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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. |
Find the range of x for which the binomial expansions of the following are valid .(4 -x/3)^(-1//2) |
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| 2. |
Express in the form a+bi (sqrt3-isqrt2)/(2sqrt3-isqrt3) |
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Answer» SOLUTION : `(sqrt3-isqrt2)/(2sqrt3-isqrt3)` `((sqrt3-isqrt2)(2sqrt3+isqrt3))/((2sqrt3-isqrt3)(2sqrt3+isqrt3))` `(6+3i-2isqrt6-sqrt(6I)^2)/(12-3i^2)` `((6+sqrt6)+(3-2sqrt6))/(15)` `(6+sqrt6)/(15)+i(3-2sqrt6)/(15)` `21=21+i0` `(1+i)/(1-i)=(1+i)^2/((1-i)(1+i))` `(i+i^2+2I)/(1-i^2)=(2i)/(1+1)=(2i)/2=i=0+ixx1` |
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| 3. |
If e^(lx)=cosx+isinx and x+iy=|(1,e^(pi i//4),e^(pi i//3)),(e^(-pi i//4),1,e^(2pi i//3)),(e^(-pi i//3),e^(-2pi i//3),e^(-2pi i //3))|, then |
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Answer» `x=-1,y=sqrt(2)` |
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| 4. |
Prove that (1+sqrt(2))/2lt int_(0)^(pi//2)(sinx)/x dx lt (pi+2sqrt(2))/4 |
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Answer» SOLUTION :`f(x)=(sin x)/x` `:. f'(x)=(xcosx-SINX)/(x^(2))=(cosx(x-tanx))/(x^(2))` For `xepsilon(0,pi//2),xlt tanx` `,.f'(x)lt0` So, `f(x)` is decreasing function. `lim_(xto0)(sinx)/x=1` and `lim_(to pi//2)(sinx)/x=2/(pi)` So the graph of the function is as shown in the FOLLOWING figure:  From the figure, Area of rectangle `OJED+` Area of rectangle `JAGH` `ltint_(0)^(pi//2)(sinx)/x dx` `lt` Area of `OJKC+` Area of `JAFE` `:.(pi)/4 . 4/(pisqrt(2))+(pi)/4 . 2/(pi) lt int_(0)^(pi//2) (sinx)/x dx lt (pi)/4 . 1+(pi)/4 . 4/(pisqrt(2))` `implies 1/(sqrt(2))+1/2 lt int_(0)^(pi//2)(sinx)/x dx lt (pi)/4 + 1/(sqrt(2))` `implies(1+sqrt(2))/2 lt int_(a)^(pi//2)(sinx)/x dx lt(pi+2sqrt(2))/4` |
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| 5. |
(2,4 )and (10 , 10 )are the ends of the latus rectum of an ellipse with eccentricity 1/2 . The length of major axis is |
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Answer» `(20 )/( 3) ` |
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| 6. |
if vector vecx satisfying vecx xx veca+ (vecx.vecb)vecc =vecd is given by vecx = lambda veca + veca xx (vecaxx(vecdxxvecc))/((veca.vecc)|veca|^(2)) |
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Answer» SOLUTION :`VECX xxveca=(vecx.vecb)vecc=vecd` |
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| 7. |
If veca=hati + 2 hatj + 2hatk, |vecb|=5 and angle between veca and vecb is (pi)/(60, then the area of the triangle formed by these two vectors as two side is : |
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Answer» `(15 SQRT3)/(2)` |
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| 8. |
lim_(trarr0) int_(0)^(2pi)(|sin(x+t)-sinx|)/(|t|)dx equals |
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Answer» 2 `=int_(0)^(2pi)(underset(trarr0)(lim)|(2COS(x+(t)/(2))sin(t)/(2))/(t)|)dt` `=int_(0)^(2pi)|cosx|dx=4` |
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| 9. |
If standard deviations of the data 1,2,……, 2n +1is sqrt (30 )then n is equal to _____________ |
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| 10. |
Statement-1 : The variance of the variates 112, 116, 120,125,132 about their A.M. is 48.8. Statement-2 : sigma=sqrt((Sigma_(i=1)^(n) f_i(x_i-barx)^2)/(Sigma_(i=1)^(n) f_i) |
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| 11. |
The tangents at the points (at_(1)^(2),2at_(1)) , (at_(2)^(2),2at_(2)) on the parabola y^(2)=4ax are al right angles if |
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Answer» `t_(1)=t_(2)` |
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| 12. |
Evaluate the following integrals (iv) int_(-a)^(a) x^(2) (a^(2)-x^(2))^(3//2)dx |
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| 13. |
