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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.8

Choose the correct or the most suitable answer from the given four alternatives.

Question 1.

A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is …….

(a) 0.2%

(b) 0.4%

(c) 0.04%

(d) 0.08%

Solution:

(b) 0.4%

Hint:

r = 10 cm

dr = 0.02

Question 2.

The percentage error of the fifth root of 31 is approximately how many times the percentage error in 31?

Solution:

\(\frac{1}{5}\)

Hint:

We know that the percentage error in the 1/5th root of a number is approximately \(\frac{1}{n}\) times the percentage error in the number.

Question 3.

Solution:

(b) 2xu

Hint:

Question 4.

Solution:

(d) 1

Hint:

Question 5.

If w (x, y) = x^{y}, x > 0, then \(\frac{\partial w}{\partial x}\) is equal to ……

Solution:

(c) yx^{y – 1}

Hint:

Question 6.

(a) xye^{xy}

(b) (1 + xy)e^{xy}

(c) (1 + y)e^{xy}

(d) (1 + x)e^{xy}

Solution:

(b) (1 + xy)e^{xy}

Hint:

Question 7.

If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is …….

(a) 0.4 cu.cm

(b) 0.45 cu.cm

(c) 2 cu.cm

(d) 4.8 cu.cm

Solution:

(d) 4.8 cu.cm

Hint.

a = 4 cm

da = 0.1 cm

v = a^{3}

dv = 3a^{2}da

= 3(4)^{2} (0.1)

= 4.8 cu. cm

Question 8.

The change in the surface area S = 6x^{2} of a cube when the edge length varies from x_{0} to x_{0} + dx is …….

(a) 12x_{0} + dx

(b) 12x_{0}dx

(c) 6x_{0}dx

(d) 6x_{0} + dx

Solution:

(b) 12x_{0}dx

Question 9.

the approximate change in the volume: V of a cube of side x meters caused by increasing the size by 1% is ……..

(a) 0.3 xdx m^{3}

(b) 0.03 x m^{3}

(c) 0.03 x^{2} m^{3}

(d) 0.03 x^{3}m^{3}

Solution:

(d) 0.03 x^{3}m^{3}

Hint:

Let the side of the cube be x units

v = x^{3}

when dx = 0.01x

dv = 3x^{2}dx

= 3x^{2}(0.01 x)

= 0.03 x^{3}m^{3}

Question 10.

If g(x, y) = 3x^{2} – 5y + 2y^{2}, x(t) = e^{t} and y(t) = cos t, then \(\frac{d g}{d t}\) is equal to ……..

(a) 6e^{2t} + 5 sin t – 4 cos t sin t

(b) 6e^{2t} – 5 sin t + 4 cos t sin t

(c) 3e^{2t} + 5 sin t + 4 cos t sin t

(d) 3e^{2t} – 5 sin t + 4 cos t sin t

Solution:

(a) 6e^{2t} + 5 sin t – 4 cos t sin t

Question 11.

Solution:

(b) \(\frac{1}{(x+1)^{2}} d x\)

Hint:

Question 12.

(a) -4

(b) -3

(c) -7

(d) 13

Solution:

(c) -7

Hint:

Question 13.

Solution:

(b) \(-x+\frac{\pi}{2}\)

Hint:

Question 14.

If w(x, y, z) = x^{2} (y – z) + y^{2} (z – x) + z^{2} (x – y), then

(a) xy + yz + zx

(a) xy + yz + zx

(b) x(y + z)

(c) y(z + x)

(d) 0

Solution:

(d) 0

Hint:

Question 15.

If f(x, y, z) = xy + yz + zx, then f_{x} – f_{z} is equal to ……….

(a) z – x

(b) y – z

(c) x – z

(d) y – x

Solution:

(a) z – x

Hint:

f_{x} = y + z

f_{z} = y + x

f_{x} – f_{z} = y + z – y – x = z – x

Additional Problems

Question 1.

