100525 + Interview Questions in GENERAL QA IN GENERAL AWARENESS

1.	In a river, a row boat is travelling down stram at a speed of 8 m/s with respected to the shore. A metor boat comes from the oposite direction at a speed of 10 m/s with respect to water. 10 seconds after crossing each other they are at 160 metres separtion. Find the speed of row boat in still water in m/s.
Answer» Correct Answer - 6 `8 xx 10 + (10 - V_(r )) 10 = 160` `V_(r) = 2 m//s` `8 = V_(b//r) + V_(r )` `V_(br) = 8 - V_(r ) 8 - 2 = 6 m//s`

Discussion

2.	A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was cut away.
Answer» Let the two parts be x and (12 – x) Given that x = \(\sqrt[3]{12 -x} \) Cubing on both side, x³ = 12 – x x³ + x – 12 = 0

2.

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was cut away.

Answer»

Let the two parts be x and (12 – x)

Given that x = \(\sqrt[3]{12 -x} \)

Cubing on both side, x³ = 12 – x

x³ + x – 12 = 0

Discussion

3.	If X is a continuous random variable then P(X > a) = (a) P (X < a) (b) 1 – P (X > a) (c) P (X > a) (d) 1 – P (x ≥ a)
Answer» (c) P (X > a)

Discussion

4.	ABCD is a square. F is the midpoint of AB. BE is one -third of BC. If the area of ∆ FBE is 108 sq.cm, find the length of AC
Answer» We know that BF=AB/2 and BE=BC/3 Also given area of triangle FBE=108 area of triangle FBE=FB×BE/2 108=AB×BC/2×3×2 (where FB=AB/2 and BE=BC/3) AB×BC=1296 We know in square all sides are equal Therefore AB²=1296 so AB=36 Now AC is a diagonal of the square so it's formula = side of square ×√2 Side of square is 36 Therefore AC=36 √2 Hope answer is what you want and if wrong please correct me

4.

ABCD is a square. F is the midpoint of AB. BE is one -third of BC. If the area of ∆ FBE is 108 sq.cm, find the length of AC

Answer»

We know that

BF=AB/2 and BE=BC/3

Also given area of triangle FBE=108

area of triangle FBE=FB×BE/2

108=AB×BC/2×3×2 (where FB=AB/2 and BE=BC/3)

AB×BC=1296

We know in square all sides are equal

Therefore AB²=1296 so AB=36

Now AC is a diagonal of the square so it's formula = side of square ×√2

Side of square is 36

Therefore AC=36 √2

Hope answer is what you want and if wrong please correct me

Discussion

5.	Evaluate \( \int e^{x} \sin x d x \).
Answer» I = ∫ e^x sin x dx By using integration by parts I = e^x (−cos x) − ∫ e^x (−cos x) dx = −e^x cos x + ∫ e^x (cos x) dx Again evaluate 2± Integral by parts, we get I = −e^xcos x + e^x sin x − ∫ e^x sin x dx I = −e^x cos x + e^x sin x − I 2I = e^x (sin x − cos x) + c I = e^x/2 (sin x − cos x) + c

5.

Evaluate \( \int e^{x} \sin x d x \).

Answer»

I = ∫ e^x sin x dx

By using integration by parts

I = e^x (−cos x) − ∫ e^x (−cos x) dx

= −e^x cos x + ∫ e^x (cos x) dx

Again evaluate 2± Integral by parts, we get

I = −e^xcos x + e^x sin x − ∫ e^x sin x dx

I = −e^x cos x + e^x sin x − I

2I = e^x (sin x − cos x) + c

I = e^x/2 (sin x − cos x) + c

Discussion

6.	tan(1/2 cos inverse root5/3
Answer» We have tan [(cos^-1(√5)/3)/2] = tan [40.8/2] (cos^(√5)/3 is approxly 40.8 degree) =tan [20.9] =0.381862....

6.

tan(1/2 cos inverse root5/3

Answer»

We have

tan [(cos^-1(√5)/3)/2]

= tan [40.8/2] (cos^(√5)/3 is approxly 40.8 degree)

=tan [20.9]

=0.381862....

