NCERT Exemplar

### Find the equation of a line which is equidistant from the lines x = – 2 and x = 6.

Answer : For the equation of line equidistant from both lines, we will find point through which line passes and is equidistant from both line. As any point lying on x = - 2 line is ( - 2, 0) and on...

### Find the equation of a vertical line passing through the point ( – 5, 6).

Answer : Equation of line parallel to y - axis (vertical) is given by x = constant, as x - coordinate is constant for every point lying on line i.e. 6. So, the required equation of line is given as...

### Find the equation of a horizontal line passing through the point (4, – 2).

Answer : Equation of line parallel to x - axis (horizontal) is y = constant, as y - coordinate of every point on the line parallel to x - axis is - 2 i.e. constant. Therefore equation of the line...

### Find the equation of a line parallel to the x – axis and having intercept – 3 on the y – axis.

Answer: Equation of line parallel to x - axis is given by y = constant, as x - coordinate of every point on the line parallel to y - axis is - 3 i.e. constant. So, the required equation of line is y...

### Find the equation of a line parallel to the x – axis at a distance of (i) 4 units above it (ii) 5 units below it

Answer : (i) Equation of line parallel to x - axis is given by y = constant, as the y - coordinate of every point on the line parallel to x - axis is 4,i.e. constant. Now the point lies above x -...

### A ray of light incident at an angle θ on a refracting face of a prism emerges from the other face normally. If the angle of the prism is 5o and the prism is made of a material of refractive index 1.5, the angle of incidence isa) 7.5ob) 5oc) 15od) 2.5o

Answer: a) 7.5o The distance between the refracting surfaces is negligible with thin prisms, thus the prism angle (A) is very small. Because A = r1 + r2, if A is tiny, both r1 and r2 will be little...

### The velocity-displacement graph of a particle is shown in the figure. a) Write the relation between v and x. b) Obtain the relation between acceleration and displacement and plot it.

a) Consider the point P(x,v) at any time t on the graph such that angle ABO is θ such that tan θ = AQ/QP = (v0-v)/x = v0/x0 When the velocity decreases from v0 to zero during the displacement, the...

### A man runs across the roof-top of a tall building and jumps horizontally with the hope of landing on the roof of the next building which is of a lower height than the first. If his speed is 9 m/s, the distance between the two buildings is 10 m and the height difference is 9 m, will he be able to land on the next building?

For a free fall at 9m, the horizontal distance covered by the man should be at least 10 m. u = 0 a = 10 m/s2 s = 9 m t = t s = ut + 1/2 at2 Substituting the values, we get t = √9/3 = 3/√5 sec The...

### A ball is dropped and its displacement vs time graph is as shown in the figure where displacement x is from the ground and all quantities are positive upwards. a) Plot qualitatively velocity vs time graph b) Plot qualitatively acceleration vs time graph

a) At t=0 and v=0 , v-t graph is: b) At x = 0, a-t graph is:

### A ball is dropped from a building of height 45 m. Simultaneously another ball is thrown up with a speed 40 m/s. Calculate the relative speed of the balls as a function of time.

V = v1 = ? U = 0 h = 45 m a = g t = t V = u + at v1 = 0 + gt v1 = gt Therefore, when the ball is thrown upward, v1 = -gt V = v2 u = 40 m/s a = g t = t V = u + at v2 = 40 – gt The relative velocity...

### It is a common observation that rain clouds can be at about a kilometre altitude above the ground. a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also, calculate in km/h b) A typical rain drop is about 4 mm diameter. Momentum is mass x speed in magnitude. Estimate its momentum when it hits ground. c) Estimate the time required to flatten the drop. d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you. e) Estimate the order of magnitude force on umbrella. Typical lateral separation between two rain drops is 5 cm.

a) Velocity attained by the rain drop which is falling freely through the height h is: v2 = u2 – 2g(-h) As u = 0 v = √2gh = 100√2 m/s = 510 km/h b) Diameter of the drop, d = 2r = 4 mm Radius of the...

### A motor car moving at a speed of 72 km/h cannot come to a stop in less than 3 s while for a truck this time interval is 5 s. On a highway the car is behind the truck both moving at 72 km/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it does not bump onto the truck. Human response time is 0.5 s.

For truck, u = 20 m/s v = 0 a = ? t = 5s v = u + at a = 4 m/s2 For car, t = 3 s u = 20 m/s v = 0 a = ac v = u + at ac = -20/3 m/s2 Let s be the distance between the car and the truck when the truck...

