XI-Physics Basic Mathematics including Vectors questions

 Understanding Basic Mathematics in XI Physics: A Guide to Vectors

Stepping into the world of XI Physics can be a thrilling adventure! But it often comes with its fair share of challenges, especially when it comes to mathematics. If you're navigating through these waters, one of the first things you'll encounter are vectors.

What Are Vectors?

Great question! Simply put, vectors are mathematical entities that have both magnitude and direction. Imagine you're a sailor and your ship's course is like a vector. It doesn't just matter how fast you're going (that's the magnitude!), but also where you're heading (that's the direction!).

Why Do Vectors Matter in Physics?

Vectors play a crucial role in Physics for several reasons. Here are a few key points where vectors show their significance:

• Helping to describe quantities like velocity, force, and displacement.
• Providing a clearer understanding of motion and direction.
• Enabling the solution of problems involving forces acting at angles.

Understanding vectors isn't just academically important—it's like having a toolkit for solving real-world physics problems!

Breaking Down Vectors: Understanding the Components

Now, let's dive a little deeper. To make sense of vectors, we break them down into components. Think of it this way, if a vector is a delicious cake, its components are the ingredients like flour, sugar, and butter. These components, typically split into x and y axes, help us understand the vector's overall behavior.

How to Deal with Vector Questions: A Step-by-Step Approach

Tackling vector problems doesn't have to feel like climbing a mountain. Here's a straightforward approach:

1. Identify the vector quantities in the problem.
2. Break them down into components if they're not already.
3. Use Pythagorean theorem or trigonometric ratios to solve for unknowns.
4. Verify your results to ensure they make sense. It's like double-checking your map while on that ship we mentioned earlier!

Suppose you're asked to find the resultant force when two forces act at right angles. By breaking down these forces into x and y components, adding them together, and then using the Pythagorean theorem, you'll have your answer in no time!

Common Mistakes and How to Avoid Them

Even the best sailors had to learn the ropes! Here are some common pitfalls and ways to steer clear of them:

• Forgetting direction: Always consider both magnitude and direction of vectors.
• Ignoring signs: Pay attention to positive and negative values; they matter!
• Misplacing components: Double check your component breakdown to ensure accuracy.

Using Vectors in Real Life

Vectors aren't just confined to textbooks. Ever think about why airplane pilots need to understand vectors? Well, flying isn't just about picking a direction and going! Pilots must account for wind speed and direction, which means using vectors to ensure they reach their destination safely.

Tips for Mastering Vectors

Like any good skill, mastering vectors comes with practice. Here are my personal tips based on experience:

• Practice regularly: The more practice questions you tackle, the more comfortable you'll become.
• Work in groups: Discussing problems with friends can offer new perspectives.
• Visualize: Drawing diagrams to visualize forces and directions can simplify complex problems.

Conclusion

Diving into the world of vectors in XI Physics is like exploring uncharted territories. But with the right approach and understanding, you can navigate through with confidence and maybe even a little excitement! Remember, vectors are more than just numbers—they're keys to unlocking a deeper understanding of the world around us.

FAQs about Vectors in XI Physics

Question	Answer
What's the difference between vectors and scalars?	Scalars have only magnitude, while vectors have both magnitude and direction.
Why are vectors crucial in physics?	They help describe motion, forces, and other physical phenomena in a comprehensive manner.
What's an example of a vector in daily life?	Wind can be described as a vector since it has a speed (magnitude) and a direction.

 

Q.1     A body is rotating with angular velocity $\overrightarrow{\omega }=(3\widehat{i}-4\widehat{j}+\widehat{k})$. The linear velocity of a point having position vector $\overrightarrow{r}=(5\widehat{i}-6\widehat{j}+6\widehat{k})$ is $–$ $[\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}]$

(1) $6\widehat{i}+2\widehat{j}-3\widehat{k}$                        

(2) $18\widehat{i}+13\widehat{j}-2\widehat{k}$

(3) $-18\widehat{i}-13\widehat{j}+2\widehat{k}$                  

(4) $6\widehat{i}-2\widehat{j}+8\widehat{k}$

Q.2     If force $\overrightarrow{F}=5\widehat{i}+3\widehat{j}+4\widehat{k}$ makes a displacement of $\overrightarrow{s}=6\widehat{i}-5\widehat{k}$ work done by the force is $–$

(1) 10 units

(2) $122\sqrt{5}$ units

(3) $5\sqrt{122}$  

(4) 20 units

Q.3     The sum of two vectors $\overrightarrow{A}and\overrightarrow{B}$ is at right angles to their difference. Then $–$

(1) A = B

(2) A = 2B

(3) B = 2A

          

(4) $\overrightarrow{A}and\overrightarrow{B}$ have the same direction

Q.4     Two vectors are perpendicular, if $–$

(1) $\overrightarrow{A}.\overrightarrow{B}=1$ 

(2) $\overrightarrow{A}\times \overrightarrow{B}=1$

(3) $\overrightarrow{A}.\overrightarrow{B}=0$

(4) $\overrightarrow{A}\times \overrightarrow{B}=AB$

Q.5     What is the vector joining the points (3, 1, 14) and ($–$2, $–$1, $–$6) ?

