Question Set B

TN 12th Standard Maths Applications of Differential Calculus Previous Year Question Papers Study Material - QB365

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Questions:

Multiple Choice Question

75 x 1 = 75

Which one of the following is not true?

(a)

Negation of a negation of a statement is the statement itself

(b)

If the last column of the truth table contains only T then it is a tautology.

(c)

If the last column of its truth table contains only F then it is a contradiction

(d)

If p and q are any two statements then p↔️q is a tautology.

Determine the truth value of each of the following statements:
(a) 4 + 2 = 5 and 6 + 3 = 9
(b) 3 + 2 = 5 and 6 + 1 = 7
(c) 4 + 5 = 9 and 1 + 2 = 4
(d) 3 + 2 = 5 and 4 + 7 = 11

(a)

(a)	(b)	(c)	(d)
F	T	F	T

(b)

(a)	(b)	(c)	(d)
T	F	T	F

(c)

(a)	(b)	(c)	(d)
T	T	F	F

(d)

(a)	(b)	(c)	(d)
F	F	T	T

The proposition p ∧ (¬p ∨ q) is

(a)

a tautology

(b)

a contradiction

(c)

logically equivalent to p ∧ q

(d)

logically equivalent to p ∨ q

The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is

(a)

ᄀ(p ∧ q) ∧ [p V (p ∧ ᄀr)]

(b)

(p ∧ q) ∧ [p ∧ (p V ᄀr)]

(c)

ᄀ(p ∧ q) ∧ [p ∧ (p ∧ r)]

(d)

ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

p	q	(p ∧ q) ⟶ ¬q
T	T	(a)
T	F	(b)
F	T	(c)
F	F	(d)

Which one of the following is correct for the truth value of (p ∧ q)⟶ ¬p?

(a)

(a)	(b)	(c)	(d)
T	T	T	T

(b)

(a)	(b)	(c)	(d)
F	T	T	T

(c)

(a)	(b)	(c)	(d)
F	F	T	T

(d)

(a)	(b)	(c)	(d)
T	T	T	F

Which one of the following is incorrect? For any two propositions p and q, we have

(a)

¬ (p∨q) ≡ ¬ p ∧ ¬q

(b)

¬ ( p∧ q)≡¬p ∨ ¬q

(c)

¬ (p ∨ q)≡¬p∨¬q

(d)

¬(¬p)≡ p

In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are

(a)

(b)

(c)

(d)

The truth table for (p ∧ q) ∨ ¬q is given below

p	q	(p ∧ q) ∨ (¬q)
T	T	(a)
T	F	(b)
F	T	(c)
F	F	(d)

Which one of the following is true?

(a)

(a)	(b)	(c)	(d)
T	T	T	T

(b)

(a)	(b)	(c)	(d)
T	F	T	T

(c)

(a)	(b)	(c)	(d)
T	T	F	T

(d)

(a)	(b)	(c)	(d)
T	F	F	F

Which one is the contrapositive of the statement (pVq)⟶r?

(a)

ㄱr➝(ㄱp∧ㄱq)

(b)

ㄱr⟶(p∨q)

(c)

r⟶(p∧q)

(d)

p⟶(q∨r)

10.

Which one is the inverse of the statement (pVq)➝(pΛq)?

(a)

(p∧q)➝(pVq)

(b)

ᄀ(pvq)➝(p∧q)

(c)

(ᄀpvᄀq)➝(ᄀp∧ᄀq)

(d)

(ᄀp∧ᄀq)➝(ᄀpVᄀq)

11.

If a compound statement involves 3 simple statements, then the number of rows in the truth table is

(a)

(b)

(c)

(d)

12.

Which one of the following statements has truth value F?

(a)

Chennai is in India or \(\sqrt 2\) is an integer

(b)

Chennai is in India or \(\sqrt 2\) is an irrational number

(c)

Chennai is in China or \(\sqrt 2\) is an integer

(d)

Chennai is in China or \(\sqrt 2\) is an irrational number

13.

Which one of the following statements has the truth value T?

(a)

sin x is an even function

(b)

Every square matrix is non-singular

(c)

The product of complex number and its conjugate is purely imaginary

(d)

\(\sqrt 5\) is an irrational number

14.

If a * b=\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) on the real numbers then * is

(a)

commutative but not associative

(b)

associative but not commutative

(c)

both commutative and associative

(d)

neither commutative nor associative

15.

In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?

