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16: 28 and 24:42

Since 16:28 = 16/28 = 4/7

and 24:42 = 24/42 = 4/7

\(\therefore\) Ratio 16: 28 and 24:42 are equal, so they form a proportion.

⇒  Perimeter of triangles is 27cm.
⇒  Sides are given in ratio 3:2:4
⇒  We know, Perimeter of triangle = Sum of all sides.
⇒  Let sides of triangle are 3x,2x and 4x.
⇒  3x+2x+4x=27
⇒  9x=27
∴  x=3
⇒  1st side =3x=3×3=9cm
⇒  2nd side =2x=2×3=6cm
⇒  3rd side =4x=4×3=12cm

396 = 2 × 2 × 3 × 11 × 3

504 = 2 × 2 × 2 × 3 × 3 × 7

636 = 2 × 2 × 3 × 53

In these three factors 2 × 2 × 3 are high common factors so when we multiply this the answer is 12.

Cubic polynomial whose zeros are 13, -5 and 2 is

P(x) = (x - 13) (x - (-51) (x - 2)

or x³ - (sum of zeros)x² + (-65 + 26 - 10)x - 130

 = x³ - 10x² - 49x - 130

Correct answer is yellow

Correct answer is red

Correct answer is blue

The factors of 396 are: 1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396

The factors of 504 are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504

The factors of 636 are: 1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636

Then the greatest common factor is 12.

Therefore, the HCF of 396, 504, 636​ is 12.

Given that a, b, y are zeros of polynomial

p(x) = 6x³ - 3x² - 5x + 1

\(\therefore\) Product of zeros = -d/a = -1/6

Sum of product of two zeros = c/a = -5/6

Sum of zeros = -b/a = \(\frac{-(-3)}{6}\) = 3/6 = 1/2

\(\because\) 1/a + 1/b + 1/y = \(\frac{by+ay+ab}{aby}\) = \(\frac{sum\,of\,product\,of\,two\,zeros}{product\,of\,zeros}\)

\(=\cfrac{\frac{-5}{6}}{\frac{-1}6}\) = 5 

Correct option is (b) 660

Given y = pq {p,q are prime numbers ≥ 3}

∴y can be generated from

3x3,3x5,3x7,3×11,3x13,3×17,3x19,5x5,5x7,5x11,7x7

⇒ Total 11 possibilties

x can be {1,2,..., 60}

Number of relations = 60 x 11 = 660

Correct option is (B) 660 

Number of possible values of a = 60, for b = pq, 

If p = 3, q = 3, 5, 7, 11, 13, 17, 19 

If p = 5 q = 5, 7, 11 

If p = 7 q = 7 

Total cases = 60 x 11 = 660

Solution:
x = p²q³ and y = p³q
HCF(x, y) = p² x q
LCM(x, y) = p³ x q³
LCM(x, y) = (p x q²) x HCF(x, y)
=> LCM is a multiple of HCF.

Suppose a = b. 

Then we get one equation: 

a² = ac + 4. 

This reduces to a(a - c) = 4. 

Therefore a = 1, a - c = 4, 

a = -1, a - c = -4, a = 4, a - c = 1, a = -4, a - c = -1, a = 2, a - c = 2, a = -2, a - c = -2. 

Thus we get 

(a, b, c) = (1, 1,-3), (-1,-1, 3), (4, 4, 3), (-4, -4,-3), (2, 2, 0), (-2, -2, 0). 

If a ≠ b, 

subtracting the second relation from the first we get 

a² - b² = c(b - a). 

This gives a + b = -c. 

Substituting this in the first equation, we get 

a² = b(-a - b) + 4.

Thus a² + b² + ab = 4. 

Multiplication by 2 gives 

(a + b)² + a² + b² = 8. 

Thus (a, b) = (2, -2), (-2, 2), (2, 0), (-2, 0), (0, 2), (0, -2). 

We get respectively 

c = 0; 0, -2, 2, -2, 2. 