The ratio of the roots of the equation ax^(2)+bx+c=0is same as the ratio of the roots of the equation px^(2)+qx+r=0.If D_(1) and D_(2) are discriminants of ax^(2)+bx+c=0and px^(2)+qx+r=0respectively, then D_(1):D_(2)= |
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Answer» `(a^(2))/(p^(2))` |
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| 14. |
If the value of the definite integral int ^(-1) ^(1) cos ^(-1) ((1)/(sqrt(1-x ^(2)))) . (cot ^(-1)""(x )/(sqrt (1-(x ^(2))^(|x|))))dx = (pi^(2)(sqrta-sqrtb))/(sqrtc)where a,b,c in N in their lowest from, then find the value of (a+b+c). |
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| 15. |
The value of theta lying in (0,pi/2) and satisfying: |(1 + sin^(2)theta""cos^(2)theta""4sin4theta),(sin^(2)theta""1 + cos^(2)theta""4sin4theta),(sin^(2)theta""cos^(2)theta" "1 + 4sin4theta)| =0 is : |
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Answer» `(3PI)/24` |
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| 16. |
Statement -1 If Delta(r)=|{:(r,r+1),(r+3,r+4):}| then sum_(r=1)^(n) Delta(r)=-3n Satement-2 If Delta(r)|{:(f_(1)(r),f_(2)(r)),(f_(3)(r),f_(4)(r)):}| Sigma_(r=1)^(n) Delta (r)={:abs((Sigma_(r=1)^(n)f_1(r),Sigma_(r=1)^(n)f_2(r)),(Sigma_(r=1)^(n)f_3(r),Sigma_(r=1)^(n) f_4(r))):} |
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| 17. |
When points A and B (-3, 4) are graphed in the standard (x,y) coordinate plane below, the midpoint of bar(AB) will be (1,2). What will be the coordinate of point A? |
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Answer» `(-7,6)` |
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| 18. |
Evaluate int ("cos" 2x - cos 2 alpha)/(cos x - cos alpha ) |
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Answer» sin x + x cos `ALPHA` + C |
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| 19. |
A differentiable function f defined for all x > 0 and satisfies f(x^2)=x^3 for all x > 0. What is the value f'(16) ? |
| Answer» SOLUTION :`F(x^2)=x^3rArrf(x^2). 2x=3X^2rArrf(x^2)=(3x)/2rArrf(4^2)=(3xx4)/2rArrf(16)=6` | |
| 20. |
Reflection of the point A (5, 12) in the line y=x tan theta is P(alpha, beta). The locus of P as theta varies is the curve x^(2)+y^(2)=k^(2), the value of k is ________ |
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| 21. |
The negation of the following statement: P : Neh!l lives in Ludhiana or she lives in Gurudaspur. |
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Answer» NEHA does not live in LUDHIANA or she does not live in Gurudaspur. |
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| 22. |
Integrate the following functions : int(x^(3))/((1+x^(2))^(2))dx |
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| 23. |
If OA and OB are the tangent from the origin to the circle x^(2)+y^(2)+2gx+2fy+c=0 and C is the centre of the circle then the area of the quadrilateral OCAB is |
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Answer» `sqrt(g^(2)+F^(2)-c)` |
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| 24. |
A square ABCD of diagonal =2a is folded along diagonal AC, so that planes DAC & BAC are at right angle. The shortest distance between DC & AB in the rew position is |
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Answer» `sqrt(2)a` `F(X)=x("ln" x-1)` `f(x) = "ln"x` `f''(x)=(1)/(x)"",""underset(1)OVERSET(e)(int)((f''+f)sinx)dx` Use integration by parts =`INTF''.sinx+intf.sinx` `=sinx.f']_(1)^(e)-intcosx.f'+f.(-cosx)]_(1)^(e)-intf'.(-cosx)=sinx. "ln"x]_(1)^(e)-x("ln"x-1).(cosx)]_(1)^(e)` |
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| 25. |
(1.2)/(1!) +(2.3)/(2!) + (3.4)/(3!)+......oo = |
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Answer» e |
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| 26. |
Find the probability of drawing 2 red balls in succession from a bag containing 4 red balls and 5 black balls when the ball that is drawn first is (i) not replaced (ii) replaced |
