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(a) Conduction is the process of flow of heat from one point to another through a substance without the actual transfer of particle.

Rate of flow of heat,

(b) \(\frac{dQ}{dT}=\frac{KA(T_1-T_2)}{d}\)  

where K is coefficient of conductivity. A is the area of cross-section of the face, \(\frac{T_1-T_2}{d}\) is temperature gradient.

Coefficient of thermal conductivity (K) is defined as the amount of heat flowing in a second between two opposite faces of a cube of side 1 m kept at a temperature difference of 1 K. S.I. unit of K is Js^-1 m^-1 K^-1 .

(i) When the balls hits the ground; their vertical velocities are equal. However the horizontal velocities will be different. Hence the resultant velocities of the two balls are different, so the balls would hit the ground with different velocities.

(ii) Both the balls would reach the ground at the same time because their initial vertical velocities, acceleration and distance covered are all equal.

Yes, a vector can have infinite number of components. But three-dimensional rectangular co-ordinates, it has only three components.

(i) If three vectors; a, b and c given zero resultant, then vector a + b  + c = 0 or a -(vector b + c) i.e vectors a ,b and c will give zero resultant, if vector a is equal to negative of vector (b + c). Because vector (b + c) lies in the plane b and c hence a will be negative of vector (b + c), vector a, b and c all lie in plane.

(ii) Form above discussion, it also follows that three vectors cannot give zero resultant, if they do not lie in a plane.

The quantity \(\frac{\theta_1-\theta_2}{x}\) or \(\frac{d\theta}{dx}\) represents the rate of fall of temperature w.r.t. distance.

The quantity \(\frac{d\theta}{dx}\) represents the rate of change of temperature w.r.t. distance and is called temperature gradient.

Q = −KA[ \(\frac{d\theta}{dx}\)] t

Q represent energy and its dimensions are : 

[Q] = [ML²T ^-2 ]

[dx] = [L]

[A] = [L² ]

[dθ] = [θ]

[t] = [T]

Dimension of K

[K] = \(\frac{[ML^2T^{-2}][L]}{[L^2][\theta][T]}\)

= [MLT^-3θ^-1]

Coefficient of thermal conductivity of a material is defined as the rate of flow of heat per unit area per unit temperature gradient when the heat flow is at right angles to the faces of a thin parallel-sided slab of material.

(i) Both the bombs will reach the ground simultaneously because the time of flight doesn’t depend upon the mass of projectile.

(ii) On throwing with different velocities, the bomb thrown with lesser velocity will reach the ground earlier.

\(T = \frac{2\,u\,sin\,\theta}{g}\)

T ∝ u

h₁/h₂ = tan² α, and R₁/R₂ = 1/1 = 1.

(a) In this case two vectors must be in the same direction i.e., vector A is the direction of B.

(b) In this case, vector B must be zero.

(c) A should be perpendicular to vector B.

S.I unit: = °C/m or K/m

Dimensions: [L^-1M⁰T⁰K¹]

(i) When the angle of projection, θ = 45^o + α , the horizontal range is

\(R=\frac{u^2\,sin\,2\,(45^0+\alpha)}{g}\)

= \(\frac{u^2sin(90^0+2\alpha)}{g}\)

= \(\frac{u^2\,cos\,2\alpha}{g}\)

(ii) When the angle of projection, θ = 45^o – α, the horizontal range is

Clearly,  R' = R

Applications of thermal conductivity :

1. 

• Thick walls are used in the construction of cold storage rooms.
• Brick being a bad conductor of heat is used to reduce the flow of heat from the surroundings to the rooms.
• Better heat insulation is obtained by using hollow bricks. 
• Air being a poorer conductor than a brick, it further avoids the conduction of heat from outside.

2. Street vendors keep ice blocks packed in saw dust to prevent them from melting rapidly.

3. The handle of a cooking utensil is made of a bad conductor of heat, such as ebonite, to protect our hand from the hot utensil. 

4. Two bedsheets used together to cover the body help retain body heat better than a single bedsheet of double the thickness. Trapped air being a bad conductor of heat, the layer of air between the two sheets reduces thermal conduction better than a sheet of double the thickness. Similarly, a blanket coupled with a bedsheet is a cheaper alternative to using two blankets.