(a)

y = \(\frac{2}{3}\)

(b)

y = \(\frac{-2}{3}\)

(c)

y = \(\frac{-3}{2}\)

(d)

y = 4

16.

The operation * defined by \(a * b =\frac{ab}{7}\) is not a binary operation on

(a)

Q⁺

(b)

(c)

(d)

17.

In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?

(a)

a*b = min (a.b)

(b)

a*b = max (a, b)

(c)

a*b = a

(d)

a*b = a^b

18.

Which one of the following is a binary operation on N?

(a)

Subtraction

(b)

Multiplication

(c)

Division

(d)

All the above

19.

Subtraction is not a binary operation in

(a)

(b)

(c)

(d)

20.

A binary operation on a set S is a function from

(a)

S ⟶ S

(b)

(SxS) ⟶ S

(c)

S⟶ (SxS)

(d)

(SxS) ⟶ (SxS)

21.

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

22.

If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is

(a)

xy + yz + zx

(b)

x(y + z)

(c)

y(z + x)

(d)

23.

Linear approximation for g(x) = cos x at \(x=\frac{\pi}{2}\) is

(a)

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

(b)

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

(c)

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

(d)

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

24.

If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to

(a)

-4

(b)

-3

(c)

-7

(d)

25.

If \(f(x)=\frac{x}{x+1}\), then its differential is given by

(a)

\(\frac { -1 }{ ({ x+1) }^{ 2 } } dx\)

(b)

\(\frac { 1 }{ ({ x+1) }^{ 2 } } dx\)

(c)

\(\frac { 1 }{ x+1 } dx\)

(d)

\(\frac {- 1 }{ x+1 } dx\)

26.

If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) 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

27.

The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is

(a)

0.3xdx m³

(b)

0.03x m³

(c)

0.03x² m³

(d)

0.03x³m³

28.

The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is

(a)

12 x_o+dx

(b)

12x_o dx

(c)

6x_o dx

(d)

6x_o+ dx

29.

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

30.

If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to

(a)

xye^xy

(b)

(1 +xy)e^xy

(c)

(1 +y)e^xy

(d)

(1 + x)e^xy

31.

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

(a)

x^y log x

(b)

y log x

(c)

yx^y-1

(d)

x log y

32.

If v (x, y) = log (e^x+ e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to

(a)

e^x + e^y

(b)

\(\frac{1}{e^x + e^y}\)

(c)

(d)

33.

If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to

(a)

\(e^{x^{2}+y^{2}}\)

(b)

2xu

(c)

x²u

(d)

y²u

34.

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

(a)

\(\frac{1}{31}\)

(b)

\(\frac15\)

(c)

(d)

35.

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%

36.

A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

(a)

\(\frac { 57 }{ { 20 }^{ 3 } } \)

(b)

\(\frac { 57 }{ { 20 }^{ 2 } } \)

(c)

\(\frac { { 19 }^{ 3 } }{ { 20 }^{ 3 } } \)

(d)

\(\frac { 57 }{ 20 } \)

37.

If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is

(a)

0.125

(b)

0.25

(c)

0.375

(d)

0.75

38.

Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

(a)

0.24

(b)

0.48

(c)

0.6

(d)

0.96

39.

The probability mass function of a random variable is defined as:

x	-2	-1	0	1	2
f(x)	k	2k	3k	4k	5k

Then E(X ) is equal to:

(a)

\(\frac { 1 }{ 15 } \)

(b)

\(\frac { 1 }{ 10 } \)

(c)

\(\frac { 1 }{ 3 } \)

(d)

\(\frac { 2 }{ 3 } \)

40.

If \(f(x)=\left\{\begin{array}{ll} 2 x & 0 \leq x \leq a \\ 0 & \text { otherwise } \end{array}\right.\) is a probability density function of a random variable, then the value of a is

(a)

(b)

(c)

(d)

41.

Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day
II. The number of customers in a queue to buy train tickets at a moment.
III. The time taken to complete a telephone call.

(a)

I and II

(b)

II only

(c)

III only

(d)

II and III

42.

Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is

(a)

(b)

(c)

(d)

43.

The random variable X has the probability density function
\(f(x)=\left\{\begin{array}{lr} a x+b & 0<x<1 \\ 0 & \text { otherwise } \end{array}\right.\) and \(E(X)=\frac { 7 }{ 12 } \), then a and b are respectively

(a)

1 and \(\frac { 1 }{ 2 } \)

(b)

\(\frac { 1 }{ 2 } \) and 1

(c)

2 and 1

(d)

1 and 2

44.