### A monkey climbs up a slippery pole for 3 seconds and subsequently slips for 3 seconds. Its velocity at time t is given by v(t) = 2t (3 – t); 0

a) For maximum velocity v(t) dv(t)/dt = 0 Substituting the value for v, we get t = 1.5 seconds b) For average velocity = total distance/time taken Average velocity = 3 m And the average velocity is...

### A man is standing on top of a building 100 m high. He throws two balls vertically, one at t = 0 and other after a time interval. The later ball is thrown at a velocity of half the first. The vertical gap between first and second ball is +15m at t = 2s. The gap is found to remain constant. Calculate the velocity with which the balls were thrown and the exact time interval between their throw.

Let the speed of ball 1 = u1 = 2u m/s Then the speed of ball 2 = u2 = u m/s The height covered by ball 1 before coming to rest = h1 The height covered by ball 2 before coming to rest = h2 We know...

### A bird is tossing between two cars moving towards each other on a straight road. One car has a speed of 18 m/h while the other has the speed of 27 km/h. The bird starts moving from first car towards the other and is moving with the speed of 36 km/h and when the two cars were separated by 36 km. What is the total distance covered by the bird? What is the total displacement of the bird?

The relative speed of the cars = 27 + 18 = 45 km/h When the two cars meet together, time t is given as t = distance between cars/relative speed of cars = 36/(27+18) t = 4/5 h Therefore, distance...

### A particle executes the motion described by x(t) = x0 (1 – e-γt) where t ≥ 0, x0 > 0 a) Where does the particles start and with what velocity? b) Find maximum and minimum values of x(t), v(t), a(t). Show that x(t) and a(t) increase with time and v(t) decreases with time.

a) x(t) = x0 (1 – e-γt) v(t) = dx(t)/dt = +x0 γ e-γt a(t) = dv/dt = x0 γ2 e-γt v(0) = x0 γ b) x(t) is minimum at t = 0 since t = 0 and [x(t)]min = 0 x(t) is maximum at t = ∞ since t = ∞ and...

### An object falling through a fluid is observed to have acceleration given by a = g – bv where g = gravitational acceleration and b is constant. After a long time of release, it is observed to fall with constant speed. What must be the value of constant speed?

The concept used in this question will be based on the behaviour of a spherical object when it is dropped through a viscous fluid. When a spherical body of radius r is dropped, it is first...

### Give example of a motion where x>0, v<0, a>0 at a particular instant.

Let the motion be represented as: x(t) = A + Be– γ t Let A>B and γ >0 Velocity is x(t) = dx/dt = -Be– γ t Acceleration is a(t) = dx/dt = B γ 2e– γ t Therefore, it can be said that x(t) > 0,...

### Give examples of a one-dimensional motion where a) the particle moving along positive x-direction comes to rest periodically and moves forward b) the particle moving along positive x-direction comes to rest periodically and moves backwardπ

When an equation has sine and cosine functions, the nature is periodic. a) When the particle is moving in positive x-direction, it is given as t > sin t When the displacement is as a function of...

### A uniformly moving cricket ball is turned back by hitting it with a bat for a very short time interval. Show the variation of its acceleration with taking acceleration in the backward direction as positive.

The force which is generated by the bat is known as impulsive force. When the effect of gravity is ignored, it can be said that the ball moves with a uniform speed horizontally and returns back to...

### Refer to the graphs below and match the following:

Graph Characteristics a) i) has v > 0 and a < 0 throughout b) ii) has x > 0 throughout and has a point with v = 0 and a point with a = 0 c) iii) has a point with zero displacement for t...

### A ball is bouncing elastically with a speed 1 m/s between walls of a railway compartment of size 10 m in a direction perpendicular to walls. The train is moving at a constant velocity of 10 m/s parallel to the direction of motion of the ball. As seen from the ground, a) the direction of motion of the ball changes every 10 seconds b) speed of ball changes every 10 seconds c) average speed of ball over any 20 seconds intervals is fixed d) the acceleration of ball is the same as from the train

The correct option is b) speed of ball changes every 10 seconds, c) average speed of ball over any 20 seconds intervals is fixed, and d) the acceleration of the ball is the same as from the train

### A spring with one end attached to a mass and the other to a rigid support is stretched and released. a) magnitude of acceleration, when just released is maximum b) magnitude of acceleration, when at equilibrium position is maximum c) speed is maximum when mass is at equilibrium position d) magnitude of displacement is always maximum whenever speed is minimum

