NIMCET Mathematics Previous Year Question Paper and Solution with PDF

Solution

Qus : 4NIMCET PYQ

If $\vec{a}=\hat{i}-\hat{k}$, $\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$, then $\begin{bmatrix}{\vec{a}} & {\vec{b}} & {\vec{c}}\end{bmatrix}$ depends on

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Solution

Qus : 5NIMCET PYQ

Angle of elevation of the top of the tower from 3 points (collinear) A, B and C on a road leading to the foot of the tower are 30°, 45° and 60°, respectively. The ratio of AB and BC is

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Solution

According to the given information, the figure should be as follows.

Let the height of tower = h

Qus : 6NIMCET PYQ

If $\vec{a}, \vec{b}$ are unit vectors such that $2\vec{a}+\vec{b} =3$ then which of the following statement is true?

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Solution

Qus : 7NIMCET PYQ

The area enclosed between the curves y² = x and y = |x| is

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Solution

Solving y² = x and y = x,

we get, y = 0, x = 0, y = 1, x = 1 Therefore,

Qus : 8NIMCET PYQ

$\int f(x)\mathrm{d}x=g(x)$, then $\int {x}^5f({x}^3)\mathrm{d}x$

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Solution

Qus : 9NIMCET PYQ

Test the continuity of the function at x = 2

$f(x)= \begin{cases} \frac{5}{2}-x & \text{ if } x<2 \\ 1 & \text{ if } x=2 \\ x-\frac{3}{2}& \text{ if } x>2 \end{cases}$

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Solution

LHL ≠ f(2)

Qus : 10NIMCET PYQ

$\lim _{{x}\rightarrow1}\frac{{x}^4-1}{x-1}=\lim _{{x}\rightarrow k}\frac{{x}^3-{k}^2}{{x}^2-{k}^2}=$, then find k

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Solution

Qus : 11NIMCET PYQ

The value of

2tan^-1[cosec(tan^-1x) - tan(cot^-1x)]

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Solution

Qus : 12NIMCET PYQ

The graph of function $f(x)=\log _e({x}^3+\sqrt[]{{x}^6+1})$ is symmetric about:

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Solution

Qus : 13NIMCET PYQ

If $3 sin x + 4 cos x = 5$, then $6tan\frac{x}{2}-9tan^2\frac{x}{2}$

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Solution

Qus : 14NIMCET PYQ

If the equation $|x^2 – 6x + 8| = a$ has four real solution then find the value of $a$?

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Solution

Qus : 15NIMCET PYQ

If A is a subset of B and B is a subset of C, then cardinality of A ∪ B ∪ C is equal to

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Solution

Qus : 16NIMCET PYQ

Largest value of $cos^2\theta -6sin\theta cos\theta+3sin^2\theta+2 $ is

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Solution

Qus : 17NIMCET PYQ

Given to events A and B such that odd in favour A are 2 : 1 and odd in favour of $A \cup B$ are 3 : 1. Consistent with this information the smallest and largest value for the probability of event B are given by

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Solution

Qus : 18NIMCET PYQ

If A and B are square matrices such that $B=-A^{-1} BA$, then $(A + B)^2$ is

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Solution

Qus : 19NIMCET PYQ

A bag contain different kind of balls in which 5 yellow, 4 black & 3 green balls. If 3 balls are drawn at random then find the probability that no black ball is chosen

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Solution

Qus : 20NIMCET PYQ

Between any two real roots of the equation $e^x sin x = 1$, the equation $e^x cos x = –1$ has

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Solution

Qus : 21NIMCET PYQ

If f(x) is a polynomial of degree 4, f(n) = n + 1 & f(0) = 25, then find f(5) = ?

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Solution

Qus : 22NIMCET PYQ

The maximum value of $f(x) = (x – 1)^2 (x + 1)^3$ is equal to $\frac{2^p3^q}{3125}$ then the ordered pair of (p, q) will be

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Solution

Qus : 23NIMCET PYQ

The coefficient of $x^{50}$ in the expression of ${(1 + x)^{1000} + 2x(1 + x)^{999} + 3x^2(1 + x)^{998} + ...... + 1001x^{1000}}$

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Solution

Qus : 24NIMCET PYQ

If ${{x}}_k=\cos \Bigg{(}\frac{2\pi k}{n}\Bigg{)}+i\sin \Bigg{(}\frac{2\pi k}{n}\Bigg{)}$ , then $\sum ^n_{k=1}({{x}}_k)=?$

