JEE Main entrance exam Question Bank Solutions

If z and ω are two non-zero complex numbers such that |zω| = 1 and Arg(z) – Arg(ω) = `π/2`, then `barzω` is equal to ______.

If x = `sum_(n = 0)^∞a^n`, y = `sum_(n = 0)^∞b^n`, z = `sum_(n = 0)^∞c^n` where a, b , c are in A.P. and |a| < 1, |b| < 1, |c| < 1 then x, y, z are in ______.

If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ²) is ______.

The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.

The value of `lim_(x → ∞) ((x^2 - 1)sin^2(πx))/(x^4 - 2x^3 + 2x - 1)` is equal to ______.

The total number of four-digit numbers such that each of first three digits is divisible by the last digit is equal to ______.

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M^TM is seven, is ______.

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 ______.