Mathematical Tools and Vectors

Q1 FREE PREVIEW

PYQ

Assertion A: If and are four points on a semicircular arc with centre at ' ' such that , then Reason R: Polygon law of vector addition yields In the light of the above statements, choose the most appropriate answer from the options given below: [JEE Main 2021, 27 Jul (Shift 1)]

ais not correct but is correct.

bis correct but is not correct.

cBoth and are correct and is the correct explanation of .

dBoth and are correct but is not the correct explanation of .

✓ Correct answer: d) Both and are correct but is not the correct explanation of .

ExplanationPolygon law is applicable in both the cases given but the equation given in the reason is not useful in explaining the assertion.

Q2 FREE PREVIEW

PYQ

Statement I : If three forces , are represented by three sides of a triangle and , then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement II : A triangle made up of three forces and as its sides taken in the same order, satisfy the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below: [JEE Main 2021, 31 Aug (Shift 2)]

aStatement I is false but Statement II is true

bStatement I is true but Statement II is false

cBoth Statement I and Statement II are false

dBoth Statement I and Statement II are true.

✓ Correct answer: d) Both Statement I and Statement II are true.

ExplanationGiven statements are: Statement: I If three forces , are represented by three sides of a triangle and , then these three forces are concurrent forces and satisfy the condition for equilibrium.Statement: II A triangle made up of three forces and as its sides taken in the same order, satisfy the condition for translatory equilibrium.Solution: Here Since (equilibrium) Hence, both the statements are correct.

Q3

PYQ

When the position vector \(\vec{r}=\text{ }x\hat{i}\text{ }+\text{ }y\hat{j}+z\hat{k}\) changes sign as \(−\vec{r},\) which one of the following vector will not flip under sign change?

a

Acceleration

b

Velocity

c

Angular momentum

d

Linear momentum

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q4

PYQ

If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at angle \(\theta\), then:

[JEE Main 2021, 24 Feb (Shift 1)]

a

\(|\vec{A}-\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)\)

b

\(|\vec{A}+\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)\)

c

\(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\frac{\theta}{2}\right)\)

d

\(|\vec{A}+\vec{B}|=2 R \sin \left(\frac{\theta}{2}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q5

PYQ

If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at angle \(\theta\), then

a

\(|\vec{A}-\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)\)

b

\(|\vec{A}+\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)\)

c

\(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\frac{\theta}{2}\right)\)

d

\(|\vec{A}+\vec{B}|=2 R \sin \left(\frac{\theta}{2}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q6

PYQ

The angle between vector \(\vec{Q}\) and the resultant of \((2\vec{Q}+2\vec{P})\) and \((2\vec{Q}-2\vec{P})\) is:

a

\(0^\circ\)

b

\({\tan }^{-1}\frac{P}{Q}\)

c

\({\tan }^{-1}\left(\frac{2\vec{Q}-2\vec{P}}{2\vec{Q}+2\vec{P}}\right)\)

d

\({\tan }^{-1}(2Q/P)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q7

PYQ

When the position vector \(\vec{r}=\text{ }x\hat{i}\text{ }+\text{ }y\hat{j}+z\hat{k}\) changes sign as \(−\vec{r},\) which one of the following vector will not flip under sign change?

[JEE Main 2023, 11 April (Shift 1)]

a

Acceleration

b

Velocity

c

Angular momentum

d

Linear momentum

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q8

PYQ

If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at angle \(\theta\), then

a

\(|\vec{A}-\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)\)

b

\(|\vec{A}+\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)\)

c

\(|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\frac{\theta}{2}\right)\)

d

\(|\vec{A}+\vec{B}|=2 R \sin \left(\frac{\theta}{2}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q9

PYQ

A light wave is propagating with plane wave fronts of the type \(\mathrm{x}+\mathrm{y}+\mathrm{z}=\) constant. The angle made by the direction of wave propagation with the x -axis is :

