NDA / CDS / AFCAT - MCQ Practice Questions

To find the sum of cosines at complementary angles, we use the sum-to-product formula to simplify the expression.

Step 1: Identify the Relationship

Notice that 75° = 90° - 15°, making these complementary angles where cos 75° = sin 15°.

Step 2: Apply Sum-to-Product Formula

We use the cosine sum-to-product identity: $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

Step 3: Substitute Standard Values

Insert the known trigonometric values: $\cos 45°=\frac{\sqrt{2}}{2}$ and $\cos 30°=\frac{\sqrt{3}}{2}$

**The answer is (A) $\frac{\sqrt{6}}{2}$

x = 2 + √3, so 1/x = 1/(2+√3) = 2-√3 (after rationalization). x + 1/x = 4. Therefore x² + 1/x² = (x + 1/x)² - 2 = 16 - 2 = 14.

By Vieta's formulas, for a cubic x³ + bx² + cx + d = 0, sum of roots = -b. Here, sum = -(-6) = 6.

Understanding:

We need to find the maximum height reached by a ball thrown vertically upward.

Formula:

Using the third equation of motion:

Step 1: Substitute known values

Step 2: Solve for $H$

Answer:

The maximum height reached by the ball is $20\text{m}$.

Quick Tip:

At maximum height, the velocity becomes zero. Always set $v=0$ when finding maximum height in vertical projectile problems.

Understanding:

We need to find the retarding force that brings a moving body to rest.

Formula:

First find acceleration using:

Then apply Newton's second law:

Step 1: Calculate acceleration

Step 2: Calculate force

Answer:

The magnitude of the force applied is $5\text{N}$.

Quick Tip:

The negative sign of acceleration indicates deceleration. The question asks for magnitude, so always take the absolute value of $a$ before computing $F$.

Understanding:

A wire of total resistance $12\Omega$ is bent into a circle. We need to find the resistance between the two ends of a diameter.

Formula:

When the wire is bent into a circle and connected at two diametrically opposite points, the wire is divided into two equal halves, each of resistance $\frac{R}{2}$, connected in parallel:

Step 1: Find resistance of each half

Step 2: Calculate parallel combination

Answer:

The effective resistance between the two ends of a diameter is $3\Omega$.

Quick Tip:

For a wire of resistance $R$ bent into a circle, the resistance between diametrically opposite points is always $\frac{R}{4}$. Here $\frac{12}{4}=3\Omega$.

Understanding:

We need to find the length of a simple pendulum given its time period.

Formula:

The time period of a simple pendulum is:

Step 1: Square both sides

Step 2: Solve for $L$

Answer:

The length of the pendulum is $1\text{m}$.

Quick Tip:

A pendulum with $T=2\text{s}$ is called a "seconds pendulum" and always has length $\approx 1\text{m}$ on Earth. This is a frequently tested fact.

Understanding:

Two resistors are in parallel across a battery. We need to find the total current.

Formula:

In a parallel circuit, the same voltage appears across each resistor. The total current is:

Step 1: Calculate individual currents

Step 2: Find total current

Verification:

Equivalent resistance $R_{\text{eq}}=\frac{4\times 6}{4+6}=\frac{24}{10}=2.4\Omega$, so $I=\frac{12}{2.4}=5\text{A}$. ✓

Answer:

The total current drawn from the battery is $5\text{A}$.

Understanding:

We need to find the time period of a body moving uniformly in a circular path.

Formula:

The time period is the time taken to complete one full revolution. The circumference of the circle is $2\pi r$:

Step 1: Substitute values

Answer:

The time period of revolution is $1\text{s}$.

Quick Tip:

Notice how choosing $v=\pi \text{m/s}$ causes $\pi$ to cancel neatly. In MCQs, look for such cancellations before doing heavy arithmetic.

Understanding:

We need to compare the resistances of two bulbs rated for the same voltage but different wattages.

Formula:

The resistance of a bulb at rated conditions is:

Step 1: Find individual resistances

Step 2: Compute the ratio

So $R_{60}:R_{100}=5:3$, meaning the $60\text{W}$ bulb has the higher resistance.

Answer:

The $60\text{W}$ bulb has higher resistance and the ratio is $5:3$.