(1) $2\widehat{i}+\widehat{j}+2\widehat{k}$                          

(2) $5\widehat{i}+2\widehat{j}+20\widehat{k}$

(3) $\widehat{i}+\widehat{j}+2\widehat{k}$                            

(4) $\widehat{i}+2\widehat{j}+2\widehat{k}$

Q.6     $\overrightarrow{A}$ is directed vertically downwards and $\overrightarrow{B}$ is directed along the north. What is the direction of $\overrightarrow{B}\times \overrightarrow{A}$ $–$

(1) east      

(2) west

(3) north    

(4) north west

Q.7     A body of 3 kg moves in the XY plane under the action of a force given by $6t\widehat{i}+4t\widehat{j}$. Assuming that the body is at rest at time t = 0, the velocity of the body at t = 3s is $–$

(1) $6\widehat{i}+6\widehat{j}$  

(2) $18\widehat{i}+6\widehat{j}$

(3) $9\widehat{i}+6\widehat{j}$  

(4) $12\widehat{i}+18\widehat{j}$

Q.8     If  $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0,$ then find the angle between $\overrightarrow{a}and\overrightarrow{b}$.

(1) π/3     

(2) π/4

(3) 2π/3   

(4) 2π/5

Q.9     A force $\overrightarrow{F}$ of magnitude 12N has non-rectangular components $\overrightarrow{P}$ and $\overrightarrow{Q}$. The sum of the magnitude of $\overrightarrow{P}$ and $\overrightarrow{Q}$ is 18N. The direction of $\overrightarrow{Q}$ is at right angles to . Find the magnitude of  $\overrightarrow{Q}$.

(1) 4N

(2) 5N

(3) 2N

(4) 7N

Q.10   If a vector $2\widehat{i}+3\widehat{j}+8\widehat{k}$ is perpendicular to the vector $4\widehat{j}-4\widehat{i}+\alpha \widehat{k}$, then the value of α is $–$

(1) 1/2

(2) $–$1/2

(3) 1   

(4) $–$ 1

Q.11   If the angle between the vectors $\overrightarrow{A}and\overrightarrow{B}$ is θ, the value of the product $(\overrightarrow{B}\times \overrightarrow{A}).\overrightarrow{A}$ is equal to $–$

(1) BA2 sin θ                        

(2) BA2 cos θ

(3) BA2 sin θ cos θ             

(4) zero

Q.12   A particle having simultaneous velocities 3m/s, 5 m/s and 7m/s is at rest. Find the angle between the first two velocities.

(1) 45°

(2) 30°

(3) 90°

(4) 60°

Q.13   A set of vectors taken in a given order gives a closed polygon. Then the resultant of these vectors is a $–$

(1) scalar quantity                  

(2) pseudo vector

(3) unit vector                        

(4) null vector

Q.14   If $0.3\widehat{i}+0.4\widehat{j}+c\widehat{k}$ is a unit vector, value of c = ?

(1) $\sqrt{0.75}$  

(2) $\sqrt{0.25}$

(3) $\sqrt{2}$      

(4) 1

Q.15   The forces, each numerically equal to 5N, are acting as  shown in the figure. Find the

           angle between forces ?  

 

 

 

 

 

     

(1) 60°

(2) 120°

(3) 30°

(4) None of these

Q.16   Find the angle between two vectors of magnitude 12 and 18 units when their resultant is 24 units.

(1)  cos θ = 1/4                     

(2)  cos θ = 1/2

(3)  cos θ = 1/ $\sqrt{2}$                 

(4)  cos θ = $\sqrt{3}/2$

Q.17   Two forces have magnitudes in the ratio 3 : 5 and the angle between their directions is 60°. If their resultant is 35N, find the sum of their magnitudes.

(1) 50 N    

(2) 60 N

(3) 30 N    

(4) 40N

Q.18   If vectors P, Q and R have magnitude 5, 12 and 13 units and $\overrightarrow{P}+\overrightarrow{Q}=\overrightarrow{R}$, find the angle between Q and R.