Discussion

7.	A uniform rod of mass m and length l is hinged at its mid point in such a way that it can rotate in the vertical plane about a horizontal axis passing through the hinge.One of its ends is attached to a string of spring constant k which is unstreched when the rod is horizontal. If this end is now given a small displacement and released then angular frequency of the resulting motion is
Answer» The correct option will be (a) sqrt(k/m) Hint:- use the spring formula to get the answer

7.

A uniform rod of mass m and length l is hinged at its mid point in such a way that it can rotate in the vertical plane about a horizontal axis passing through the hinge.One of its ends is attached to a string of spring constant k which is unstreched when the rod is horizontal. If this end is now given a small displacement and released then angular frequency of the resulting motion is

Answer»

The correct option will be (a) sqrt(k/m)

Hint:- use the spring formula to get the answer

Discussion

8.	‘A’ can complete a piece of work in 12 days. ‘A’ and ‘B’ together can complete the same piece of work in 8 days. In how many days can ‘B’ alone complete the same piece of work?(a) 15 days (b) 18 days (c) 24 days (d) 28 days (e) None of these
Answer» (c) Number of days = 12x8/12-8 = 24 days

8.

‘A’ can complete a piece of work in 12 days. ‘A’ and ‘B’ together can complete the same piece of work in 8 days. In how many days can ‘B’ alone complete the same piece of work?(a) 15 days (b) 18 days (c) 24 days (d) 28 days (e) None of these

Answer»

(c) Number of days = 12x8/12-8

= 24 days

Discussion

9.	It takes 4 hrs for pipe A to empty a 100 liter tank; if another pipe B which is double the size of pipe A is employed, how long will it take to empty the tank? Sol. Since the Pipe is double the size,
Answer» Since the Pipe is double the size, it will take1/2 time of the time taken by the smaller pipe Therefore 1/2x4 =2 hrs

Discussion

10.	http://www.sarthaks.com/?qa=blob&qa_blobid=7938989254442030424 plz answer the question in image as soon as possibleTwo identically charged pith balls are suspended from the same point by two massless identical threads density of each ball is ρ. If system is immersed in a medium of density σ, balls remain undeflected, then the dielectric constant of medium is
Answer» If the balls are suspended in a medium of density σ and the distance between the balls remains same, then the dielectric constant of the medium is given by: So The dielectric constant of the medium(K) = ρ/(ρ-σ) So the option 1 is correct

10.

http://www.sarthaks.com/?qa=blob&qa_blobid=7938989254442030424 plz answer the question in image as soon as possibleTwo identically charged pith balls are suspended from the same point by two massless identical threads density of each ball is ρ. If system is immersed in a medium of density σ, balls remain undeflected, then the dielectric constant of medium is

Answer»

If the balls are suspended in a medium of density σ and the distance between the balls remains same, then the dielectric constant of the medium is given by:

So The dielectric constant of the medium(K) = ρ/(ρ-σ)

So the option 1 is correct

Discussion

11.	Milk is an animal produce. How would you categories a person who consumes milk and milk products such as cheese and yoghurt - vegetarian or non - vegetarian?
Answer» Milk is vegetarian, and you can still drink milk on a vegetarian diet and call yourself a vegetarian. So, in short, yes, milk is vegetarian.

Discussion

12.	why is food present in the roots of some plants, although it is made in the leaves?
Answer» Leaves are all about photosynthesis, creating food molecules and absorbing carbon dioxide for the plant. Root systems also provide support for plants in the form of an anchor in the soil. In these plants, roots act like a storage of food. They support the plant and transport food. So, food is also present in roots of some plants not only in leaves.

12.

why is food present in the roots of some plants, although it is made in the leaves?

Answer»

Leaves are all about photosynthesis, creating food molecules and absorbing carbon dioxide for the plant.

Root systems also provide support for plants in the form of an anchor in the soil.

In these plants, roots act like a storage of food. They support the plant and transport food.

So, food is also present in roots of some plants not only in leaves.

