2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Miscellaneous Exercise

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Karnataka 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Miscellaneous Exercise

Question 1.
Using differentials, find the approximate value of each of the following

(a) \(\left(\frac{17}{81}\right)^{1 / 4}\)
Answer:

(b) (33)^-1/5
Answer:

Question 2.
Show that the function given by log has maximum at \(f(x)=\frac{\log x}{x}\) has maximum at x = e
Answer:

Question 3.
The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base ?
Answer:

Question 4.
Find the equation of the normal to curve y² = 4x which passes through the point (1, 2).
Answer:

Question 5.
Show that the normal at any point 6, to the curve x = a cos θ +a θ sin θ ,y = a sin θ –
a θ cos θ is at a constant distance from the origin.
Answer:

∴ equation of normal is
y – (a sin θ – a θ cos θ)
= – cot θ (x – a cos θ – a θ sin θ)
y + x cot θ= a cot θ cos θ+a θ cot θ sin θ – a θ cos θ
\(y+\frac{x \cos \theta}{\sin \theta}=\frac{a \cos ^{2} \theta}{\sin \theta}+\frac{a \sin ^{2} \theta}{\sin \theta}\)
x cos θ a cos²θ, a sin²θ
normal form is x cos θ+ y sin θ = p when p is the normal from the normal to the given curve is at a consistent distance ‘a’ from the origin.

Question 6.
Find the intervals in which the function f given by
\(f(x)=\frac{4 \sin x-2 x-x \cos x}{2+\cos x}\)
(i) increasing
(ii) decreasing
Answer:

Question 7.
Find the intervals in which the function f given by
x³ + -1/x³, x ≠ 0
(i) increasing
(ii) decreasing
Answer:

Question 8.
Find the maximum area of an isosceles triangle inscribed in the ellipse
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with its vertex at one end of the major axis.
Answer:

Question 9.
‘A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m³. If building of tank costs ₹ 70 per sq metres for the base and ₹ 45 per square metre for sides. What is the cost of least expensive tank?
Answer:

Question 10.
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
Answer:

Question 11.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Answer:

Question 12.
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is
\(\left(a^{\frac{3}{2}}+b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
Answer:

Question 13.
Find the points at which the function f given by f (x) = (x – 2)⁴ (x + 1)³ has
(i) local maxima
(ii) local minima
(iii) point of inflexion
Answer:
f (x) = (x – 2)⁴ (x + 1)³
f’ (x) = (x – 2)⁴ 3 (x + 1)² + (x + 1)³ 4(x – 2)³
= (x – 2)³ (x + 1) [3 (x -2) + 4 (x + 1)]
= (x – 2)³ (x + 1)² [3x – 6 + 4x + 4]
= (x – 2)³ (x + 1)² [7x – 2]
f’ (x) = 0 ⇒ x = 2, x = -1, x = 2/7
f’ (x)=(x – 2)³ (x+1)² x 7 + (x – 2)³ (7x – 2) 2 (x +1) + (x + 1)² (7x – 2)3 (x – 2)²
here second demivalive test fails as f” (x) = 0 then we apply first demivalive test
At x = 2 f'(x) changes from negative to positive.
∴ function has minimum value and the local minimum value is f (2) = 0
At x = -1
f’ (x) does not change (positive to negative)
∴ f^x has neither maximum value nor minimum value.
∴ x = -1 is the point of inflection at x = 2/7
f’ (x) change from positive to negative
hence function has been maximum at x = 2/7 and local maximum value is

Question 14.
Find the absolute maximum and minimum values of the function f given by
f (x) = cos² x + sin x, x ∈ [o,π]
Answer:

Question 15.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \(\frac{4 r}{3}\)
Answer:
Let the radius of sphere be ‘r’

Question 16.
Let f be a function defined on [a, b] such that f’ (x) > 0, for all xe (a, b). Then prove that f is an increasing function on (a, b).
Answer:
Let x₁, x₂ be any two real number [a, b] such that x₁ <  x₂, then f (x) satisfies the conditions of L.M.V theorem. ∋ C ∈  (a, b)

Question 17.
Show that the height of the cylinder of maximum volume that can be inscribed in sphere of radius R is \(\frac{2 \mathbf{R}}{\sqrt{3}}\). Also find the maximum volume.
Answer:
Let r be the radius of the cylinder and height 2x

Question 18.
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle a is one-third that of the cone and α the greatest volume of cylinder is \(\frac{4}{27} \pi \mathrm{h}^{3}\)
Answer:
Height of cone – h
Radius of cone – r
Height of cylinder – y
Radius of cylinder – x



Choose the correct answer in the Exercises from 19 to 24.

Question 19.
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
(A) 1 m³/h
(B) 0.1 m³h
(C) 1.1 m³/h
(D) 0.5 m³/h
Answer:

Question 20.
The slope of the tangent to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (2,-1) is
(A) \(\frac{22}{7}\)
(B) \(\frac{6}{7}\)
(C) \(\frac{7}{6}\)
(D)\(\frac{-6}{7}\)
Answer:

Question 21.
The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is  ………..
(A) 1
(B) 2
(C) 3
(D) \(\frac{1}{2}\)
Answer:

Question 22.
The normal at the point (1,1) on the curve 2y + x² = 3 is ……………….
(A) x + y = 0
(B) x – y = 0
(C) x + y = 1
(D) x – y = 1
Answer:

Question 23.
The normal to the curve x² = 4y passing (1,2) is
(A) x + y = 3
(B) x – y = 3
(C) x + y = 1
(D) x – y = 1
Answer:

Question 24.
The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are ………………
(A) \(\left(4, \pm \frac{8}{3}\right)\)
(B) \(\left(4, \frac{-8}{3}\right)\)
(C) \(\left(4,+\frac{3}{8}\right)\)
(D) \(\left(\pm 4, \frac{8}{3}\right)\)
Answer:

2nd PUC Maths Application of Derivatives Miscellaneous Exercise Additional Questions and Answers

Question 1.
If the length of the three sides of a trapezium other than the base is 10cm, find the area of the trapezium when it is maximum (CBSE 2010)
Answer:

Question 2.
Find the intervals in which the function sinx + cos x x ∈ [0,π] is (1) strictly increasing (2) strictly decreasing. (CBSE 2010)
Answer:

Question 3.
If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, find the apps error in. Calculating its surface area, (CBSE 2011)
Answer:

Question 4.
Find the relationship between a and b so that the function of defined by
\(f(x)=\left\{\begin{array}{l}{a x+1, \text { If } x \leq 3 \text { is continous at } x=3} \\{b x+3 \text { if } x>3} \end{array}\right.\)
Answer:

Question 5.
An open box with a square box is to be made out of a given quantity of sheet of area a². Show that the maximum volume of the box is \(a^{3} / 6 \sqrt{3}\) (PUC 2011)
Answer:

Question 6.
Show that the rectangle of maximum perimeter which can be insenibed in a circle of radius a is a square of side \(\sqrt{2}\) a. (CBSE 2008)
Answer:

Question 7.
Discuss the continuity of
\(f(x)=\left\{\begin{array}{cl}{\frac{1-\cos x}{x^{2}}} & {x \neq 0} \\{1} & {x=0} \end{array}\right.\) (KPUC 2008)
Answer: