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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 (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 / 4. Then the value of d [f (θ)] / d (tan θ) is 1) 2 2) 3 3) 1... 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 1) 5 2) 6 3) 8 4) 7 Solution: (3) 8... 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 + x2) Q: There exists some x ∈ IR such that 2 f (x) + 1 = 2x (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, 2) and decreasing on (2, ∞) 4) g is decreasing on... 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) (0, 1 / e) If k > 0, then for 2 distinct roots local minima... 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 / e 2) 1 3) e 4) loge 2 Solution: (1) 1 / e If k > 0,... 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) = kex – x f’ (x) = kex - 1 < 0 if... 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 (- ∞, 0) and positive on (0, ∞) 3) g’ (x) changes sign 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) is increasing on (-1, 1) and has a local minimum at x = 1 3) f... 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) - (2 + a)2 f’’ (-1) = 0 3) f’ (1) f’ (-1) = (2 - a)2 4) f’ (1) 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 = 1 / 2 and b = - 1 / 4 1) Statement 1 is true, Statement 2 is true;... View Article

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Archives

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

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

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

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

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

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

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

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