Discussion

13.	why are potatoes considered stems and not roots?
Answer» We find several nodes and buds (eyes) on the body of a potato which are the characteristics of a stem. Thus potato is a modified stem.

Discussion

14.	http://www.sarthaks.com/?qa=blob&qa_blobid=3488500208727684347For a given incident ray as shown in figure, to under go total internal reflection at the vertical face, the refractive index of slab will be
Answer» We know the two conditions for total internal reflection : 1. The ray of light must travel from a denser medium into a rarer medium. 2. The angle of incidence in the denser medium must be greater than the critical angle for that pair of media. Therefore, the refractive index of slab will be √(3/2)

14.

http://www.sarthaks.com/?qa=blob&qa_blobid=3488500208727684347For a given incident ray as shown in figure, to under go total internal reflection at the vertical face, the refractive index of slab will be

Answer»

We know the two conditions for total internal reflection :

1. The ray of light must travel from a denser medium into a rarer medium.

2. The angle of incidence in the denser medium must be greater than the critical angle for that pair of media.

Therefore, the refractive index of slab will be √(3/2)

Discussion

15.	A body is rotating in steady rate. What is torque acting on the body?
Answer» Zero. Torque is required only for producing angular acceleration.

Discussion

16.	If the density of material of a square plate and a circular plate shown in figure is same, the center of mass of the composite system will be(a) inside the square plate(b) inside the circular plate(c) at the point of contact(d) outside the system
Answer» Correct answer is (a) inside the square plate

Discussion

17.	A flywheel is revolving with constant angular velocity. A chip of its rim breaks and flies away. What will be the effect on the angular velocity?
Answer» The reduction in mass will decrease moments of inertia. Hence angular velocity will be increased in order to conserve angular momentum.

Discussion

18.	Ram is twice as good as Shyam in work; Shyam will do a piece of work in 30 days; in how many days Ram will do the work?
Answer» Since Ram is twice as good, he will do the work in 30/2 =15 days

Discussion

19.	Is radius of gyration of a body constant quantity?
Answer» No. It changes with change in position of the axis of rotation.

Discussion

20.	The dimension of angular momentum is (a) M°L1 T-1 (b) M1 L2 T2(c) M1 L2 T-1(d) M1 L2 T-2
Answer» Correct answer is (c) M¹ L²T^-1 Angular momentum = Moment of inertia x angular velocity (Angular momentum) = [M¹ L² ][T^-1] = [M¹ L² T^-1].

20.

The dimension of angular momentum is (a) M°L1 T-1 (b) M1 L2 T2(c) M1 L2 T-1(d) M1 L2 T-2

Answer»

Correct answer is (c) M¹ L²T^-1

Angular momentum = Moment of inertia x angular velocity (Angular momentum) = [M¹ L² ][T^-1] = [M¹ L² T^-1].

Discussion

21.	What is another name for angular momentum?
Answer» Moment of momentum is also named as angular momentum.

Discussion

22.	A platform diver holds his hands and legs straight and makes loops in air before entering into water.1. State the principle behind this.2. What happens when he tries to land in the pool by stretching his arms and legs?3. In the above situation, rotational kinetic energy is not conserved. Explain.
Answer» 1. Conservation of angular momentum Statement Conservation of angular momentum states that, if the total torque acting on a system is zero, its angular momentum will be conserved. 2. Angular velocity decreases. 3. Initial kinetic energy K.E₁ = \(\frac{L^2}{2I_1}\) ____(1) where I₁ is the moment of inertia of diver when he makes loops in air final kinetic energy K.E₂ = \(\frac{L^2}{2I_2}\) _____(2) where I₂ is the moment of inertia of diver when he stretches his hands But I₁ < I₂ Hence from eq(1) and eq(2), we get KE₁ > KE₂ which means that rotational kinetic is not conserved.

22.

A platform diver holds his hands and legs straight and makes loops in air before entering into water.1. State the principle behind this.2. What happens when he tries to land in the pool by stretching his arms and legs?3. In the above situation, rotational kinetic energy is not conserved. Explain.