If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is

(a)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 6 }\left( \frac { 2 }{ 5 } \right) ^{ 4 }\)

(b)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 10 }\)

(c)

\(\left( \frac { 10 }{ 5 } \right) { \left( \frac { 3 }{ 5 } \right) }^{ 4 }\left( \frac { 2 }{ 5 } \right) ^{ 6 }\)

(d)

\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 5 }\left( \frac { 2 }{ 5 } \right) ^{ 5 }\)

45.

If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).

(a)

\(\frac { 2 }{ 3 } \)

(b)

\(\frac { 2 }{ 5 } \)

(c)

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

(d)

\(\frac { 1 }{ 3 } \)

46.

On a multiple-choice exam with 3 possible destructives for each of the 5 questions, the probability that a student will get 4 or more correct answers just by guessing is

(a)

\(\frac { 11 }{ 243 } \)

(b)

\(\frac { 3 }{ 8 } \)

(c)

\(\frac { 1 }{ 243 } \)

(d)

\(\frac { 5 }{ 243 } \)

47.

Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E(X) is

(a)

0.11

(b)

1.1

(c)

(d)

48.

Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are

(a)

50,40

(b)

40,50

(c)

40.75,40

(d)

41,41

49.

If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?

(a)

0 and 12

(b)

5 and 17

(c)

7 and 19

(d)

16 and 24

50.

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

(a)

i + 2n, i = 0,1,2... n

(b)

2i- n, i = 0,1,2... n

(c)

n - i, i = 0,1,2... n

(d)

2i + 2n, i = 0, 1, 2...n

51.

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

(b)

(c)

(d)

52.

A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

(a)

(b)

(c)

(d)

53.

Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k², where k is the face that comes up k = {1, 2, 3, 4, 5}.
The expected amount to win at this game in Rs. is

(a)

\(\cfrac { 19 }{ 6 } \)

(b)

\(-\cfrac { 19 }{ 6 } \)

(c)

\(\cfrac { 3 }{ 2 } \)

(d)

\(-\cfrac { 3 }{ 2 } \)

54.

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.

(a)

\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 3 } \)

(b)

\( \frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 6 } \)

(c)

\(l,\frac { { l }^{ 2 } }{ 12 } \)

(d)

\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 12 } \)

55.

Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct

(a)

both mean and variance exist

(b)

mean exists but variance does not exist

(c)

both mean and variance do not exist

(d)

variance exists but Mean does not exist

56.

The point of inflection of the curve y = (x - 1)³ is

(a)

(0, 0)

(b)

(0, 1)

(c)

(1, 0)

(d)

(1, 1)

57.

The curve y= ax⁴ + bx² with ab > 0

(a)

has, no horizontal tangent

(b)

is concave up

(c)

is concave down

(d)

has no points of inflection

58.

The maximum product of two positive numbers, when their sum of the squares is 200, is

(a)

100

(b)

\(25\sqrt { 7 } \)

(c)

(d)

\(24\sqrt { 14 } \)

59.

One of the closest points on the curve x² - y² = 4 to the point (6, 0) is

(a)

(2,0)

(b)

\(\left( \sqrt { 5 } ,1 \right) \)

(c)

\(\left( 3,\sqrt { 5 } \right) \)

(d)

\(\left( \sqrt { 13 } ,-\sqrt { 3 } \right) \)

60.

The maximum value of the function \(x^{2} e^{-2 x}, x>0\) is

(a)

\(\frac { 1 }{ e } \)

(b)

\(\frac { 1 }{ 2e } \)

(c)

\(\frac { 1 }{ { e }^{ 2 } } \)

(d)

\(\frac { 4 }{ { e }^{ 4 } } \)

61.

The maximum slope of the tangent to the curve y = e^x sin x, x ∈ [0, 2π] is at

(a)

\(x=\frac { \pi }{ 4 } \)

(b)

\(x=\frac { \pi }{ 2 } \)

(c)

\(x=\pi \)

(d)

\(x=\frac { 3\pi }{ 2 } \)

62.

The minimum value of the function |3 - x| + 9 is

(a)

(b)

(c)

(d)

63.

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

(a)

(b)

2.5

(c)

(d)

3.5

64.

The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is

(a)

(b)

\(\sqrt { 2 } \)

(c)

\(\frac { 3 }{ 2 } \)

(d)

65.

