NIMCET 2025 Previous Year Question Paper and Solution with PDF

The scores of students in a national level examination are normally distributed with a mean of 500 and a standard deviation of 100. If the value of the cumulative distribution of the standard normal random variable at 0.5 is 0.691, then the probability that a randomly selected student scored between 450 and 500 is

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Solution

Probability Between 450 and 500 (Normal Distribution)

The exam scores are normally distributed with a mean $\mu = 500$ and standard deviation $\sigma = 100$ .
Given: $P(Z \leq 0.5) = 0.691$
We need to find: $P(450 \leq X \leq 500)$

Step 1: Convert scores to Z-scores

Use the formula: $Z = \frac{X - \mu}{\sigma}$

For $X = 500$ : $Z = \frac{500 - 500}{100} = 0$ For $X = 450$ : $Z = \frac{450 - 500}{100} = -0.5$

Step 2: Find the probability using cumulative values

We calculate: $P(450 \leq X \leq 500) = P(-0.5 \leq Z \leq 0) = P(Z \leq 0) - P(Z \leq -0.5)$

From symmetry: $P(Z \leq -0.5) = 1 - P(Z \leq 0.5) = 1 - 0.691 = 0.309$ and $P(Z \leq 0) = 0.5$

Step 3: Final Calculation

$P(-0.5 \leq Z \leq 0) = 0.5 - 0.309 = \boxed{0.191}$

✅ Final Answer:

The probability that a randomly selected student scored between 450 and 500 is: 0.191

Qus : 6NIMCET PYQ 2025

Number of permutations of the letters of the word BANGLORE such that the string ANGLE appears together in all permutations, is

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Solution

Qus : 7NIMCET PYQ 2025

Let A and B be two square matrices of same order satisfying

$A^2+5A+5I =0$ and

$B^2+3B+I=0$ repectively. Where I is the identity matrix. Then the inverse of the matrix

$C= BA+2B+2A+4I$ is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 8NIMCET PYQ 2025

The captains of five cricket teams, including India and Australia, are lined up randomly next to one other for a group photo. What is the probability that the captains of India and Australia will stand next to each other?

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 9NIMCET PYQ 2025

The value of

$\frac{d}{dx}\int ^{2\sin x}_{\sin {x}^2}{e}^{{t}^2}dt$ at

$x=\pi$

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 10NIMCET PYQ 2025

There are two coins, say blue and red. For blue coin, probability of getting head is 0.99 and for red coin, it is 0.01. One coin is chosen randomly and is tossed. The probability of getting head is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 11NIMCET PYQ 2025

The number of all even integers between 99 and 999 which are not multiple of 3 and 5 is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 12NIMCET PYQ 2025

Let A = {1,2,3, ... , 20}. Let

$R\subseteq A\times A$ such that R = {(x,y): y = 2x - 7}. Then the number of elements in R, is equal to

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Solution

Qus : 13NIMCET PYQ 2025

$\vec{a}, \vec{b}$ and

$\vec{c}$ are three vectors such that

$\vec{a} \times \vec{b}=\vec{c}$ ,

$\vec{a}.\vec{c} = 2$ and

$\vec{b}.\vec{c} = 1$ . If

$|\vec{b}| = 1$ , then the value of

$|\vec{a}|$ is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 14NIMCET PYQ 2025

If x, y and z are three cube roots of 27, then the determinant of the matrix

$\begin{bmatrix}{x} & {y} & {z} \\ {y} & {z} & {x} \\ {z} & {x} & {y}\end{bmatrix}$ is