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Solution

Qus : 25NIMCET PYQ

Number of point of which f(x) is not differentiable $f(x)=|cosx|+3$ in $[-\pi, \pi]$

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Solution

Qus : 26NIMCET PYQ

If $n_1$ and $n_2$ are the number of real valued solutions $x = | sin^{–1} x |$ & $x = sin (x)$ respectively, then the value of $n_2– n_1$ is

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Solution

Qus : 27NIMCET PYQ

The negation of $\sim S\vee(\sim R\wedge S)$ is equivalent to

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Solution

Qus : 28NIMCET PYQ

A point P in the first quadrant, lies on $y^2 = 4ax$, a > 0, and keeps a distance of 5a units from its focus. Which of the following points lies on the locus of P?

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Solution

Qus : 29NIMCET PYQ

If $\int x\, \sin x\, sec^3x\, dx=\frac{1}{2}\Bigg{[}f(x){se}c^2x+g(x)\Bigg{(}\frac{\tan x}{x}\Bigg{)}\Bigg{]}+C$, then which of the following is true?

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Solution

Qus : 30NIMCET PYQ

Let a, b, c, d be no zero numbers. If the point of intersection of the line 4ax + 2ay + c = 0 & 5bx + 2by + d=0 lies in the fourth quadrant and is equidistance from the two are then

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Solution

Qus : 31NIMCET PYQ

$\theta={\cos }^{-1}\Bigg{(}\frac{3}{\sqrt[]{10}}\Bigg{)}$ is the angle between $\vec{a}=\hat{i}-2x\hat{j}+2y\hat{k}$ & $\vec{b}=x\hat{i}+\hat{j}+y\hat{k}$ then possible values of (x,y) that lie on the locus

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Solution

Qus : 32NIMCET PYQ

Let R be reflexive relation on the finite set a having 10 elements and if m is the number of ordered pair in R, then

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Solution

Qus : 33NIMCET PYQ

If $| x - 6|= | x - 4x | -| x^2- 5x +6 |$ , where x is a real variable

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Solution

Qus : 34NIMCET PYQ

The range of values of $\theta$ in the interval $(0,\pi)$ such that the points (3, 2) and $(cos\theta ,sin\theta)$ lie on the samesides of the line x + y – 1 = 0, is

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Solution

Qus : 35NIMCET PYQ

Which of the following number is the coefficient of $x^{100}$ in the expansion of $\log _e\Bigg{(}\frac{1+x}{1+{x}^2}\Bigg{)},\, |x|{\lt}1$ ?

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Solution

Qus : 36NIMCET PYQ

A real valued function f is defined as $f(x)=\begin{cases}{-1} & {-2\leq x\leq0} \\ {x-1} & {0\leq x\leq2}\end{cases}$. Which of the following statement is FALSE?

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Solution

Qus : 37NIMCET PYQ

A line segment AB of length 10 meters is passing through the foot of the perpendicular of a pillar, which is standing at right angle to the ground. Top of the pillar subtends angles $tan^{–1}$ 3 and $tan^{–1} 2$ at A and B respectively. Which of the following choice represents the height of the pillar?

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Solution

Qus : 38NIMCET PYQ

If a vector having magnitude of 5 units, makes equal angle with each of the three mutually perpendicular axes,then the sum of the magnitude of the projections on each of the axis is

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Solution

Qus : 39NIMCET PYQ

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ballsis transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred is red, is:

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Solution

Qus : 40NIMCET PYQ

Let $f(x)=\frac{x^2-1}{|x|-1}$. Then the value of $lim_{x\to-1} f(x)$ is

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Solution

Qus : 41NIMCET PYQ

The mean of 5 observation is 5 and their variance is 12.4. If three of the observations are 1,2 and 6; then the mean deviation from the mean of the data is:

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Solution

Qus : 42NIMCET PYQ

In a beauty contest, half the number of experts voted Mr. A and two thirds voted for Mr. B 10 voted for both and 6 did not for either. How may experts were there in all.