Thus we get the triples: 

(a, b, c) = (1, 1, -3), (-1,-1, 3), (4, 4, 3), (-4, -4, -3), (2, 2, 0),(-2,-2, 0), (2,-2, 0), (-2, 2, 0), (2, 0,-2), (-2, 0, 2), (0, 2,-2), (0,-2, 2).

We have P(0) = b. Since b is rational and b² = P(0)² is an integer, we conclude that b is an integer. Observe that 

P(1)² = (1 + a + b)² = a² + 2a(1 + b) + (1 + b)² ∈ Z 

P(2)² = (4 + 2a + b)² = 4a² + 4a(4 + b) + (4 + b)² ∈ Z 

Eliminating a², we see that 4a(b - 2) + 4(1 + b)² - (4 + b)² ∈ Z. Since b ≠ 2, it follows that a is rational. 

Hence the equation x² + 2x(1 + b) + (1 + b)² - (a² + 2a(1 + b) + (1 + b)²) = 0 is a quadratic equation with integer coeffcients and has rational solution a. It follows that a is an integer.

The correct answer is Australia.

• ICC men's T20 rankings
  • The ICC Men's T20 Team Rankings is an international Twenty 20 cricket rankings system of the International Cricket Council.
  • After every T20 match, the two teams involved receive points based on a mathematical formula.
  • The total of each team's points total is divided by the total number of matches to give a rating, and all teams are ranked on a table in order of rating.

1.If one is involved in manufacturing activity, one starts with buying raw materials and packing materials etc. These materials are then converted into the end product. During the time required to convert raw material to the end product, various expenses like wages, rent, salary, utility bill, insurance etc are to be paid. Once the end product is ready, it has to be sold and money received from the customer. 

2.Items like vegetables, fruits, flowers and fish are perishable in nature. This calls for special storage conditions and equipment – cold storage, freezers etc. These have financial implications. Similarly, some of the manufactured food or medicinal products have expiry dates – beyond which they are not fit for consumption. This imposes certain constraints on inventory management.

Steps involved in marketing plan are: 

(1) Business situation analysis: 

'Where we have been?' – is the question responded to as the first step in designing the marketing plan.

 I have completed my MBA from a reputed institute in India.

I have experience in this field for 10 years and I have been quite successful and completely understand the needs of the customers. 

(2) Identify the target market: 

For a new venture, it's very essential to define clearly the specific group of potential customers whose needs the enterprise aims to fulfill. The target market in this case are teenagers. 

(3) Conduct SWOT analysis: 

It is important for the entrepreneur to consider in the 'target market' his/her enterprise's: 

(a) Strengths (b) Weaknesses (c) Threats (d) Opportunities 

Strengths: I am already in the field of fastmoving consumer goods which gives me knowledge about various aspects 

Weakness: I might not be able to concoct the right combination to satisfy the needs of the customer. 

Threats: There is lot of competition already in the market 

Opportunities: There is market for chocolates in the country

Correct answer is: (a) 13/100

The smallest number by which 1/13 should be multiplied so that its decimal expansion terminates after two decimal points is 13/100 as

\(\cfrac{1}{13}\times \cfrac{13}{100}=\cfrac{1}{100}=0.01\)

D. Division of labour

B. India is a peace loving country

The state in which ‘Silent valley movement’ started is Kerala.

a = 5 + 2√6

b = 1/a = \(\frac{1}{5+2√6}\) = \(\frac{5-2\sqrt6}{25-24}\) = 5 - 2√6

∴ a - b = (5 + 2√6) - (5 - 2√6) = 4√6

It is a simple wooden or metallic machine that uses a wheel and rope to lift heavy loads. Nowadays, plastic pulleys are also available in the market to carry small loads. This can be rotated freely about an axis passing through its center. It can change the direction of a force which makes it much easier for people to lift anything. With this, you can pull down on one end to lift the 10 kgs and one-meter high object.

Types of Pulley:

• Fixed Pulley

When the block of the pulley is fixed on a high platform, it is known as fixed. An extensible string passes over the groove where its one end is attached to the body to be lifted while the other end is free.

