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Q4.6(c):
Establish the following vector inequalities geometrically or otherwise : (c) $|a-b| < |a| + |b|$. When does the equality sign above apply?
Solution :
Initial Theoretical Foundation
To establish the vector inequality $| \vec{a} - \vec{b} | \leq | \vec{a} | + | \vec{b} |$, we utilize the properties of vector addition and the Triangle Inequality theorem in vector space. Consider two vectors $\vec{a}$ and $\vec{b}$. The vector subtraction $\vec{a} - \vec{b}$ is defined as $\vec{a} + (-\vec{b})$.
Step 1: Geometric Representation
In a triangle formed by vectors $\vec{a}$ and $-\vec{b}$ placed head-to-tail, the resultant vector is $\vec{R} = \vec{a} + (-\vec{b}) = \vec{a} - \vec{b}$. By the Triangle Inequality Theorem, the length of any one side of a triangle is strictly less than the sum of the lengths of the other two sides, unless the vectors are collinear.
Step 2: Analytical Derivation via Magnitude Formula
The magnitude of the difference of two vectors is given by the law of cosines [Per the Parallelogram Law of Vector Addition]:
$| \vec{a} - \vec{b} | = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos\theta}$
Where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
We compare this to the scalar sum $(|\vec{a}| + |\vec{b}|)$:
$(|\vec{a}| + |\vec{b}|)^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|$
Since $-2|\vec{a}||\vec{b}|\cos\theta \leq 2|\vec{a}||\vec{b}|$ (because $\cos\theta \geq -1$), it follows that:
$|\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos\theta \leq |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|$
Taking the square root of both sides, we obtain $| \vec{a} - \vec{b} | \leq | \vec{a} | + | \vec{b} |$.
Step 3: Determination of Equality Condition
The equality $| \vec{a} - \vec{b} | = | \vec{a} | + | \vec{b} |$ holds when the triangle collapses into a straight line. This occurs when the vectors $\vec{a}$ and $-\vec{b}$ are parallel and in the same direction.
- The direction of $-\vec{b}$ is opposite to $\vec{b}$.
- For $\vec{a}$ and $-\vec{b}$ to be in the same direction, $\vec{a}$ must be in the opposite direction of $\vec{b}$.
- Thus, $\vec{a}$ and $\vec{b}$ must be anti-parallel (i.e., the angle between them $\theta = 180^{\circ}$).
| Condition | Resulting Inequality |
|---|---|
| General case ($\theta \neq 180^{\circ}$) | $| \vec{a} - \vec{b} | < | \vec{a} | + | \vec{b} |$ |
| Anti-parallel ($\theta = 180^{\circ}$) | $| \vec{a} - \vec{b} | = | \vec{a} | + | \vec{b} |$ |
Final Solution: The inequality $| \vec{a} - \vec{b} | \leq | \vec{a} | + | \vec{b} |$ is a direct consequence of the triangle inequality. The equality sign applies if and only if the vectors $\vec{a}$ and $\vec{b}$ are anti-parallel to each other.
More Questions from Class 11 Physics Motion in a Plane EXERCISES
- Q3.1: State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- Q3.10: On an open ground, a motorist follows a track that turns to his left by an angle of (60^{\circ}) after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
- Q3.11(a): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,
- Q3.11(b): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?
- Q3.12: The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of (40 m s^{-1}) can go without hitting the ceiling of the hall ?
- Q3.13: A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball ?
- Q3.14: A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone ?
- Q3.15: An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.
- Q3.16(a): Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
- Q3.16(b): Read each statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
- Q3.16(c): Read each statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
- Q3.17(a): The position of a particle is given by (r = 3.0t \hat{i} - 2.0t^{2} \hat{j} + 4.0 \hat{k} m) where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find the (v) and (a) of the particle?
- Q3.17(b): The position of a particle is given by (r = 3.0t \hat{i} - 2.0t^{2} \hat{j} + 4.0 \hat{k} m) where t is in seconds and the coefficients have the proper units for r to be in metres. (b) What is the magnitude and direction of velocity of the particle at (t = 2.0 s) ?
- Q3.18(a): A particle starts from the origin at (t = 0 s) with a velocity of (10.0 \hat{j} m/s) and moves in the x-y plane with a constant acceleration of ((8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}). (a) At what time is the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
- Q3.18(b): A particle starts from the origin at (t = 0 s) with a velocity of (10.0 \hat{j} m/s) and moves in the x-y plane with a constant acceleration of ((8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}). (b) What is the speed of the particle at the time ?
