Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 1:

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

(a)   (b) 

Answer:

(a) Consider

Then,

Now, dy is approximately equal to Δy and is given by,

Hence, the approximate value of  = 0.667 + 0.010

= 0.677.

(b) Consider . Let x = 32 and Δx = 1.

Then,

Now, dy is approximately equal to Δy and is given by,

Hence, the approximate value of 

= 0.5 − 0.003 = 0.497.

Question 2:

Show that the function given by has maximum at x = e.

Answer:

Now,

1 − log x = 0

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:

Let ΔABC be isosceles where BC is the base of fixed length b.

Let the length of the two equal sides of ΔABC be a.

Draw AD⊥BC.

Now, in ΔADC, by applying the Pythagoras theorem, we have:

∴ Area of triangle

The rate of change of the area with respect to time (t) is given by,

It is given that the two equal sides of the triangle are decreasing at the rate of 3 cm per second.

∴

Then, when a = b, we have:

Hence, if the two equal sides are equal to the base, then the area of the triangle is decreasing at the rate of .

Question 4:

Find the equation of the normal to curve y² = 4x at the point (1, 2).

Answer:

The equation of the given curve is .

Differentiating with respect to x, we have:

Now, the slope of the normal at point (1, 2) is 

∴Equation of the normal at (1, 2) is y − 2 = −1(x − 1).

⇒ y − 2 = − x + 1

⇒ x + y − 3 = 0

Question 5:

Show that the normal at any point θ to the curve

 is at a constant distance from the origin.

Answer:

We have x = a cos θ + a θ sin θ.

∴ Slope of the normal at any point θ is .

The equation of the normal at a given point (x, y) is given by,

Now, the perpendicular distance of the normal from the origin is

Hence, the perpendicular distance of the normal from the origin is constant.

Question 6:

Find the intervals in which the function f given by

is (i) increasing (ii) decreasing

Answer:

Now, 

cos x = 0 or cos x = 4

But, cos x ≠ 4

∴cos x = 0

 divides (0, 2π) into three disjoint intervals i.e.,

In intervals , 

Thus, f(x) is increasing for

In the interval

Thus, f(x) is decreasing for .

Question 7:

Find the intervals in which the function f given by is

(i) increasing (ii) decreasing

Answer:

Now, the points x = 1 and x = −1 divide the real line into three disjoint intervals i.e.,

In intervals  i.e., when x < −1 and x > 1,

Thus, when x < −1 and x > 1, f is increasing.

In interval (−1, 1) i.e., when −1 < x < 1, 

Thus, when −1 < x < 1, f is decreasing.

Question 8:

Find the maximum area of an isosceles triangle inscribed in the ellipse  with its vertex at one end of the major axis.

Answer:

The given ellipse is .

Let the major axis be along the x −axis.

Let ABC be the triangle inscribed in the ellipse where vertex C is at (a, 0).

Since the ellipse is symmetrical with respect to the x−axis and y −axis, we can assume the coordinates of A to be (−x₁, y₁) and the coordinates of B to be (−x₁, −y₁).

Now, we have .

∴Coordinates of A are  and the coordinates of B are 

As the point (x₁, y₁) lies on the ellipse, the area of triangle ABC (A) is given by,

But, x₁ cannot be equal to a.

Also, when , then

Thus, the area is the maximum when 

∴ Maximum area of the triangle is given by,

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 Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Answer:

Let l, b, and h represent the length, breadth, and height of the tank respectively.

Then, we have height (h) = 2 m

Volume of the tank = 8m³

Volume of the tank = l × b × h

∴ 8 = l × b × 2

Now, area of the base = lb = 4

Area of the 4 walls (A) = 2h (l + b)

However, the length cannot be negative.

Therefore, we have l = 4.

Thus, by second derivative test, the area is the minimum when l = 2.

We have l = b = h = 2.

∴Cost of building the base = Rs 70 × (lb) = Rs 70 (4) = Rs 280

Cost of building the walls = Rs 2h (l + b) × 45 = Rs 90 (2) (2 + 2)

= Rs 8 (90) = Rs 720

Required total cost = Rs (280 + 720) = Rs 1000

Hence, the total cost of the tank will be Rs 1000.

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:

Let r be the radius of the circle and a be the side of the square.

Then, we have:

The sum of the areas of the circle and the square (A) is given by,

Hence, it has been proved that the sum of their areas is least when the side of the square is double the radius of the circle.

Question 11:

A window is in the form of 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:

Let x and y be the length and breadth of the rectangular window.

Radius of the semicircular opening

It is given that the perimeter of the window is 10 m.

∴Area of the window (A) is given by,

Thus, when 

Therefore, by second derivative test, the area is the maximum when length .

Hence, the required dimensions of the window to admit maximum light is given by

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 minimum length of the hypotenuse is 

Answer:

Let ΔABC be right-angled at B. Let AB = x and BC = y.

Let P be a point on the hypotenuse of the triangle such that P is at a distance of a and b from the sides AB and BC respectively.

Let ∠C = θ.

We have,

Now,

PC = b cosec θ

And, AP = a sec θ

∴AC = AP + PC

⇒ AC = b cosec θ + a sec θ … (1)

Therefore, by second derivative test, the length of the hypotenuse is the maximum when 

Now, when , we have:

Hence, the maximum length of the hypotenuses is .