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Solution

Qus : 15NIMCET PYQ 2025

Let

$A=\{{5}^n-4n-1\colon n\in N\}$ and

$B=\{{}16(n-1)\colon n\in N\}$ be sets. Then

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Solution

Qus : 16NIMCET PYQ 2025

Let

$\vec{a}$ ,

$\vec{b}$ and

$\vec{c}$ be unit vectors such that the angle between them is

${\cos }^{-1}\Bigg{\{}\frac{1}{4}\Bigg{\}}$ . If

$\vec{b}=2\vec{c}+\lambda \vec{a}$ , where

$\lambda$ > 0 and

$\vec{b}=4$ , then

$\lambda$ is equal to

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 17NIMCET PYQ 2025

A tower subtends angles

$\alpha, 2\alpha$ and

$3\alpha$ respectively at points A, B and C which are lying on a

horizontal line through the foot of the tower. Then

$\frac{AB}{BC}$ is equal to

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Solution

Qus : 18NIMCET PYQ 2025

$\vec{a}$ and

$\vec{b}$ are twp vectors such that |

$\vec{a}$ |=3, |

$\vec{b}$ |=4 and |

$\vec{a}+\vec{b}$ |=1, then the value of

$|\vec{a}-\vec{b}|$ is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 19NIMCET PYQ 2025

$\vec{a}=\hat{i}+\hat{j}+\hat{k}$ ,

$\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and

$\vec{c}=\hat{i}-2\hat{j}+\hat{k}$ , then a vector of magnitude

$\sqrt{22}$ which is parallel to

$2\vec{a}-\vec{b}+3\vec{c}$ is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 20NIMCET PYQ 2025

Consider the sample space

$\Omega={\{(x,y):x,y\in{\{1,2,3,4\}\}}}$ where each outcome is equally likely. Let A = {x ≥ 2} and B = {y > x} be two events. Then which of the following is NOT true?

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 21NIMCET PYQ 2025

Let the line

$\frac{x}{4}+\frac{y}{2}=1$ meets the x-axis and y-axis at A and B, respectively. M is the midpoint of side AB, and M' is the image of the point M across the line x + y = 1. Let the point P lie on the line x + y = 1 such that the

$\Delta$ ABP is an isosceles triangle with AP = BP. Then the distance between M' and P is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 22NIMCET PYQ 2025

Which one of the following is NOT a correct statement?

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 23NIMCET PYQ 2025

An equilateral triangle is inscribed in the parabola

$y^2 = x$ . One vertex of the triangle is at the vertex of the parabola. The centroid of triangle is

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Solution

Qus : 24NIMCET PYQ 2025

The angles of depression of the top and bottom of an 8m tall building from the top of a multi storied building are 30° and 45°, respectively. What is the height of the multistoried building and the distance between the two buildings?

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 25NIMCET PYQ 2025

The number of accidents per week in a town follows Poisson distribution with mean 2. If the probability that there are three accidents in two weeks time is

$ke^{-6}$ , then the value of k is

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Solution

Qus : 26NIMCET PYQ 2025

$B=sin^2 y+cos^4 y$ , then for all real y

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Solution

Qus : 27NIMCET PYQ 2025

Let

$F_1, F_2$ be foci of hyperbola

$\frac{{x}^2}{{a}^2}-\frac{{y}^2}{{b}^2}=1$ , a>0, b>0, and let O be the origin. Let M be an arbitrary point on curve C and above X-axis and H be a point on

$MF_1$ such that

$MF_2\perp{{F}}_1{{F}}_2$ ,

$MF_1\perp{{O}}{{H}}$ ,

$|OH|=\lambda |OF_2|$ with

$\lambda \in(2/5, 3/5)$ , then the range of the eccentricity

$e$ is

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Solution

Qus : 28NIMCET PYQ 2025

A circle with its center in the first quadrant touches both the coordinate axes and the line x -y - 2 = 0. Then the area of the circle is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 29NIMCET PYQ 2025

$\alpha$ and

$\beta$ are the two roots of the quadratic equation

$x^2 + ax + b = 0, (ab \ne 0)$ then the quadratic roots whose roots

$\frac{1}{\alpha^3+\alpha}$ and

$\frac{1}{\beta^3+\beta}$ is

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Solution

Qus : 30NIMCET PYQ 2025

The maximum value of

$sinx+sin(x+1)$ is

$k cos \frac{1}{2}$ . Then the value of k is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 31NIMCET PYQ 2025

Let

$\mathbb{R}\rightarrow\mathbb{R}$ be any function defined as

$f(x)=\begin{cases}{{x}^{\alpha}\sin \frac{1}{{x}^{\beta}}} & {,x\ne0} \\ {0} & {,x=0}\end{cases}$ ,

