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The function x² log x in the interval (1,e) has

1) A point of maximum 2) A point of minimum 3) Point of maximum as well as of a minimum 4) Neither a point of maximum nor minimum... View Article

The maximum value of the function x³ + x² + x – 4 is

1) 127 2) 4 3) Does not have a maximum value 4) None of the above Solution: (3) Does not have a maximum value f (x) = x3 + x2 + x... View Article

The minimum value 4e^2x + 9e^–2x is

1) 11 2) 12 3) 10 4) 14 Solution: (2) 12 f (x) = 4e2x + 9e-2x f’ (x) = 8e2x – 18e-2x f ‘(x) = 0 8e2x – 18e-2x = 0 Taking... View Article

The function x⁵ – 5x⁴ + 5x³ – 10 has a maximum where x =

1) 3 2) 2 3) 1 4) 0 Solution: (3) 1 f (x) = x5 – 5x4 + 5x3 – 10 The function will have a maximum value at x = 1.... View Article

Local maximum and local minimum values of the function (x – 1)(x + 2)² are

1) –4, 0 2) 0, –4 3) 4, 0 4) None of these Solution: (2) 0, –4 f (x) = (x – 1)(x + 2)2 f’ (x) = 0 x = 0, - 2 f (-2) = (-2 -... View Article

The maximum and minimum values of the function |sin 4x + 3| are

1) 1, 2 2) 4, 2 3) 2, 4 4) –1, 1 Solution: (2) 4, 2 f (x) = |sin 4x + 3| The minimum value of sin x is -1 and maximum is 1.... View Article

The value of the function (x – 1)(x – 2)² at its maxima is

1) 1 2) 2 3) 0 4) 4/27 Solution: (4) 4/27 f (x) = (x – 1)(x – 2)2 f (x) = (x - 1) (x2 + 4 - 4x) = x3 - 5x2 + 8x - 4 Now f'(x) = 3x2 - 10x... View Article

In the section each question has some statements (A, B,C,D,…) given in Column – I and some statements (p,q,r,s,t,…) in column – II. Any given statement is column – I can have correct matching with ONE OR MORE statement(s) in column – II for example, if for a given question, statement B matches with the statements given in q and r, then for that particular question against statement B, darken the bubbles corresponding to q and r in the ORS. i.e., answer r will be q and r

1) A → p; B → r 2) A → r; B → q 3) A → p; B → q 4) A → q; B → r Solution: (1) A → p; B → r A → p; B → r (A) f (x) = x + sin x... View Article

In the section, each question has some statements (A, B, C, D,…) given in Column – I and some statements (p,q,r,s,t,…) in column – II. Any given statement is column – I can have correct matching with ONE OR MORE statement(s) in the column – II for example, if for a given questions, statement B matches with the statements given in q and r, then for that particular question against statement B, darken the bubbles corresponding to q and r in the ORS. i.e., answer r will be q and r. In the following [x] denotes the greatest integer less than or equal to x. Match the functions in Column – I with the properties in the column – II

In the following [x] denotes the greatest integer less than or equal to x. Match the functions in Column – I with the properties in the column –... View Article

In the section, each question has some statements (A, B, C, D,…) given in Column– I and some statements (p,q,r,s,t,…) in the column – II. Any given statement is column – I can have correct matching with ONE OR MORE statement(s) in the column – II for example, if for a given questions, statement B matches with the statements given in q and r, then for that particular question against statement B, darken the bubbles corresponding to q and r in the ORS. i.e., answer r will be q and r. Match that statements are given in Column – I with the intervals/union of intervals given in Column – II

Match that statements are given in Column – I with the intervals/union of intervals given in Column – II 1) A→r; B→q, s; C→r, s; D→p, r... 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 p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p’(0) is

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 and p(3) = 2, then... 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 p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p’(0) is

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 and p(3) = 2, then... 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 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 3) 4 4) 1 Solution: (4) 1 f’ (x) = 2010 (x -... 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 / 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

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