(Hint : This problem can be approached in two different ways:

(i) Refer to the diagram: target T is at horizontal distance x = R + Δx and below point of projection y = – h.

(ii) From point P in the diagram: Projection at speed v_o at an angle θ below horizontal with height h and horizontal range Δx.)

No. The rectangular components of a vector A has values A cosθ and A sinθ. Since the values of cosθ and sinθ can never be greater than one, hence the value of any rectangular components of a vector can never be greater than the given vector.

u²/2g is the sum of the height reached by them.

At an angle of 45° to the horizontal.

Correct Option is (D) \(tan^2 \theta : 1\)

Maximum vertical height \(H = \frac{u^2 sin^2 \theta}{g}\)

\(H_1 = \frac{u^2 sin^2 \theta}{g}\)

\(H_2 = \frac{u^2 sin^2 (90 - \theta)}{g}\)            .... \(\because \theta = (90 - \theta)\)

\(\frac{H_1}{H_2} = \frac{Sin^2 \theta}{Cos^2 \theta}\)

\(\frac{H_1}{H_2} = \frac{tan^2 \theta}{1}\)

Correct option is: (D) tan² θ : 1

Given: u = 40 ms^-1, t = 2 s, y = 20.4 m,

a_y = -9.8 m/s²

To find: 

(i) Angle of projection (θ)

(ii) Horizontal range of projectile (R)

Formulae: (i) y = u_y t + \(\frac{1}{2}\) a_y t²

(ii) R = \(\frac{u^2\,sin\,2\theta}{g}\)

Calculation: Taking vertical upward motion of the projectile from point of projection up to the top of vertical pole we have

u_y = 40 sinθ,

From formula (i),

∴ 20.4 = 40 sinθ × 2 + \(\frac{1}{2}\) (-9.8) × 2²

∴ 20.4 = 80 sinθ – 19.6

or sinθ = \(\frac{(20.4+19.6)}{80}=\frac{1}{2}\)

or θ = 30°.

From formula (ii),

Horizontal range = \(\frac{40^2}{9.8}\) sin 2 × 30°

= 141.4 m

The angle of projection is 30°. The horizontal range of projection is 141.4 m

Both the balls will hit the ground together. It is because, the two balls fall through the same height and the initial velocity of both the balls along the vertical is zero.

Correct Option is (A) 9 m

Given,

V = kt

\(\because v = \frac{dx}{dt}\)

\( \frac{dx}{dt} = kt\)

\(\int\limits_0^x dx = \int\limits_ 0^3 kt \ dt\)

\(x = k \left[\frac{t}{2}\right]_0^3\)

\(x = k \left[\frac{9}{2}\right]\)

\(x = 2 \times \frac{9}{2}\).

x = 9 m

Correct option is: (A) 9 m

At the projection point.

The coordinates of first projectile as seen from the second projectile are

X = x₁ - x₂

= u₁cosθ₁ t - u₂cosθ₂ t

X = (u₁cosθ₁ - u₂cosθ₂)t

Y = y₁ - y₂

= u₁sinθ₁ t - 1/2 gt² - (u₂sinθ₂ t - 1/2 gt²)

Y = (u₁ sin θ₁ - u₂sin θ₂)t

∴ = a constant, say m

This equation is of the form Y = mX, which is the equation of a straight line. Thus the path of a projectile as seen from another projectile is a straight line.

Because the projectile fired horizontally constantly is being acted upon by acceleration due to gravity acting vertically downwards.

At highest point of projectile path, the angle between velocity and acceleration is 90°.

Correct Answer is: (b) 1/3 s

T = 2π/ω = 2 or ω = π.

x = acos ωt

or a/2 = acos ωt

or ωt = π/3

or t = π/3π = 1/3 s.

Correct Answer is: (b) F ∝ v² , (c) F ∝ A ,  (d) F ∝ ρ

Correct Answer is: (a, b)

F ∝ r if r < R, and F ∝ 1/r² if r > R.

∴ for r₁, r₂ < R, F₁/F₂ = r₁/r₂

for r₁, r₂ > R, F₁/F₂ = r₂²/r₁² .

Correct Answer is: (a, c, & d)

Use arguments of symmetry as the yz plane divides the object symmetrically.

