Mathematical Reasoning & Induction · JEE Advanced previous year questions

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be defined by $$f(x)=\{\begin{array}{ll}\frac{6x+\sin x}{2x+\sin x}& \text{if }x\ne 0,\\ \frac{7}{3}& \text{if }x=0.\end{array}$$ Then which of the following statements is (are) TRUE?

1. A.The point $x=0$ is a point of local maxima of $f$
2. B.The point $x=0$ is a point of local minima of $f$
3. C.Number of points of local maxima of $f$ in the interval $[\pi ,6\pi ]$ is 3
4. D.Number of points of local minima of $f$ in the interval $[2\pi ,4\pi ]$ is 1

Let $S=\left\{a+b\sqrt{2}:a,b\in \mathbb{Z}\right\}$, $T_1=\left\{(-1+\sqrt{2}{)}^n:n\in \mathbb{N}\right\}$, and $T_2=\left\{(1+\sqrt{2}{)}^n:n\in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

1. A.$\mathbb{Z}\cup T_1\cup T_2\subset S$
2. B.$T_1\cap \left(0,\frac{1}{2024}\right)=\varphi$, where $\varphi$ denotes the empty set.
3. C.$T_2\cap (2024,\infty )\ne \varphi$
4. D.For any given $a,b\in \mathbb{Z}$, $\cos \left(\pi (a+b\sqrt{2})\right)+i\sin \left(\pi (a+b\sqrt{2})\right)\in \mathbb{Z}$ if and only if $b=0$, where $i=\sqrt{-1}$.

Let ${\mathbb{R}}^2$ denote $\mathbb{R}\times \mathbb{R}$. Let $$S=\{(a,b,c):a,b,c\in \mathbb{R}\text{ and }a{x}^2+2bxy+c{y}^2>0\text{ for all }(x,y)\in {\mathbb{R}}^2-\{(0,0)\}\}.$$ Then which of the following statements is (are) TRUE?

1. A.$\left(2,\frac{7}{2},6\right)\in S$
2. B.If $\left(3,b,\frac{1}{12}\right)\in S$, then $|2b|<1$.
3. C.For any given $(a,b,c)\in S$, the system of linear equations $ax+by=1$, $bx+cy=-1$ has a unique solution.
4. D.For any given $(a,b,c)\in S$, the system of linear equations $(a+1)x+by=0$, $bx+(c+1)y=0$ has a unique solution.

Let ${\mathbb{R}}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $dist(X,Y)$ denote the distance between two points $X$ and $Y$ in ${\mathbb{R}}^3$. Let $$S=\left\{X\in {\mathbb{R}}^3:(dist(X,P){)}^2-(dist(X,Q){)}^2=50\right\}\text{ and}$$ $$T=\left\{Y\in {\mathbb{R}}^3:(dist(Y,Q){)}^2-(dist(Y,P){)}^2=50\right\}.$$ Then which of the following statements is (are) TRUE?

1. A.There is a triangle whose area is $1$ and all of whose vertices are from $S$.
2. B.There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $LM$ is also in $T$.
3. C.There are infinitely many rectangles of perimeter $48$, two of whose vertices are from $S$ and the other two vertices are from $T$.
4. D.There is a square of perimeter $48$, two of whose vertices are from $S$ and the other two vertices are from $T$.

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $$f(x)=\{\begin{array}{ll}{x}^2\sin \left(\frac{\pi }{x^2}\right),& \text{if }x\ne 0,\\ 0,& \text{if }x=0.\end{array}$$ Then which of the following statements is TRUE?

1. A.$f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}},\infty \right)$.
2. B.$f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi },\infty \right)$.
3. C.The set of solutions of $f(x)=0$ in the interval $\left(0,\frac{1}{10^{10}}\right)$ is finite.
4. D.$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{{\pi }^2},\frac{1}{\pi }\right)$.

Let $f:[0,1]\to [0,1]$ be the function defined by $f(x)=\frac{x^3}{3}-x^2+\frac{5}{9}x+\frac{17}{36}$. Consider the square region $S=[0,1]\times [0,1]$. Let $G=\{(x,y)\in S:y>f(x)\}$ be called the green region and $R=\{(x,y)\in S:y<f(x)\}$ be called the red region. Let $L_h=\{(x,h)\in S:x\in [0,1]\}$ be the horizontal line drawn at a height $h\in [0,1]$. Then which of the following statements is(are) true?

1. A.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the green region below the line $L_h$
2. B.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the red region below the line $L_h$
3. C.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the green region above the line $L_h$ equals the area of the red region below the line $L_h$
4. D.There exists an $h\in \left[\frac{1}{4},\frac{2}{3}\right]$ such that the area of the red region above the line $L_h$ equals the area of the green region below the line $L_h$