Answer»

1. Conservation of angular momentum

Statement

Conservation of angular momentum states that, if the total torque acting on a system is zero, its angular momentum will be conserved.

2. Angular velocity decreases.

3. Initial kinetic energy K.E₁ = \(\frac{L^2}{2I_1}\) ____(1)

where I₁ is the moment of inertia of diver when he makes loops in air

final kinetic energy K.E₂ = \(\frac{L^2}{2I_2}\) _____(2)

where I₂ is the moment of inertia of diver when he stretches his hands

But I₁ < I₂

Hence from eq(1) and eq(2), we get

KE₁ > KE₂

which means that rotational kinetic is not conserved.

Discussion

23.	A girl standing on a turn table. What happens to the rotation speed, if she stretches her hand?
Answer» lf a girl rotating with a uniform speed on turn table, it’s angular momentum (L = Iω) will be a constant. When she suddenly stretches her hand, I Increases and ω decreases to keep L as constant.

Discussion

24.	Explain why friction is necessary to make the disc in roll in the direction indicated.Give the direction of the frictional force at B, and the sense of frictional torque, before perfect rolling begins.What is the force of friction after perfect rolling begins?
Answer» Friction is necessary for rolling. 1. Frictional force at B opposes velocity of B. So, the frictional force is in the same direction as the arrow. The sense of frictional torque is such as to oppose angular motion, \(\vec {ω_0}\) and \(\vec {τ_0}\) are both normal to the paper, the first into the paper and the second coming out of the paper. 2. Frictional force decreases the velocity of the point of contact B. Perfect rolling ensures when this velocity is zero. Once this is so, the force of friction is zero.

24.

Explain why friction is necessary to make the disc in roll in the direction indicated.Give the direction of the frictional force at B, and the sense of frictional torque, before perfect rolling begins.What is the force of friction after perfect rolling begins?

Answer»

Friction is necessary for rolling.

1. Frictional force at B opposes velocity of B. So, the frictional force is in the same direction as the arrow. The sense of frictional torque is such as to oppose angular motion, \(\vec {ω_0}\) and \(\vec {τ_0}\) are both normal to the paper, the first into the paper and the second coming out of the paper.

2. Frictional force decreases the velocity of the point of contact B. Perfect rolling ensures when this velocity is zero. Once this is so, the force of friction is zero.

Discussion

25.	An economy is in equilibrium. Calculate the Investment Expenditure from the following : National Income=800 Marginal Propensity to Save=0.3 Autonomous Consumption=100
Answer» Given Y=800 MPS(s)=0.3 i.e. MPC (c )=1-MPS=1-0.3=0.7 C=100 We know that at equilibrium, Y=C+I C=ab+by =100+0.7y By putting the value of Y & C 800=100+0.7(800)+I 800=100+560+I I=800-660 I=Rs.140 Thus, the Investment expenditure is Rs.140.

Discussion

26.	A girl has to lean towards right when carrying a bag in her left hand. Why?
Answer» When a girl carries her bag in her left hand, the center of gravity of system will shift towards left. In order to bring it in the middle, the girl has to lean towards right.

Discussion

27.	If the earth loses the atmosphere what will happen to the length of the day?
Answer» For earth, the angular momentum (L = Iω) is a constant, because there is no torque acting on it. When earth loses the atmosphere, I decreases and ω increases to keep L as constant. Hence length of the day decreases.

Discussion

28.

Moments of inertia of some bodies with axis are given in the table below. Fill in the blanksBodyAxisMoment of inertiathe circular ringPerpendicular to plane, through center...........Circular disc..............1/4 MR2Hollow cylinder............MR2solid sphereDiameter.........

Answer»

Body	Axis	Moment of inertia
the circular ring	Perpendicular to plane, through center	MR²
Circular disc	Diameter	1/4 MR²
Hollow cylinder	Axis of cylinder	MR²
solid sphere	Diameter	\(\frac{2}{5}\)MR²

Discussion

29.	Match the following:a) Moment of forceΤ ∆ θ(b) F∆rLinear motion(c) CoupleTorque(d) 1/2 Iω2Τ ∆ rRotational motion1/2 MR2L2/2I
Answer» (a) Torque (b) τ ∆ θ (c) Rotational motion (d) \(\frac{1}{2} \frac{L^2}{1}.\)

29.

