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If the sum of squares of all real values of "alpha" ,for which the lines `2x-y+3=0,6x+3y+1=0` and `[alpha x+2y-2=0` do not form a triangle is "p" ,then the greatest integer less than or equal to "p" is___________

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JEE MAINS PREVIOUS YEAR-JEE MAIN 2024 ACTUAL PAPER-Question
  1. A is the area of region 0leylemin(2x,6x-x^2), then find 12A

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  2. The lines (x-2)/(2)=(y)/(-2)=(z-7)/(16) and (x+3)/(4)=(y+2)/(3)=(z+2)/...

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  3. If the sum of squares of all real values of "alpha" ,for which the lin...

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  4. f(x)=int(0)^(x)g(t)log(e)((1-t)/(1+t))dt ,where "g" is a continuous od...

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  5. The coefficient of x^2012 in the expansion of (1 - x)^2008 (1+x+x^2)...

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  6. Let "A" be a 2times2 real matrix and "I" be the identity matrix of ord...

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  7. Let A=[[1,0,0],[0,alpha,beta],[0,beta,alpha]] and abs(2A^(3))=2^(21) w...

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  8. Let "PQR" be a triangle with "R(-1,4,2)" .Suppose "M(2,1,2)" is the mi...

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  9. Let "R" be a relation on Z times Z defined by (a,b)R(c,d) if and only ...

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  10. Let "O" be the origin and the position vectors of "A" and "B" be 2 hat...

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  11. lim(x rarr(pi)/(2))((1)/((x-(pi)/(2))^(2))int(x^(3))^(3)cos(t^((1)/(3)...

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  12. A function "y=f(x)" satisfies f(x)sin2x+sin x-(1+cos^(2)x)f'(x)=0 with...

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  13. If f(x)={(2+2x,-1 le x lt 0),(1-x/(3),0 le x le 3)) g(x)={(-x,-3 le x ...

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  14. If the value of the integral int(-(pi)/(2))^((pi)/(2))((x^(2)cos x)/(1...

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  15. If alpha,-(pi)/(2) lt alpha lt (pi)/(2) is the solution of 4cos theta+...

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  16. For x in(-(pi)/(2),(pi)/(2)), if y(x)=int(cosec x+sin x)/(cosec x sec ...

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  17. Consider the function f:[(1)/(2),1]rarr R defined by f(x)=4sqrt(2)x^(3...

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  18. Let (5,(a)/(4)) be the circumcenter of a triangle with vertices A(a,-2...

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  19. If z=(1)/(2)-2i is such that |z+1|=alpha z+beta(1+i),i=sqrt(-1) and al...

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  20. If "AA"^T=I where A^T is transpose of matrix A then, 1/2A[(A+A^T)^2+(A...

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