The function sin⁴ x + cos⁴x is increasing in the interval

(a)

\(\left[ \frac { 5\pi }{ 8 } ,\frac { 3\pi }{ 4 } \right] \)

(b)

\(\left[ \frac { \pi }{ 2 } ,\frac { 5\pi }{ 8 } \right] \)

(c)

\(\left[ \frac { \pi }{ 4 } ,\frac { \pi }{ 2 } \right] \)

(d)

\(\left[ 0,\frac { \pi }{ 4 } \right] \)

66.

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)

(a)

(b)

(c)

(d)

∞

67.

Angle between y² = x and x²= y at the origin is

(a)

\({ tan }^{ -1 }\cfrac { 3 }{ 4 } \)

(b)

\({ tan }^{ -1 }\left( \cfrac { 4 }{ 3 } \right) \)

(c)

\(\cfrac { \pi }{ 2 } \)

(d)

\(\cfrac { \pi }{ 4 } \)

68.

The tangent to the curve y² - xy + 9 = 0 is vertical when

(a)

y = 0

(b)

\(\\ \\ y=\pm \sqrt { 3 } \)

(c)

\(y=\frac { 1 }{ 2 } \)

(d)

\(y=\pm 3\)

69.

The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is

(a)

\(-4\sqrt { 3 } \)

(b)

-4

(c)

\(\cfrac { \sqrt { 3 } }{ 12 } \)

(d)

\(4\sqrt { 3 } \)

70.

The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

(a)

-8

(b)

-4

(c)

-2

(d)

71.

The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is

(a)

(4, 11)

(b)

(4, -11)

(c)

(-4, 11)

(d)

(-4,-11)

72.

A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

(a)

(b)

2.5

(c)

(d)

3.5

73.

The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is

(a)

t = 0

(b)

\(\\ \\ \\ t=\cfrac { 1 }{ 3 } \)

(c)

t =1

(d)

t = 3

74.

A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon's angle of elevation in radian per second when the balloon is 30 metres above the ground.

(a)

\(\frac{3}{25} \text { radians } / \mathrm{sec}\)

(b)

\(\frac{4}{25} \text { radians } / \mathrm{sec}\)

(c)

\(\frac{1}{5} \text { radians } / \mathrm{sec}\)

(d)

\(\frac{1}{3} \text { radians } / \mathrm{sec}\)

75.

The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm

(a)

3 cm/s

(b)

2 cm/s

(c)

1 cm/s

(d)

\(\cfrac { 1 }{ 2 } cm/s\)

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Answers Key:

Multiple Choice Question

75 x 1 = 75

(d)
If p and q are any two statements then p↔️q is a tautology.

(a)

(a) (b) (c) (d)

F T F T

(d)
ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

(b)

(a) (b) (c) (d)

F T T T

(b)
ㄱr⟶(p∨q)

10.

11.

(b)
8

12.

13.

(d)
\(\sqrt 5\) is an irrational number

14.

(a)
commutative but not associative

15.

(b)
y = \(\frac{-2}{3}\)

16.

(b)
Z

17.

(d)
a*b = a^b

18.

(b)
Multiplication

19.

20.

(b)
(SxS) ⟶ S

21.

(b)
y - z

22.

(d)
0

23.

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

24.

25.

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

26.

(a)
6e^2t+ 5 sin t - 4 cos t sin t

27.

