If a vector \(2\hat{i}+3\hat{j}+8\dot{k}\) is perpendicular to the vector \(2\hat{i}+3\hat{j}+8\dot{k}\) , then the value of α is

1/2
–1/2
1
–1

Solution

If \(\left | \overrightarrow{A}\times \overrightarrow{B} \right |=\sqrt{3}\, \overrightarrow{A}.\overrightarrow{B}\) then the value of \(\left | \overrightarrow{A}\times \overrightarrow{B} \right |=\sqrt{3}\, \overrightarrow{A}.\overrightarrow{B}\) is

(A²+B²√3AB)½
(A²+B²+AB)½
(A²+B²+^AB⁄_√3)½
A + B

Solution

A particle moves along a circle of radius \(\left ( \frac{20}{\pi } \right )\)m with constant tangential acceleration. It the velocity of particle is 80 m/sec at end of second revolution after motion has begun, the tangential acceleration is

40 π m/sec²
40 m/sec²
640 π m/sec²
160 π m/sec²

Solution

Circumference of circle is 2πr = 40m

Total distance travelled in two revolution is 80m. Initial velocity u =0, final veloctiy v = 80 m/sec

so from

v²=u² + 2as

⇒ (80)² = 0² + 2 × 80 × a

⇒ a = 40m/sec²

The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

cannot be predicted
are equal to each other
are equal to each other in magnitude
are not equal to each other in magnitude

Solution

From a 10 m high building a stone A is dropped, and simultaneously another identical stone B is thrown horizontally with an initial speed of 5 ms–1. Which one of the following statements is true?

It is not possible to calculate which one of the two stones will reach the ground first
Both the stones (A and B) will reach the ground simultaneously
As tone reaches the ground earlier than B
B stone reaches the ground earlier than A

Solution

s=¹⁄₂gt²,s and g are same for both the balls, so time of fall‘t’ will also be the same for both of them (s is vertical height)

A body of 3kg. moves in X-Y plane under the action of force given by \(6t\hat{i}+4t\hat{j}\). Assuming that the body is at rest at time t = 0, the velocity of body at t = 3 sec is

\(6t\hat{i}+4t\hat{j}\)
\(6t\hat{i}+4t\hat{j}\)
\(6t\hat{i}+4t\hat{j}\)
\(6t\hat{i}+4t\hat{j}\)

Solution

Two projectiles are fired from the same point with the same speed at angles of projection 60º and 30º respectively.Which one of the following is true?

Their maximum height will be same
Their range will be same
Their landing velocity will be same
Their time of flight will be same

Solution

A ball whose kinetic energy is E is thrown at an angle of 45º with horizontal. Its kinetic energy at highest point off light will be

E
E/2
\(\frac{E}{\sqrt{2}}\)
zero

Solution

Since E=¹⁄₂mv²⇒v=\(\sqrt{\frac{2E}{m}}\)

Now at highest point of flight, the vertical component of velocity is zero & only horizontal component is nonzero. So K.E. at highest point is E'=¹⁄₂m(v cos 45º)²= E / 2

The position vector of a particle is \(\overrightarrow{r}=(a\, cos\, \omega t)\hat{i}+(a\, sin\, \omega t)\hat{j}\).The velocity of the particle is

directed towards the origin
directed away from the origin
parallel to the position vector
perpendicular to the position vector

Solution

A person swims in a river aiming to reach exactly on the opposite point on the bank of a river. His speed of swimming is 0.5 m/s at an angle of 120º with the direction of flow of water. The speed of water is

1.0 m/s
0.5 m/s
0.25 m/s
0.43 m/s

Solution

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