[JEE Main 2025, 2 Apr (Shift 1)]

a

\({\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

b

\({\cos }^{-1}\left(\frac{2}{3}\right)\)

c

\({\cos }^{-1}\left(\frac{1}{3}\right)\)

d

\({\cos }^{-1}\left(\sqrt{\frac{2}{3}}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q10

PYQ

The angle between vector \(\vec{Q}\) and the resultant of \((2\vec{Q}+2\vec{P})\) and \((2\vec{Q}-2\vec{P})\) is

a

\(0^\circ\)

b

\({\tan }^{-1}\frac{P}{Q}\)

c

\({\tan }^{-1}\left(\frac{2\vec{Q}-2\vec{P}}{2\vec{Q}+2\vec{P}}\right)\)

d

\({\tan }^{-1}(2Q/P)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q11

PYQ

A light wave is propagating with plane wave fronts of the type \(\mathrm{x}+\mathrm{y}+\mathrm{z}=\) constant. The angle made by the direction of wave propagation with the x -axis is :

[JEE Main 2025, 2 Apr (Shift 1)]

a

\({\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

b

\({\cos }^{-1}\left(\frac{2}{3}\right)\)

c

\({\cos }^{-1}\left(\frac{1}{3}\right)\)

d

\({\cos }^{-1}\left(\sqrt{\frac{2}{3}}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q12

PYQ

If two vectors \(\vec{P}=\hat{i}+2 m \hat{j}+m \hat{k}\) and \(\vec{Q}=4 \hat{i}-2 \hat{j}+m \hat{k}\) are perpendicular to each other. Then, the value of \(m\) will be:

[JEE Main 2023, 24 Jan (Shift 2)]

a

1

b

–1

c

–3

d

2

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q13

PYQ

If \(\vec{A}\) and \(\vec{B}\) are two vectors satisfying the relation \(\vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|\). Then the value of \(|\vec{A}-\vec{B}|\) will be:

a

\(\sqrt{{A}^{2}+{B}^{2}-\sqrt{2}AB}\)

b

\(\sqrt{{A}^{2}+{B}^{2}}\)

c

\(\sqrt{{A}^{2}+{B}^{2}+2AB}\)

d

\(\sqrt{{A}^{2}+{B}^{2}+\sqrt{2}AB}\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q14

PYQ

If two vectors \(\vec{P}=\hat{i}+2m\hat{j}+m\hat{k}\) and \(\vec{Q}=4\hat{i}-2 \hat{j}+m \hat{k}\) are perpendicular to each other. Then, the value of \(m\) will be:

[JEE Main 2022, 28 July (Shift 2)]

a

1

b

–1

c

–3

d

2

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q15

PYQ

Two vectors \(\vec{P}\) and \(\vec{Q}\) have equal magnitudes. If the magnitude of \(\vec{P}+\vec{Q}\) is \(n\) times the magnitude of \(\vec{P}-\vec{Q}\), then angle between \(\vec{P}\) and \(\vec{Q}\) is:

[JEE Main 2019, 12 Jan (Shift 2)]

a

\({\sin }^{-1}\left(\frac{{n}^{2}-1}{{n}^{2}+1}\right)\)

b

\({\cos }^{-1}\left(\frac{{n}^{2}-1}{{n}^{2}+1}\right)\)

c

\({\sin }^{-1}\left(\frac{n-1}{n+1}\right)\)

d

\({\cos }^{-1}\left(\frac{n-1}{n+1}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q16

PYQ

If \(\vec{A}\) and \(\vec{B}\) are two vectors satisfying the relation \(\vec{A} \cdot \vec{B}=|\vec{A} \times \vec{B}|\). Then the value of \(|\vec{A}-\vec{B}|\) will be:

[JEE Main 2021, 31 August (Shift 2)]

a

\(\sqrt{{A}^{2}+{B}^{2}-\sqrt{2}AB}\)

b

\(\sqrt{{A}^{2}+{B}^{2}}\)

c

\(\sqrt{{A}^{2}+{B}^{2}+2AB}\)

d

\(\sqrt{{A}^{2}+{B}^{2}+\sqrt{2}AB}\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q17