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Answer» `(II) (16)/(81)` |
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| 27. |
""^(14)C_(4)+sum_(j=1)^(4)""^((18-j))C_(3)is equal to |
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Answer» `""^(14)C_(5)` |
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| 28. |
If A= [[0,6,7],[-6,0,8],[7,-8,0]], B= [[0,1,1],[1,0,2],[1,2,0]],C= [[2],[-2],[3]] calculate AC, BC and (A+B)C. Also verify that (A+B)C=AC+BC |
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| 29. |
Define a function phi : N to N as follows : phi(1)=1,phi(P^(n))=P^(n-1)(P-1) if P is prime and n in N and phi("mn")=phi(m)phi(n) if m & n are relatively prime natural numbers The number of natural numbers 'n' such that phi(n) is odd is |
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Answer» 1 |
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| 30. |
The sum of the series (x^(2))/(2)+(2)/(3)x^(3)+(3)/(4)x^(4)+(4)/(5)x^(5)+….= |
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Answer» `(X)/(1+x)+LOG(1+x)` |
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| 31. |
Find the radical centre of the following circles x^2+y^2-4x-6y+5=0 x^2+y^2-2x-4y-1=0 x^2+y^2-6x-2y=0 |
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| 32. |
Assume X, Y, Z, W and P are matrices of order2xx n, 3xxk, 2xxp, n xx 3 and p xx k, respectively. The restriction on n, k and p so that PY+WY will be defined are: |
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Answer» <P>`K=3, p=n` |
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| 33. |
Algebraic sum of the intercepts by the plane 3x-4y+7z=84 on the axes is |
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Answer» 6 |
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| 34. |
Euler's substitution: Integrals of the form intR(x, sqrt(ax^(2)+bx+c))dx are claculated with the aid of one of the followingthree Euler substitutions: i.sqrt(ax^(2)+bx+c)=t+-x sqrt(a)if a gt 0 ii.sqrt(ax^(2)+bx+c)=tx+-x sqrt(c)if c gt 0 iii.sqrt(ax^(2)+bx+c)=(x-a)t if ax^(2)+bx+c=a(x-a)(x-b) i.e., if alpha is real root ofax^(2)+bx+c=0 Which of the followingfunctions does not appear in the primitive of(1)/(1+sqrt(x^(2)+2x+2)) if t is a function of x ? |
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Answer» `log_(e)|t+1|` `2x+2tx=t^(2)-2 or x=(t^(2)-2)/(2(1+t)) or dx=(t^(2)+2t+2)/(2(1+t)^(2))dt` `1+sqrt(x^(2)+2x+2)=1+t-(t^(2)-2)/(2(1+t))=(t^(2)+4t+4)/(2(1+t)).` Substitutinginto the integral, we get `I=int(2(1+t)(t^(2)+2t+2))/((t^(2)+4t+4)2(1+t)^(2))dt=int((t^(2)+2t+2)dt)/((1+t)(t+2)^(2))` Now, let us EXPAND the obtained proper rational fraction into partial fractions: `(t^(2)+2t+2)/((t+1)(t+2)^(2))=(A)/(t+1)+(B)/(t+2)+(D)/((t+2)^(2)).` |
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| 35. |
The sum of coefficients in the expansion of (x^(2) - 1/3)^(199) xx (x^(3) + 1/2)^(200) is |
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Answer» `1/3` |
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| 36. |
Euler's substitution: Integrals of the form intR(x, sqrt(ax^(2)+bx+c))dx are claculated with the aid of one of the followingthree Euler substitutions: i.sqrt(ax^(2)+bx+c)=t+-x sqrt(a)if a gt 0 ii.sqrt(ax^(2)+bx+c)=tx+-x sqrt(c)if c gt 0 iii.sqrt(ax^(2)+bx+c)=(x-a)t if ax^(2)+bx+c=a(x-a)(x-b) i.e., if alpha is real root ofax^(2)+bx+c=0 Which of the following functions does not apear in the primitive of (dx)/(x+sqrt(x^(2)-x+1)) if t is a function of x? |
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Answer» `log_(e)|t|` SINCE here `c=1 gt 0, ` we can APPLY the SECOND Euler substitution: `sqrt(x^(2)-x+1)=tx-1` ` or (2t-1)x=(t^(2)-1)x^(2),x=(2t-1)/(t^(2)-1)` Substitutinginto I, we GET an integral of a RATIONAL fraction: `int (dx)/(x+sqrt(x^(2)-x+1))=int(-2t^(2)+2t-2)/(t(t-1)(t+1)^(2))dt` Now, `(-2t^(2)+2t-2)/(t(t-1)(t+1))=(A)/(t)+(B)/(t-1)+(D)/((t+1)^(2))+(E)/(t+1)` |
