Consider the function $$f(x)=\begin{cases}{-{x}^3+3{x}^2+1,} & {if\, x\leq2} \\ {\cos x,} & {if\, 2{\lt}x\leq4} \\ {{e}^{-x},} & {if\, x{\gt}4}\end{cases}$$  Which of the following statements about f(x) is true:

1

The value of ${{Lt}}_{x\rightarrow0}\frac{{e}^x-{e}^{-x}-2x}{1-\cos x}$ is equal to 

Topic: NIMCET 2024

2

Consider the function $f(x)={x}^{2/3}{(6-x)}^{1/3}$. Which of the following statement is false?

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3

A coin is thrown 8 number of times. What is the probability of getting a head in an odd number of throw?

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4

Let $f(x)=\begin{cases}{{x}^2\sin \frac{1}{x}} & {,\, x\ne0} \\ {0} & {,x=0}\end{cases}$Then which of the …

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5

Given a set A with median $m_1 = 2$ and set B with median $m_2 = 4$What can we say about the median of the combined set?

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6

The value of $f(1)$ for $f\Bigg{(}\frac{1-x}{1+x}\Bigg{)}=x+2$ is

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7

Let Z be the set of all integers, and consider the sets $X=\{(x,y)\colon{x}^2+2{y}^2=3,\, x,y\in Z\}$ and $Y=\{(x,y)\co…

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8

Let A and B be two events defined on a sample space $\Omega$. Suppose $A^C$ denotes the complement of A relative to the…

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9

The value of $\sum ^n_{r=1}\frac{{{{}^nP}}_r}{r!}$ is:

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10

Among the given numbers below, the smallest number which will be divided by 9, 10, 15 and 20, leaves the remainders 4, …

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1

If one AM (Arithmetic mean) 'a' and two GM's (Geometric means) p and q be inserted between any two positive numbers, th…

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2

The equation $3x^2 + 10xy + 11y^2 + 14x + 12y + 5 = 0$ represents

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3

The points (1,1/2) and (3,-1/2) are

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4

How much work does it take to slide a crate for a distance of 25m along a loading dock by pulling on it with a 180 N fo…

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5

The vector $\vec{A}=(2x+1)\hat{i}+(x^2-6y)\hat{j}+(xy^2+3z)\hat{k}$ is a

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6

Region R is defined as region in first quadrant satisfying the condition $x^2 + y^2 < 4$. Given that a point P=(r,s)…

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7

Lines $L_1, L_2, .., L_10 $are distinct among which the lines $L_2, L_4, L_6, L_8, L_{10}$ are parallel to each other a…

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8

If the line $a^2 x + ay +1=0$, for some real number $a$, is normal to the curve $xy=1$ then

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9

Out of a group of 50 students taking examinations in Mathematics, Physics, and Chemistry, 37 students passed Mathematic…

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10

Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f(0)=\frac{1}{\pi}$ and $f(x)=\frac{x}{e^{\p…

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