JEE Main entrance exam Question Bank Solutions for Mathematics (JEE Main)

Let S_k = `sum_(r = 1)^k tan^-1(6^r/(2^(2r + 1) + 3^(2r + 1)))`. Then `lim_(k→∞)` S_k = is equal to ______.

The Boolean expression (p ∧∼ q) ⇒ (q ∨∼ p) is equivalent to ______.

If the truth value of the Boolean expression ((p ∨ q) ∧ (q→r) ∧ (∼r))→(p ∧ q) is false, then the truth values of the statements p, q, r respectively can be ______.

Let α > 0, β > 0 be such that α³ + β² = 4. If the maximum value of the term independent of x in the binomial expansion of (αx^1/9 + βx^–1/6)¹⁰ is 10k, then k is equal to ______.

If cos(α + β) = `(3/5)`, sin(α – β) = `5/13` and 0 < α, β < `π/4`, then tan (2α) is equal to ______.

`int(logx)^2dx` equals ______.

Let f(x) = sinx.cos³x and g(x) = cosx.sin³x, then the value of `7((f(π/7) + g(π/7))/(g((5π)/14) + f((5π)/14)))` is ______.

If a² + b² + c² = –2 and f(x) = `|(1 + a^2x, (1 + b^2)x, (1 + c^2)x),((1 + a^2)x, 1 + b^2x, (1 + c^2)x),((1 + a^2)x, (1 + b^2)x, (1 + c^2)x)|` then f(x) is a polynomial of degree ______.

If f(x) = `3cos(x + (5π)/6) - 5sinx + 2`, then maximum value of f(x) is ______.

Let tan9° = `(1 - sqrt((sqrt(5)k)/m))k` where k = `sqrt(5) + 1` then m is equal to  ______.

The maximum value of the expression 5cosα + 12sinα – 8 is equal to ______.

If b = `(3 + cot  π/8 + cot  (11π)/24 - cot  (5π)/24)`, then the value of `|bsqrt(2)|` is ______.

If L = `lim_(x→∞)(x^2sin  1/x - x)/(1 - |x|)`, then value of L is ______.

If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.

If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.

If `lim_(x→∞) 1/(x + 1) tan((πx + 1)/(2x + 2)) = a/(π - b)(a, b ∈ N)`; then the value of a + b is ______.

If `θ∈[(5π)/2, 3π]` and 2cosθ + sinθ = 1, then the value of 7cosθ + 6sinθ is ______.

If `lim_(n→∞)sum_(k = 2)^ncos^-1(1 + sqrt((k - 1)(k + 2)(k + 1)k)/(k(k + 1))) = π/λ`, then the value of λ is ______.

If sinθ = `1/sqrt(2)` and `π/2 < θ < π`. Then the value of `(sinθ + cosθ)/tanθ` is ______.

The statement `∼ ("p"leftrightarrow∼"q")` is ______.