Quick Tip:

Since $R=V^2/P$ and $V$ is the same, resistance is inversely proportional to power. Lower wattage $\Rightarrow$ higher resistance.

Understanding:

We must find the stopping distance of a train under a braking force.

Formula:

First find deceleration using Newton's second law:

Then use the equation of motion:

Step 1: Calculate deceleration

Step 2: Find stopping distance

Answer:

The train travels $200\text{m}$ before stopping.

Quick Tip:

Always convert $\text{km/h}$ to $\text{m/s}$ by multiplying by $\frac{5}{18}$. Here $72\times \frac{5}{18}=20\text{m/s}$.

Understanding:

We need to find the speed of sound using its wavelength and frequency.

Formula:

The wave speed relation is:

Step 1: Substitute values

Answer:

The speed of sound in air is $340\text{m/s}$.

Quick Tip:

The speed of sound in air at $0°\text{C}$ is approximately $332\text{m/s}$ and at room temperature ($\approx 25°\text{C}$) is about $346\text{m/s}$. The standard value $340\text{m/s}$ is widely used in competitive exams.

Understanding:

We need to find the potential energy at height $h$ and the kinetic energy just before impact, using energy conservation.

Formula:

Potential energy:

By conservation of mechanical energy (no air resistance), all PE converts to KE:

Step 1: Calculate potential energy

Step 2: Apply energy conservation

Just before hitting the ground, height $=0$, so all potential energy converts to kinetic energy:

Answer:

The potential energy at $5\text{m}$ is $100\text{J}$ and the kinetic energy just before impact is also $100\text{J}$.

Quick Tip:

By the law of conservation of mechanical energy, in the absence of friction or air resistance, the total mechanical energy remains constant. Therefore $K{E}_{\text{bottom}}=P{E}_{\text{top}}$.

Understanding:

The question asks about a joint India-US defence exercise focused on counter-terrorism and jungle warfare, held in Uttarakhand in 2023.

Step 1: Identify the exercise

Exercise Yudh Abhyas is the flagship joint military training exercise between the Indian Army and the United States Army. It has been held annually since 2004 and alternates between India and the United States.

Step 2: Confirm the 2023 edition details

The 19th edition of Exercise Yudh Abhyas 2023 was held at Auli, Uttarakhand, India. It focused on counter-terrorism operations, jungle warfare, and high-altitude combat training, reflecting the challenging terrain of the region.

Step 3: Eliminate other options

Exercise Shakti is a joint exercise between India and France. Exercise Malabar is a trilateral naval exercise between India, the USA, and Japan. Exercise Tasman Saber is conducted between the USA and Australia.

Answer:

Exercise Yudh Abhyas is the joint India-US military exercise held in Uttarakhand in 2023 focused on counter-terrorism and jungle warfare.

Understanding:

The question asks for the full form of MIRV in the context of India's Agni-V missile test under Mission Divyastra in 2024.

Step 1: Define MIRV

MIRV stands for Multiple Independently targetable Re-entry Vehicle. A MIRV-equipped missile can carry several warheads, each of which can be directed at a different target.

Step 2: Context of Mission Divyastra

India conducted Mission Divyastra in March 2024, successfully test-firing the Agni-V missile equipped with MIRV technology. This made India one of a select group of nations (USA, Russia, UK, France, China) to possess MIRV capability, significantly enhancing India's strategic deterrence.

Step 3: Eliminate other options

The other options are incorrectly expanded forms that do not correspond to the standard defence/aerospace terminology used for MIRV technology.

Answer:

MIRV stands for Multiple Independently targetable Re-entry Vehicle.

Understanding:

The question asks about the LCA variant inducted into the Indian Air Force in February 2024.

Step 1: Background of Tejas variants

Tejas Mark 1 is the baseline version already in service with the Indian Air Force in two squadrons. Tejas Mark 1A is an upgraded version featuring an Active Electronically Scanned Array (AESA) radar, enhanced electronic warfare suite, air-to-air refuelling capability, and modern avionics.