The correct answer is a) magnitude of acceleration, when just released is maximum and c) speed is maximum when mass is at equilibrium position

### For the one-dimensional motion, describe by x = t – sint a) x(t)>0 for all t>0 b) v(t)>0 for all t>0 c) a(t)>0 for all t>0 d) v(t) lies between 0 and 2

The correct answer is a) x(t)>0 for all t>0 and d) v(t) lies between 0 and 2

### A graph of x versus t is shown in the figure. Choose correct alternatives from below. a) the particle was released from rest at t=0 b) at B, the acceleration a>0 c) at C, the velocity and the acceleration vanish d) average velocity for the motion A and D is positive e) the speed at D exceeds that at E

The correct answer is a) the particle was released from rest at t=0, c) at C, the velocity and the acceleration vanish and e) the speed at D exceeds that at E

### The variation of quantity A with quantity B, plotted in figure describes the motion of a particle in a straight line. a) quantity B may represent time b) quantity A is velocity if motion is uniform c) quantity A is displacement if motion is uniform d) quantity A is velocity if motion is uniformly accelerated

The correct answer is a) quantity B may represent time, c) quantity A is displacement if motion is uniform, and d) quantity A is velocity if motion is uniformly accelerated

### At a metro station, a girl walks up a stationary escalator in time t1. If she remains stationary on the escalator, then the escalator take her up in time t2. The time taken by her to walk up on the moving escalator will be a) (t1 + t2)/2 b) t1t2/(t2 – t1) c) t1t2/(t2 + t1) d) t1 – t2

The correct answer is c) t1t2/(t2 + t1)

### The displacement of a particle is given by x = (t-2)2 where x is in metres and t is seconds. The distance covered by the particle in first 4 seconds is a) 4 m b) 8 m c) 12 m d) 16 m

The correct answer is b) 8 m

### A vehicle travels half the distance L with speed V1 and the other half with speed V2, then its average speed is a) (V1+V2)/2 b) (2V1+V2)/(V1+V2) c) (2V1V2)/(V1+V2) d) L(V1+V2)/V1V2

The correct answer is c) (2V1V2)/(V1+V2)

### A lift is coming from 8th floor and is just about to reach 4th floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct? a) x<0, v<0, a>0 b) x>0, v<0, a<0 c) x>0, v<0, a>0 d) x>0, v>0, a<0

The correct answer is a) x<0, v<0, a<0 The value of x and v becomes negative as the lift is moving from the 8th floor to the 4th floor whereas acceleration is acting upwards and stays...

### What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?

Solution: It is given that a cube with 2 units edge, one of whose vertices coincides with the origin and the 3 edges passing through the origin, coincides with the positive direction of the axes...

### Choose the correct answer from the given four options indicated against each of the Exercises if the distance between the points and is , then the value of is (A) 5 (B) (C) 5 (D) none of these

Solution: Option(B) $\pm 5$ Explanation: Suppose $P$ be the point whose coordinate is $(a, 0,1)$ and $Q$ represents the point $(0,$, $(1,2) .$ It is given that, $\mathrm{PQ}=\sqrt{27}$ From the...

### Choose the correct answer from the given four options indicated against each of the Exercises distance of the point from the origin is (A) (B) 3 (C) 4 (D) 5

Solution: Option (A) $\sqrt{50}$ Explanation: Suppose $\mathrm{P}$ be the point whose coordinate is $(3,4,5)$ and $\mathrm{Q}$ represents the origin. From the distance formula it can be written as...

### Choose the correct answer from the given four options indicated against each of the Exercises what is the length of foot of perpendicular drawn from the point on -axis (A) (B) (C) 5 (D) none of these

Solution: Option(B) $\sqrt{34}$ Explanation: As it is known that $y$-axis lies on $x$ y plane and $y z$. Therefore, its distance from $x y$ and $y z$ plane is 0 . $\therefore$ By the basic...

### Let A and C be the vertices of a triangle. The internal bisector of the angle meets at the point . Find the coordinates of .

Solution: It is given $A(2,2,-3), B(5,6,9)$ and $C(2,7,9)$ are the vertices of a triangle. And it is also given that the internal bisector of the angle A meets BC at the point D....

### If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (– 2, b, – 5) and C (4, 7, c), find the values of a, b, c.

Solution: It is given that the triangle ABC having vertices $A(a, 1,3), B(-2, b,-5)$ and $C(4,7, c)$ and origin is the centroid. The coordinates of the centroid for a triangle is given by the...