\({ f }_{ y }=\frac { 1 }{ 1+\frac { { x }^{ 2 } }{ { y }^{ 2 } } } \left( \frac { -x }{ { y }^{ 2 } } \right) =\frac { -\frac { x }{ { y }^{ 2 } } }{ \frac { { x }^{ 2 }+{ y }^{ 2 } }{ { { x }{ x }^{ 2 } } } } \)
= \(\frac { -x }{ { x }^{ 2 }+{ y }^{ 2 } } \)
\({ f }_{ x }=\frac { 1 }{ 1+\frac { { x }^{ 2 } }{ { { y }^{ 2 } } } } \left( \frac { 1 }{ y } \right) =\frac { \frac { 1 }{ y } }{ \frac { { x }^{ 2 }+{ y }^{ 2 } }{ { { y }^{ 2 } } } } \)
\(=\frac { y }{ { { x }^{ 2 }+{ y }^{ 2 } } } \)
\({ f }_{ xy }=\frac { \partial }{ \partial x } ({ f }_{ y })\)
\({ f }_{ xy }=-\left[ \frac { { (x }^{ 2 }+{ y }^{ 2 })(1)-x{ (2x) } }{ { ({ x }^{ 2 }+{ y }^{ 2 }) }^{ 2 } } \right] \)
\(=-\left[ \frac { { x }^{ 2 }+{ y }^{ 2 }-2{ x }^{ 2 } }{ { ({ x }^{ 2 }+{ y }^{ 2 }) }^{ 2 } } \right] \)
\(=-\left[ \frac { { y }^{ 2 }-{ x }^{ 2 } }{ { ({ x }^{ 2 }+{ y }^{ 2 }) }^{ 2 } } \right] \)
\(=\frac { x^{ 2 }-{ y }^{ 2 } }{ { ({ x }^{ 2 }+{ y }^{ 2 }) }^{ 2 } } \) ...(1)
\({ f }_{ xy }=\frac { \partial }{ \partial y } ({ f }_{ x })\)
\({ f }_{ xy }=\frac { ({ x }^{ 2 }+{ y }^{ 2 })(1)-y(2y) }{ ({ x }^{ 2 }+{ { y }^{ 2 }) }^{ 2 } } \)
\(=\frac { { x }^{ 2 }+{ y }^{ 2 }-2{ y }^{ 2 } }{ ({ x }^{ 2 }+{ { y }^{ 2 }) }^{ 2 } } \)
\(=\frac { { x }^{ 2 }-{ y }^{ 2 } }{ ({ x }^{ 2 }+{ { y }^{ 2 }) }^{ 2 } } \) ...(2)
∴ From (1) and (2), f_xy = f_yz

28.

(b)
12x_o dx

29.

(d)
4.8 cu.cm

30.

(b)
(1 +xy)e^xy

31.

32.

(d)
1

33.

(b)
2xu

34.

(b)
\(\frac15\)

35.

(b)
0.4%

36.

(b)
\(\frac { 57 }{ { 20 }^{ 2 } } \)

37.

(b)
0.25

38.

(d)
0.96

39.

(d)
\(\frac { 2 }{ 3 } \)

40.

(a)
1

41.

(a)
I and II

42.

(b)
2

43.

(a)
1 and \(\frac { 1 }{ 2 } \)

44.

(d)
\(\left( \frac { 10 }{ 5 } \right) \left( \frac { 3 }{ 5 } \right) ^{ 5 }\left( \frac { 2 }{ 5 } \right) ^{ 5 }\)

45.

(a)
\(\frac { 2 }{ 3 } \)

46.

(d)
\(\frac { 5 }{ 243 } \)

47.

(b)
1.1

48.

49.

(d)
16 and 24

50.

(b)
2i- n, i = 0,1,2... n

51.

(d)
2

52.

(d)
4

53.

(d)
\(-\cfrac { 3 }{ 2 } \)

54.

(d)
\(\frac { l }{ 2 } ,\frac { { l }^{ 2 } }{ 12 } \)

55.

(b)
mean exists but variance does not exist

56.

57.

(d)
has no points of inflection

58.

(a)
100

59.

60.

61.

(b)
\(x=\frac { \pi }{ 2 } \)

62.

(d)
9

63.

64.

(d)
2

65.

66.

(a)
0

67.

68.

(d)
\(y=\pm 3\)

69.

Given equation of the curve is y = 1 + x³ and the line is x + 12y = 12
Slope of the tangent to the curve
m₁ = \(\frac { dy }{ dx } \) = 3x² and the
Slope of the line = m₂
= \(\frac{-1}{2}\) \(\left[ \because m=\frac { co-efficient\quad of\quad x }{ co-efficient\quad of\quad y } \right] \)
Since the slope of the tangent to the curve and the line are orthogonal, m₁ m₂ = - 1.
∴ 3x²\(\left( \frac { -1 }{ 2 } \right) \) = -1
⇒ \(\frac{x^2}{4}\) = 1
⇒ x² = 4
⇒ x = ±2
When x = 2, y = 1 + 2³ = 9
When x = -2, y = 1+ (-2)³
= 1-8 = -7
∴ Equation of the tangent at (2, 9) is
y-9 = 12(x-2) [∵ m₁= 3x²= 3(2)² = 12]
∴ y - 9 = 12x - 24
∴ 12x - y = 15
Equation of the tangent at (-2, -7) is
y + 7= 12(x + 2)
⇒ y + 7 = 12x + 24
⇒ 12x - y = -17