Match the following:a) Moment of forceΤ ∆ θ(b) F∆rLinear motion(c) CoupleTorque(d) 1/2 Iω2Τ ∆ rRotational motion1/2 MR2L2/2I

Answer»

(a) Torque

(b) τ ∆ θ

(c) Rotational motion

(d) \(\frac{1}{2} \frac{L^2}{1}.\)

Discussion

30.	If the polar ice cap melts what will happen to the length of the day?
Answer» For earth, angular momentum is a constant (Lω = constant, ie no torque acts on the earth). When the polar ice cap melts, the water thus formed will flow down to the equatorial region. The accumulation of water in equatorial line will increase the moment of inertia I of earth. In order to keep the angular momentum as a constant, ω will decrease. The decrease in ‘ω’will increase the length of day.

30.

If the polar ice cap melts what will happen to the length of the day?

Answer»

For earth, angular momentum is a constant (Lω = constant, ie no torque acts on the earth). When the polar ice cap melts, the water thus formed will flow down to the equatorial region.

The accumulation of water in equatorial line will increase the moment of inertia I of earth. In order to keep the angular momentum as a constant, ω will decrease. The decrease in ‘ω’will increase the length of day.

Discussion

31.	A cat is able to function its feet after a fall, taking the advantage of principle of conservation of angular momentum.1. Give the law of conservation of a angular momentum.2. Explain how cat is able to do so
Answer» 1. When there is no external torque, the total angular momentum of a body or a system of bodies are a constant. τ = \(\frac{dL}{dt}\) (when τ = 0 , we get \(\frac{dL}{dt}\) = 0). ie L = constant. But L = Iω ∴ Iω = a constant. 2. When a cat falls, it stretches its body. So that the moment of inertia becomes large. As Iω = constant, the value of angular speed will be decreased due to the increased value of moment of inertia. So cat lands on its feet gently.

31.

A cat is able to function its feet after a fall, taking the advantage of principle of conservation of angular momentum.1. Give the law of conservation of a angular momentum.2. Explain how cat is able to do so

Answer»

1. When there is no external torque, the total angular momentum of a body or a system of bodies are a constant.

τ = \(\frac{dL}{dt}\) (when τ = 0 , we get \(\frac{dL}{dt}\) = 0).

ie L = constant.

But L = Iω

∴ Iω = a constant.

2. When a cat falls, it stretches its body. So that the moment of inertia becomes large. As Iω = constant, the value of angular speed will be decreased due to the increased value of moment of inertia. So cat lands on its feet gently.

Discussion

32.	If y = 3 cos (logex) + 4 sin (logex) show that x2y2 + xy1 + y = 0.
Answer» y = 3 cos (log_ex) + 4sin (log_ex) Diff wrt x y₁ = 3(-sin(logx).1/x) + 4 cos (log x). 1/2 xy₁ = -3 sin (logx) + 4 cos(logx) diff wrt x xy₂ y₁. 1 = -3cos(logx).1/x + 4(-sin(logx)).1/x xy₂ + y₁ = 1/2-(3cos(logx) + 4sin(logx)) x²y₂ + xy₁ = 1/2-(3cos(logx) + 4sin(logx)) x²y₂ + xy₁ = -y ∴ x²y₂ + xyj₁ + y = 0

32.

If y = 3 cos (logex) + 4 sin (logex) show that x2y2 + xy1 + y = 0.

