12 - Maths Unit 5 - Differential Calculus Applications - I

1. If the length of the diagonal of a square is increasing at the rate of

What is the rate of increase of its area when the side is

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The rate of increase of the area of a square is directly proportional to the rate of increase of its side length. Therefore, if the length of the diagonal is increasing at a certain rate, the side length of the square will also be increasing at the same rate. Since the area of a square is calculated by squaring its side length, the rate of increase of the area will be the square of the rate of increase of the side length. Therefore, the correct answer is (1).

Explanation

The rate of increase of the area of a square is directly proportional to the rate of increase of its side length. Therefore, if the length of the diagonal is increasing at a certain rate, the side length of the square will also be increasing at the same rate. Since the area of a square is calculated by squaring its side length, the rate of increase of the area will be the square of the rate of increase of the side length. Therefore, the correct answer is (1).

2. The rate of change of area

of a circle of radius

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The rate of change of the area of a circle with respect to its radius is given by the formula 2πr. This means that for every unit increase in the radius, the area of the circle increases by 2π times the radius. Therefore, the correct answer is (2), as it represents the correct formula for the rate of change of the area of a circle.

Explanation

The rate of change of the area of a circle with respect to its radius is given by the formula 2πr. This means that for every unit increase in the radius, the area of the circle increases by 2π times the radius. Therefore, the correct answer is (2), as it represents the correct formula for the rate of change of the area of a circle.

3. The radius of a cylinder is increasing at the rate of

and its altitude is decreasing at the rate of

. The rate of change of volume when the radius is

and the altitude is

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The rate of change of volume of a cylinder can be found using the formula dV/dt = πr^2dh/dt + 2πrhdr/dt. In this case, the radius is increasing at a rate of and the altitude is decreasing at a rate of . Plugging these values into the formula, we get dV/dt = π( )^2( ) + 2π( )( ). Simplifying this expression gives us dV/dt = 2π( )^2( ) - 2π( )( ). Therefore, the rate of change of volume when the radius is and the altitude is is given by option (2).

Explanation

The rate of change of volume of a cylinder can be found using the formula dV/dt = πr^2dh/dt + 2πrhdr/dt. In this case, the radius is increasing at a rate of and the altitude is decreasing at a rate of . Plugging these values into the formula, we get dV/dt = π( )^2( ) + 2π( )( ). Simplifying this expression gives us dV/dt = 2π( )^2( ) - 2π( )( ). Therefore, the rate of change of volume when the radius is and the altitude is is given by option (2).

4. If the velocity of a particle moving along a straight line is directly proportional to the square of its distance from a fixed point on the line. Then its acceleration is proportional to (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The given statement states that the velocity of a particle is directly proportional to the square of its distance from a fixed point on the line. This implies that as the distance from the fixed point increases, the velocity of the particle also increases. Acceleration, on the other hand, is the rate at which the velocity of an object changes. Since the velocity is directly proportional to the square of the distance, the rate at which the velocity changes (acceleration) will also be directly proportional to the distance. Therefore, the acceleration is proportional to the distance, which corresponds to answer choice (3).

Explanation

The given statement states that the velocity of a particle is directly proportional to the square of its distance from a fixed point on the line. This implies that as the distance from the fixed point increases, the velocity of the particle also increases. Acceleration, on the other hand, is the rate at which the velocity of an object changes. Since the velocity is directly proportional to the square of the distance, the rate at which the velocity changes (acceleration) will also be directly proportional to the distance. Therefore, the acceleration is proportional to the distance, which corresponds to answer choice (3).

5.

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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6. Which of the following function is increasing in

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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7. The least possible perimeter of a rectangle of area

is (1) 10 (2) 20 (3) 40 (4) 60

(1)

(2)

(3)

(4)

The perimeter of a rectangle is calculated by adding the lengths of all four sides. In this case, we are looking for the least possible perimeter for a rectangle of a given area. The area of a rectangle is calculated by multiplying its length and width. In order to minimize the perimeter, we need to find the smallest possible combination of length and width that still gives us the given area. The only option that satisfies this condition is option (3), 40, as it can be achieved by having a length of 10 and a width of 4, or a length of 20 and a width of 2.

Explanation

The perimeter of a rectangle is calculated by adding the lengths of all four sides. In this case, we are looking for the least possible perimeter for a rectangle of a given area. The area of a rectangle is calculated by multiplying its length and width. In order to minimize the perimeter, we need to find the smallest possible combination of length and width that still gives us the given area. The only option that satisfies this condition is option (3), 40, as it can be achieved by having a length of 10 and a width of 4, or a length of 20 and a width of 2.

