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If x √(1+y) + y√(1+x) = 0, then dy/dx is equal to

(1) 1/(1+x)2 (2) -1/(1+x)2 (3) 1/(1+x2) (4) 1/(1-x2) Solution: Given x √(1+x) + y√(1+x) = 0 x √(1+x) = - y√(1+x) Squaring both... View Article

If (1 / x) + (1 / y) = 1, the function a³x² + b³y² is minimum, then x and y are

1) a + b, a - b 2) a, b 3) (a / (a + b)), b / (a + b) 4) (a + b) / a, (a + b) / b Solution: (4) (a + b) / a, (a + b) / b (1 / x) + (1 / y)... View Article

If two sides of a triangle are given, then the area of the triangle will be maximum if the angle between the given sides be

1) π / 3 2) π / 4 3) π / 6 4) π / 2 Solution: (4) π / 2 Let a and b are given, then area A = (1 / 2) ab sin C dA / dC = (1 / 2)... View Article

The function x√1 – x², (x > 0) has

1) A local maxima 2) A local minima 3) Neither local maxima nor a local minima 4) None of the above Solution: (1) A local maxima f... View Article

For x ∈ (0, 5π / 2), defined f (x) = integral from 0 to x √t sint dt. Then f has

1) Local maximum at π and 2 π 2) Local minimum at π and 2 π 3) Local minimum at π and local maximum at 2 π 4) Local maximum at π and a... View Article

Local maximum value of the function log x / x is

1) e 2) 1 3) 1 / e 4) 2e Solution: (3) 1 / e f (x) = log x / x f’ (x) = 1 / x2 - (log x / x2) For maximum or minimum value of f... View Article

The minimum value of |x| + |x + (1 / 2)| + |x – 3| + |x – (5 / 2)| is

1) 0 2) 2 3) 4 4) 6 Solution: (4) 6 f (x) = |x| + |x + (1 / 2)| + |x - 3| + |x - (5 / 2)| = - 4x + 5, for x ≤ - 1 / 2 = -... View Article

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

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