Step 2: 2024 induction

The first Tejas Mark 1A aircraft was handed over to the Indian Air Force on February 28, 2024, by Hindustan Aeronautics Limited (HAL) at a ceremony in Bengaluru, in the presence of Prime Minister Narendra Modi. The Indian Air Force placed an order for 83 Tejas Mark 1A jets.

Step 3: Eliminate other options

Tejas Mark 2 and Tejas Navy Mark 2 are still under development and had not been inducted by February 2024.

Answer:

Tejas Mark 1A was inducted into the Indian Air Force in February 2024.

Understanding:

The question asks about the displacement of INS Vikrant, India's first indigenously built aircraft carrier, commissioned in 2022.

Step 1: INS Vikrant overview

INS Vikrant (IAC-1) was commissioned on September 2, 2022, by Prime Minister Narendra Modi at Cochin Shipyard Limited, Kochi. It is the first aircraft carrier to be designed and built in India.

Step 2: Key specifications

INS Vikrant has a full load displacement of approximately 45,000 tonnes. It is about 262 metres long and 62 metres wide, and can carry up to 30 aircraft including MiG-29K fighter jets and helicopters such as Kamov Ka-31 and MH-60R.

Step 3: Eliminate other options

A displacement of 25,000 tonnes is too small for a fleet carrier. 35,500 tonnes is the displacement of older carriers. 65,000 tonnes is closer to US supercarriers like USS Gerald R. Ford class.

Answer:

INS Vikrant has a displacement of approximately 45,000 tonnes.

Understanding:

The question asks about the two countries that participate in Exercise Tasman Saber.

Step 1: Identify Exercise Tasman Saber

Exercise Tasman Saber is a major bilateral military exercise conducted between the United States and Australia. It is one of the largest combined exercises in the Indo-Pacific region.

Step 2: Nature of the exercise

The exercise involves joint training in warfighting skills, coordination, and interoperability between the US and Australian defence forces across land, sea, and air domains. It is conducted biennially.

Step 3: Eliminate other options

India and Australia conduct Exercise AUSTRAHIND (Army) and Exercise AUSINDEX (Navy). The USA and New Zealand conduct Exercise Kaikōura. Australia and the UK are partners but do not conduct an exercise named Tasman Saber.

Answer:

Exercise Tasman Saber is conducted between the United States and Australia.

Understanding:

The question asks about the helicopter-launched ATGM developed by DRDO that was successfully test-fired in 2023.

Step 1: Identify Helina

Helina (Helicopter-launched Nag) is an air-launched version of the Nag ATGM, developed by the Defence Research and Development Organisation (DRDO). It is designed to be launched from Advanced Light Helicopters (ALH-WSI) and is an infrared image-seeking, fire-and-forget missile.

Step 2: 2023 test details

Helina was successfully test-fired multiple times in 2023, demonstrating its capability against armoured targets. Its range is approximately 7–8 km and it can defeat tanks equipped with explosive reactive armour.

Step 3: Eliminate other options

Nag is the ground-launched version fired from a Namica vehicle, not helicopters. Prahar is a surface-to-surface ballistic missile. Rudram is an anti-radiation missile launched from aircraft, not helicopters targeting armoured vehicles.

Answer:

Helina (Helicopter-launched Nag) is the ATGM developed by DRDO that was test-fired from a helicopter platform in 2023.

Understanding:

The question asks about the country with which India signed a defence partnership in 2023 that involved the GE-414 engine technology transfer for Tejas Mk2.

Step 1: The GE-414 deal

During Prime Minister Narendra Modi's state visit to the United States in June 2023, India and the USA agreed on a landmark deal for GE Aerospace (General Electric) to co-produce the F414 jet engines with Hindustan Aeronautics Limited (HAL) in India. This was a historic technology transfer of a cutting-edge fighter jet engine.

Step 2: Significance

The F414 engine will power the Tejas Mark 2 fighter jet. This deal was considered a major breakthrough as the USA had not previously shared such sensitive engine technology with India. It falls under the India-US Defence Technology and Trade Initiative (DTTI).

Step 3: Eliminate other options

Russia supplied engines for older aircraft. France's Safran engines power helicopters. Israel is a partner mainly in aerospace and missile systems. None of these were involved in the GE-414 deal.

Answer:

India signed the GE-414 jet engine technology transfer deal with the United States of America in 2023.