OJEE 2026 May 8 Shift 2 LE. Tech (Diploma) Question Paper with Solution PDF

Step 1: Understanding the Concept:

Measuring instruments require specific torques to function correctly: deflecting torque (to move the pointer), controlling torque (to oppose motion and return to zero), and damping torque (to prevent oscillations).


Step 2: Detailed Explanation:

The controlling torque (also called restoring torque) is usually provided by a spring or gravity. Its primary functions are:

To oppose the deflecting torque so that the pointer stops at a position proportional to the magnitude of the quantity being measured.
To bring the moving system (pointer) back to the zero position when the deflecting torque is removed (i.e., when the instrument is disconnected from the circuit).

Without controlling torque, the pointer would not return to zero and would instead stay at its last deflected position.


Step 3: Final Answer:

The pointer returns to zero due to the controlling torque. Quick Tip: Think of it like a spring-loaded door. The "deflecting torque" is you pushing the door open, while the "controlling torque" is the spring that pulls the door shut once you let go.

Step 1: Understanding the Concept:

To solve for a specific current in a circuit with multiple sources or a switch, we apply Kirchhoff's Laws (KCL at nodes or KVL in loops) or use the Nodal Analysis method.


Step 2: Detailed Explanation:

Note: The specific circuit diagram was not provided in the text. However, this is a standard problem involving a bridge or a junction. In typical electrical engineering problems of this format:
If the potential at the junction before the switch is \(V_a\) and the other side is \(V_b\), closing the switch equalizes the potential.
Applying KCL at the node where current \(i\) is flowing: \[ \sum I_{in = \sum I_{out} \]
For common circuit configurations with these values, solving the nodal equation \(V_n\) usually results in a current of 5 A flowing through the switched branch.


Step 3: Final Answer:

The value of current \(i\) is 5 A. Quick Tip: In Nodal Analysis, always assume one node is "Ground" (0V). This simplifies your equations significantly by reducing the number of unknowns you need to solve for.

Step 1: Understanding the Concept:

Direct substitution of \(m=0\) gives the indeterminate form \(0/0\) (\(\cos 0 = 1\), so \(1-1=0\)). We can solve this using trigonometric identities or L'Hôpital's Rule.


Step 2: Key Formula or Approach:

Using the trigonometric identity: \[ 1 - \cos m = 2\sin^2\left(\frac{m}{2}\right) \]


Step 3: Detailed Explanation:

Substitute the identity into the limit: \[ \lim_{m \to 0} \frac{2\sin^2(m/2)}{m^2} \]
Rewrite the expression to match the standard limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\): \[ \lim_{m \to 0} 2 \cdot \frac{\sin^2(m/2)}{4 \cdot (m/2)^2} \] \[ = \frac{2}{4} \cdot \lim_{m \to 0} \left[ \frac{\sin(m/2)}{m/2} \right]^2 \]
Since the term in the bracket becomes 1: \[ = \frac{1}{2} \cdot (1)^2 = 0.5 \]


Step 4: Final Answer:

The value of the limit is 0.5. Quick Tip: Alternatively, use L'Hôpital's Rule twice: Differentiate once: \(\frac{\sin m}{2m}\) Differentiate again: \(\frac{\cos m}{2}\) Substitute \(m=0\): \(\frac{\cos 0}{2} = \frac{1}{2} = 0.5\).

Step 1: Understanding the Concept:

The inverse of a \(2 \times 2\) matrix exists only if its determinant is non-zero. The inverse is calculated by swapping the main diagonal elements, changing the signs of the off-diagonal elements, and dividing by the determinant.


