OJEE 2026 May 8 Shift 1 LE. Tech (Diploma) Question Paper with Solution PDF

Step 1: Understanding the Concept:

The capacitance of a parallel plate capacitor depends on the geometry of the plates (Area) and the distance between them. If the linear dimensions (length and width) of the plates change, the area changes quadratically.


Step 2: Key Formula or Approach:

The capacitance \(C\) is given by: \[ C = \frac{\epsilon_0 A}{d} \]
where \(A\) is the area of the plates and \(d\) is the separation distance.


Step 3: Detailed Explanation:

Let the initial linear dimensions be \(L\) and \(W\). Initial Area \(A = L \times W\).
Initial capacitance: \[ C = \frac{\epsilon_0 (L \times W)}{d} = 50\muF \]
When linear dimensions are doubled, the new length \(L' = 2L\) and new width \(W' = 2W\).
New Area \(A' = 2L \times 2W = 4(L \times W) = 4A\).
The new separation \(d' = 4d\).
The new capacitance \(C'\) is: \[ C' = \frac{\epsilon_0 A'}{d'} = \frac{\epsilon_0 (4A)}{4d} \] \[ C' = \frac{\epsilon_0 A}{d} = C \]
Since \(C' = C\), the value remains \(50\muF\).


Step 4: Final Answer:

The new value of the capacitor is 50μF. Quick Tip: Linear dimensions refer to length and width. If linear dimensions are scaled by \(k\), the area \(A\) scales by \(k^2\). Here, \(2^2 = 4\), which exactly cancels the 4-fold increase in distance.

Step 1: Understanding the Concept:

This problem typically refers to a Wheatstone Bridge circuit. A bridge is said to be balanced when the ratio of the resistances in the two arms is equal, resulting in no potential difference across the central galvanometer or resistor.


Step 2: Detailed Explanation:

In a standard balanced Wheatstone bridge configuration where resistors \(P, Q, R, S\) form a loop and a resistor (in this case \(1\Omega\)) is connected across the junctions:
If \(\frac{P}{Q} = \frac{R}{S}\), then the bridge is balanced.
In a balanced state, the potential at the two ends of the central resistor is the same.
Potential Difference (\(V\)) = 0.
By Ohm's Law: \[ I = \frac{V}{R} = \frac{0}{1} = 0A \]
Since most competitive questions of this format use balanced bridge values, the current through the central branch is zero.


Step 3: Final Answer:

The current flowing through the 1Ω resistor is 0A. Quick Tip: Always check for the balanced Wheatstone bridge condition (\(R_1/R_2 = R_3/R_4\)) in complex-looking resistor networks. It often simplifies the problem by allowing you to remove the central branch.

Step 1: Understanding the Concept:

The two given lines have the same coefficients for \(x\) and \(y\) (3 and 4), which means they are parallel. The distance between two parallel lines is the constant perpendicular distance between them.


Step 2: Key Formula or Approach:

The distance \(d\) between two parallel lines \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is: \[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]


Step 3: Detailed Explanation:

Given lines:
Line 1: \(3x + 4y + 5 = 0 \implies C_1 = 5\)
Line 2: \(3x + 4y - 7 = 0 \implies C_2 = -7\)
Here \(A = 3\) and \(B = 4\).
Applying the formula: \[ d = \frac{|5 - (-7)|}{\sqrt{3^2 + 4^2}} \] \[ d = \frac{|5 + 7|}{\sqrt{9 + 16}} \] \[ d = \frac{12}{\sqrt{25}} \] \[ d = \frac{12}{5} \]


Step 4: Final Answer:

The distance between the lines is 12/5. Quick Tip: Before using the parallel distance formula, ensure the \(A\) and \(B\) coefficients are identical in both equations. If one is a multiple of the other, divide or multiply to make them match.

Step 1: Understanding the Concept:

The Section Formula is used to find the coordinates of a point that divides a line segment joining two given points in a specific ratio.


Step 2: Key Formula or Approach:

For points \((x_1, y_1)\) and \((x_2, y_2)\) divided in ratio \(m:n\) internally, the coordinates \((x, y)\) are: \[ x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n} \]


Step 3: Detailed Explanation:

Given: \((x_1, y_1) = (1, 2)\), \((x_2, y_2) = (5, 6)\), and \(m:n = 1:3\).

Calculating \(x\): \[ x = \frac{1(5) + 3(1)}{1+3} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \]
Calculating \(y\): \[ y = \frac{1(6) + 3(2)}{1+3} = \frac{6 + 6}{4} = \frac{12}{4} = 3 \]
The point is \((2, 3)\).


Step 4: Final Answer:

The point dividing the segment is (2, 3). Quick Tip: To avoid mixing up the numbers, remember the "cross-multiplication" rule: the first part of the ratio (\(m\)) multiplies the second point's coordinates, and the second part of the ratio (\(n\)) multiplies the first point's coordinates.

Step 1: Understanding the Concept:

This problem uses the fundamental trigonometric limit property which states that as the angle approaches zero, the ratio of the sine of the angle to the angle itself tends to unity.


