` |
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| 36. |
A moving body is opposed by a force per unit mass of value cx and resistance per unit mass of value bv^(2) where x and v are displacement and velocity of the particle at that instant. Find the velocity of the particle in terms of x, if its initial velocity is v_(0). |
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| 37. |
IFsqrt( x^2 + 4a +5 )+ sqrt( x^2 + 4b+ 5 )=2 (a-b)thenx= |
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Answer» `((a-b)^2 -5)/(2(a+b))` |
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| 38. |
If A and B are square matrices of the same order , then (i) (AB) = ……… (ii)(kA) ' = …….. (iii)[k(A-B)]' =…… k is any scalar) |
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Answer» (II)`(kA)'=K*A`' (III)`[k(A-B)]=k(A'-B)` |
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| 39. |
Let ABCD be a square such that the side AB is on the line 2x-y=17 and two vertices C and D are on the parabola y=x^(2). If lengthof side of ABCD is less than 10 and its area is A, then A is equal to |
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| 40. |
Find the radius of the circle passing through the foci of an ellipse 9x^2+16y^2=144 and having least radius. |
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| 41. |
If f:RrarrR is defined by f(x) = x^2-3x+2, find f(f(x)) |
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Answer» SOLUTION :F(f(X)) = `f(x^2-3x+2)` `= (x^2-3x+2)^2-3(x^2-3x+2)+2` `= x^4+9x^2+4-6x^3+4x^2-12x-3x^2+9x-6+2` `= x^4-6x^3+10x^2-3x` |
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| 42. |
If z = x + iy and if the point P in the argand plane represents z , then the locus of P satisfying the equation |z-2-3i|=5 |
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| 43. |
underset(x to 0)"Lt" {1-x+[x-1}+[1-x]}" is " |
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Answer» 0 |
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| 44. |
a={1,2,3,4,5}, Relation R on A is defined by R={(x,y)//x lt y" and "|x^(2)-y^(2)|lt 9} then R = |
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Answer» `{(1,1),(2,2),(3,3),(4,4),(5,5)}` |
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| 45. |
Match the following lists: |
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Answer» `=cot^(-1)(-1-(x-1)^(2))` Now` -OO lt -1 -(x-1)^(2) lt -1` ` :. Cot^(-1) (-1-(x-1)^(2)) in [(3pi)/(4),pi)` Also, `f(0)=f(2)` So, `f(x)` is onto and many-one. B.  Clearly, from the graph, `f(x)` is many-one and onto. c.  d. Clearly, `f(x)` is INVERSE to itself. So, `f(x)` is BIJECTIVE. |
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| 46. |
Match the following lists: |
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Answer» `or 2 tan^(-1)x in (-(pi)/(2),(pi)/(2))` `or tan^(-1)x in (-(pi)/(4),(pi)/(4))` `ortan^(-1)x in (-1,1)` b. `f(x)=sin^(-1)(sinx) and g(x)=sin(sin^(-1)x)` `f(x)` is DEFINED if `sin x in [-1,1]` which is true for all `x in R`. But g(x) is defined for only `x in [-1,1]`. Hence, `f(x)` and g(x) are identical if `x in [-1,1]`. C.`f(x)=log_(x^(2))25 and g(x)=log_(x)5` `f(x)` is defined ` AA x in R - {0,1} and g(x) ` is defined for `(0, oo) -{1}.` Hence, `f(x)`and g(x) are identical if `x in (0,1) cup (1,oo).` d.`f(x)=sec^(-1)x+"cosec"^(-1)x,g(x)=sin^(-1)x+cos^(-1)x` `f(x)` has domain `R -(-1,1) and g(x)` has domain [-1, 1] Hence, both the functions are identical only if `x=-1,1.` |
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| 47. |
Find maximum and minimum value of the following functions in the given interval.f(x)=2x^(3)-3x^(2)-12x+1, x in [-2, (5)/(2)] |
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Answer» MINIMUM value `-19` |
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| 48. |
Match the following lists: |
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Answer» `={([(1)^(1)]^(n)",",x gt 0),([(-1)^(-1)]^(n) ",",x lt 0):}={(1",",x gt 0),(-1",",x lt 0):}.` Hence, `f(x)` is an ODD function. b. `f(x)=(x)/(e^(x)-1)+(x)/(2)+1` `or f(-x)=(-x)/(e^(-x)-1)-(x)/(2)+1` `=(xe^(x))/(e^(x)-1)-(x)/(2)+1` `=(xe^(x)-x+x)/(e^(x)-1)-(x)/(2)+1` `=x+(x)/(e^(x)-1)-(x)/(2)+1` `=(x)/(e^(x)-1)+(x)/(2)+1` `=f(x)` C.`f(x)={(0",","if x is rational"),(1",","if x is irrational"):}` `f(-x)={(0",","if -x is rational"),(1",","if -x is irrational"):}` `={(0",","if x is rational"),(1",","if x is irrational"):}` `=f(x)` d. `f(x)=max [tanx, cotx}` From the graph of function, it can be verified that `f(x)` is neither odd nor even. ALSO, `f(x+pi)=max {TAN(x+pi),cot(x+pi)` `=max {tanx, cot x}` Hence, `f(x)` is periodic with period `pi`. |
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| 49. |
If z = x + iy and if the point P in the argand plane represents z , then the locus of P satisfying the equation |z - 3 + i| = 4 |
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| 50. |
Find the area of the region bounded by the ellipse x^(2)/16+y^(2)/9=1. |
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