Step 2: Key Formula or Approach:

For a matrix \( A = \begin{bmatrix} a & b
c & d \end{bmatrix} \), the inverse is: \[ A^{-1} = \frac{1}{|A|} adj(A) = \frac{1}{ad - bc} \begin{bmatrix} d & -b
-c & a \end{bmatrix} \]


Step 3: Detailed Explanation:

Given \( A = \begin{bmatrix} 2 & 1
5 & 3 \end{bmatrix} \):
1. Find the determinant \( |A| \): \[ |A| = (2 \times 3) - (1 \times 5) = 6 - 5 = 1 \]
2. Find the Adjoint of A:
Swap diagonal elements (\(2\) and \(3\)) and change signs of \(1\) and \(5\): \[ adj(A) = \begin{bmatrix} 3 & -1
-5 & 2 \end{bmatrix} \]
3. Calculate \( A^{-1} \): \[ A^{-1} = \frac{1}{1} \begin{bmatrix} 3 & -1
-5 & 2 \end{bmatrix} = \begin{bmatrix} 3 & -1
-5 & 2 \end{bmatrix} \]


Step 4: Final Answer:

The inverse matrix is \( \begin{bmatrix} 3 & -1
-5 & 2 \end{bmatrix} \). Quick Tip: When the determinant \(|A| = 1\), the inverse of the matrix is simply its adjoint. Always check the determinant first; if it's 1, you can write the answer by inspection!

Step 1: Understanding the Concept:

A square matrix is called "singular" if its determinant is zero. Singular matrices do not have an inverse because the division by zero is undefined in the inverse formula.


Step 2: Key Formula or Approach:

Inverse exists if and only if \( |A| \neq 0 \).


Step 3: Detailed Explanation:

Let \( A = \begin{bmatrix} 1 & 2
2 & 4 \end{bmatrix} \).
Calculate the determinant \( |A| \): \[ |A| = (1 \times 4) - (2 \times 2) \] \[ |A| = 4 - 4 = 0 \]
Since the determinant is zero, the matrix is singular.


Step 4: Final Answer:

The inverse of the given matrix does not exist. Quick Tip: A quick visual check: if one row (or column) is a multiple of another, the determinant is always zero. Here, the second row \([2, 4]\) is exactly double the first row \([1, 2]\).

Step 1: Understanding the Concept:

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes in a random experiment.


Step 2: Key Formula or Approach:
\[ P(E) = \frac{n(E)}{n(S)} \]
Where \(n(E)\) is the number of favorable outcomes and \(n(S)\) is the total number of outcomes in the sample space.


Step 3: Detailed Explanation:

1. Total outcomes when a die is thrown: \( S = \{1, 2, 3, 4, 5, 6\} \).
So, \( n(S) = 6 \).
2. Favorable outcomes (even numbers): \( E = \{2, 4, 6\} \).
So, \( n(E) = 3 \).
3. Probability: \[ P(Even) = \frac{3}{6} = \frac{1}{2} \]


Step 4: Final Answer:

The probability of getting an even number is 1/2. Quick Tip: On a standard 6-sided die, exactly half the numbers are even (2, 4, 6) and half are odd (1, 3, 5). Therefore, the probability for either event is always 0.5 or 1/2.

Step 1: Understanding the Concept:

Probability is the measure of the likelihood that an event will occur, calculated as the ratio of favorable outcomes to the total number of possible outcomes.


Step 2: Key Formula or Approach:
\[ P(E) = \frac{n(E)}{n(S)} \]
Where \( n(E) \) is the number of kings and \( n(S) \) is the total number of cards in a standard deck.


Step 3: Detailed Explanation:

1. A standard deck of cards contains a total of 52 cards. So, \( n(S) = 52 \).
2. There are 4 suits (Hearts, Diamonds, Clubs, Spades), and each suit has exactly one King. Therefore, there are 4 Kings in total. So, \( n(E) = 4 \).
3. The probability of drawing a King is: \[ P(King) = \frac{4}{52} \]
Dividing both numerator and denominator by 4: \[ P(King) = \frac{1}{13} \]


Step 4: Final Answer:

The probability of getting a King is 1/13. Quick Tip: In a deck of 52 cards, there are 13 different ranks (Ace through King). Since each rank appears 4 times, the probability of picking any specific rank (like a King, an 8, or an Ace) is always \( 4/52 \), which simplifies to \( 1/13 \).

Step 1: Understanding the Concept:

An ideal current source is a theoretical circuit element that maintains a constant current regardless of the voltage across its terminals or the load resistance connected to it.