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| 37. |
No.of divisors of the form 4n+2(nge0) of the integer 240 is |
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Answer» 3 |
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| 38. |
Find the value of p so that the lines (1-x)/3=(7y-14)/(2p)=(z-3)/2 and (7-7x)/(3p)=(y-5)/1=(6-z)/5 are at right angles . |
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| 39. |
Areaof the regionboundedby thecurvey= sinx andx -axis- pi/2 lex le(pi)/(2) ……… |
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Answer» 1 |
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| 40. |
{:(I. "Unit vector perpendicular to the plane of" 2i - 6j - 3k. 4i + 3j - k, a. (5)/(sqrt(3)) (i + j + k)),(II. "Unit vector perpendicular to the plane determined by the points" (1.-1.2).(2.0.-1).(0.2.1),b. j - k),(III. "Vector perpendicular to the plane of" i - j - k. i + j + k, c. (1)/(sqrt(6)) (2i + j + k)),(IV. "Vector of length 5 and perpendicular to both" a = 2i + j - 3k, d. (1)/(7) (3i - 2j + 6k)):} |
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Answer» a,c,d,B |
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| 41. |
f(x) ={{:((x)/(2x^(2)+|x|),xne0),( 1, x=0):}then f(x) is |
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Answer» continuousbut non- differentiableat x=0 |
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| 42. |
A and B are teo candidates seeking admission to IIT. Probability of A getting selected is 0.5 and that of A and B both being selected is 0.3. What is the probability of B being selected ? |
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| 43. |
Show that ""^(10)C_(2)+""^(11)C_(2)+""^(12)C_(2)+""^(13)C_(2)+......+""^(20)C_(2)=""^(21)C_(3)-""^(10)C_(3) |
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| 44. |
Product of two positive integers a and b is 192. Let 8 = hef (a, b) and I = lem (a, b). If the ratio of A.M. between g and I to the H.M. between g and l is (169)/(48), then smaller of a,b is |
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Answer» 8,24 |
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| 45. |
Find the centre and radius of the following circles : ax^2 + ay^2 + 2gx + 2fy + K = 0 |
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Answer» Solution : `ax^2 + ay^2 + 2gx + 2FY + K = 0` or, `x^2 + y^2 + (2G)/a x + (2f)/a y + k/a = 0` `THEREFORE` Centre of (-g/a, -f/a) and radius = `sqrt(g^2/a^2 + f^2/a^2 - k/a) = sqrt((g^2 + f^2 - AK)/a)` |
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| 46. |
Differentiate the functions given in Exercises 1 to 11 w.r.t. x. (x cos x)+ (1/x). |
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| 47. |
If the lines x+2y+k=-, x+y-3=0 are conjugate w.r.t. the circle x^(2)+y^(2)-4x+3y-1=0 then k= |
| Answer» ANSWER :A | |
| 48. |
Find the equation of the circle passing through (2,-1) having the centre at (2,3). |
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| 49. |
f(x) = {{:((1-coskx)/(x sinx), if x ne 0),(1/2, if x = 0):} at x= 0 |
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Answer» At `x=0, LHL = underset(hrarr0^(-))(lim)(1-coskx)/(xsinx)=underset(hrarr0)lim(1-cos(0-h))/((0-h)sin(0-h))` `= underset(hrarr0)(lim)(1-cos(-kh))/(-hsin(-h))` `= underset(hrarr0)(lim)(1-coskh)/(hsinh) ,[:' cos (-theta)=cos theta, sin(-theta)= - sin theta]` `= underset(hrarr0)lim(1-1+2SIN^(2)'(kh)/(2))/(hsinh), [:' cos theta = 1 - 2 sin^(2) ' (theta)/(2)]` `= underset(hrarr0)(lim) (2sin^(2)'(kh)/(2))/(h sinh)` `= underset(hrarr0)lim(2sin'(kh)/(2))/((kh)/(2)).(sin'(kh)/(2))/((kh)/(2)).(1)/((sinh)/(h)).(K^(2)h//4)/(h)` `= (2k^(2))/(4) = (k^(2))/(2)` , [`:' underset(hrarr0)(lim)' (sinh)/(h) = 1]` Also, `f(0) = 1/2 rArr(k^(2))/(2) = 1/2rArr k +- 1` |
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