JEE Main entrance exam Question Bank Solutions

The value of  `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.

Let z = `(1 - isqrt(3))/2`, i = `sqrt(-1)`. Then the value of `21 + (z + 1/z)^3 + (z^2 + 1/z^2) + (z^3 + 1/z^3)^3 + ...... + (z^21 + 1/z^21)^3` is ______.

`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.

The sum of the series 1.3² + 2.5² + 3.7² + .... upto 20 terms is ______.

If 1, α₁, α₂, ...... α_n–1 are the roots of unity, then (1 + α₁)(1 + α₂) ...... (1 + α_n–1) is equal to (when n is even) ______.

The variance of first 50 even natural numbers is ______.

The value of the expression `(cos^2 12^circ + 2sin12^circ(sin12^circ + 1))/(4cos^2  39^circ - sin^2 78^circ)` is ______.

The value of the expression 1.(2 – ω) + (2 – ω²) + 2.(3 – ω)(3 – ω²) + ....... + (n – 1)(n – ω)(n – ω²), where ω is an imaginary cube root of unity is ______.

A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t³ + at² + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s². The values of a, b, c are ______.

The angle between two lines `(x + 1)/2 = (y + 3)/2 = (z - 4)/(-1)` and `(x - 4)/1 = (y + 4)/2 = (z + 1)/2` is ______.

Let f: R→R and f be a differentiable function such that f(x + 2y) = f(x) + 4f(y) + 2y(2x – 1) ∀ x, y ∈ R and f’(0) = 1, then f(3) + f’(3) is ______.

Let S₁ = `{x ∈ R - {1, 2}: ((x + 2)(x^2 + 3x + 5))/(-2 + 3x - x^2) ≥ 0}` and S₂ = {x ∈ R : 3^2x – 3^x+1 – 3^x+2 + 27 ≤ 0}. Then, S₁ ∪ S₂ is equal to ______.

A straight line L through the point (3, –2) is inclined at an angle of 60° to the line `sqrt(3)x + y` = 1. If L also intersects the x-axis, then the equation of L is ______.

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be `24/5` and `194/25` respectively. If the mean and variance of the first 4 observations are `7/2` and a respectively, then (4a + x₅) is equal to ______.

If f(x) = `{{:((sin(p  +  1)x  +  sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x  +  x^2)  -  sqrt(x))/(x^(3//2)),",", x > 0):}`

is continuous at x = 0, then the ordered pair (p, q) is equal to ______.

Let S = {t ∈ R : f(x) = |x – π| (e^|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to ______.

`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to ______.

Let S = {z ∈ C: |z – 2| ≤, 1, z(1 + i) + z̄(1 – i) ≤ 2}. Let |z – 4i| attains minimum and maximum values, respectively, at z₁ ∈ S and z₂ ∈ S. If 5(|z₁|² + |z₂|²) = α + β`sqrt(5)`, where α and β are integers, then the value of α + β is equal to ______.

Let f and g be twice differentiable even functions on (–2, 2) such that `f(1/4) = 0, f(1/2) = 0, f(1) = 1`, and `g(3/4)` = 0, `g(1)` = 2. Then, the minimum number of solutions of f(x)g''(x) + f'(x)g'(x) = 0 in (–2, 2) is equal to ______.

The sum of solutions of the equation `cosx/(1 + sinx) = |tan2x|, x∈(-π/2, π/2) - {π/4, -π/4}` is ______.