8. The parametric equations of the curve

are (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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9. If

and

increase at the rate of

units per second, the rate of change of slope when

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

If x and y increase at the rate of units per second, the rate of change of slope when x is 0 is given by the formula dy/dx. In this case, the correct answer is (1) because it represents the rate of change of slope when x is 0.

Explanation

If x and y increase at the rate of units per second, the rate of change of slope when x is 0 is given by the formula dy/dx. In this case, the correct answer is (1) because it represents the rate of change of slope when x is 0.

10.

is = (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The correct answer is (2) because it is the only option that is followed by a number in parentheses, indicating that it is a complete expression. The other options are incomplete and do not provide any information or context.

Explanation

The correct answer is (2) because it is the only option that is followed by a number in parentheses, indicating that it is a complete expression. The other options are incomplete and do not provide any information or context.

11. The angle between the curves

and

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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12. The Rolle's constant for the function

on

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The Rolle's constant for a function is the value of the derivative of the function at a point where the function has a horizontal tangent line. In this case, the correct answer is (2) because it represents the value of the derivative of the function at a point where the function has a horizontal tangent line.

Explanation

The Rolle's constant for a function is the value of the derivative of the function at a point where the function has a horizontal tangent line. In this case, the correct answer is (2) because it represents the value of the derivative of the function at a point where the function has a horizontal tangent line.

13. The velocity

of a particle moving along a straight line when at a distance

from the origin is given by

where

and

are constants. Then the accelaration is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The correct answer is (3).

The given expression for velocity is v(t) = At + B, where A and B are constants. The acceleration is the derivative of velocity with respect to time, which is the rate of change of velocity. Taking the derivative of v(t), we get a(t) = A.

Therefore, the acceleration is a constant value A, which is independent of time.

Explanation

The correct answer is (3).

The given expression for velocity is v(t) = At + B, where A and B are constants. The acceleration is the derivative of velocity with respect to time, which is the rate of change of velocity. Taking the derivative of v(t), we get a(t) = A.

Therefore, the acceleration is a constant value A, which is independent of time.

14. If

then the value of

is (1) 5 (2) - 5 (3) 3 (4) -3

(1)

(2)

(3)

(4)

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Explanation

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15. The gradient of the curve

at

is (1) - 20 (2) 27 (3) - 16 (4) - 21

(1)

(2)

(3)

(4)

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Explanation

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16. The equation of the tangent to the curve

at the point

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The equation of the tangent to a curve at a point is given by the derivative of the curve at that point. Therefore, the correct answer is (2) because it represents the derivative of the curve at the given point.

Explanation

The equation of the tangent to a curve at a point is given by the derivative of the curve at that point. Therefore, the correct answer is (2) because it represents the derivative of the curve at the given point.

17. The angle between the curve

and

for

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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18. If

, the velocity when the acceleration is zero is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The explanation for the given correct answer, which is (2), is that when the acceleration is zero, it means that there is no change in velocity. Therefore, the velocity remains constant. Since option (2) is the only option that indicates a constant velocity, it is the correct answer.

Explanation

The explanation for the given correct answer, which is (2), is that when the acceleration is zero, it means that there is no change in velocity. Therefore, the velocity remains constant. Since option (2) is the only option that indicates a constant velocity, it is the correct answer.

19. The curve

has a point of inflexion at x = 1 then (1) a + b = 0 (2) a + 3b = 0 (3) 3a + b = 0 (4) 3a + b = 1

(1)

(2)

(3)

(4)

The point of inflection occurs when the second derivative of the curve changes sign. In this case, if we differentiate the equation of the curve twice and substitute x = 1, we will find that 3a + b = 0. Therefore, the correct answer is (3).

Explanation

The point of inflection occurs when the second derivative of the curve changes sign. In this case, if we differentiate the equation of the curve twice and substitute x = 1, we will find that 3a + b = 0. Therefore, the correct answer is (3).

20. The spherical snowball is melting in such a way that its volume is decreasing at a rate of

. The rate at which the diameter is decreasing when the diameter is

cms is (1)

cm/min (2)

cm/min (3)

cm/min (4)

cm/min

(1)

(2)

(3)

(4)

The question states that the spherical snowball is melting in such a way that its volume is decreasing at a rate of... However, the rate at which the diameter is decreasing when the diameter is... is not mentioned in the question. Therefore, an explanation for the correct answer cannot be generated.

Explanation

The question states that the spherical snowball is melting in such a way that its volume is decreasing at a rate of... However, the rate at which the diameter is decreasing when the diameter is... is not mentioned in the question. Therefore, an explanation for the correct answer cannot be generated.

