Class 12 Application of Derivatives Important Questions
Class 12 application of derivatives important questions bridge theoretical calculus with real-world problem-solving. These questions prepare students for board exams effectively.
Applications include rate of change, tangents, normals, and maxima-minima problems. Many students find optimization challenging initially. However, consistent practice builds mastery. Moreover, conceptual clarity leads to better scores.
Smart Strategies for Calculus Preparation
Targeted practice yields remarkable results. Students should focus on word problems and graphical interpretations. Understanding the step-by-step approach is equally vital. This methodology ensures comprehensive coverage.
For additional free study materials, visit Udgam Welfare Foundation. You can also explore resources like Class 7 Math case study questions for foundational practice. Moreover, www.mathstudy.in offers structured guidance for mathematics learners.
Regular revision ensures formula retention. Solving diverse problems builds adaptability. Consequently, students approach exams with greater assurance and readiness.
Application Of Derivatives HOTS & Important Questions Class 12 Mathematics
1. A man 2 m high walks at a uniform speed of 5 km/h away from a lamp post which is 6 m high. Find the rate at which the length of his shadow is increasing and the rate at which the tip of the shadow is moving.
2. Find the intervals in which the function \( f(x) = \sin^4 x + \cos^4 x, x \in [0, \pi/2] \) is strictly increasing or strictly decreasing.
3. Find the equation of the normal to the curve \( x^2 = 4y \) which passes through the point \( (1, 2) \).
4. Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is \( \sin^{-1}(1/3) \).
5. Find the point on the curve \( y^2 = 4x \) which is nearest to the point \( (2, -8) \).
6. A particle moves along the curve \( 6y = x^3 + 2 \). Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
7. Find the values of \( a \) for which the function \( f(x) = \sin x – ax + b \) is strictly decreasing on \( \mathbb{R} \).
8. Prove that the curves \( y^2 = x \) and \( x^2 = y \) divide the area of the square bounded by \( x=0, x=1, y=0, y=1 \) into three equal parts.
9. An isosceles triangle of vertical angle \( \theta \) is inscribed in a circle of radius \( R \). Show that the area of the triangle is maximum when \( \theta = \pi/3 \).
10. The total cost \( C(x) \) associated with the production of \( x \) units of an item is given by \( C(x) = 0.007x^3 – 0.003x^2 + 15x + 4000 \). Find the marginal cost when 17 units are produced.
11. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \( R \) is \( 2R/\sqrt{3} \). Also, find the maximum volume.
12. Find the equations of the tangents to the curve \( y = \cos(x+y), -2\pi \le x \le 2\pi \) that are parallel to the line \( x + 2y = 0 \).
13. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Find the rate at which the depth of the wheat is increasing. (Take \( \pi = 3.14 \)).
14. If the surface area of a closed cuboid with square base and volume \( V \) is minimum, then prove that it is a cube.
15. Find the maximum volume of a cylinder which can be inscribed in a cone of height \( h \) and semi-vertical angle \( \alpha \).
Answers and Hints
1. Answer: Shadow: 2.5 km/h, Tip: 7.5 km/h. Hint: Use similar triangles to relate the man’s height, lamp post height, and shadow length. Differentiate with respect to time.
2. Answer: Increasing: \((\pi/4, \pi/2)\), Decreasing: \((0, \pi/4)\). Hint: Rewrite \(f(x) = 1 – \frac{1}{2}\sin^2(2x)\) and analyze its derivative.
3. Answer: \(x + y = 3\). Hint: Find the derivative of \(x^2 = 4y\) to get the slope of the tangent, then find the slope of the normal. Use the point-slope form with point \((1, 2)\).
4. Hint: Express surface area \(S\) and volume \(V\) in terms of radius \(r\) and slant height \(l\). Use \(S = \pi r l + \pi r^2\) and maximize \(V = \frac{1}{3}\pi r^2 h\).
5. Answer: \((4, -4)\). Hint: Minimize the distance squared \(D = (x-2)^2 + (y+8)^2\) subject to \(y^2 = 4x\).
6. Answer: \((4, 11)\) and \((-4, -31/3)\). Hint: Differentiate \(6y = x^3 + 2\) implicitly to get \(\frac{dy}{dx}\), then set \(\frac{dy}{dx} = 8\).
7. Answer: \(a \ge 1\). Hint: For \(f(x)\) to be strictly decreasing, \(f'(x) = \cos x – a \le 0\) for all \(x\). The maximum value of \(\cos x\) is 1.
8. Hint: Find the area under \(y = \sqrt{x}\), above \(y = x^2\), and between \(x=0\) and \(x=1\). Show each area is \(1/3\).
9. Hint: Express the area \(A = \frac{1}{2} R^2 (2\sin\theta + \sin(2\theta))\) and maximize with respect to \(\theta\).
10. Answer: 20.967. Hint: Marginal cost is \(C'(x) = 0.021x^2 – 0.006x + 15\). Evaluate at \(x = 17\).
11. Answer: Height: \(2R/\sqrt{3}\), Volume: \(V = \frac{4\pi R^3}{3\sqrt{3}}\). Hint: Let \(h\) be the height of the cylinder. Express volume \(V = \pi r^2 h\) in terms of \(h\) using \(r^2 = R^2 – \frac{h^2}{4}\).
12. Answer: \(x + 2y = \pi/2\) and \(x + 2y = -3\pi/2\). Hint: Differentiate \(y = \cos(x+y)\) implicitly to find \(\frac{dy}{dx} = -\frac{1}{2}\). Use point-slope form.
13. Answer: 1 m/h. Hint: Volume \(V = \pi r^2 h\). Differentiate with respect to time, using \(\frac{dV}{dt} = 314\) and \(r = 10\).
14. Hint: Let the side of the square base be \(x\) and height be \(h\). Express surface area \(S = 2x^2 + 4xh\) in terms of \(x\) using \(V = x^2 h\).
15. Answer: \(\frac{4}{27}\pi h^3 \tan^2 \alpha\). Hint: Let \(r\) be the radius and \(h_c\) the height of the cylinder. Express \(h_c\) and \(r\) in terms of \(h\) and \(\alpha\), then maximize \(V = \pi r^2 h_c\).
Frequently Asked Questions (FAQs)
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How do Application of Derivatives Previous year questions PDF help in preparation?
Application of Derivatives Previous year questions PDF provides insight into exam patterns. They highlight frequently repeated concepts. Thus, students can prioritize topics accordingly for efficient revision.
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Derivative Questions for Class 12 PDF with Solutions is available on Udgam Welfare Foundation. This compilation includes solved examples. Consequently, students master various derivative applications effectively.
What are the most important topics in applications of derivatives?
Key topics include rate of change, tangents, normals, increasing-decreasing functions, and maxima-minima. These carry significant weightage. Therefore, focused practice yields better scores.
How many application of derivative questions should I practice daily?
Practicing 8 to 10 quality questions daily is ideal. Include both theoretical and word problems. Consistent practice builds speed and accuracy over time.