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Let `f:R to R` be a function defined by `f(x)=(4^x)/(4^x+2)` and `M=int_(f(a))^(f(1-a))x sin^(4)(x(1-x))dx`, `N=int_(f(a))^(f(1-a))sin^(4)(x(1-x))dx`,`a!=(1)/(2)` .If `alpha M=beta N,alpha,beta in N` ,then the least value of `alpha^(2)+beta^(2)` is equal to________

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JEE MAINS PREVIOUS YEAR-JEE MAIN 2024 ACTUAL PAPER-Question
  1. If the integral 525int(0)^((pi)/(2))sin2x cos^((11)/(2))x(1+Cos^((5)/(...

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  2. Let the foci and length of the latus rectum of an ellipse (x^(2))/(a^(...

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  3. Let f:R to R be a function defined by f(x)=(4^x)/(4^x+2) and M=int(f(a...

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  4. Let Q and R be the feet of perpendiculars from the point P(a,a,a) on t...

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  5. If F :(- oo, -1] rightarrow (a, b] is defined as f(x) = e^(x^3-3x+1) s...

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  6. The area of the region enclosed by the parabolas y=4x-x^(2) and 3y=(x-...

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  7. Let A(a,b),B(3,4) and C(-6,-8) respectively denote the centroid,circum...

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  8. Consider the function f:(0,oo)rarr R defined by f(x)=e^(-|log(e)x|) .I...

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  9. Let z(1) and z(2) be two complex numbers such that z(1)+z(2)=5 and z(1...

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  10. If for some m,n,^(6)C(m)+2(""^(6)C(m+1))+^(6)C(m+2)>^(8)C(3) and ^(n-1...

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  11. If a = sin^(-1) (sin 5) and b = cos^(-1) (cos 5) then value of a^2 + b...

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  12. If the mean and variance of 6 observations a, b, 68, 44, 48, 60 are 55...

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  13. Let a variable line passing through the centre of the circle x^(2)+y^(...

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  14. Let 2^("nd"),8^("th") and 44^("th") terms of a non-constant A.P. be re...

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  15. An ellipse frac{x^2}{a^2}+frac{y^2}{b^2}=1 (a> b) whose eccentircity i...

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  16. Let f,g:(0,oo)rarr R be two functions defined by f(x)=int(-x)^(x)(|t|-...

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  17. Let f:rarr R rarr(0,oo) be strictly increasing function such that lim(...

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  18. The temperature T(t) of a body at time t=0 is 160^(@)F and it decrease...

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  19. The shortest distance,between lines L(1) and L(2) ,where L(1):(x-1)/(2...

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  20. The number of ways in which 10 identical apples can be distributed amo...

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