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Solution

Let the total number of experts be

N

E

is the set of experts who voted for miss

A

F

is the set of experts who voted for miss

B

.
Since

6

did not vote for either, n

(E \cup F) = N - 6

n (E) = \frac{N}{2}, n (F) = \frac{2}{3} N

and

n (E \cap F) = 10

.
So,

N - 6 = \frac{N}{2} + \frac{2}{3} N - 10

Solving the above equation gives

\frac{N}{6} = 4 \Rightarrow N = 24

Qus : 43NIMCET PYQ

A circle touches the x–axis and also touches the circle with centre (0, 3) and radius 2. The locus of the centre of the circle is

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Solution

Qus : 44NIMCET PYQ

The value of non-zero scalars α and β such that for all vectors

and

such that

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Solution

Qus : 45NIMCET PYQ

A force of 78 grams acts at the point (2,3,5). The direction ratios of the line of action being 2,2,1 . The magnitude of its moment about the line joining the origin to the point (12,3,4) is

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Solution

Qus : 46NIMCET PYQ

Number of real solutions of the equation

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Solution

Qus : 47NIMCET PYQ

The sum of infinite terms of a decreasing GP is equal to the greatest value of the function

in the interval [-2,3] and the difference between the first two terms is f'(0). Then the common ratio of GP is

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Solution

Qus : 48NIMCET PYQ

Number of onto (surjective) functions from A to B if n(A)=6 and n(B)=3, is

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Solution

Qus : 49NIMCET PYQ

If $|z|<\sqrt{3}-1$, then $|z^{2}+2z cos \alpha|$ is

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Solution

Qus : 50NIMCET PYQ

A computer producing factory has only two plants T₁ and T₂ produces 20% and plant T₂ produces 80% of the total computers produced. 7% of the computers produced in the factory turn out to be defective. It is known that P (computer turns out to be defective given that it is produced in plant T₁ 10P(computer turns out to be defective given that it is produced in plant T₂ ). A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant T₂ is

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Solution

Qus : 51NIMCET PYQ

If A > 0, B > 0 and A + B = $\frac{\pi}{6}$ , then the minimum value of $tanA + tanB$

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NIMCET Previous Year PYQNIMCET NIMCET 2022 PYQ

Solution

On differentiating

x= tanA + tan(π/6-A)

we get :

dx/dA = sec²A-sec²(π/6-A)

now putting

dx/dA=0

we get

cos²(A) = cos²(π/6-A) so 0≤A≤π/6

therefore

A=π/6-A from here we get A = π/12 = B

so minimum value of that function is

2tanπ/12 which is equal to 2(2-√3)

Qus : 52NIMCET PYQ

Inverse of the function $f(x)=\frac{10^x-10^{-x}}{10^{x}+10^{-x}}$ is

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Solution

Let $f(x)=y$, then

⇒ $\frac{10^x-10^-x}{10^x+10^-x}=y$

⇒ $\frac{10^{2x}-1}{10^{2x}+1}=y$

⇒ $10^{2x}=\frac{1+y}{1-y}$ By Componendo Dividendo Rule

⇒ $x=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+y}{1-y}\Bigg{)}$

⇒ ${f}^{-1}(y)=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+y}{1-y}\Bigg{)}$

⇒ ${f}^{-1}(x)=\frac{1}{2}\log _{10}\Bigg{(}\frac{1+x}{1-x}\Bigg{)}$

Qus : 53NIMCET PYQ

The tangent at the point (2, -2) to the curve $x^2 y^2-2x=4(1-y)$ does not passes through the point

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Solution

Qus : 54NIMCET PYQ

The integral $\int \sqrt{1+2 cot x(cosec x+cotx)} dx$ , $(0<x<\frac{\pi}{2})$ (where C is a constant of integration) is equal to

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Solution

Qus : 55NIMCET PYQ

If all the words, with or without meaning, are written using the letters of the word QUEEN add are arranged as in English Dictionary, then the position of the word QUEEN is

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Solution

Letters of the word QUEEN are E,E,N,Q,U

Words beginning with E (4!) = 24

Words beginning with N (4!/2!)=12

Words beginning with QE (3!) = 6

Words beginning with QN (3!/2!)= 3

Total words = 24+12+6+9=45

QUEEN is the next word and has rank 46th.

Qus : 56NIMCET PYQ

The curve satisfying the differential equation

and passing through the point (1,1) also passes through the point __________

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Solution

Qus : 57NIMCET PYQ

is equal to

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Solution

Qus : 58NIMCET PYQ

If S and S' are foci of the ellipse

, B is the end of the minor axis and BSS' is an equilateral triangle, then the eccentricity of the ellipse is

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Solution

Qus : 59NIMCET PYQ

The equation of the circle passing through the point (4,6) and whose diameters are along x + 2y - 5 =0 and 3x - y - 1=0 is

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Solution

Qus : 60NIMCET PYQ

In a parallelogram ABCD, P is the midpoint of AD. Also, BP and AC intersect at Q. Then AQ : QC =

