JEE Main entrance exam Question Bank Solutions

The number of ordered pairs (a, b), (where a, b ∈ R) satisfying the equation a²⁰⁰⁸ + b²⁰⁰⁸ = 2008 |a||b| – 2006 is equal to ______.

Let right angled isosceles triangle ABC be inscribed in a circle according to adjacent diagram vertex A is moved along the circle to reach at A' such that are \[{\stackrel\frown{AA'}}\] = `(πr)/3`, if r = `sqrt(3) + 1` then (A'C)² is ______.

If |x – 1| + |x – 2| + |x – 3| ≥ 6 then ______.

The points A(5, –1, 1); B(7, –4, 7); C(1, –6, 10) and D(–1, –3, 4) are vertices of a ______.

The middle term in the expansion of (1 – 3x + 3x² – x³)⁶ is ______.

The value of cos 255° + sin 195° is ______.

The number of solution of tan x + sec x = 2 cos x in [0, 2π] is ______.

Let for the 9^th term in the binomial expansion of (3 + 6x)ⁿ, in the increasing powers of 6x, to be the greatest for x = `3/2`, the least value of n is n₀. If k is the ratio of the coefficient of x⁶ to the coefficient of x³, then k + n₀ is equal to ______.

If the coefficient of x¹⁰ in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5^kl, where l, k ∈ N and l is coprime to 5, then k is equal to ______.

Let the coefficients of the middle terms in the expansion of `(1/sqrt(6) + βx)^4, (1 - 3βx)^2` and `(1 - β/2x)^6, β > 0`, common difference of this A.P., then `50 - (2d)/β^2` is equal to ______.

The term independent of x in the expansion of `[(x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2))]^10`, x ≠ 1 is equal to ______.

The sum of the real values of x for which the middle term in the binomial expansion of `(x^3/3 + 3/x)^8` equals 5670 is ______.

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to ______.

A rectangle R with endpoints of the one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x – y + 4 = 0, then the area of R is ______.

The value of `""^50"C"_4 + sum_("r" = 1)^6 ""^(56 - "r")"C"_3` is ______.

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed is ______.

The value of –¹⁵C₁ + 2.¹⁵C₂ – 3.¹⁵C₃ + ... –15.¹⁵C₁₅ + ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ... + ¹⁴C₁₁ is  ______.

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsmen and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsmen and 1 wicketkeeper, is ______.

If ²⁰C_r is the coefficient of x^r in the expansion of (1 + x)²⁰, then the value of `sum_(r = 0)^20 r^2  ""^20C_r` is equal to ______.

The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d² is equal to ______.