• Movable Pulley

When the block of the pulley is not fixed but carries the load, it is known as Movable. An inextensible string is tied around the groove where its one end is fixed to fixed support while the other end is kept free to apply the effort. As the effort is applied, the block together with the load moves upward.

A mechanical device that changes the direction or magnitude of a force is known as a simple machine. In general terms, they are defined as simple mechanisms that make use of leverage or mechanical advantage to multiply force. Simple machines have few or no moving parts to modify motion and force. 

Given system of equations are 2x + 3y = 1 and 4x + 6y = 4. 

Comparing given system of equations with a₁x + b₁y = c₁ and a₂x + b₂y = c₂, we get 

a₁ = 2, b₁ = 3, c₁ = 1 and a₂ = 4, b₂ = 6, c₂ = 4.

Now, \(\frac{a_1}{a_2}= \frac{2}{4} = \frac{1}{2},\)

\(\frac{b_1}{b_2}\) = \(\frac{3}{6} = \frac{1}{2}\) & \(\frac{c_1}{c_2}\) = \(\frac{1}{4}\)

Hence, \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) = \(\frac{1}{2} \neq \frac{1}{4} = \frac{c_1}{c_2}\)

Hence, \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) \(\neq\) \(\frac{c_1}{c_2}\), which is condition of No Solution. 

Hence, given system of equations has no solution. 

Hence proved

For the equation ax²+ bx + c = 0, a ≠ 0, 

The discriminant D is given by D =b² – 4ac.

And we know that roots of given equation is given by x = \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-b\pm\sqrt{D}}{2a}.\)

(i) If D > 0 then roots are real and distinct. 

(ii) If D = 0 then roots are real and equal. 

(iii) If D < 0 then rots are distinct but not real (imaginary).

Around the world, agriculture is the leading cause of water degradation. In the United States, agricultural pollution is the top source of contamination in rivers and streams, the second-biggest source in wetlands, and the third main source in lakes.

There is no division of voltage among the appliances when connected in parallel. The potential difference across each appliance is equal to the supplied voltage. The total effective resistance of the circuit can be reduced by connecting electrical appliances in parallel.

Given that probability of winning a game is 0.7 = P(A) (Let) We know that for a game, there are only two possibility losing and winning.

Let the probability of losing the game is P(B). 

We know that, the sum of probabilities for all possible outcomes is equal to 1. 

∴ P(A) + P(B) = 1

⇒ P(B) = 1 – P(A) = 1– 0.7 = 0.3. (∵ P(A) = 0.7)

Hence, the probability of losing the game is 0.3.

This is a witty line more than anything alse, as it is literal. Trinculo is speaking about the fact that in order to avoid the horrific storm, he's forced to share his sleep or area with Caliban, who is for lack of any other description terrified.

The strangeness Trinculo speaks of can be seen in a few different ways, as their meaning isn't defined. It can mean a "strange bedfellow," which they were, or it can also mean "foreign," "unknown," or "odd."

Correct option is: (B) -cosθ

Correct option is: (B) 1 – cos2θ

Let P be (x,y). Then

PA : PB = 2 : 3 

⇒ 3PA = 2 PB

⇒ 9(PA)² =  4(PB)² 

⇒ 9[(x – 5)² + (y + 4)²] = 4 [(x - 7)² + (y - 6)²]

⇒  5x² + 5y² - 34x + 120y + 29 = 0

Hence, the equation of the locus is 5x² + 5y² 34x + 120y + 29 = 0. This equation represent circle

Let P = (x, y). Distance of P from the x-axis is |y|. Therefore

AP = 2|y| ⇔ (AP)² = 4y²

⇔ (x - 4)² + y² = 4y²

⇔ x² - 3y² - 8x + 16 = 0

Hence the equation of the locus is x² - 3y² - 8x + 16 = 0

We have 

OP = 2PA 

(OP)² = 4(PA)²

x² + y² = 4[(x - 1)² + (y - 2)²]

x² + y² = 4x² + 4y² - 8x - 16y + 20

3x² + 3y² - 8x - 16y + 20 = 0 

Hence, the equation of the locus is 3x² + 3y² - 8x + 16y + 20 = 0 

The location and flow of patients, staff and various equipment decides the OT complex to be divided into different zones. These zoning are based on the need of variable level of cleanliness and sterility. 

