NATA Sample Papers & Mock Test - 8 Free Online Test 2026

The centre of the circle is (-a, -b) and the radius is √(a² - b²).
So, the equation of the circle is (x + a)² + (y + b)² = (√(a² - b²))².
⇒ x² + y² + 2ax + 2by + 2b² = 0
Hence, the correct answer is A: x² + y² + 2ax + 2by + 2b² = 0.

Let the height of the building be AB and its shadow be BC. Let the angle of elevation of the sun be ∠C = θ.
Given: θ = 30°, AB = 50 m.
To find BC, we have:
⇒ tan(θ) = AB / BC
⇒ tan(30°) = 50 / BC
⇒ 1/√3 = 50 / BC
∴ BC = 50√3 m
Hence, the length of the shadow is 50√3 m.

Since the triangle is right angled so the circumcenter will be the middle point of hypotenuse i.e., (2, 1).

Let 

Let, x⁶ = t
Differentiate w.r.t.  x,

Using formula

Now, 

Hence, correct option is (A)

Given, r = 0.8 and bᵧₓ = 0.2
∴ r² = bₓᵧ * bᵧₓ⇒ (0.8)² = bₓᵧ * (0.2)
⇒ 0.64 = bₓᵧ * 0.2
⇒ bₓᵧ = 0.64 / 0.2 = 3.2
Hence, the regression coefficient bₓᵧ = 3.2.

Given:
Divisor = 12 times the quotient
Divisor = 5 times the remainder
Remainder = 48
Thus, Divisor = 48 × 5 = 240
Now, Quotient = Divisor ÷ 12 = 240 ÷ 12 = 20
Finally, the dividend is calculated as:
Dividend = Quotient × Divisor + Remainder
Dividend = 20 × 240 + 48 = 4800 + 48 = 4848
So, the correct answer is D) 4848.

Let the given point be P = (3, 4, 5).

The distance of point P from the x-axis is given by the formula:

√(y² + z²) = √(4² + 5²)
= √(16 + 25)
= √41

Therefore, the correct answer is D) √41.

If log₁₀a, log₁₀b, log₁₀c are in A.P.,
Then, by the formula:
y = (x + z) / 2,
where:
x = log₁₀a,
y = log₁₀b,
z = log₁₀c.

This leads to:
log₁₀b = (log₁₀a + log₁₀c) / 2
⇒ 2log₁₀b = log₁₀a + log₁₀c
⇒ log₁₀(b²) = log₁₀(ac)
⇒ b² = ac

This is the condition for the variables to be in G.P.

Hence, the correct answer is B.

Checking by option one by one:

(i) For n = 1, the value of the expression 47^n + 16^n - 1 is 62, which is not divisible by 4. Hence, this option is discarded.

(ii) For n = 1, the value of the expression 2(42^n + 1) - 33^n + 1 is 47, which is not divisible by 9. Hence, this option is discarded.

(iii) For n = 2, the value of the expression 4^n - 3^n - 1 is 9, which is not divisible by 11. Hence, this option is discarded.

(iv) For any value of n, this option always satisfies the condition for divisibility by 17, and hence this is the correct option.

Given that, the trigonometric expression is Δ 

We know that 

= s²
Thus the value of the given trigonometric expression is s².

using the properties


= 9 + 36 − 343 = −298.

According to the sum rule,

We are given the function y = f(ax² + b).
To differentiate y with respect to x, we apply the chain rule:
dy/dx = f′(ax² + b) * d/dx (ax² + b)
Now, differentiate the inner function (ax² + b) with respect to x:
d/dx (ax² + b) = 2ax
Thus, we have:
dy/dx = f′(ax² + b) * 2ax
Therefore, the correct answer is: B) 2ax f′(ax² + b).

Given,


Vectors  and  are parallel in the same direction and vectors  and  are also parallel but are in the opposite direction.
Hence,  and  are in opposite directions.
So, the angle between them is π.

In the given question we have to find the relation between N and P.

• M’s son is Q’s brother
• P and Q are sisters

It means M is the father of P. 
M and N are brothers
Thus, N is the uncle of P.
So, the correct answer is option (B).

In the given figure series
An element is moving 90ºClock-wise and every time a new element is getting added alternately. Then newly added element is moved clockwise, and new elements is added. 

• In figure 2, square is moves clockwise in figure and new element circle is added.

Similarly,

• In figure 3, circle moves clockwise and triangle is added.
• In figure 4, triangle moves clockwise and a plus sign is added.

To complete the sequence , in option figure plus sign should be moved clockwise from its current position and a new element is to be added.

• In option B, plus sign move clockwise, but the element added i.e triangle is a repeated element.
• In option C,  both the elements are not as per the description 
• In option D, plus sign moves clockwise but two places clockwise and triangle is repeated element.

In option A. plus sign moves clockwise and new element a shaded star is added. Thus, it fulfils the requirement to complete the series. 

Hence, the answer is option .