Correct Answer is: (a) △T = 3/2 T △r/r

T² ∝ r³ or T² = cr³

∴ 2T △T = 3cr² △r

Dividing, 2T △T/T² = 3cr² △r/cr³.

Correct Answer is: (b, d)

Let A = area of cross-section of the tube, ρ = density of liquid.

If one surface is pushed down by x, the other surface moves up by x.

Net unbalanced force on the liquid column = 2xAρg. 

Mass of the liquid column = lAρ.

Let a = acceleration of liquid column.

∴ -2xAρg = (lAρ)a

or a = - (2g/l) x

Put ω² = 2g/l.

∴ a = - ω² x . This is SHM with T = 2π/ω = 2π √l/2g.

Correct Answer is: (c) I/2 

Correct Answer is: (d) The particle will remain stationary.

The gravitational intensity inside a spherical shell is zero.

Law of conservation of momentum is a general law which is applicable to all systems.

The pull of the cable is minimum (zero) when the lift is falling freely.

The pull of the cable is maximum when the lift is moving with same acceleration.

A. Rough endoplasmic reticulum

The Rough endoplasmic reticulum is involved in the synthesis of proteins and the Smooth endoplasmic reticulum is involved in the synthesis of lipids. All the other organs do not synthesize these molecules and have different functions.

Electrical appliances in a house are connected in parallel.

Hint: On connecting the electrical appliances in parallel, each appliance works independently without being affected whether the other appliance is switched on or off.

A fuse is thus a safety device which prevent damages to electric circuit and possible fire so when excess current flow the wire of the fuse melts breaks the circuit

The key or Switch can be placed any¬where in the electric circuit.

(ii) circuit is open.

(iii) no current flows through it.

(c) There is a minimum limit on the current which can safely flow in the electric circuits.

Nutrient, their sources, effects deficiency and pleaty (excursiveness) diseases caused by them:

A compact fluorescent light (CFL) is just like the standard florescent lights that are common in use. The CFL has a smaller tube that’s manufactured in a swirl shape so it fits the basic form of an incandescent bulb. The CFL work by exciting gases with electricity, due to this feature, in CFL little heat is generated, most of the electricity used goes directly into providing light. It’s less wasteful, so you end up spending less on your regular electric bill by making your home more energy efficient, and the bulbs last much longer. 

Advantages of CFLs: 

(a) High Efficiency: CFLs use 20% to 33% less electricity than incandescent bulbs. 

(b) Long Life: With no filament to burn out, florescent lights tend to last 8 to 15 times longer than incandescent bulbs. 

(c) Cost effective: the Up-front price of CFL is higher, but by using less electricity and helping you avoid changing bulbs, you’ll end up spending less in the long run. 

(d) Safety: CFLs are even safer than incandescent bulbs since lower temperature means the risk of home fire is less.

(c) all the bulbs A,B and C will glow at the same time

No, a geyser and a television set require different amount of current. Therefore the fuse used in these will be of different ratings.

1. Wrong 

2. Right 

3. Right 

4. Wrong 

5. Right

(a) Proteins

Fats are those organic compounds found in cells which are insoluble in water. These are esters of glycerol. Fats are formed of elements like carbon, hydrogen and oxygen. Fats work as reserved energy sources in our body. Our body gets maximum energy from fats. In our body, solid fats is known as lipid and liquid fat is known as oil. It is slippery and is a source of energy for the body.

Types of fats and sources: 

1. Fats obtained from plants: Mustard, ground nut, sesame seeds, coconut, cashew nut, walnut. 

2. Fats obtained from animals: milk, eggs, meat, butter etc.

Functions of fats: 

1. Fats provide energy to the body. 

2. Fats accumulate beneath the body and gives strength. Too much consumption of fats makes body clumsy. 

3. Fats protects the internal organs of the body from outer shocks and gives strength to the muscles.

Fiber rich food can be included in diet as: 

1. Include fiber rich food like maize, kidney beans, pulses etc. 

2. Fruits like apple, pear, guava etc, should be eaten with peels because its fiber content is quite high. 

3. Radish, cabbage, peas, cucumber etc., have high quantity of fibers. Therefore, these should be included in our diet. 

4. Salad, dalia, dry Suits, ground nuts are also good sources of fibers.

5. Use flour instead of fine flour (maida).

6. Consume brown rice or oats.