$\alpha , \beta \in \mathbb{R}$ . Which of the following is true? (

$\mathbb{R}$ denotes the set of all real numbers)

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 32NIMCET PYQ 2025

The number of 3-digit integers that are multiple of 6 which can be formed by using the digits 1,2,3,4,5,6 without repetition is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 33NIMCET PYQ 2025

The circle

$x^2 + y^2+ \alpha x+ \beta y+ \gamma=0$ is the image of the circle

$x^2 + y^2- 6x- 10y+ 30=0$ across the line 3x + y = 2. The value of

$[\alpha+ \beta+ \gamma]$ is (where [.] represents the floor function.)

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Solution

Qus : 34NIMCET PYQ 2025

Let

$\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ ,

$\vec{b}=\hat{i}+2\hat{j}-\hat{k}$ and

$\vec{c}=3\hat{i}+\hat{j}+\lambda \hat{k}$ be the co-terminal edges

of a parallelopiped whose volume is 5 units. Then the value of

$\lambda$ is

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Solution

Qus : 35NIMCET PYQ 2025

The area enclosed between the curve y = sin x, y = cosx,

$0\leq x\leq\frac{\pi}{2}$ is

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 36NIMCET PYQ 2025

Given the equation

$x+y =1$ ,

$x^2+y^2 =2$ ,

$x^5 +y^5 =A$ . Let N be the number of solution pairs (x,y) to this system of equations. Then AN is equal to

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Solution

Qus : 37NIMCET PYQ 2025

Number of three digit numbers that can be formed using 0, 1, 2, 3 and 5 where these digits are allowed to repeat any number of times, is equal to

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NIMCET Previous Year PYQNIMCET NIMCET 2025 PYQ

Solution

Qus : 38NIMCET PYQ 2025

Let

$g:\mathbb{R}\rightarrow \mathbb{R}$ and

$h:\mathbb{R}\rightarrow \mathbb{R}$ , be two functions such that

$h(x) = sgn(g(x))$ . Then select which of the following is not true?(

$\mathbb{R}$ denotes the set of all real numbers, sgn stands for signum function)

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Solution

Qus : 39NIMCET PYQ 2025

An airplane, when 4000m high from the ground, passes vertically above another airplane at an instant when the angles of elevation of the two airplanes from the same point on the ground are 60° and 30°, respectively. Find the vertical distance between the two airplanes.

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Solution

Qus : 40NIMCET PYQ 2025

Suppose

$t_1, t_2, ...t_55$ are in AP such that

$\sum ^{18}_{l=0}{{t}}_{3l+1}=1197$ and

${{t}}_7+{{3}}t_{22}=174$ . If

$\sum ^9_{l=1}{{{t}}_l}^2=947b$ , then the value of

$b$ is

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Solution

Qus : 41NIMCET PYQ 2025

Let E and F be two events such that P(E) > 0 and P(F) > 0. Which one of the following is NOT equivalent to the condition that

$P(E) =P(E|F)$ ?

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Solution

Qus : 42NIMCET PYQ 2025

What is the general solution of the equation

$tan \theta + cot \theta = 2$ ?

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Solution

Qus : 43NIMCET PYQ 2025

$cos^2(10°)cos(20°)cos(40°)cos(50°) cos(70°) = \alpha+\frac{\sqrt{3}}{16} cos(10°)$ , then

$3\alpha^{-1}$ is equal to

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Solution

Qus : 44NIMCET PYQ 2025

The slope of the normal line to the curve

$x = t^2 + 3t - 8$ and

$y = 2t^2 - 2t - 5$ at the point (2,-1) is

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Solution

Qus : 45NIMCET PYQ 2025

A tower subtends an angle of 30° at a point on the same level as the foot of the tower. At a second point h meters above the first, the depression of the foot of the tower is 60°. What is the horizontal distance of the tower from the point?

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Solution

Qus : 46NIMCET PYQ 2025

There are 40 female and 20 male students in a class. If the average heights of female and male students are 5.15 feet and 5.66 feet, respectively, then the average height (in feet) of all the students in the class equals

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