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This section contains some integer type questions. The answers to each of the questions is a single – digit integer, ranging from 0 to 9. Let f be a function defined on R (the set of all real numbers) such that

Let f be a function defined on R (the set of all real numbers) such that 1) 2 2) 3... View Article

This section contains some integer type questions. The answers to each of the questions is a single – digit integer, ranging from 0 to 9. Let f (θ) = sin [tan^-1 (sin θ) / √cos 2θ)], where – π / 4 < θ < π / 4. Then the value of d [f (θ)] / d (tan θ) is

Let f (θ) = sin [tan-1 (sin θ) / √cos 2θ)], where – π / 4 < θ < π... View Article

This section contains some integer type questions. The answers to each of the questions is a single – digit integer, ranging from 0 to 9. The minimum value of the sum of real numbers a^–5, a^–4, 3a^–3,1, a⁸ and a¹⁰, where a > 0 is

The minimum value of the sum of real numbers a–5, a–4, 3a–3,1, a8 and a10, where a > 0 is... View Article

Let f (x) = (1 – x)² + sin² x + x² for all x ∈ IR, and g (x) = integral from 1 to x [2 (t – 1)] / [t + 1] – ln t] f (t) dt for all x ∈ (1, ∞]. Consider the statements. P: There exists some x ∈ IR such that f (x) + 2x = 2 (1 + x²). Q: There exists some x ∈ IR such that 2 f (x) + 1 = 2x (1 + x). Then,

Consider the statements P: There exists some x ∈ IR such that f (x) + 2x = 2 (1 +... View Article

Let f (x) = (1 – x)² sin² x + x² for all x ∈ R, and let g (x) = integral from 1 to x [2 (t – 1)] / [t + 1] – ln t] f (t) dt for all x ∈ (1, ∞]. Which of the following is true

1) g is increasing on (1, ∞) 2) g is decreasing on (1, ∞) 3) g is increasing on (1,... View Article

It is a continuous function f defined on the real line R, assume positive and negative values in R then the equation f(x) = 0 has root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative then the equation f(x) = 0 has a root in R. Consider f(x) = ke^x – x for all real x where k is a real constant. For k > 0, the set of all values of k for which ke^x – x = 0 has two distinct roots is

1) (0, 1 / e) 2) (1 / e, 1) 3) (1 / e, ∞) 4) (0, 1) Solution: (1)... View Article

It is a continuous function f defined on the real line R, assume positive and negative values in R then the equation f(x) = 0 has root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative then the equation f(x) = 0 has a root in R. Consider f(x) = ke^x – x for all real x where k is a real constant. The positive value of k for which ke^x – x = 0 has only one root is

The positive value of k for which kex – x = 0 has only one root is 1) 1 /... View Article

It is a continuous function f defined on the real line R, assume positive and negative values in R then the equation f(x) = 0 has root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative then the equation f(x) = 0 has a root in R. Consider f (x) = ke^x – x for all real x where k is a real constant.The line y = x meets y = ke^x for k ≤ 0 at

1) No point 2) One point 3) Two points 4) More than two points Solution: (2) One point f (x)... View Article

Let g (x) = Integral from 0 to e^x f’ (t) / [1 + t²] dt. Which of the following is true

1) g’ (x) is positive on (- ∞, 0) and negative on (0, ∞) 2) g’ (x) is negative on... View Article

Consider the function f : (-∞, ∞) → (-∞, ∞) defined by f (x) = [x² – ax + 1] / [x² + ax + 1], 0 < a < 2. Which of the following is true

1) f (x) is decreasing on (-1, 1) and has a local minimum at x = 1 2) f (x)... View Article

Consider the function f : (-∞, ∞) → (-∞, ∞) defined by f (x) = [x² – ax + 1] / [x² + ax + 1], 0 < a < 2. Which of the following is true

1) (2 + a)2 f’’ (1) + (2 – a)2 f’’ (-1) = 0 2) (2 – a)2 f’’ (1)... View Article

Let a, b ∈ R be such that the function f given by f (x) = ln |x| + bx² + ax, x ≠ 0 has extreme values at x = -1 and x = 2. Statement 1: f has local maximum at x = – 1 and at x = 2. Statement 2: a = 1 / 2 and b = – 1 / 4

Statement 1: f has local maximum at x = – 1 and at x = 2 Statement 2: a =... View Article

Is lemon juice acidic or basic

Lemon has a high amount of citric acid, so in its natural state, it is acidic with the pH value... View Article

How can we increase the efficiency of a heat engine?

The efficiency of the heat engine is given by the formula η = 1 – (TL/TH) TL is the temperature... View Article

Which compound is diamagnetic?

The magnetic field induced in the diamagnetic material is opposite to the direction of the magnetic field. Therefore, diamagnetic materials... View Article

How do you use the ideal gas law?

The ideal gas law is given by the equation PV = nRT P is the pressure V is the volume... View Article

Why are gases not ideal?

No real gas can be an ideal gas. There are certain assumptions made for an ideal gas. In the ideal... View Article

Why is DC current not used in homes?

Direct current is not used at home because for the same value of the voltage, DC is more lethal than... View Article

Do electrons have a fixed wavelength

The wavelength of the electron depends on its momentum. The momentum of the electron is determined using the formula p=mv... View Article

Is the base the opposite of acid

The base can be considered as the chemical opposite of the acid. The reaction between the acid and the base... View Article

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