x ≡ 13 (mod 17)

Let p be the required number … (1)

7x – 3 ≡ p (mod 17) … (2)

From (1),

x – 13 = 17n for some integer M.

x – 13 is a multiple of 17.

x must be 30.

∴ 30 – 13 = 17

Which is a multiple of 17.

From (2),

7 x 30 – 3 ≡ p (mod 17)

210 – 3 ≡ p (mod 17)

207 ≡ p (mod 17)

207 ≡ 3 (mod 17)

∴ P ≡ 3

Given 3x – 2 = 0 (mod 11) 

3x – 2 = 11n (n may be any integer) 

3x = 2 + 11n 

x = (11n + 2)/3 

Substitute the value of n = 2, 5, 8, 11 …

When n ≡ 2 ⇒ x = (22 + 2)/3 = 24/3 = 8

When n = 5 ⇒ x = (55 + 2)/3 = 57/3 = 19 

When n = 8 ⇒ x = (88 + 2)/3 = 90/3 = 30 

When n = 11 ⇒ x = (121 + 2)/3 = 123/3 = 41 

∴ The value of x is 8, 19, 30, 41

5x – 4 = 6n (n may be any integer)

5x = 6n + 4

x = (6n + 4)/5

The value of n 1, 6, 11, 16 …

∴ The value of x is 2, 8, 14, 20 …

5² = 25 = 1 (mod 8)

5⁴ = (5²)² = l² (mod 8) = 1 

5⁴ = 1 (mod 8)

सही विकल्प है (क) 16.36

सही विकल्प है (घ) 94.83

If a body in translatory motion moves along a straight line then the motion is called rectilinear motion.

Here a = 8, S_∞ = 32/3 

a/(1 - r) = 32/3 

8/(1 - r) = 32/3

32 – 32 r = 24 ⇒ 32 r = 8 

r = 8/32 = 1/4 

Common ratio = 1/4

Given, a = 6, d = 5 

General term t_n = a + (n – 1) d

= 6 + (n – 1) 5 

= 6 + 5 n – 5 

= 5 n + 1 

The general form of the A.P. is a, a + d, a + 2d, 

The A.P. is 6, 11, 16, 21, … 5n + 1.

Applying Euclid’s divison lemma to 32 and 60, we get 

60 = 32 x 1 + 28 … (i)

The remainder is 28 ≠ 0.

Again applying division lemma

32 = 28 x 1 + 4 … (ii)

The remainder 4 ≠ 0.

Again applying division lemma

28 = 4 x 7 + 0 … (iii)

The remainder zero.

∴ H.C.F. of 32 and 60 is 4.

From (ii), we get

32 = 28 x 1 + 4

⇒ 4 = 32 – 28 x 1

⇒ 4 = 32 – (60 – 32 x 1) x 1

⇒ 4 = 32 – 60 + 32

⇒ 4 = 32 x 2 + (-1) x 60

∴ x = 2 and y = -1

Narsinh Mehta wrote devotional songs in Gujarati.

Gangasati
he was a medieval saint poet of bhakti tradition of western India who composed several devotional songs in Gujarati language

f ′ (x) = –3x³ – 24x² – 45x

= – 3x (x² + 8x + 15) = – 3x (x + 5) (x + 3)

f ′ (x) = 0 ⇒ x = –5, x = –3, x = 0

f ″(x) = –9x² – 48x – 45

= –3 (3x² + 16x + 15)

f ″(0) = – 45 < 0. Therefore, x = 0 is point of local maxima

f ″(–3) = 18 > 0. Therefore, x = –3 is point of local minima

f ″(–5) = –30 < 0. Therefore x = –5 is point of local maxima.

Correct option is  B) 4

9 is the smallest odd composite number.

4 is the smallest composite number.

1 is neither a prime number nor a composite number.

We have,

a⁴ – 16b⁴

= (a²)² – (4b²)²

= (a² + 4b²) (a² – 4b²)

= (a² + 4b²) (a + 2b) (a – 2b)

We have,

64 – (a + 1)²

= 8² – (a + 1)²

= [8 + (a + 1)] [8 – (a + 1)]

= (a + 9) (7 – a)

We have,

x⁸ – 1

= (x⁴)²–(1)²

= (x⁴ + 1) (x⁴ – 1)

= (x⁴ + 1) (x² + 1) (x – 1) (x + 1)

f is one-one since each element of A is assigned to distinct element of the set A. Also, f is onto since f (A) = A. Moreover, f ^–1 = {(b, a), (d, b), (a, c), (c, d)}.

f is a function since each element of A in the first place in the ordered pairs is related to only one element of A in the second place while g is not a function because 1 is related to more than one element of A, namely, 2 and 3.

The range of f = {2, 3} and the range of g = {5, 6}.

No. Since for 1, 2 ∈ Z, 1 * 2 = 1 – 2 + 1.2 = 1 while 2 * 1 = 2 – 1 + 2.1 = 3 so that 1 * 2 ≠ 2 * 1.