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Solution

Qus : 61NIMCET PYQ

The median AD of ΔABC is bisected at E and BE is extended to meet the side AC in F. The AF : FC =

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Solution

Qus : 62NIMCET PYQ

Let $X_i, i = 1,2,.. , n$ be n observations and $w_i = px_i +k, i = 1,2, ,n$ where p and k are constants. If the mean of $x_i 's$ is 48 and the standard deviation is 12, whereas the mean of $w_i 's$ is 55 and the standard deviation is 15, then the value of p and k should be

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Solution

Qus : 63NIMCET PYQ

Not Available

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Solution

Qus : 64NIMCET PYQ

, then the values of A₁, A₂, A₃, A₄ are

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Solution

Qus : 65NIMCET PYQ

then x =

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Solution

Qus : 66NIMCET PYQ

Equation of the tangent from the point (3,−1) to the ellipse 2x² + 9y² = 3 is

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Solution

Qus : 67NIMCET PYQ

The position vectors of the vertices

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Solution

Qus : 68NIMCET PYQ

The $sin^2 x tanx + cos^2 x cot x-sin2x=1+tanx+cotx $, $x \in (0 , \pi)$, then x

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Solution

Qus : 69NIMCET PYQ

Not Available right now

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Solution

Qus : 70NIMCET PYQ

In a chess tournament, n men and 2 women players participated. Each player plays 2 games against every other player. Also, the total number of games played by the men among themselves exceeded by 66 the number of games that the men played against the women. Then the total number of players in the tournament is

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Solution

Qus : 71NIMCET PYQ

Suppose A₁ , A₂ , A₃ , …..A₃₀ are thirty sets each having 5 elements with no common elements across the sets and B₁ , B₂ , B₃ , ..... , B_n are

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Solution

Qus : 72NIMCET PYQ

Let

,where [x]denotes the greatest integer

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Solution

Qus : 73NIMCET PYQ

Let U and V be two events of a sample space S and P(A) denote the probability of an event A. Which of the following statements is true?

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Solution

Qus : 74NIMCET PYQ

If a man purchases a raffle ticket, he can win a first prize of Rs.5,000 or a second prize of Rs.2,000 with probabilities 0.001 and 0.003 respectively. What should be a fair price to pay for the ticket?

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Solution

Qus : 75NIMCET PYQ

If the mean deviation 1, 1+d, 1+2d, … , 1+100d from their mean is 255, then d is equal to

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Solution

Qus : 76NIMCET PYQ

and

, then a possible value of n is among the following is

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Solution

Qus : 77NIMCET PYQ

Let S be the set $\{a\in Z^+:a\leq100\}$.If the equation $[tan^2 x]-tan x - a = 0$ has real roots (where [ . ] is the greatest integer function), then the number of elements is S is

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Solution

Qus : 78NIMCET PYQ

The solution set of the inequality

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Solution

Qus : 79NIMCET PYQ

If $a, b, c$ are in GP and $log a - log 2b$, $log 2b - log 3c$ and $log 3c - log a$ are in AP, then $a, b, c$are the lengths of the sides of a triangle which is

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Solution

Qus : 80NIMCET PYQ

Not Available

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Solution

Qus : 81NIMCET PYQ

If (1 + x – 2x²)⁶= 1 + a₁x + a₂x²+ ... + a₁₂x¹², then the value a₂+ a₄+ a₆+ ... + a₁₂

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Solution

Qus : 82NIMCET PYQ

Not Available

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Solution

Qus : 83NIMCET PYQ

A man takes a step forward with probability 0.4 and backward with probability 0.6. The probability that at the end of eleven steps, he is one step away from the starting point is

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Solution

Qus : 84NIMCET PYQ

Not Available

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Solution

Qus : 85NIMCET PYQ

If x, y, z are distinct real numbers then

= 0, then xyz=

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Solution

Qus : 86NIMCET PYQ

Not Available

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Solution

Qus : 87NIMCET PYQ

If $a, a, a_2, ., a_{2n-1},b$ are in AP, $a, b_1, b_2,...b_{2n-1}, b $are in GP and $a, c_1, c_2,... c_{2n-1}, b $ are in HP, where a, b are positive, then the equation $a_n x^2-b_n+c_n$ has its roots

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Solution

Qus : 88NIMCET PYQ

For the two circles $x^2+y^2=16$ and $x^2+y^2-2y=0$, there is/are

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Solution

Qus : 89NIMCET PYQ

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