Answer»

y = 3 cos (log_ex) + 4sin (log_ex)

Diff wrt x

y₁ = 3(-sin(logx).1/x) + 4 cos (log x). 1/2

xy₁ = -3 sin (logx) + 4 cos(logx)

diff wrt x

xy₂ y₁. 1 = -3cos(logx).1/x + 4(-sin(logx)).1/x

xy₂ + y₁ = 1/2-(3cos(logx) + 4sin(logx))

x²y₂ + xy₁ = 1/2-(3cos(logx) + 4sin(logx))

x²y₂ + xy₁ = -y

∴ x²y₂ + xyj₁ + y = 0

Discussion

33.	\( \frac{x}{2}-\frac{y}{3}=\frac{1}{4} \)
Answer» y=3/2x-3/4. eq-1 y=mx+c. eq-2 by equating this two equations we get m=3/2

Discussion

34.	6th maths data handling 9.1 question 2
Answer» SFGRDF8D yfebhlhtid6fhv7tsy yrig6rv fugugy8 ft 6dy f9ugdyfudf YR6R6RR7TPF9YW5 5e5shdy7tst7td4e57dyuGstde5Rsttdt8dt77TDY8CISHL dydyicv7ydt8c8y

Discussion

35.	Rewrite the following sentences as illustrated through examples I and II :Ex I. You have been compelled to do it.II. You have had to do it.(a) The old lady has been compelled to sell her house.(b) Boys have been compelled to break the window.(c) He has been compelled to dismiss his Secretary.(d) I have been compelled to find a new car.(e) I have been compelled to do the work.
Answer» (a) The old lady has had to sell her house. (b) Boys have had to break the window. (c) He has had to dismiss his Secretary. (d) I have had to find a new car. (e) I have had to do the work.

35.

Rewrite the following sentences as illustrated through examples I and II :Ex I. You have been compelled to do it.II. You have had to do it.(a) The old lady has been compelled to sell her house.(b) Boys have been compelled to break the window.(c) He has been compelled to dismiss his Secretary.(d) I have been compelled to find a new car.(e) I have been compelled to do the work.

Answer»

(a) The old lady has had to sell her house.

(b) Boys have had to break the window.

(c) He has had to dismiss his Secretary.

(d) I have had to find a new car.

(e) I have had to do the work.

Discussion

36.	Fill in the blanks with appropriate prepositions given in brackets :(over, for, after, of, on, above, in)(a) My servant looks ............... my house. (b) I feel pity ............... the poor.(c) The dog was run ............... by the bus. (d) I called ............... my friend yesterday. (e) She is well aware ............... the fact.(f) A soldier should value honour ............... life.(g) He deals ............... rice.
Answer» (a) after (b) for (c) over (d) on (e) of (f) above (g) in

36.

Fill in the blanks with appropriate prepositions given in brackets :(over, for, after, of, on, above, in)(a) My servant looks ............... my house. (b) I feel pity ............... the poor.(c) The dog was run ............... by the bus. (d) I called ............... my friend yesterday. (e) She is well aware ............... the fact.(f) A soldier should value honour ............... life.(g) He deals ............... rice.

Answer»

(a) after

(b) for

(c) over

(d) on

(e) of

(f) above

(g) in

Discussion

37.	Pick out the example of simile / metaphor : (a) The stars looked like stupid little fish.(b) Time is money.(c) He is strong as an ox.
Answer» (a) Simile (b) Metaphor (c) Simile.

37.

Pick out the example of simile / metaphor : (a) The stars looked like stupid little fish.(b) Time is money.(c) He is strong as an ox.

Answer»

(a) Simile

(b) Metaphor

(c) Simile.

Discussion

38.	Please explain how the formula of (Si2O5-2)n generate for sheet silicate and formula (SiO3-2)n for single chain silicate?
Answer» Silicate minerals containing chains are termed inosilicates. In single chains \((SiO_3^{-2})_n\) the silicon to oxygen ratio is 1:3, while in double chains \((Si_4O_{11}^{-6})_n\) , the silicon to oxygen ratio is 4:11. The formula of silicate sheet is \((Si_2O_5^{-2})_n\) .

Discussion

39.	Why do some people not remember to take medicine on time?
Answer» We don't wish to be dependent on medicine. We want to lead our life without medicine. The bitterness of the medicine makes antipathy to take medicine on time. So some people do not remember to take medicine in time.