Conventionally, the OT complex has been divided into four zones: 

1. Protective Zone: It is the outermost entry/exit zone of the OT complex. It includes: 

(a) Changing rooms for OT staff (doctors, nursing staff and other support staff). 

(b) Patient’s waiting area and reception 

(c) Rooms for administrative Staff 

(d) Stores and records room. 

(e) Receiving area for various materials and equipment 

2. Clean zone: It connects protective zone to aseptic zone. It includes: 

(a) Pre-operating room 

(b) Recovery room

(c) Store room for sterile equipment and consumables.

3. Sterile/Aseptic zone: It includes operation rooms which are kept sterile. This zone includes: 

(a) Operating room/suite in particular 

(b) Scrubbing station/ room and gowning area/ room. 

(c) Pre-Anaesthesia room 

(d) Sterile Instruments trolley area. 

4. Disposal Zone: Areas in this zone include dirty utility and disposal corridor. Disposal areas from operating room and connecting corridors leads to Disposal zone. The connecting corridors are outside the aseptic zone.

Given

Number of students in section A = 16

Number of students in section B = 24

Calculation:

Let the number of students in section C is x

One student is from each student, so remaining one student is selected from the remaining students such that (16 + 24 + x – 3) = 37 + x

⇒¹⁶C₁  × ²⁴C₁  × ^xC₁  × ^{37 + X}C₁ = 437760

⇒ 16  × 24  × X  × (37 + X) × (36 + X)!/(36 + X)! = 437760

⇒ X(37 + X) = 1140

⇒ X² + 37X – 1140 = 0

using discriminant method:

D = b² -4ac

= (37)² – 4(1)(-1140)

= 5929

X = (-b + √D)/2a

= -(37 + 77 )/2

= 20

∴ The number of students is 20.

Correct answer is (B) R₁ + R₂ = R

Given

Amount is divided between Shbanam and Savita is Rs.49,000

Ratio of Shabnam and Savita = 2 : 3

Calculation

Shabnam's share = Rs. [49000 × (2/5)] ⇒Rs.19,600

Savita’s share = Rs. [49000 × (3/5)] ⇒Rs.29,400

Thrice the share of Shabnab = Rs.58,800

Twice the share of Savita = Rs.58,800

⇒Difference between thrice the share of Shabnab and twice the share of Savita = Rs.58,800 – Rs.58,800

= Rs.0

Study of cells

Given

Income and expenditure of A = Rs.1,50,000 and Rs.80,000

Income and expenditure of B = Rs.2,00,000 and Rs.40,000

Calculation:

Income of A = Expenditure + Saving

⇒ 1,50,000 =  80,000 + Saving

⇒ Saving of A = Rs.70,000

Income of Rajan = Rs. 1,00,000

Expenditure on food = 80,000

Expenditure on entertainment = Rs 8000

Saving = Rs. 12,000

Income of B = Expenditure + Savings

⇒ 2,00,000 = 40,000 + Savings

⇒ Saving of B = 1,60,000

According to the question:

⇒ Saving of A : Saving of B

⇒ 70,000 : 1,60,000

⇒ 7 : 16

Correct option is D) came round

Ascorbic acid

Correct answer is  pilots

Benefits of Drills: 

a) Drills help develop teamwork and help to develop self-confidence. 

b) Drills help to prepare crew for responding rapidly and effectively in an emergency situation. 

c) Drills can help prepare the crew to make decisions under pressure and Drills can help to identify how procedures might be improved. 

d) Drills help the crew to become familiar with the equipment and procedures and whether they are working properly.

The benefits of drill are as under: 

1. It helps a team to respond quickly and effectively to a situation 

2. It helps an individual and group to get familiar with standard procedure and use an equipment or weapon effectively 

3. It helps in developing self-confidence and perfection in performing duties

Correct answer is  mail carriers

Correct option is C) closing