### Find the coordinate of the points which trisect the line segment joining the points A and

Solution: It is given the line segment joining the points are A $(2,1,-3)$ and $B(5,-8,3)$ Now suppose $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ be the points which...

### Three vertices of a Parallelogram ABCD are A (1, 2, 3), B (– 1, – 2, – 1) and C (2, 3, 2). Find the fourth vertex D.

Solution: It is given that the three consecutive vertices of a parallelogram ABCD are A $(1,2,3), B(-1,$, $-2,-1)$ and $C(2,3,2)$ Suppose the fourth vertex be $D(x, y, z)$. By using midpoint...

### In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0 (b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...

### Find the angle between the planes whose vector equations are

Solution: It is given that The eq. of the given planes are $\vec{r}(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 \text { and } \vec{r}(3 \hat{i}-3 \hat{j}+5 \hat{k})=5$ If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$...

### Lifetimes of the molecules in the excited states are often measured by using pulsed radiation source of duration nearly in the nanosecond range. If the radiation source has the duration of and the number of photons emitted during the pulse source is , calculate the energy of the source.

Frequency of radiation $(\nu)$, $\nu=\frac{1}{2.0 \times 10^{-9} s}$ $\nu=5.0 \times 10^{8} s^{-1}$ Energy $(E)$ of source $=$ Nhv Where, $N$ is the no. photons emitted $\mathrm{h}$ is Planck's...

### Arrange the following type of radiations in increasing order of frequency: (a) radiation from microwave oven (b) amber light from traffic signal (c) radiation from FM radio (d) cosmic rays from outer space and (e) X-rays.

The following is the frequency order in ascending order: Radiation from FM radio < amber light < radiation from microwave oven < X- rays < cosmic rays The following is the increasing...

### Explain, giving reasons, which of the following sets of quantum numbers are not possible.a) b) c) d) e) f)

a) This is not possible. The number n cannot be zero. (b) Possible. (c) This is not possible. The value of l can't be the same as the value of n. (d) This is not possible. Because mt can't be 1 when...

### What is the lowest value of n that allows g orbitals to exist?

For g-orbitals, l = 4. The possible values of ‘l’ range from 0 to (n-1),. For any given value of ‘n’, Hence,   least value of n = 5, l = 4 (g orbital),

### How can the production of dihydrogen, obtained from ‘coal gasification’, be increased?

Solution: By the course of coal gasification, dihydrogen is created as $C_{(g)}+H_{2} O_{(g)} \rightarrow C O_{(g)}+H_{2(g)}$ [C-Coal] Response with carbon monoxide with steam within the sight of an...

### Write the negation of the following simple statements(i) All similar triangles are congruent.(ii) Area of a circle is same as the perimeter of the circle.

(i) "Not p" is the negation of the assertion p. The negation of p is represented by $\sim p$. The truth value of $\sim p$ is the opposite of the truth value of p. The negation of the statement is...

### Write the negation of the following simple statements(i) 2 is not a prime number.(ii) Every real number is an irrational number.

(i) "Not p" is the negation of the assertion p. The negation of p is represented by $\sim p$. The truth value of $\sim p$ is the opposite of the truth value of p. The negation of the statement is “2...

### Write the negation of the following simple statements(i) The number 17 is prime.(ii) 2 + 7 = 6.

(i) "Not p" is the negation of the statement p. The negation of p is represented by the "$\sim p$." The truth value of $\sim p$ is the opposite of the truth value of p. The negation of the statement...

### Find the component statements of the following compound statements.(i) √7 is a rational number or an irrational number.(ii) 0 is less than every positive integer and every negative integer.

(i) A compound statement is made up of two or more statements (Components). As a result, the components of the given statement 7are a rational or irrational number, respectively. p: √7is a rational...

### Find the component statements of the following compound statements.(i) Number 7 is prime and odd.(ii) Chennai is in India and is the capital of Tamil Nadu.

(i) A compound statement is made up of two or more statements (Components). As a result, the elements of the provided statement "Number 7 is prime and odd" are as follows: p: The number 7 is prime....

### Which of the following sentences are statements? Justify(i) Where is your bag?(ii) Every square is a rectangle.

(i) If a statement is true or false but not both, it is a declarative sentence. "Where is your bag?" is a question in this context. As a result, it is not a statement. (ii) Every square is a...

### Which of the following sentences are statements? Justify(i) A triangle has three sides.(ii) 0 is a complex number.

(i) A statement is a declarative sentence if it is either true or false but not both. Hence, it is a true statement (ii) If a statement is true or false but not both, it is a declarative sentence....