 

(1) $\theta ={\cos }^{-1}\left(\frac{12}{13}\right)$                  

(2)  $\theta ={\cos }^{-1}\left(\frac{5}{13}\right)$

(3) $\theta ={\cos }^{-1}\left(\frac{1}{13}\right)$                  

(4) None of these

Q.19   The resultant of two vectors of magnitudes 2A and $\sqrt{2}A$ acting at an angle θ is $\sqrt{10}A.$ Find the value of  θ.

(1) 90°

(2) 60°

(3) 45°

(4) 30°

Q.20   The resultant of two vectors P and Q acting at a point inclined to each other at an angle θ is R. If the magnitude of vector Q is doubled, R is also doubled. If the vector Q is reversed in direction R is again doubled. Find the ratio between P, Q and R.

(1) $\sqrt{1}:\sqrt{3}:\sqrt{1}$                       

(2) $\sqrt{2}:\sqrt{3}:\sqrt{2}$

(3) $\sqrt{2}:\sqrt{5}:\sqrt{2}$                     

(4) $\sqrt{1}:\sqrt{2}:\sqrt{3}$

Q.21   If the resultant of two forces of magnitudes P and Q acting at a point at an angle of 60° is $\sqrt{13}Q$, then find P/Q.

(1) 6   

(2) 1

(3) 3   

(4) 5

Q.22   The resultant of $\overrightarrow{P}$ and $\overrightarrow{Q}$

           is perpendicular to $\overrightarrow{P}$.

           What is the angle between       $\overrightarrow{P}$ and $\overrightarrow{Q}$. 

 

 

 

 

          

(1) $\theta ={\cos }^{-1}\left(\frac{-P}{Q}\right)$                 

(2) $\theta ={\cos }^{-1}\left(\frac{-P}{2Q}\right)$             

(3) $\theta ={\cos }^{-1}\left(\frac{Q}{2P}\right)$                 

(4) None of these

Q.23   A vector of modulus a is turned through θ. Find the change in the vector.

(1) a sin (θ/2)                        

(2) 2a sin (θ/2)

(3) 2a cos (θ/2)                     

(4) 3a sin (θ/2)

Q.24   Two forces of ${\overrightarrow{F}}_1=250\text{N}$ due east and ${\overrightarrow{F}}_2=250\text{N}$ due north have their common initial point. Find ${\overrightarrow{F}}_2-{\overrightarrow{F}}_1$

(1) $250\sqrt{2}\text{}\text{N}$                          

(2) $150\sqrt{2}\text{}\text{N}$

(3) $350\sqrt{2}\text{}\text{N}$                          

(4) None

Q.25   One of the rectangular components of a velocity of

           60 kmh$–$1 is 30 kmh$–$1. Find other rectangular component ?

(1) $10\sqrt{3}km{h}^{-1}$                      

(2) $25\sqrt{3}km{h}^{-1}$

(3) $2\sqrt{3}km{h}^{-1}$                        

(4) $30\sqrt{3}km{h}^{-1}$

Q.26   A woman walks 250m in the direction 30° east of north, then 175m directly east. Find the magnitude of the displacement.

(1) 170m   

(2) 235m

(3) 370m   

(4) 145m

Q.27   A force $\overrightarrow{F}=6\widehat{i}+x\widehat{j}$ acting on a particle displaces it from the point A (3, 4) to the point B (1, 1). If the work done is 3 units, then find value of x.

(1) 6   

(2) $–$1

(3) 3   

(4) $–$5

Q.28   Find the angle between the vectors $\overrightarrow{A}=\widehat{i}+\widehat{j}-2\widehat{k}$ and $\overrightarrow{B}=-\widehat{i}+2\widehat{j}-\widehat{k}$.

(1) 90°

(2) 60°

(3) 45°

(4) 30°

Q.29   A vector $\overrightarrow{a}$ of magnitude 10 units and another vector $\overrightarrow{b}$ of magnitude 6.0 units differ in directions by 60°. Find the magnitude of the vector product $\overrightarrow{a}\times \overrightarrow{b}$.

(1) 16 

(2) 12

(3) 32 

(4) 52

Q.30   Considering two vectors, $\overrightarrow{F}=(4\overrightarrow{i}-10\overrightarrow{j})\text{newton}$  and $\overrightarrow{r}=(-5\overrightarrow{i}-3\overrightarrow{j})\text{m}$ compute  $\overrightarrow{r}\times \overrightarrow{F}$.

