12 - Maths Unit 5 - Differential Calculus Applications - I
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Prepard by, R VISVANATHAN, PG ASST IN MATHS, GHSS, PERIYATHACHUR, TINDIVANAM TK-605651
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Questions and Answers
- 1.
The gradient of the curve at is (1) - 20 (2) 27 (3) - 16 (4) - 21
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4) -
- 2.
The rate of change of area of a circle of radius is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)Explanation
The rate of change of the area of a circle with respect to its radius is given by the formula 2Ï€r. This means that for every unit increase in the radius, the area of the circle increases by 2Ï€ 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.Rate this question:
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- 3.
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)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (3)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.Rate this question:
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- 4.
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
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
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- 5.
The slope of the tangent to the curve at is (1) 3 (2) 2 (3) 1 (4) - 1
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1)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.Rate this question:
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- 6.
The slope of the normal to the curve at the point whose coordinate is 2 is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (3) -
- 7.
The point on the curve at which the tangent is parallel to the - axis is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4)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.Rate this question:
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- 8.
The equation of the tangent to the curve at the point is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
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- 9.
The equation of the normal to the curve at the point is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (3) -
- 10.
The angle between the curves and is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4) -
- 11.
The angle between the curve and for is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 12.
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)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (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).Rate this question:
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- 13.
If the normal to the curve makes an angle with the axis then the slope of the normal is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (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).Rate this question:
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- 14.
The parametric equations of the curve are (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2) -
- 15.
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)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (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).Rate this question:
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- 16.
For what value of is the rate of increase of is twice the rate of increase of (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4)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).Rate this question:
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- 17.
What is the surface area of a sphere when the volume is increasing at the same rate as its radius? (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (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).Rate this question:
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- 18.
If and increase at the rate of units per second, the rate of change of slope when is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1)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.Rate this question:
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- 19.
The gradient of the tangent to the curve at the point where the curve cuts the axis is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
-
- 20.
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)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 21.
The angle between the parabolas and at the origin is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (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).Rate this question:
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- 22.
For the curve ; the tangent line is parallel to the axis when is equal to (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 23.
The value of so that the curves and intersect orthogonally is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (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).Rate this question:
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- 24.
The Rolle's constant for the function on is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
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- 25.
The value of of Lagranges Mean Value Theorem for when and is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 26.
If a normal makes an angle with positive axis then the slope of the curve at the point where the normal is drawn is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 27.
is = (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
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- 28.
If , the velocity when the acceleration is zero is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
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- 29.
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)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (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).Rate this question:
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- 30.
If then the value of is (1) 5 (2) - 5 (3) 3 (4) -3
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 31.
The of Lagranges Mean Value Theorem for the function is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4) -
- 32.
The value of in Rolle's Theorem for the function on is (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2) -
- 33.
(1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4) -
- 34.
The function is increasing in (1) (2) (3) (4) everywhere
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (3)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.Rate this question:
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- 35.
Which of the following function is increasing in (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1) -
- 36.
The function is decreasing in (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)Explanation
The function is decreasing in (2) because as the input values increase, the output values of the function decrease.Rate this question:
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- 37.
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
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1)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.Rate this question:
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- 38.
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
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
B. (2)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.Rate this question:
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- 39.
The least possible perimeter of a rectangle of area is (1) 10 (2) 20 (3) 40 (4) 60
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (3)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.Rate this question:
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- 40.
If on then the absolute maximum value is (1) 2 (2) 3 (3) 4 (4) 5
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4)Explanation
The given question is incomplete and not readable.Rate this question:
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- 41.
The curve is (1) concave upward for (2) concave downward for (3) everywhere concave upward (4) everywhere concave downward
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4)Explanation
The correct answer is (4) everywhere concave downward. This means that the curve is curving downwards at every point along its length.Rate this question:
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- 42.
Which of the following curves is concave down? (1) (2) (3) (4)
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
A. (1)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.Rate this question:
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- 43.
The point of inflexion of the curve is at (1) x=0 (2) x=3 (3) x=12 (4) nowhere
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
D. (4)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.Rate this question:
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- 44.
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
- A.
(1)
- B.
(2)
- C.
(3)
- D.
(4)
Correct Answer
C. (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).Rate this question:
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