If u = x^{y} then is equal to ……..

(a) yx^{y – 1}

(b) u log x

(c) u log y

(d) xy^{x – 1}

Solution:

(a) yx^{y – 1}

Hint:

Question 2.

(a) 0

(b) 1

(c) 2

(d) 4

Solution:

(c) 2

Hint:

Question 3.

Solution:

(d) -u

Hint:

Question 4.

The curve y^{2} (x- 2) = x^{2} (1 +x) has …….

(a) an asymptote parallel to x-axis

(b) an asymptote parallel to y-axis

(c) asymptotes parallel to both axis

(d) no asymptotes

Solution:

(b) an asymptote parallel to y-axis

Hint:

Question 5.

If x = r cos θ, y = r sin θ, then \(\frac{\partial r}{\partial x}\) is equal to …….

(a) sec θ

(b) sin θ

(c) cos θ

(d) cosec θ

Solution:

(c) cos θ

Hint:

Question 6.

(a) 0

(b) u

(c) 2u

(d) u – 1

Solution:

(a) 0

Hint:

Question 7.

The percentage error in the 11^{th} root of the number 28 is approximately …. times the percentage error is 28.

Solution:

\(\frac{1}{11}\)

Hint:

Take log on both sides

The percentage error is approximately \(\frac{1}{11}\) times the percentage error is 28.

Question 8.

The curve a^{2}y^{2} = x^{2} (a^{2} – x^{2}) has ……

(a) only one loop between x = 0 and x = a

(b) two loops between x = 0 and x = a

(c) two loops between x = -a and x = a

(d) no loop

Solution:

(c) two loops between x = -a and x = a

Hint.

Question 9.

An asymptote to the curve y^{2} (a + 2x) = x^{2} (3a – x) is ………

(a) x = 3a

(b) x = – a/2

(c) x = a/2

(d) x = 0

Solution:

(b) x = – a/2

Hint.

Question 10.

In which region the curve y^{2} (a + x) = x^{2} (3a – x) does not lie?

(a) x > 0

(b) 0 < x < 3a

(c) x ≤ – a and x > 3a

(d) – a < x < 3a

Solution:

(c) x ≤ – a and x > 3a

Hint.

y^{2} (a + x) = x^{2} (3a – x)

Taking y = 0 (1)

⇒ 0 = x^{2} (3a – x)

⇒ x = 0, x = 3a

∴ The points are (0, 0) (3a, 0)

There is a loop between x = 0 and x = 3a

when x > 3 a, y ➝ imaginary

the curves does not exist beyond x = 3a i.e., x > 3a

the curve has asymptote at x = -a

the curve does not exist when x < -a

the curve exists in the region – a < x < 3a

Question 11.

If M = y sin x, then \(\frac{\partial^{2} u}{\partial x \partial y}\) is equal to …….

(a) cos x

(b) cos y

(c) sin x

(d) 0

Solution:

(a) cos x

Hint:

Question 12.

(a) 0

(b) 1

(c) 2u

(d) u

Solution:

(a) 0

Hint:

Question 13.

The curve 9y^{2} = x^{2} (4 – x^{2}) is symmetrical about …….

(a) y-axis

(b) x-axis

(c) y = x

(d) both the axes

Solution:

(d) both the axes

Hint.

Replace x by – x and y by -y

9 (-y^{2}) = (-x)^{2}(4-(-x)^{2})

The equation is unaltered

∴ the curve is symmetrical about both the axes.

Question 14.

The curve ay^{2} = x^{2} (3a – x) cuts the y-axis at …….

(a) x = -3a, x = 0

(b) x = 0, x = 3a

(c) x = 0, x = a

(d) x = 0

Solution:

(d) x = 0

Hint:

Given ay^{2} = x^{2} (3a – x)

The point of intersection with y-axis by putting x = 0

In (1) = ay^{2} = 0 (3a – 0)

ay = 0; y = 0

∴ The curve intersects y-axis at the origin is x = 0