Discussion

40.	Who wrote the poem 'Everyone Sans'? (a) Siegfried Sass(b) W. B. Yeats(c) T.S. Eliot(d) William Wordsworth
Answer» (a) Siegfried Sass

Discussion

41.	7. Show that: \(\cfrac{2\,cos^2A+1}{2\,cos^2A-1}\) = tan(60° + A) tan(60°- A)8. Show that: cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1
Answer» (7) R.H.S = tan(60° + A) tan(60°-A) = \(\cfrac{tan\,60°+tan\,A}{1-tan\,60°\,tan\,A}\) x \(\cfrac{tan\,60°-tan\,A}{1+tan\,60°\,tan\,A}\) = \(\cfrac{\sqrt3+tan\,A}{1-\sqrt3\,tan\,A}\) x \(\cfrac{\sqrt3-tan\,A}{1+\sqrt3\,tan\,A}\) = \(\cfrac{(\sqrt3)^2+tan^2\,A}{1-(\sqrt3\,tan\,A)^2}\) (\(\because\) (a + b)(a - b) = a² - b²) = \(\cfrac{3-tan^2A}{1-3\,tan^2A}\) = \(\cfrac{\cfrac{3-sin^2A}{cos^2A}}{\cfrac{1-3\,sin^2A}{cos^2A}}\) = \(\cfrac{3\,cos^2A-sin^2A}{cos^2A-3\,sin^2A}\) = \(\cfrac{2\,cos^2A+1-sin^2A-sin^2A}{cos^2A-2sin^2A-(1-cos^2A)}\) = \(\cfrac{2(cos^2A-sin^2A)+1}{2(cos^2A-sin^2A)-1}\) = \(\cfrac{2\,cos^2A+1}{2\,cos^2A-1}\) = L.H.S Hence proved (8) cot x cot 2x - cot 2x cot 3x - cot 3x cot x = cot x cot 2x - cot 3x (cot 2x + cot x) = cot x cot 2x - cot (2x + x) (cot 2x + cot x) = cot x cot 2x - \(\left(\cfrac{cot \,2x\,cot \,x-1}{cot\,x+cot\,2x}\right)\)(cot x + cot 2x) (\(\because\) cot (A - B)) = \(\cfrac{cot\,A\,cotB-1}{cot\,B+cot\,A}\) = cot x cot 2x - (cot x cot 2x - 1) = cot x cot 2x - cot x cot 2x + 1 = 1 Hence proved

41.

7. Show that: \(\cfrac{2\,cos^2A+1}{2\,cos^2A-1}\) = tan(60° + A) tan(60°- A)8. Show that: cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1

Answer»

(7) R.H.S = tan(60° + A) tan(60°-A)

= \(\cfrac{tan\,60°+tan\,A}{1-tan\,60°\,tan\,A}\) x \(\cfrac{tan\,60°-tan\,A}{1+tan\,60°\,tan\,A}\)

= \(\cfrac{\sqrt3+tan\,A}{1-\sqrt3\,tan\,A}\) x \(\cfrac{\sqrt3-tan\,A}{1+\sqrt3\,tan\,A}\)

= \(\cfrac{(\sqrt3)^2+tan^2\,A}{1-(\sqrt3\,tan\,A)^2}\) (\(\because\) (a + b)(a - b) = a² - b²)

= \(\cfrac{3-tan^2A}{1-3\,tan^2A}\)

= \(\cfrac{\cfrac{3-sin^2A}{cos^2A}}{\cfrac{1-3\,sin^2A}{cos^2A}}\)

= \(\cfrac{3\,cos^2A-sin^2A}{cos^2A-3\,sin^2A}\)

= \(\cfrac{2\,cos^2A+1-sin^2A-sin^2A}{cos^2A-2sin^2A-(1-cos^2A)}\)

= \(\cfrac{2(cos^2A-sin^2A)+1}{2(cos^2A-sin^2A)-1}\)

= \(\cfrac{2\,cos^2A+1}{2\,cos^2A-1}\)

= L.H.S

Hence proved

(8) cot x cot 2x - cot 2x cot 3x - cot 3x cot x

= cot x cot 2x - cot 3x (cot 2x + cot x)