Total number of members: Seven
R likes Orange and his favourite sport is Cricket. The one who likes Pink does not like Chess. Therefore, his favourite sport is Football.
Q's favourite sport is shooting, and he likes Black. S likes Red. W likes Basketball. P likes Volleyball. T likes Pink.

The one who likes Volleyball does not like Brown and Green. Therefore, he likes blue.
The one who likes Basketball does not like Green. Therefore, he likes brown.
V's choice of colour is neither Blue nor Black. Therefore, he likes green. T's favourite sport is neither Hockey nor Basketball. 
The final arrangement is:

Therefore, V likes green colour.
Hence, 'green' is the correct answer.

According to the question, the given figure:

In the first figure, the hexagon is inscribed in a circle and in the second figure the circle is inscribed in the hexagon.
Similarly,
In the third circle, the pentagon is inscribed in a triangle and in the fourth figure the triangle must be inscribed in a pentagon.
Hence, Option D is the correct option.

Organic architecture is a type of architectural design wherein buildings are inspired by, built around, and blend in with their natural surroundings.
The term organic architecture was coined by the American architect Frank Lloyd Wrigh.

• The idea behind organic architecture is to put human life, nature and the built environment on the same level.
• To build a sustainable ecosystem where all components support one another and thrive as a result.

Nanoparticles of chemical elements have properties like accumulation in the environment because of their small size and complex structure. They also have the capability to enter food chains and to trigger the production of free radicals which are chemical species that affect the environment (especially ozone layer) adversely. They can be easily inhaled in the body and can cause lung inflammation and heart problems.

Colour theory can be defined as practical guidance to colour mixing and the visual effects of a specific colour combination.
The collection of rules and guidelines which designers use to communicate with users through appealing colour schemes in visual interfaces is known as colour theory.
The science and art of using colours, which explains the visual effects of colours mix, match and contrast.
Colour theory is also used to explain how humans perceive colour. 
A formalization of "color theory" began in the 18th century, initially within a partisan controversy over Isaac Newton's theory of color

Red is used in depicting many things, all of them are potent.
Red is popularly used to specify romantic love, and its physical passion ,at the same time it even depicts violence, anger, and aggression, that indicates danger.
Paradoxically–for a colour associated with action and energy–red is universally used as the colour that means "stop."

Earthquakes contribute to environmental pollution by rendering a large number of dust particles into the air while the automobiles and industries contribute to pollution by the emission of harmful gases and chemicals into the air and water bodies. Sewage also is responsible for the pollution of water bodies.
All these sources, other than Earthquake are artificial sources of pollution while the earthquake is a natural source.

The Visual Elements of art are Line, Shape, Tone, Colour, Pattern, Texture and Form. When we analyse any drawing, painting, sculpture or design, we examine these component parts to see how they combine to create the overall effect of the artwork.

Example- 

• Lines cross over one another to form shapes.
• Shapes can be filled with tone and Colour, or repeated to create pattern
• A shape may be rendered with a rough surface to create a texture.

The indefinite article 'a' is should be used here.
They are used to refer to a person or a thing that needs to be represented as a group or a single entity even though the group is formed up using singular individuals.
As in this example, we refer to a dozen of oranges as a group or single entity.
The article 'a' connects the noun (a dozen oranges) to the verb (bought).

A retarder is used to increase the initial setting time of the cement.
The substance added as a retarder usually slow down the rate of a chemical change.
Gypsum retarder contain 4 types-

1. Organic acids
2. Phosphates
3. Proteins
4. Composite retarder

Piazza del Campo is the principal public space of the historic center of Siena, Tuscany, Italy.

• It is regarded as one of Europe's greatest medieval squares.
• It is renowned worldwide for its beauty and architectural integrity.
• Piazza del Campo was intended as an area where the entire population could meet for activities, games and political events, thus it was built to hold the entire town.

Vincent Willem van Gogh was a Dutch Post-Impressionist painter who posthumously became one of the most famous and influential figures in Western art history.
The Starry Night is an impressionist style of oil-on-canvas painting, painted in June 1889.
The painting was painted in the year 1889 during his stay in Saint-Paul asylum in Saint-Remy, France during his breakdown and the mutilation of his left year.
The painting is a scene showing just before sunrise, with the addition of an imaginary village.

The Finger Plan is an urban plan from 1947 which provides a strategy for the development of the Copenhagen metropolitan area, Denmark.

• According to the plan, Copenhagen is to develop along five 'fingers', centered on S-train commuter rail lines, which extend from the 'palm', that is the dense urban fabric of central Copenhagen. In between the fingers, green "wedges" are intended to provide land for agriculture and recreational purposes.
• The Finger Plan for Greater Copenhagen was introduced in 1947 when a group of town planners realized that Greater Copenhagen was beginning to spread uncontrollably.

Any colour that is vivid or bold in nature is termed as warm colour.

• They contrast with cool colours.
• They tend to be advanced in space.
• Warm colours are stimulating and they heighten emotions.
• They are used to conveying messages of happiness, sociability and energy.