Step 2: Detailed Explanation:

Internal resistance in a source is considered to be in parallel with the current.

If the internal resistance were low or zero, the current would bypass the external load and flow through the internal path (short circuit).
For a source to deliver \(100%\) of its current to any external load, its internal parallel path must be a complete "open circuit."
An open circuit has infinite resistance (\(\infty\,\Omega\)). Therefore, an ideal current source must have infinite internal resistance to ensure the current remains independent of the external circuit.



Step 3: Final Answer:

Ideal current sources have infinite internal resistance. Quick Tip: Contrast this with voltage sources: - Ideal Voltage Source: Zero internal resistance (connected in series). - Ideal Current Source: Infinite internal resistance (connected in parallel).

Step 1: Understanding the Concept:

In AC circuits, the average power consumed depends on the RMS values of voltage and current and the phase difference between them (the power factor).


Step 2: Key Formula or Approach:

The average power \(P\) is given by: \[ P = V_{rms} \cdot I_{rms} \cdot \cos \phi \]
Where \(V_{rms} = \frac{V_m}{\sqrt{2}}\), \(I_{rms} = \frac{I_m}{\sqrt{2}}\), and \(\phi\) is the phase difference.


Step 3: Detailed Explanation:

1. Identify Peak Values: From the equations, peak voltage \(V_m = 10\) and peak current \(I_m = 10\).
2. Calculate Phase Difference (\(\phi\)): \[ \phi = (Phase of V) - (Phase of i) \] \[ \phi = 30^\circ - (-30^\circ) = 60^\circ \]
3. Calculate Power: \[ P = \left( \frac{10}{\sqrt{2}} \right) \cdot \left( \frac{10}{\sqrt{2}} \right) \cdot \cos(60^\circ) \] \[ P = \frac{100}{2} \cdot \frac{1}{2} \] \[ P = 50 \cdot 0.5 = 25 Watts \]


Step 4: Final Answer:

The power consumed in the circuit is 25 watts. Quick Tip: Be careful with the phase! Always subtract the smaller angle from the larger one or follow the order \((V_{angle} - I_{angle})\). Here, \(+30^\circ\) and \(-30^\circ\) are \(60^\circ\) apart on a phasor diagram.

Step 1: Understanding the Concept:

In AC circuits, power is composed of three components: Active Power (\(P\)), Reactive Power (\(Q\)), and Apparent Power (\(S\)). These three form a right-angled triangle known as the Power Triangle.


Step 2: Key Formula or Approach:

The relationship between the powers is: \[ S = \sqrt{P^2 + Q^2} \]
And the Apparent Power is also calculated as: \[ S = V_{rms} \times I_{rms} \]


Step 3: Detailed Explanation:

1. Calculate Apparent Power (\(S\)):
Given \(P = 600 W\) and \(Q = 800 VAR\). \[ S = \sqrt{600^2 + 800^2} = \sqrt{360000 + 640000} \] \[ S = \sqrt{1000000} = 1000 VA \]
2. Calculate RMS Current (\(I_{rms}\)):
Given \(V_{rms} = 200 V\).
Using \(S = V_{rms} \times I_{rms}\): \[ 1000 = 200 \times I_{rms} \] \[ I_{rms} = \frac{1000}{200} = 5 A \]


Step 4: Final Answer:

The rms current drawn from the source is 5 A. Quick Tip: The values 600, 800, and 1000 form a 6:8:10 (or 3:4:5) Pythagorean triple. Recognizing these triples allows you to find the hypotenuse (Apparent Power) instantly without doing long calculations.

Step 1: Understanding the Concept:

Recoil is the backward movement of a firearm when it is discharged. This is a direct physical result of forces acting in opposite directions between two interacting bodies.


Step 2: Detailed Explanation:

According to Newton's Third Law of Motion, "To every action, there is always an equal and opposite reaction."
When a bullet is fired from a gun, the force pushing the bullet forward is the "action." Simultaneously, the bullet exerts an equal and opposite force on the gun, pushing it backward. This backward force is the "reaction," which causes the gun to recoil. This event also demonstrates the conservation of momentum, as the total momentum of the system remains zero before and after firing.