Step 2: Key Formula or Approach:

The standard limit is: \[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]


Step 3: Detailed Explanation:

The expression is: \[ \lim_{m \to 0} \frac{5 \sin m}{m} \]
By the constant multiple rule of limits: \[ 5 \times \left( \lim_{m \to 0} \frac{\sin m}{m} \right) \]
Applying the standard limit: \[ 5 \times 1 = 5 \]
Correction Note: Based on the provided options, if "1" is intended as the answer, the "5" in the numerator was likely a typo in the original question paper. If "5" is not present, the answer is 1. Given the strict options, (B) 1 is the closest conceptual answer for the base identity.


Step 4: Final Answer:

The value of the limit is 5. (If the question was intended as \(\lim \frac{\sin m{m}\), the answer is 1). Quick Tip: This limit only holds true when the angle \(m\) is measured in {radians}. If the angle were in degrees, the limit would be \(\pi/180\).

Step 1: Understanding the Concept:

Order is the highest derivative present in the equation. Degree is the power of that highest derivative, provided the equation is a polynomial in its derivatives.


Step 2: Detailed Explanation:

In the given equation: \[ \left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + y = 0 \]
1. The derivatives present are \(\frac{d^2y}{dx^2}\) (second order) and \(\frac{dy}{dx}\) (first order). The highest order is 2.
2. The power raised to this highest derivative (\(\frac{d^2y}{dx^2}\)) is 3.
Thus, Order = 2 and Degree = 3.


Step 3: Final Answer:

The order and degree are 2 and 3 respectively. Quick Tip: Always find the Order first. Once you identify the "boss" (the highest derivative), the Degree is simply the power that specific derivative is carrying. Ignore the powers of all lower-order derivatives.

Step 1: Understanding the Concept:

The area of a triangle formed by three points can be calculated using their coordinates. If the calculated area is zero, it implies that the three points do not form a triangle because they lie on a single straight line (collinear).


Step 2: Key Formula or Approach:

For points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), the area is: \[ Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]


Step 3: Detailed Explanation:

Given points: \((1, 2), (3, 6), (5, 10)\). \[ Area = \frac{1}{2} |1(6 - 10) + 3(10 - 2) + 5(2 - 6)| \] \[ Area = \frac{1}{2} |1(-4) + 3(8) + 5(-4)| \] \[ Area = \frac{1}{2} |-4 + 24 - 20| \] \[ Area = \frac{1}{2} |0| = 0 \]
Alternatively, observe the slope between \((1,2)\) and \((3,6)\) is \(\frac{6-2}{3-1} = 2\), and between \((3,6)\) and \((5,10)\) is \(\frac{10-6}{5-3} = 2\). Since the slopes are equal, the points are collinear.


Step 4: Final Answer:

The area of the triangle is 0. Quick Tip: Before calculating the full area, check if the points follow a linear pattern. Here, for every increase of 2 in \(x\), \(y\) increases by 4 (\(y = 2x\)). Since all points satisfy this, they are collinear and the area is 0.

Step 1: Understanding the Concept:

Resistors can be combined in series to increase total resistance (\(R_s = R_1 + R_2\)) or in parallel to decrease total resistance (\(1/R_p = 1/R_1 + 1/R_2\)). Mixed circuits allow for specific intermediate values.


Step 2: Detailed Explanation:

Let's evaluate option (D):
1. Parallel Part: Two \(50\Omega\) resistors in parallel: \[ R_p = \frac{50 \times 50}{50 + 50} = \frac{2500}{100} = 25\Omega \]
2. Series Part: This \(25\Omega\) combination is connected in series with another \(50\Omega\) resistor: \[ R_{total} = R_p + 50 = 25 + 50 = 75\Omega \]
This matches the required effective resistance.


Step 3: Final Answer:

To get \(75\Omega\), two \(50\Omega\) resistors should be in parallel, then connected in series with a third \(50\Omega\) resistor. Quick Tip: When resistors of equal value \(R\) are in parallel, the resistance is \(R/n\). For two \(50\Omega\) resistors, it's \(50/2 = 25\Omega\). Adding \(50\Omega\) in series gives \(25 + 50 = 75\Omega\).

Step 1: Understanding the Concept:

A transformer works on alternating magnetic fields. These fields induce circulating currents within the conductive iron core itself, leading to energy loss in the form of heat.


Step 2: Detailed Explanation:

Eddy currents are loops of electrical current induced within conductors by a changing magnetic field. In a solid transformer core, these currents have a large area to flow, causing significant "Eddy current loss." To prevent this, the core is made of thin sheets (laminations) insulated from each other. This breaks the large circular paths into much smaller loops, significantly increasing resistance to these currents and reducing total power loss.


Step 3: Final Answer:

Transformer cores are laminated to reduce eddy current loss. Quick Tip: Remember: Laminations target Eddy currents (by breaking the path). Soft iron/Silicon steel targets Hysteresis (by reducing magnetic friction).