- Q3.19: (\hat{i}) and (\hat{j}) are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors (\hat{i} + \hat{j}), and (\hat{i} - \hat{j})? What are the components of a vector (A= 2 \hat{i} + 3 \hat{j}) along the directions of (\hat{i} + \hat{j}) and (\hat{i} - \hat{j})? [You may use graphical method]
- Q3.2: Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
- Q3.20(a): For any arbitrary motion in space, which of the following relations are true : (a) (v_{average} = (\frac{1}{2}) (v(t_{1}) + v(t_{2})))
- Q3.20(b): For any arbitrary motion in space, which of the following relations are true : (b) (v_{average} = \frac{[r(t_{2}) - r(t_{1})]}{t_{2} – t_{1}})
- Q3.20(c): For any arbitrary motion in space, which of the following relations are true : (c) (v(t) = v(0) + a t)
- Q3.20(d): For any arbitrary motion in space, which of the following relations are true : (d) (r(t) = r(0) + v(0)t + (\frac{1}{2}) a t^{2})
- Q3.20(e): For any arbitrary motion in space, which of the following relations are true : (e) (a_{average} = \frac{[v(t_{2}) - v(t_{1})]}{t_{2} – t_{1}}) (The ‘average’ stands for average of the quantity over the time interval (t_{1}) to (t_{2}))
- Q3.21(a): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process
- Q3.21(b): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
- Q3.21(c): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
- Q3.21(d): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
- Q3.21(e): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
- Q3.22: An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s a part is (30^{\circ}), what is the speed of the aircraft ?
- Q3.3: Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
- Q3.4(a): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,
- Q3.4(b): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions ,
- Q3.4(c): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar,
- Q3.4(d): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,
- Q3.4(e): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,
- Q3.4(f): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
- Q3.5(a): Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,
- Q3.5(b): Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,
- Q3.5(c): Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
- Q3.5(d): Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
- Q3.5(e): Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
- Q3.6(a): Establish the following vector inequalities geometrically or otherwise : (a) (|a+b| < |a| + |b|) When does the equality sign above apply?
- Q3.6(b): Establish the following vector inequalities geometrically or otherwise : (b) (|a+b| > ||a| −|b||) When does the equality sign above apply?
- Q3.6(c): Establish the following vector inequalities geometrically or otherwise : (c) (|a−b| < |a| + |b|) When does the equality sign above apply?
- Q3.6(d): Establish the following vector inequalities geometrically or otherwise : (d) (|a−b| > ||a| − |b||) When does the equality sign above apply?
- Q3.7(a): Given (a + b + c + d = 0), which of the following statements are correct : (a) (a), (b), (c), and (d) must each be a null vector,
- Q3.7(b): Given (a + b + c + d = 0), which of the following statements are correct : (b) The magnitude of ((a + c)) equals the magnitude of ((b + d)),
- Q3.7(c): Given (a + b + c + d = 0), which of the following statements are correct : (c) The magnitude of (a) can never be greater than the sum of the magnitudes of (b), (c), and (d),
- Q3.7(d): Given (a + b + c + d = 0), which of the following statements are correct : (d) (b + c) must lie in the plane of (a) and (d) if (a) and (d) are not collinear, and in the line of (a) and (d), if they are collinear ?
- Q3.8: Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 3.19. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
- Q3.9(a): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (a) net displacement,
- Q3.9(b): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (b) average velocity,
- Q3.9(c): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?
- Q4.1: State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
- Q4.10: On an open ground, a motorist follows a track that turns to his left by an angle of $60^{\circ}$ after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
- Q4.11(a): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi,
- Q4.11(b): A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (b) the magnitude of average velocity ? Are the two equal ?
- Q4.12: Rain is falling vertically with a speed of $30 m s^{-1}$. A woman rides a bicycle with a speed of $10 m s^{-1}$ in the north to south direction. What is the direction in which she should hold her umbrella ?
- Q4.13: A man can swim with a speed of $4.0 km/h$ in still water. How long does he take to cross a river $1.0 km$ wide if the river flows steadily at $3.0 km/h$ and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank ?
- Q4.14: In a harbour, wind is blowing at the speed of $72 km/h$ and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of $51 km/h$ to the north, what is the direction of the flag on the mast of the boat ?
- Q4.15: The ceiling of a long hall is $25 m$ high. What is the maximum horizontal distance that a ball thrown with a speed of $40 m s^{-1}$ can go without hitting the ceiling of the hall ?
- Q4.16: A cricketer can throw a ball to a maximum horizontal distance of $100 m$. How much high above the ground can the cricketer throw the same ball ?
- Q4.17: A stone tied to the end of a string $80 cm$ long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in $25 s$, what is the magnitude and direction of acceleration of the stone ?
- Q4.18: An aircraft executes a horizontal loop of radius $1.00 km$ with a steady speed of $900 km/h$. Compare its centripetal acceleration with the acceleration due to gravity.
- Q4.19(a): Read each statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre
- Q4.19(b): Read each statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point
- Q4.19(c): Read each statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector
- Q4.2: Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
- Q4.20(a): The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (a) Find the $\boldsymbol{v}$ and $\boldsymbol{a}$ of the particle?
- Q4.20(b): The position of a particle is given by $\boldsymbol{r} = 3.0t \hat{i} - 2.0t^2 \hat{j} + 4.0 \hat{k} m$ where $t$ is in seconds and the coefficients have the proper units for $\boldsymbol{r}$ to be in metres. (b) What is the magnitude and direction of velocity of the particle at $t = 2.0 s$ ?
- Q4.21(a): A particle starts from the origin at $t = 0 s$ with a velocity of $10.0 \hat{j} m/s$ and moves in the x-y plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (a) At what time is the x- coordinate of the particle $16 m$? What is the y-coordinate of the particle at that time?