### Find the sixth term of the expansion if the binomial coefficient of the third term from the end is

Given function is $\left(y^{1 / 2}+x^{1 / 3}\right)^{n}$ Given the binomial coefficient of the third phrase from the beginning, $=45$ So, ${ }^{n} C_{n-2}=45$ The above expression can be re-written...

### Find the coefficient of in the expansion of

Given function is $\left(x-x^{2}\right)^{10}$ $T_{r+1}={ }^{10} C_{r} x^{10-r}\left(-x^{2}\right)^{r}=(-1)^{r 10} C_{r} x^{10-r} x^{2 r}=(-1)^{r 10} C_{r} x^{10+r}$ For the coefficient of $x^{15}$,...

### Find the middle term (terms) in the expansion of(i) (ii)

(i) Given function is $\left(\frac{x}{a}-\frac{a}{x}\right)^{10}$ Here, Index of $n$ is $10$ which is even number. So, there is one middle term which is $(10 / 2+1)^{\text {th }}$ term that is...

### Why does a child feel more pain when falls down on a hard cement floor, than when she falls on the soft muddy ground in the garden?

When a kid falls on a hard cement floor, the child feels more pain than when the child falls on soft muddy ground in the yard, since the time it takes the child to stop on the cemented ground is...

### A block placed on a rough horizontal surface is pulled by a horizontal force F. Let f be the force applied by the rough surface on the block. Plot a graph of f versus F.

F1 is the force exerted on the heavy box, which is equal to F1 and is resisted by the lesser frictional force f1. F = Fs, which is the maximum static frictional force, is required for the box to...

### Evaluate .

We are given, $\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^2}2x}}{{{{\sin }^2}4x}}$ To simplify we have to multiply and divide the numerator and denominator by $\frac{{4{x^2}}}{{16{x^2}}}$ to...

### Evaluate .

We solve the given limit by using L. Hospital’s rule which is, If $\mathop {\lim }\limits_{{\text{x}} \to {\text{a}}} \frac{{{\text{f}}({\text{x}})}}{{{\text{g}}({\text{x}})}} = \frac{0}{0}$. Then,...

### The earth has a radius of 6400 km. The inner core of the 1000 km radius is solid. Outside it, there is a region from 1000 km to a radius of 3500 km which is in a molten state. Then again from 3500 km to 6400 km the earth is solid. Only longitudinal (P) waves can travel inside a liquid. Assume that the P wave has a speed of 8 km/s in solid parts and of 5 km/s in liquid parts of the earth. An earthquake occurs at someplace close to the surface of the earth. Calculate the time after which it will be recorded in a seismometer at a diametrically opposite point on the earth if wave travels along diameter?

Answer: According to the question, r1 = 1000 km, r2 = 3500 km, r3 = 6400 km and d1 = 1000 km And we can calculate, d2 = 3500 – 1000 d2 = 2500 km d3 = 6400 – 3500 d3 = 2900 km Expression for the...

### Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops, and 4 loops, the frequencies are in the ratio 1:2:3:4.

Answer: When n = 1, f1 = v/2L This is known as the fundamental frequency. When n = 2, f2 = 2(v/2L) This is known as the first overtone. When n = 3, f3 = 3(v/2L) This is known as the second overtone....

### A tuning fork vibrating with a frequency of 512 Hz is kept close to the open end of a tube filled with water. The water level in the tube is gradually lowered. When the water level is 17 cm below the open end, the maximum intensity of sound is heard. If the room temperature is 20oC, calculate

c) if the water in the tube is replaced with mercury, will there be any difference in your observations? Answer: (c) Sound is reflected into the air column by water and mercury in the tube, forming...

### A tuning fork vibrating with a frequency of 512 Hz is kept close to the open end of a tube filled with water. The water level in the tube is gradually lowered. When the water level is 17 cm below the open end, the maximum intensity of sound is heard. If the room temperature is 20oC, calculate

a) speed of sound in air at room temperature b) speed of sound in air at 0oC Answer: According to the question, the frequency of the tuning fork is f = 512 Hz a)  When the first maxima are taken...

### The pattern of standing waves formed on a stretched string at two instants of time are shown in the figure. The velocity of two waves superimposing to form stationary waves is 360 m/s and their frequencies are 256 Hz.

a) calculate the time at which the second curve is plotted b) mark nodes and antinodes on the curve c) calculate the distance between A’ and C’ Answer: According to the quetion, the frequency of the...