(1) $62\widehat{k}\text{N-m}$                         

(2) $62\widehat{i}\text{N-m}$

(3) $52\overrightarrow{k}\text{N-m}$                         

(4) None of these

Q.31   Two particles A and B are moving in x-y plane. Their positions vary with time t according to relation xA(t) = 3t, xB(t) = 6 and yA(t) = t,

           yB(t) =2+3t2. The distance between these particles at t = 1 is:-

(1) 5   

(2) 3

(3) 4   

(4) 12

Q.32   For the given (y $–$ x) graph, find average value of y over an interval 0 ≤ x ≤ 3 :-

 

(1) 10 

(2) 50/3

(3) 20/3     

(4) 20

Q.33   If θ1 + θ2 = π/2 and θ1 = 2θ2, then the value of sin2θ1 + cos2θ2 is :-

(1) 1/2

(2) 1

(3) 3/2

(4) 2

Q.34   If the ratio of maximum and minimum magnitudes of the resultant of two vectors and is 3 : 1 then $\left|\overrightarrow{a}\right|$ is equal to :

(1) $\left|\overrightarrow{a}\right|$      

(2) $2\left|\overrightarrow{b}\right|$

(3) $3\left|\overrightarrow{b}\right|$    

(4) $4\left|\overrightarrow{b}\right|$

Q.35   At point P, the value of $\frac{dy}{dx}$ is :

 

 

 

 

 

 

 

(1) Zero     

(2) Positive

(3) Negative                           

(4) Infinite

Q.36   The angle between two vectors $\overrightarrow{R}=-\widehat{i}+\frac{\widehat{j}}{3}+\widehat{k}$

            and $\overrightarrow{S}=X\widehat{i}+3\widehat{j}+(X-1)\widehat{k}$

(1) Is obtuse angle                  

(2) Is acute angle

(3) Is right angle                     

(4) Depend on X

Q.37   The component of vector $2\widehat{i}-3\widehat{j}+2\widehat{k}$ perpendicular to $\widehat{i}+\widehat{j}+\widehat{k}$ is $–$

(1) $\frac{5}{3}(\widehat{i}-2\widehat{j}+\widehat{k})$                      

(2) $\frac{1}{3}(\widehat{i}+\widehat{j}-2\widehat{k})$

(3) $\frac{(7\widehat{i}-10\widehat{j}+7\widehat{k})}{3}$                   

(4) $\frac{(5\widehat{i}-8\widehat{j}+5\widehat{k})}{3}$

Q.38   The side of a square is increasing at rate of

           0.2cm/s. The rate of increase of perimeter w.r.t. time is :

(1) 0.2 cm/s                            

(2) 0.4 cm/s

(3) 0.6 cm/s                            

(4) 0.8 cm/s

Q.39   If  $\overrightarrow{A}=2\widehat{i}+p\widehat{j}+q\widehat{k},\overrightarrow{B}=5\widehat{i}+7\widehat{j}+3\widehat{k}$ are parallel then the value of p and q are $–$

(1) 14/5 and 6/5                      

(2) 14/3 and 6/5

(3) 6/5 and 1/3                        

(4) 3/4 and 1/4

Q.40   In the figure shown below the angle in between $\overrightarrow{A}and\overrightarrow{B}$ is : (C = B/2)

 

 

(1) 30°

(2) 60°

(3) 120°     

(4) 150°

Q.41   A physical quantity which has a direction :-

(1) must be a vector

          

(2) must be a scalar

(3) may be scalar or vector

          

(4) none of the above

Q.42   Sun rays cast 16m long shadow of a pole, when Sun is 37° above horizontal. When Sun rises to 53° above horizontal, length of shadow become:-

(1) 8 m      

(2) 16 m

(3) 9 m      

(4) 4 m

Q.43   The unit vector along $\widehat{i}+\widehat{j}$ is :-

(1) $\widehat{k}$ 

(2) $\widehat{i}+\widehat{j}$

(3) $\frac{\widehat{i}+\widehat{j}}{\sqrt{2}}$     

(4) $\frac{\widehat{i}+\widehat{j}}{2}$

Q.44   100 coplanar forces each equal to 10 N acting on a body. Each force makes angle π/50 with the preceding force, what is the resultant of the forces:

(1) 1000 N

(2) 500 N

(3) 250 N  

(4) zero

Q.45   Magnitude of resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$  is equal to magnitude of $\overrightarrow{P}$. Find the angle between $\overrightarrow{Q}$ and resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$.

(1) 30°

(2) 45°

 

(3) 60°

(4) 90°

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