- Q4.21(b): A particle starts from the origin at $t = 0 s$ with a velocity of $10.0 \hat{j} m/s$ and moves in the x-y plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) m s^{-2}$. (b) What is the speed of the particle at the time ?
- Q4.22: $\hat{i}$ and $\hat{j}$ are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors $\hat{i} + \hat{j}$ , and $\hat{i} - \hat{j}$ ? What are the components of a vector $A= 2 \hat{i} + 3 \hat{j}$ along the directions of $\hat{i} + \hat{j}$ and $\hat{i} - \hat{j}$ ? [You may use graphical method]
- Q4.23(a): For any arbitrary motion in space, which of the following relations are true : (a) $v_{average} = (1/2) (v (t_1) + v (t_2))$
- Q4.23(b): For any arbitrary motion in space, which of the following relations are true : (b) $v_{average} = [r(t_2) - r(t_1) ] /(t_2 - t_1)$
- Q4.23(c): For any arbitrary motion in space, which of the following relations are true : (c) $v (t) = v (0) + a t$
- Q4.23(d): For any arbitrary motion in space, which of the following relations are true : (d) $r (t) = r (0) + v (0) t + (1/2) a t^2$
- Q4.23(e): For any arbitrary motion in space, which of the following relations are true : (e) $a_{average} =[ v (t_2) - v (t_1 )] /( t_2 - t_1)$ (The ‘average’ stands for average of the quantity over the time interval $t_1$ to $t_2$)
- Q4.24(a): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process
- Q4.24(b): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
- Q4.24(c): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
- Q4.24(d): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
- Q4.24(e): Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
- Q4.25: An aircraft is flying at a height of $3400 m$ above the ground. If the angle subtended at a ground observation point by the aircraft positions $10.0 s$ apart is $30^{\circ}$, what is the speed of the aircraft ?
- Q4.3: Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
- Q4.4(a): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars,
- Q4.4(b): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions ,
- Q4.4(c): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar,
- Q4.4(d): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars,
- Q4.4(e): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors,
- Q4.4(f): State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
- Q4.5(a): Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar,
- Q4.5(b): Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar,
- Q4.5(c): Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
- Q4.5(d): Read each statement below carefully and state with reasons, if it is true or false : (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
- Q4.5(e): Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
- Q4.6(a): Establish the following vector inequalities geometrically or otherwise : (a) $|a+b| < |a| + |b|$. When does the equality sign above apply?
- Q4.6(b): Establish the following vector inequalities geometrically or otherwise : (b) $|a+b| > ||a| -|b||$. When does the equality sign above apply?
- Q4.6(d): Establish the following vector inequalities geometrically or otherwise : (d) $|a-b| > ||a| - |b||$. When does the equality sign above apply?
- Q4.7(a): Given $a + b + c + d = 0$, which of the following statements are correct : (a) $a$, $b$, $c$, and $d$ must each be a null vector,
- Q4.7(b): Given $a + b + c + d = 0$, which of the following statements are correct : (b) The magnitude of $(a + c)$ equals the magnitude of $( b + d)$,
- Q4.7(c): Given $a + b + c + d = 0$, which of the following statements are correct : (c) The magnitude of $a$ can never be greater than the sum of the magnitudes of $b$, $c$, and $d$,
- Q4.7(d): Given $a + b + c + d = 0$, which of the following statements are correct : (d) $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of $a$ and $d$, if they are collinear ?
- Q4.8: Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 4.20. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
- Q4.9(a): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement,
- Q4.9(b): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (b) average velocity,
- Q4.9(c): A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 4.21. If the round trip takes 10 min, what is the (c) average speed of the cyclist ?
CBSE Solutions for Class 11 Physics Motion in a Plane
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I am an M.Sc. in Physics and B.Ed. with over 22 years of teaching experience. I have extensive experience teaching CBSE/NCERT Physics, particularly Classes 11 and 12, with a strong focus on conceptual clarity, numerical problem-solving and examination preparation. I have taught at the secondary and senior-secondary levels and have also worked in the corporate sector. My professional certifications include Microsoft Certified Educator, DOEACC PGDCA, and certification in “A to Z of Online Teaching” from upEducators. I am also CBSE-certified in Design Thinking. I secured 3rd rank in the CENTA Teaching Professionals’ Olympiad in 2018, reflecting my strong subject knowledge and teaching expertise. My teaching approach combines clear explanations, real-life applications, structured practice, regular assessments and personalised feedback to help students develop confidence and perform consistently well in Physics
Her teaching is nice and understand the subject with easy way for children. I will refer her to some of my circle friend for their children
Top Tutor
Pandeypur, Varanasi
11 yrs of
Exp
7 students
2131 hrs of classes completed
600
per hour
Online Classes
I am Mathematics teacher, passionate to teach Mathematics of different states of India. I am teaching class 11th since 2011 only in a reputed CBSE BOARD school. Several students who are taught by me are studying in different IITs.
Excellent Maths teacher, sir taught 9,10,11,12 cbse, My son improved a lot. Even gives test, Top board tutor.