1. Total internal reflecting prisms are used in a camera, binoculars, periscope. 

2. Total internal reflection of light is used in optical fibres.

[Note: Total internal reflection of light plays an important role in sparkling brilliance of a diamond.]

₂n₁ = 0.5 = sin i

∴ Critical angle i = 30°.

Price Floor: A price floor is the lowest legal price of a commodity at which it can be sold, fixed by the government. Price floors are used by the government to prevent prices from being too low.

The main reason for imposing the price floor policy is the welfare of the producers/farmers. Eg., the minimum wages, minimum support price.

Consequence:

Buffer Stock: In order to maintain the minimum support price, the government may have to build buffer stocks to enable producers to dispose of their surplus stocks. The government purchases the surplus stocks available with the farmers/producers; these stocks are released in case the production of the supported commodity suffers.

Maximum price of a good refers to the maximum price of a good a seller is allowed to charge by government.

Since such a price is below the equilibrium price, it leads to shortage because demand is greater than supply at this price. The shortage in turn may lead to black marketing.

Detailed Answer:

Price Ceiling or Maximum Price: It refers to fixing of the maximum price of a commodity at a level lower than the equilibrium price. Government imposes price ceiling in case of essential commodities (Wheat, Sugar, Kerosene etc.) when the equilibrium price determined by free market forces of demand and supply is high.

Implications of price ceiling:

(i) Shortage (excess demand): At this controlled price quantity demanded is more than quantity supplied. It creates a shortage and to overcome this shortage government may enforce the rationing system.

(ii) Black marketing: If rationing is not administered by the government effectively it results in black marketing. Due to excess demand buyers will compete and they would be willing to buy a commodity at a higher price than the price fixed by the government.

Schedule

Explanation:

In the above schedule the market equilibrium is established at price of Rs.4 where the quantity demanded and quantity supplied are equal, with equilibrium quantity of 10 units.

For certain goods and services, Government, sets minimum price. This minimum price is called minimum price ceiling.

This price is normally set at a level higher than the equilibrium price. This leads to axcess supply. Since producers are not able to sell all they want to sell, they illegally sell the good or service below the minimum price.

When quantity supplied is less than quantity demanded of a product.

No. In case of excess demand, market price is less than equilibrium price. Excess demand will push the market price back to its equilibrium level. i.e., equilibrium price is restored in the economy.

When quantity demanded is less than quantity supplied of a product.

Oligopoly is a form of the market in which there is a large number of buyers, but only a few big sellers of a commodity. 

Duopoly is a form of market in which there are two sellers of a commodity with many buyers.

Black marketing may be termed as a direct consequence of price ceiling as it implies a situation whereby the commodity under the government's control policy is illegally sold at a higher price than the one fixed by the government, it may primarily arise due to the presence of consumer who may be willing to pay higher price for the commodity than to go without it.

Economic viability of an industry refers to the situation when demand and supply curves of the industry meet at some positive level of output. 

Non-viability of an industry refers to a situation when demand curve and supply curve do not intersect each other at any positive quantity. In such a situation, supply curve lies above the demand curve.

It refers to a market situation in which demand equals supply.

When market demand exceeds market supply of a commodity at a given price it is known as excess demand. 

Excess supply means market supply of a commodity is more than market demand for a commodity at the given price.

Equilibrium price is the price which corresponds to the equality between market demand and market supply of a commodity. 

Equilibrium quantity is the quantity which corresponds to the equilibrium price in the market.

It refers to a situation in which a particular commodity is sold at a price higher than the price fixed by the government (control price). 

Patent rights is the official recognition of the originators of a new product or technology. No one else can use their technology without obtaining a license. A cartel is a formal collusive agreement among rival firms in the market under oligopoly. Firms collude to avoid competition.

Market refers to the mechanism of sale and purchase of goods and services. Market equilibrium is a situation of zero excess demand and zero excess supply. Or, it is a situation where: market demand = market supply.

The quantity sold at the equilibrium price.

When the supply is perfectly inelastic during very short period.

Price which is equal to average cost is known as break-even price. 

Market price is the price that exists in the market at a particular point of time. 

Normal price is the price that prevails in the long period. 

Profits are said to be normal when: TR = TC or AR = AC. 

Profits are said to be extra-normal or abnormal when: TR > TC or AR > AC. 

Extra-normal or abnormal losses occur when: TR < TC or AR < AC.

Homogeneous product refers to a product of which all units are identical in all respects. Product differentiation is a situation when different producers in the market try to differentiate their product (with respect to size, weight, packaging, etc.) with a view to attracting the buyers and exercising partial control over price.

The price fixed by the government in case of essential commodities.

True

As a single seller he can fix whatever price he wishes to fix for his product. But he can sell more only by lowering the price of his product.

Yes. Because of product differentiation, each firm can decide its price policy independently. So that each firm has a partial control over price of its product.

A) 1 – B, 2 – A, 3 – D, 4 – C