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Is water an acid or base

Pure water is neither an acid nor a base. Water is neutral. In water, there is a balance between the hydroniums and hydroxyls so they cancel each... View Article

How do you create a truth table?

A truth table is a basic method to describe the Boolean function. The Boolean variables can either be 0 or 1. The inputs and the output in a... View Article

Do electrons have a size?

The electrons are extremely small and nobody has determined its size but the radius of the electron was calculated to be equal to 2.8179 x 10-15... View Article

For the function f (x) = e^cosx, Rolle’s theorem is

1) applicable when π / 2 ≤ x ≤ 3π / 2 2) applicable when 0 ≤ x ≤ π / 2 3) applicable when 0 ≤ x ≤ π 4) applicable when π / 4 ≤ x ≤ π / 2... View Article

The function f(x) = (x – 3)² satisfies all the conditions of mean value theorem in [3, 4]. A point on y = (x – 3)², where the tangent is parallel to the chord joining (3, 0) and (4, 1) is

1) (7 / 2, 1 / 2) 2) (7 / 2, 1 / 4) 3) (1, 4) 4) (4, 1) Solution: (2) (7 / 2, 1 / 4) Let the point be (x1, y1). y1 = (x1 - 3)2... View Article

The abscissa of the points of the curve y = x3 in the interval [–2, 2], where the slope of the tangents can be obtained by mean value theorem for the interval [–2, 2], are

1) ± 2 / √3 2) ± √3 3) ± √3 / 2 4) 0 Solution: (1) ± 2 / √3 y = x3 = f (x) f (2) = 8 and f (-2) = - 8 f’ (x) = 3x2 f’ (x) =... View Article

If the mean value theorem is f (b) – f (a) = (b – a) f’ (c). Then for the function x² – 2x + 3 in [1, (3 / 2)], the value of c is

1) 6 / 5 2) 5 / 4 3) 4 / 3 4) 7 / 6 Solution: (2) 5 / 4 f’ (c) = [f (b) – f (a)] / [b – a] f (x) = x2 – 2x + 3 f’ (x) = 2x – 2... View Article

If the function f (x) = x³ – 6x² + ax + b satisfies Rolle’s theorem in the interval [1, 3] and f’ [(2√3 + 1) / √2] = 0, then

1) a = – 11 2) a = – 6 3) a = 6 4) a = 11 Solution: (4) a = 11 f (x) = x3 - 6x2 + ax + b f’ (x) = 3x2 - 12x + a f’ (c) = 0 f’... View Article

If f (x) satisfies the conditions of Rolle’s theorem in [1, 2] and f (x) is continuous in [1, 2] then integral from 1 to 2 f’ (x) dx is =

1) 3 2) 0 3) 1 4) 2 Solution: (2) 0 ∫12 f’ (x) dx = [f (x)]12 = f (2) - f (1) = 0 [because f (x) satisfies the conditions of... View Article

Let f (x) = √x – 1 + √x + 24 – 10 √x – 1; 1 < x < 26 be real valued function. Then f’ (x) for a < x < 26

1) 0 2) 1 / √x - 1 3) 2√x - 1 - 5 4) None of these Solution: (1) 0 f (x) = √(x – 1) + √{x + 24 – 10√(x – 1)} = √(x – 1) + √{25 +... View Article

A function f is defined by f (x) = e^x sin x in [0, π]. Which of the following is not correct

1) f is continuous in [0, π] 2) f is differentiable in [0, π] 3) f(0) = f(π) 4) Rolle’s theorem is not true in [0, π] Solution: (4)... View Article

A function f (x) = 1 + (1 / x) is defined on the closed interval [1, 3]. A point in the interval, where the function satisfies the mean value theorem, is

1) x = 1 / 3 2) x = 1 / √3 3) x = √3 4) None of these Solution: (3) x = √3 For mean value theorem f’ (c) = [f (b) - f (a)] / [b -... View Article

Let f (x) satisfy all the conditions of mean value theorem in [0, 2] if f (0) = 0 and |f’ (x)| ≤ 1 / 2 for all x in [0, 2] then

1) f (x) ≤ 2 2) |f (x)| ≤ 1 3) f (x) = 2x 4) f (x) = 3 for at least on x in [0, 2] Solution: (2) |f (x)| ≤ 1 f’ (c) = [f (x) - f... View Article

The point where the function f (x) = x² – 5x – 6 satisfies the condition of Rolle’s theorem is

1) x = 5 2) x = 5 / 2 3) x = 6 4) None of these Solution: (2) x = 5 / 2 f (x) = x2 - 5x - 6 f’ (x) = 2x - 5 f’ (x) = 0 x = 5 /... View Article

For the function x + (1 / x), x ∈ [1, 3], the value of c for the mean value theorem is

1) 1 2) √3 3) 2 4) None of these Solution: (2) √3 f (x) = x + (1 / x) f’ (x) = 1 - (1 / x2) f’ (c) = 1 - (1 / c2) 1 - (1 / c2)... View Article

A value of c for which the conclusion of mean value theorem holds for the function f (x) = log_e x on the interval (1, 3) is

1) 2 log3 e 2) (1 / 2) loge 3 3) log3 e 4) loge 3 Solution: (1) 2 log3 e f’ (c) = [f (b) - f (a)] / [b - a] = f (3) - f (1) / [3... View Article

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