Question 13:

Find the points at which the function f given by has

(i) local maxima (ii) local minima

(ii) point of inflexion

Answer:

The given function is

Now, for values of x close to and to the left of  Also, for values of x close to   and to the right of

Thus,  is the point of local maxima.

Now, for values of x close to 2 and to the left of  Also, for values of x close to 2 and to the right of 2,

Thus, x = 2 is the point of local minima.

Now, as the value of x varies through −1, does not changes its sign.

Thus, x = −1 is the point of inflexion.

Question 14:

Find the absolute maximum and minimum values of the function f given by

Answer:

Now, evaluating the value of f at critical points and at the end points of the interval  (i.e., at x = 0 and x = π), we have:

Hence, the absolute maximum value of f is  occurring at   and the absolute minimum value of f is 1 occurring at

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 .

Answer:

A sphere of fixed radius (r) is given.

Let R and h be the radius and the height of the cone respectively.

The volume (V) of the cone is given by,

Now, from the right triangle BCD, we have:

∴h

∴ The volume is the maximum when

Hence, it can be seen that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is .

Question 16:

Let f be a function defined on [a, b] such that f ‘(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Answer:

Let

x1, x2∈(a,b)such that

x1

x1, x2]. Since f (x) is differentiable on (a, b) and

[x1, x2]⊂(a,b). Therefore, f(x) is continous on [

x1, x2] and differentiable on

(x1, x2). By the Lagrange’s mean value theorm, there exists

c∈(x1, x2)such that

f'(c)=f(x2)-f(x1)x2-x1          …(1)Since f‘(x) > 0 for all

x∈(a,b), so in particular, f‘(c) > 0

f'(c)>0⇒f(x2)-f(x1)x2-x1>0          [Using (1)]

⇒f(x2)-f(x1)>0        [∵

x2-x1>0 when x1

⇒f(x2)>f(x1)⇒f(x1)

x1, x2are arbitrary points in

(a,b). Therefore,

x1f (x) is increasing on (a,b).

Question 17:

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is . Also find the maximum volume.

Answer:

A sphere of fixed radius (R) is given.

Let r and h be the radius and the height of the cylinder respectively.

From the given figure, we have

The volume (V) of the cylinder is given by,

Now, it can be observed that at .

∴The volume is the maximum when

When , the height of the cylinder is

Hence, the volume of the cylinder is the maximum when the height of the cylinder is .

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 α is one-third that of the cone and the greatest volume of cylinder is tan²α.

Answer:

The given right circular cone of fixed height (h) and semi-vertical angle (α) can be drawn as:

Here, a cylinder of radius R and height H is inscribed in the cone.

Then, ∠GAO = α, OG = r, OA = h, OE = R, and CE = H.

We have,

r = h tan α

Now, since ΔAOG is similar to ΔCEG, we have:

Now, the volume (V) of the cylinder is given by,

And, for , we have:

∴By second derivative test, the volume of the cylinder is the greatest when

Thus, the height of the cylinder is one-third the height of the cone when the volume of the cylinder is the greatest.

Now, the maximum volume of the cylinder can be obtained as:

Hence, the given result is proved.

Question 19:

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of

(A) 1 m/h (B) 0.1 m/h

(C) 1.1 m/h (D) 0.5 m/h

Answer:

Let r be the radius of the cylinder.

Then, volume (V) of the cylinder is given by,

Differentiating with respect to time t, we have:

The tank is being filled with wheat at the rate of 314 cubic metres per hour.

∴

Thus, we have:

Hence, the depth of wheat is increasing at the rate of 1 m/h.

The correct Answer is A.

Question 20:

The slope of the tangent to the curve at the point (2, −1) is

(A)   (B)   (C)   (D) 

Answer:

The given curve is

The given point is (2, −1).

At x = 2, we have:

The common value of t is 2.

Hence, the slope of the tangent to the given curve at point (2, −1) is

The correct Answer is B.

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) 

Answer:

The equation of the tangent to the given curve is y = mx + 1.

Now, substituting y = mx + 1 in y² = 4x, we get:

Since a tangent touches the curve at one point, the roots of equation (i) must be equal.

Therefore, we have:

Hence, the required value of m is 1.

The correct Answer is A.

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 = 0 (D) x − y = 1

Answer:

The equation of the given curve is 2y + x² = 3.

Differentiating with respect to x, we have:

The slope of the normal to the given curve at point (1, 1) is

Hence, the equation of the normal to the given curve at (1, 1) is given as:

The correct Answer is B.

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:

The equation of the given curve is x² = 4y.

Differentiating with respect to x, we have:

The slope of the normal to the given curve at point (h, k) is given by,

∴Equation of the normal at point (h, k) is given as:

Now, it is given that the normal passes through the point (1, 2).

Therefore, we have:

Since (h, k) lies on the curve x² = 4y, we have h² = 4k.

From equation (i), we have:

Hence, the equation of the normal is given as:

The correct Answer is A.

Question 24:

The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are

(A)  (B) 

(C)  (D) 

Answer:

The equation of the given curve is 9y² = x³.

Differentiating with respect to x, we have:

The slope of the normal to the given curve at point  is

∴ The equation of the normal to the curve at  is

It is given that the normal makes equal intercepts with the axes.

Therefore, We have:

Also, the point lies on the curve, so we have

From (i) and (ii), we have:

From (ii), we have:

Hence, the required points are

The correct Answer is A.