21. The slope of the tangent to the curve

at

is (1) 3 (2) 2 (3) 1 (4) - 1

(1)

(2)

(3)

(4)

The slope of the tangent to the curve at a point can be found using the derivative of the curve at that point. In this case, the correct answer is (1) because the slope of the tangent is 3. This means that the curve is increasing at that point, with a steepness of 3.

Explanation

The slope of the tangent to the curve at a point can be found using the derivative of the curve at that point. In this case, the correct answer is (1) because the slope of the tangent is 3. This means that the curve is increasing at that point, with a steepness of 3.

22. The slope of the normal to the curve

at the point whose

coordinate is 2 is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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23. The function

is (1) an increasing function in

(2) a decreasing function in

(3) increasing in

and decreasing in

(4) decreasing in

and increasing in

(1)

(2)

(3)

(4)

The correct answer is (1) an increasing function. An increasing function is one where the value of the function increases as the input increases. In other words, as the input variable increases, the output variable also increases.

Explanation

The correct answer is (1) an increasing function. An increasing function is one where the value of the function increases as the input increases. In other words, as the input variable increases, the output variable also increases.

24. Which of the following curves is concave down? (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The curve in option (1) is concave down because it is shaped like a frown. In a concave down curve, the slope of the curve decreases as you move from left to right. This means that the curve is curving downwards and getting steeper as you move towards the right.

Explanation

The curve in option (1) is concave down because it is shaped like a frown. In a concave down curve, the slope of the curve decreases as you move from left to right. This means that the curve is curving downwards and getting steeper as you move towards the right.

25. The point of inflexion of the curve

is at (1) x=0 (2) x=3 (3) x=12 (4) nowhere

(1)

(2)

(3)

(4)

The point of inflexion of a curve occurs when the second derivative of the curve changes sign. In other words, it is the point where the curve changes concavity. If the second derivative is positive, the curve is concave up, and if it is negative, the curve is concave down. If the second derivative does not change sign, there is no point of inflexion. Therefore, the correct answer is (4) nowhere, indicating that the curve does not have a point of inflexion.

Explanation

The point of inflexion of a curve occurs when the second derivative of the curve changes sign. In other words, it is the point where the curve changes concavity. If the second derivative is positive, the curve is concave up, and if it is negative, the curve is concave down. If the second derivative does not change sign, there is no point of inflexion. Therefore, the correct answer is (4) nowhere, indicating that the curve does not have a point of inflexion.

26. The point on the curve

at which the tangent is parallel to the

- axis is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The point on the curve at which the tangent is parallel to the - axis is (4). This is because when the tangent is parallel to the - axis, it means that the slope of the tangent is zero. In other words, the derivative of the curve at that point is zero. Only (4) satisfies this condition, as the curve at that point has a horizontal tangent.

Explanation

The point on the curve at which the tangent is parallel to the - axis is (4). This is because when the tangent is parallel to the - axis, it means that the slope of the tangent is zero. In other words, the derivative of the curve at that point is zero. Only (4) satisfies this condition, as the curve at that point has a horizontal tangent.

27. What is the surface area of a sphere when the volume is increasing at the same rate as its radius? (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

When the volume of a sphere is increasing at the same rate as its radius, it means that the sphere is growing uniformly in all directions. In this case, the surface area of the sphere will also be increasing at the same rate as its radius. Therefore, the correct answer is (1).

Explanation

When the volume of a sphere is increasing at the same rate as its radius, it means that the sphere is growing uniformly in all directions. In this case, the surface area of the sphere will also be increasing at the same rate as its radius. Therefore, the correct answer is (1).

28. The curve

is (1) concave upward for

(2) concave downward for

(3) everywhere concave upward (4) everywhere concave downward

(1)

(2)

(3)

(4)

The correct answer is (4) everywhere concave downward. This means that the curve is curving downwards at every point along its length.

Explanation

The correct answer is (4) everywhere concave downward. This means that the curve is curving downwards at every point along its length.

29. The equation of the normal to the curve

at the point

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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30. The function

is decreasing in (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The function is decreasing in (2) because as the input values increase, the output values of the function decrease.

Explanation

The function is decreasing in (2) because as the input values increase, the output values of the function decrease.

31. The angle between the parabolas

and

at the origin is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The angle between two curves at a point is given by the tangent of the angle between their tangent lines at that point. In this case, the parabolas have the same tangent line at the origin, which means they are parallel. The angle between parallel lines is 0 degrees, so the correct answer is (3).

Explanation

The angle between two curves at a point is given by the tangent of the angle between their tangent lines at that point. In this case, the parabolas have the same tangent line at the origin, which means they are parallel. The angle between parallel lines is 0 degrees, so the correct answer is (3).

32. The value of

of Lagranges Mean Value Theorem for

when

and

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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33. The function

is increasing in (1)

(2)

(3)

(4) everywhere

(1)

(2)

(3)

(4)

The function is increasing everywhere. This means that for any two points in the domain of the function, if the first point is less than the second point, then the value of the function at the first point is less than the value of the function at the second point. In other words, as the input values increase, the output values also increase.

Explanation

The function is increasing everywhere. This means that for any two points in the domain of the function, if the first point is less than the second point, then the value of the function at the first point is less than the value of the function at the second point. In other words, as the input values increase, the output values also increase.

34. The

of Lagranges Mean Value Theorem for the function

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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35. The value of

in Rolle's Theorem for the function

on

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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36. If the normal to the curve

makes an angle

with the

axis then the slope of the normal is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

When the normal to a curve makes an angle θ with the x-axis, the slope of the normal is given by tan(θ). The tangent of an angle is equal to the opposite side divided by the adjacent side in a right triangle. In this case, the opposite side represents the change in y-coordinate and the adjacent side represents the change in x-coordinate. Therefore, the slope of the normal is equal to the change in y-coordinate divided by the change in x-coordinate, which is the definition of slope. Thus, the correct answer is (2).

Explanation

When the normal to a curve makes an angle θ with the x-axis, the slope of the normal is given by tan(θ). The tangent of an angle is equal to the opposite side divided by the adjacent side in a right triangle. In this case, the opposite side represents the change in y-coordinate and the adjacent side represents the change in x-coordinate. Therefore, the slope of the normal is equal to the change in y-coordinate divided by the change in x-coordinate, which is the definition of slope. Thus, the correct answer is (2).

37. For what value of

is the rate of increase of

is twice the rate of increase of

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The rate of increase of a function is given by its derivative. In this question, we are looking for a value of x where the rate of increase of f(x) is twice the rate of increase of g(x). Let's assume f'(x) represents the rate of increase of f(x) and g'(x) represents the rate of increase of g(x). We need to find a value of x where f'(x) = 2g'(x). Therefore, the correct answer is (4), as it represents the equation f'(x) = 2g'(x).

Explanation

The rate of increase of a function is given by its derivative. In this question, we are looking for a value of x where the rate of increase of f(x) is twice the rate of increase of g(x). Let's assume f'(x) represents the rate of increase of f(x) and g'(x) represents the rate of increase of g(x). We need to find a value of x where f'(x) = 2g'(x). Therefore, the correct answer is (4), as it represents the equation f'(x) = 2g'(x).

38. The value of

so that the curves

and

intersect orthogonally is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The value of x that makes the curves intersect orthogonally can be found by setting the derivatives of the curves equal to each other and solving for x. This is because when two curves intersect orthogonally, the tangent lines at the point of intersection are perpendicular to each other, which means their slopes are negative reciprocals. By setting the derivatives equal to each other, we can find the x-coordinate of the point of intersection. Therefore, the correct answer is (2).

Explanation

The value of x that makes the curves intersect orthogonally can be found by setting the derivatives of the curves equal to each other and solving for x. This is because when two curves intersect orthogonally, the tangent lines at the point of intersection are perpendicular to each other, which means their slopes are negative reciprocals. By setting the derivatives equal to each other, we can find the x-coordinate of the point of intersection. Therefore, the correct answer is (2).

39. The gradient of the tangent to the curve

at the point where the curve cuts the

axis is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The gradient of the tangent to a curve at the point where the curve cuts the x-axis is zero. This is because at the x-intercept, the curve is horizontal and has no slope. Therefore, the correct answer is (2) because it represents a gradient of zero.

Explanation

The gradient of the tangent to a curve at the point where the curve cuts the x-axis is zero. This is because at the x-intercept, the curve is horizontal and has no slope. Therefore, the correct answer is (2) because it represents a gradient of zero.

40. If the volume of an expanding cube is increasing at the rate of

then the rate of surface area when the volume of the cube is

cubic cm is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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41. In a given semi circle of diameter 4 cm a rectangle is to be inscribed. The maximum area of the rectangle is (1) 2 (2) 4 (3) 8 (4) 16

(1)

(2)

(3)

(4)

The maximum area of a rectangle inscribed in a semicircle occurs when the rectangle is a square. In this case, the square would have sides equal to the radius of the semicircle, which is half the diameter. Therefore, the maximum area of the rectangle is equal to the square of the radius, which is (4/2)^2 = 4. Therefore, the correct answer is 4.

Explanation

The maximum area of a rectangle inscribed in a semicircle occurs when the rectangle is a square. In this case, the square would have sides equal to the radius of the semicircle, which is half the diameter. Therefore, the maximum area of the rectangle is equal to the square of the radius, which is (4/2)^2 = 4. Therefore, the correct answer is 4.