PYQ

Two vectors \(\vec{X}\) and \(\vec{Y}\) have equal magnitude. The magnitude of \((\vec{X}-\vec{Y})\) is \(n\) times the magnitude of \((\vec{X}+\vec{Y})\). The angle between \(\vec{X}\) and \(\vec{Y}\) is:

[JEE Main 2014, 12 April]

a

\({\cos }^{-1}\left(\frac{{n}^{2}+1}{{n}^{2}-1}\right)\)

b

\({\cos }^{-1}\left(\frac{-{n}^{2}-1}{{n}^{2}-1}\right)\)

c

\({\cos }^{-1}\left(\frac{{n}^{2}-1}{-{n}^{2}-1}\right)\)

d

\({\cos }^{-1}\left(\frac{{n}^{2}+1}{-{n}^{2}-1}\right)\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q18

PYQ

A mosquito is moving with a velocity \( \vec{v}=0.5 t^{2} \hat{i}+3 t \hat{j}+9 \hat{k} \mathrm{~m} / \mathrm{s} \) and accelerating in uniform conditions. What will be the direction of mosquito after \( 2 \mathrm{~s} \)?

[JEE Main 2021, 16 Mar (Shift 1)]

a

\({\tan }^{-1}\left(\frac{\sqrt{2}}{3}\right)\) from X-axis

b

\({\tan }^{-1}\left(\frac{\sqrt{117}}{2}\right)\) from X-axis

c

\({\tan }^{-1}\left(\frac{5}{2}\right)\) from Y-axis

d

\({\tan }^{-1}\left(\frac{5}{2}\right)\) from X-axis

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q19

PYQ

Two forces having magnitude \(A\) and \(\frac{A}{2}\) are perpendicular to each other. The magnitude of their resultant is:

[JEE Main 2023, 08 Apr (Shift 2)]

a

\(\frac{\sqrt{5}A}{4}\)

b

\(\frac{5A}{2}\)

c

\(\frac{\sqrt{5}{A}^{2}}{2}\)

d

\(\frac{\sqrt{5}A}{2}\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q20

PYQ

If two vectors \(\vec{P}=\hat{i}+2m\hat{j}+m\hat{k}\) and \(\vec{Q}=4\hat{i}-2\hat{j}+m\hat{k}\) are perpendicular to each other. Then, the value of m will be:

[JEE Main 2023, 24 Jan (Shift 2)]

a

1

b

-1

c

-3

d

2

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q21

PYQ

When vector \(\vec{A}=2 \hat{i}+3 \hat{j}+2 \hat{k}\) is subtracted from vector \(\vec{B}\), it gives a vector equal to \(2 \hat{j}\). Then the magnitude of vector \(\vec{B}\) will be:

[JEE Main 2023, 11 Apr (Shift 1)]

a

\(\sqrt{33}\)

b

3

c

\(\sqrt{6}\)

d

\(\sqrt{5}\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q22

PYQ

What will be the projection of vector \(\vec{A}=\hat{i}+\hat{j}+\hat{k}\) on vector \(\vec{B}=\hat{i}+\hat{j}\)?

[JEE Main 2019, 11 January (Shift 1)]

a

\(2(\hat{i}+\hat{j}+\hat{k})\)

b

\(\sqrt{2}(\hat{i}+\hat{j})\)

c

\((\hat{i}+\hat{j})\)

d

\(\sqrt{2}(\hat{i}+\hat{j}+\hat{k})\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Q23

PYQ

Two forces having magnitude \(A\) and \(\frac{A}{2}\) are perpendicular to each other. The magnitude of their resultant is

a

\(\frac{\sqrt{5}A}{4}\)

b

\(\frac{5A}{2}\)

c

\(\frac{\sqrt{5}{A}^{2}}{2}\)

d

\(\frac{\sqrt{5}A}{2}\)

🔒

Answer & explanation — PYQ Pass

Options are free to see. Unlock the correct answer and full explanation with Pass.

Unlock · ₹149

Practice more JEE Physics PYQs