Step 1: Understanding the Concept:

When an electric current flows through an inductor (coil), it creates a magnetic field. The work done to establish this current is stored in the inductor as magnetic potential energy.


Step 2: Key Formula or Approach:

The energy \(U\) stored in an inductor is given by the formula: \[ U = \frac{1}{2} LI^2 \]
where \(L\) is the inductance and \(I\) is the current.


Step 3: Detailed Explanation:

Given: \(I = 3 A\) \(L = 32 mH = 32 \times 10^{-3} H\)
Placing values in the formula: \[ U = \frac{1}{2} \times (32 \times 10^{-3}) \times (3)^2 \] \[ U = 16 \times 10^{-3} \times 9 \] \[ U = 144 \times 10^{-3} \] \[ U = 0.144 J \]


Step 4: Final Answer:

The energy stored in the coil is 0.144 J. Quick Tip: Always convert units to the standard SI system before calculating. Here, "mH" (millihenry) must be converted to "H" (Henry) by multiplying by \(10^{-3}\) to ensure the answer is in Joules.

Step 1: Understanding the Concept:

When an object is on an inclined plane, gravity acts vertically downward. This force is resolved into two components: one perpendicular to the plane and one parallel to the plane.


Step 2: Key Formula or Approach:

According to Newton's Second Law, \(F = ma\). The net force acting along the direction of motion (down the plane) determines the acceleration.


Step 3: Detailed Explanation:

Consider a body of mass \(m\) on a frictionless plane inclined at an angle \(\theta\).
1. The weight \(mg\) acts straight down.
2. The component of weight perpendicular to the plane is \(mg \cos \theta\) (balanced by the Normal force).
3. The component of weight parallel to the plane (acting downwards) is \(mg \sin \theta\).
Using \(F = ma\): \[ mg \sin \theta = ma \]
Dividing both sides by \(m\): \[ a = g \sin \theta \]


Step 4: Final Answer:

The acceleration is g sin θ. Quick Tip: If the surface has friction, the acceleration decreases to \(g(\sin \theta - \mu \cos \theta)\). However, in standard ideal cases where friction isn't mentioned, we consider only the gravity component.

Step 1: Understanding the Concept:

The equations of linear motion (also known as kinematic equations) describe the behavior of an object moving in a straight line under constant acceleration.


Step 2: Detailed Explanation:

There are three fundamental equations for uniformly accelerated motion:

First Equation: \(v = u + at\) (relates velocity and time).
Second Equation: \(s = ut + \frac{1}{2}at^2\) (relates displacement and time).
Third Equation: \(v^2 = u^2 + 2as\) (relates velocity and displacement).

Since all options (A), (B), and (C) are valid kinematic equations, (D) is the correct choice.


Step 3: Final Answer:

All the provided options are equations of linear motion. Quick Tip: These equations are only valid when acceleration (\(a\)) is constant. If acceleration changes over time, you must use calculus (integration/differentiation) to find the motion parameters.

Step 1: Understanding the Concept:

Newton's First Law of Motion, often called the Law of Inertia, describes the behavior of objects when the net external force acting on them is zero.


Step 2: Detailed Explanation:

The law states that an object will not change its motion unless a force acts on it.

If the object is at rest, it stays at rest.
If the object is in motion, it stays in motion with the same speed and in the same direction.

This resistance to change in the state of motion is known as inertia. Option (B) describes the second law, and option (C) describes the third law.


Step 3: Final Answer:

The statement in option (A) correctly defines Newton's first law of motion. Quick Tip: Inertia is directly proportional to the mass of an object. The more mass an object has, the more it resists changes to its state of motion.

Step 1: Understanding the Concept:

Newton's Second Law provides a quantitative definition of force. It relates the net force acting on a body to the resulting change in its motion (momentum).


Step 2: Key Formula or Approach:

Mathematically, the law is expressed as: \[ F \propto \frac{dp}{dt} \]
Where \(p = mv\) (momentum). For a constant mass, this simplifies to the famous equation: \[ F = ma \]


Step 3: Detailed Explanation:

The law implies that to change the momentum of an object, a force must be applied. The faster you want to change the momentum, the greater the force required. The direction of the acceleration (or change in momentum) is always the same as the direction of the applied force.


Step 4: Final Answer:

The statement describes Newton's second law of motion. Quick Tip: Momentum (\(p\)) is a vector quantity. This means that a force can change the speed of an object, its direction, or both.

Step 1: Understanding the Concept:

Newton's Third Law states that forces always exist in pairs. Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force back on the first.


Step 2: Detailed Explanation:

When a gun is fired, the gunpowder explosion exerts a forward force on the bullet (Action). According to Newton's Third Law, the bullet exerts an equal and opposite backward force on the gun (Reaction). This backward force causes the gun to "recoil" or jerk backward into the shooter's shoulder. This phenomenon is also a perfect example of the Law of Conservation of Momentum.


Step 3: Final Answer:

The recoil of a gun is based on Newton's third law of motion. Quick Tip: Even though the forces are equal in magnitude, the acceleration of the gun is much less than the bullet because the gun has a much larger mass (\(a = F/m\)).