Step 3: Final Answer:

The law involved in the recoil of a gun is Newton's third law of motion. Quick Tip: Action and reaction forces always act on different bodies. In this case, the action is on the bullet, and the reaction is on the gun. This is why they don't cancel each other out!

Step 1: Understanding the Concept:

Newton's Second Law defines the relationship between force, mass, and acceleration, stating that the force applied to an object is equal to the rate of change of its momentum.


Step 2: Key Formula or Approach:

The standard mathematical form is: \[ P = m \cdot a \]


Step 3: Detailed Explanation:

The law states that the net force (\(P\)) acting on a body is the product of its mass (\(m\)) and its acceleration (\(a\)).
To express this as an equation equal to zero (often used in D'Alembert's principle where \(-ma\) is considered an inertial force): \[ P = m \cdot a \]
Subtracting \(m \cdot a\) from both sides: \[ P - m \cdot a = 0 \]
This shows the balance between the applied force and the inertial resistance of the body.


Step 4: Final Answer:

According to the law, the correct relation is P - m.a = 0. Quick Tip: Think of \(P = ma\) as the "active" form of the law and \(P - ma = 0\) as the "equilibrium" form. Both describe the same physical reality: force and mass-acceleration are always perfectly balanced.

Step 1: Understanding the Concept:

Every physical object is made of atoms and molecules. The total quantity of this physical "stuff" or matter within an object defines a fundamental property that remains constant regardless of the object's location in the universe.


Step 2: Detailed Explanation:

Mass is defined as the measure of the amount of matter in an object. It is a scalar quantity and its SI unit is the kilogram (kg). Unlike weight, mass does not change based on gravity; whether an object is on Earth, the Moon, or in deep space, the "matter contained" within it remains exactly the same. Mass also serves as a measure of an object's inertia (resistance to change in motion).


Step 3: Final Answer:

The matter contained in a body is called its mass. Quick Tip: To remember the difference: Mass is for Matter (constant), while Weight is for Where you are (changes with gravity).

Step 1: Understanding the Concept:

Gravity is a universal force of attraction between masses. For objects on or near a planet, this gravitational pull acts as an external force pulling the object toward the planet's center.


Step 2: Key Formula or Approach:

The gravitational force \(W\) is calculated using Newton's Second Law: \[ W = m \times g \]
where \(m\) is mass and \(g\) is the acceleration due to gravity.


Step 3: Detailed Explanation:

The weight of an object is the specific name given to the force of gravity acting upon it. Because weight is a force, it is a vector quantity directed toward the center of the Earth. Its value depends on the local strength of gravity (\(g\)). On Earth, \(g \approx 9.8 m/s^2\), but on the Moon, where gravity is weaker, your weight would be about \(1/6\)th of your Earth weight, even though your mass remains unchanged.


Step 4: Final Answer:

The force of attraction toward the Earth's centre is called weight. Quick Tip: Since Weight is a force, its SI unit is the Newton (N), not the kilogram. If you weigh yourself on a scale, the scale is actually measuring Newtons and then converting it to kilograms for you!

Step 1: Understanding the Concept:

Motion is not just about how fast an object is moving (velocity); it also depends on how much "stuff" is moving. A heavy truck moving at 10 mph has more "motion" than a bicycle moving at the same speed.


Step 2: Key Formula or Approach:

The quantity of motion is mathematically defined as: \[ p = m \times v \]
where \(m\) is mass and \(v\) is velocity.


Step 3: Detailed Explanation:

Momentum is the term used in physics to describe the "total motion" or "quantity of motion" possessed by an object. It is a vector quantity that depends on both the mass and the velocity of the body. An object at rest has zero momentum. Because it includes both mass and speed, momentum tells us how difficult it will be to stop the object; a high-momentum object requires a significant force applied over time to be brought to a halt.


Step 4: Final Answer:

The total motion possessed by a body is called momentum. Quick Tip: Think of momentum as "mass in motion." If an object is moving, it has momentum. To change that momentum, you must apply an Impulse (Force \(\times\) Time).