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

Consider

Statement 1: (p∧∼q) ∧ (∼p ∧ q) is a fallacy.

Statement 2: (p→q) ↔ (∼q→∼p) is a tautology.

The sum of n terms of an AP is 3n² + 5n. The number of term which equals 164 is ______.

If arg(z) < 0, then arg(–z) – arg(z) = ______.

(p ⇒ q) ∩ (q ⇒ r) ⇒ (p ⇒ r) is ______.

If a₁, a₂, a₃, .......... are an A.P. such that a₁ + a₅ + a₁₀ + a₁₅ + a₂₀ + a₂₄ = 225, then a₁ + a₂ + a₃ + ...... + a₂₃ + a₂₄ is equal to ______.

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.

If 100 times the 100^th term of an A.P. with non zero common difference equals the 50 times its 50^th term, then the 150th term of this A.P. is ______.

The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.

Let f(x) be a polynomial of degree 4 having extreme values at x = 1 and x = 2. If `lim_(x rightarrow 0) ((f(x))/x^2 + 1)` = 3 then f(–1) is equal to ______.

The conditional statement ((p ∧ q) `rightarrow` ((∼p) ∨ r)) v (((∼p) ∨ r) `rightarrow` (p ∧ q)) is ______.

Which of the following Boolean expressions is not a tautology?

If q is false and p ∧ q `leftrightarrow` r is true, then which one of the following statements is a tautology?

`tan(2tan^-1  1/5 + sec^-1  sqrt(5)/2 + 2tan^-1  1/8)` is equal to ______.

The set of all values of k for which (tan^–1 x)³ + (cot^–1 x)³ = kπ³, x ∈ R, is the internal ______.

If `cos^-1(2/(3x)) + cos^-1(3/(4x)) = π/2(x > 3/4)`, then x is equal to ______.

If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.

Let `{a_n}_(n = 0)^∞` be a sequence such that a₀ = a₁ = 0 and a_n+2 = 2a_n+1 – a_n + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.

Let y = y(x) be the solution of the differential equation `(dy)/(dx) + (sqrt(2)y)/(2cos^4x - cos2x) = xe^(tan^-1(sqrt(2)cost2x)), 0 < x < π/2` with `y(π/4) = π^2/32`. If `y(π/3) = π^2/18e^(-tan^-1(α))`, then the value of 3α² is equal to ______.

The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.

If y = y(x) is the solution of the differential equation `(1 + e^(2x))(dy)/(dx) + 2(1 + y^2)e^x` = 0 and y(0) = 0, then `6(y^'(0) + (y(log_esqrt(3))))^2` is equal to ______.