NTA JEE Main 22nd July 2021 Shift 1

Three particles P, Q and R are moving along the vectors $$\vec{A} = \hat{i} + \hat{j}$$, $$\vec{B} = \hat{j} + \hat{k}$$ and $$\vec{C} = -\hat{i} + \hat{j}$$, respectively. They strike on a point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $$\vec{A}$$ and $$\vec{B}$$. Similarly particle Q is moving normal to the plane which contains vector $$\vec{A}$$ and $$\vec{C}$$. The angle between the direction of motion of P and Q is $$\cos^{-1}\left(\dfrac{1}{\sqrt{x}}\right)$$. Then the value of $$x$$ is ___.

Three students $$S_1$$, $$S_2$$ and $$S_3$$ perform an experiment for determining the acceleration due to gravity $$(g)$$ using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

(Least count of length = 0.1 m, least count for time = 0.1 s)
If $$E_1$$, $$E_2$$ and $$E_3$$ are the percentage errors in $$g$$ for students 1, 2 and 3, respectively, then the minimum percentage error is obtained by student no ___.

The position of the centre of mass of a uniform semi-circular wire of radius $$R$$ placed in $$x-y$$ plane with its centre at the origin and the line joining its ends as $$x$$-axis is given by $$\left(0, \frac{xR}{\pi}\right)$$. Then, the value of $$|x|$$ is ___.

The centre of a wheel rolling on a plane surface moves with a speed $$v_0$$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $$\sqrt{x}v_0$$. Then the value of $$x$$ is ___.

The area of cross-section of a railway track is 0.01 m$$^2$$. The temperature variation is 10 $$^\circ$$C. Coefficient of linear expansion of material of track is $$10^{-5}$$ $$^\circ$$C$$^{-1}$$. The energy stored per meter in the track is J m$$^{-1}$$. (Young's modulus of material of track is $$10^{11}$$ N m$$^{-2}$$)

In 5 minutes, a body cools from 75 $$^\circ$$C to 65 $$^\circ$$C at room temperature of 25 $$^\circ$$C. The temperature of body at the end of next 5 minutes is ___ $$^\circ$$C.

The total charge enclosed in an incremental volume of $$2 \times 10^{-9}$$ m$$^3$$ located at the origin is ___ nC, if electric flux density of its field is found as $${D} = e^{-x}\sin y \hat{i} - e^{-x}\cos y \hat{j} + 2z\hat{k}$$ C m$$^{-2}$$

In an electric circuit, a call of certain emf provides a potential difference of 1.25 V across a load resistance of 5 $$\Omega$$. However, it provides a potential difference of 1 V across a load resistance of 2 $$\Omega$$. The emf of the cell is given by $$\frac{x}{10}$$ V. Then the value of $$x$$ is ___.

A ray of light passing through a prism $$\left(\mu = \sqrt{3}\right)$$ suffers minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ___ (in degrees).

In a given circuit diagram, a 5 V zener diode along with a series resistance is connected across a 50 V power supply. The minimum value of the resistance required, if the maximum zener current is 90 mA will be ___ $$\Omega$$.