= cot x cot 2x - cot (2x + x) (cot 2x + cot x)

= cot x cot 2x - \(\left(\cfrac{cot \,2x\,cot \,x-1}{cot\,x+cot\,2x}\right)\)(cot x + cot 2x)

(\(\because\) cot (A - B))

= \(\cfrac{cot\,A\,cotB-1}{cot\,B+cot\,A}\)

= cot x cot 2x - (cot x cot 2x - 1)

= cot x cot 2x - cot x cot 2x + 1

= 1

Hence proved

Discussion

42.	Can you please give me explanation of this physics question.
Answer» Please send complete ques.using options given in the menu bar.

Discussion

43.	Why was letter writing done more carefully in the past?
Answer» In the past, only means of communication of man was letter writing. Modern facilities were not available in the past. When man began to write letter, he tried to reflect his taste, temper and ideas in the letter. So, he used to write letter very meticulously and carefully.

Discussion

44.	Why was letter-writing done more meticulously in the past?
Answer» The letter-writers of the past poured the feelings of their heart in their letters : Letters were the imbodiments of their imagination, emotion, personal taste and temper. They spent needed time and energy in writing letters. Modern facilities were not avaialable to them. So, they wrote letter meticlously and beautifully.

Discussion

45.	Letter writing was done more carefully in the......(a) Present(b) Pre-historic period(c) past(d) future
Answer» Letter writing was done more carefully in the past.

Discussion

46.	Not for one moment ____ my friend’s innocence. A) I doubted B) I did doubt C) did I doubt D) I doubt
Answer» Correct option is C) did I doubt

Discussion

47.	He failed the examination three times but ________ he passed. A) at the end B) at finish C) at last D) at least
Answer» Correct option is C) at last

Discussion

48.	Origin is the centroid of a triangle ABC. If A =(4, –3) and B = (–5, 2), then find the coordinates of C.
Answer» Suppose C = (x,y) Then (0,0) = (4 - 5 + x/3, -3 + 2 + y) ⇒ 4 - 5 + x/3 = 0 and -3 + 2 + y/3 = 0 ⇒ x = 1, y = 1 So the coordinates of C are (1,1).

48.

Origin is the centroid of a triangle ABC. If A =(4, –3) and B = (–5, 2), then find the coordinates of C.

Answer»

Suppose C = (x,y) Then

(0,0) = (4 - 5 + x/3, -3 + 2 + y)

⇒ 4 - 5 + x/3 = 0 and -3 + 2 + y/3 = 0

⇒ x = 1, y = 1

So the coordinates of C are (1,1).

Discussion

49.	A line segment (bar)AB is of length 10 units and A = (2, 3). If the abscissa of B is 10, then there will be two values for the coordinate of B whose sum is equal to (A) 3 (B) –3(C) 6(D) –6
Answer» Correct option (d) -6 Explanation : Suppose B = (10, y). Then AB = 10 ⇒ (10 – 2)² + (y + 3)² 10² ⇒ (y + 3)² + 36 ⇒ y = 3 = ±6 ⇒ y = 3 or -9 Therefore, the sum of the values of y = 3 - 9 = -6.

49.

A line segment (bar)AB is of length 10 units and A = (2, 3). If the abscissa of B is 10, then there will be two values for the coordinate of B whose sum is equal to (A) 3 (B) –3(C) 6(D) –6

Answer»

Correct option (d) -6

Explanation :

Suppose B = (10, y). Then

AB = 10 ⇒ (10 – 2)² + (y + 3)² 10²

⇒ (y + 3)² + 36

⇒ y = 3 = ±6

⇒ y = 3 or -9

Therefore, the sum of the values of y = 3 - 9 = -6.

Discussion

50.

Factorise: 4(a+b) - 6(a+b)2.

Answer»

\(4(a + b) -6(a + b)^2\)

\(= 2(a + b) (2 - 3(a + b))\)

\(= 2(a+ b) (2 - 3a -3b)\)

Discussion

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