Multiple Topics Questions - JEE

Match the following: **Column-I:** I) Distance from origin to plane through $(1,1,1)$ perpendicular to $x/3=y/0=z/4$ is $p/q$ (coprime). Value of $(p-q)$. II) $\displaystyle\lim_{\delta x\to 0}\sum_{i=1}^n x_i\,\delta x$ over $[0,4]$ divided into $n$ equal parts. III) $f(x)\geq M$ for all $x$, where $xf(x)+(1-x)f(-x)=x^2+x+1$. $M=p/q$ (coprime). Value of $(q-p)$. IV) Trapezium $ABCD$ with $AB\parallel CD$, $AD\perp AB$, $AB=3CD=4$, inscribed circle radius $r$. Value of $4r^2$. **Column-II:** P) 1, Q) 2, R) 3, S) 6, T) 8

Match the following: **Column-I:** I) Number of values of $x$ of the form $1/n$ ($n\in\mathbb{N}$) in $[1/15,1/10]$ satisfying $\{x\}+\{2x\}+\cdots+\{12x\}=78x$. II) $(2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_r x^r$. Value of $a_9/a_{11}$. III) $G$ = centroid of $\triangle ABC$ (sides $a,b,c$). $P$ with $PA=1$, $PB=2$, $PC=1$, $PG=1$. Value of $(a^2+b^2+c^2)/9$. IV) $f: \mathbb{R}^+\to\mathbb{R}$ satisfies $f(xy)=f(x)+f(y)$, $f(16)=3$. Value of $f(1/2)+f(1/4)$. **Column-II:** P) 1, Q) 2, R) 3, S) 4, T) 10

Match the following: **Column-I:** I) In $\triangle ABC$ with $A(-2,3)$, $B(4,-6)$, $C(1,1)$: $CS_1$, $CS_2$ divide the triangle into 3 equal areas ($S_1,S_2$ on $AB$). Lines through origin parallel to $CS_1$, $CS_2$ form $y^2+\lambda xy-\mu x^2=0$. Value of $(\lambda+\mu)$. II) Area enclosed by $y=x$ and $x^2+y-2=0$ is $p/q$ (coprime). Value of $(p-q)$. III) Curve satisfying $dy/dx=(x^2+y^2)/(2xy)$, passing through $(2,1)$ is a hyperbola with eccentricity $e$. Value of $e^4$. IV) $\displaystyle\lim_{n\to\infty}\frac{1}{n^{2m}}[(n^2+1^2)^m(n^2+2^2)^m\cdots(n^2+n^2)^m]^{1/n}=\left(\frac{a}{e^2}\cdot\frac{e^{\pi/2}}{2}\right)^m$. Value of $a$. **Column-II:** P) 2, Q) 4, R) 6, S) 7, T) 8

Match the following: **Column-I:** I) Number of values of $x$ of the form $1/n$ ($n\in\mathbb{N}$) in $[1/15,1/10]$ satisfying $\{x\}+\{2x\}+\cdots+\{12x\}=78x$. II) $(2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_r x^r$. Value of $a_9/a_{11}$. III) $G$ = centroid of $\triangle ABC$ (sides $a,b,c$). $P$ with $PA=1$, $PB=2$, $PC=1$, $PG=1$. Value of $(a^2+b^2+c^2)/9$. IV) $f: \mathbb{R}^+\to\mathbb{R}$ satisfies $f(xy)=f(x)+f(y)$, $f(16)=3$. Value of $f(1/2)+f(1/4)$. **Column-II:** P) 1, Q) 2, R) 3, S) 4, T) 10

Match the following: **Column-I:** I) Distance from origin to plane through $(1,1,1)$ perpendicular to $x/3=y/0=z/4$ is $p/q$ (coprime). Value of $(p-q)$. II) $\displaystyle\lim_{\delta x\to 0}\sum_{i=1}^n x_i\,\delta x$ over $[0,4]$ divided into $n$ equal parts. III) $f(x)\geq M$ for all $x$, where $xf(x)+(1-x)f(-x)=x^2+x+1$. $M=p/q$ (coprime). Value of $(q-p)$. IV) Trapezium $ABCD$ with $AB\parallel CD$, $AD\perp AB$, $AB=3CD=4$, inscribed circle radius $r$. Value of $4r^2$. **Column-II:** P) 1, Q) 2, R) 3, S) 6, T) 8

Match the following: **Column-I:** I) In $\triangle ABC$ with $A(-2,3)$, $B(4,-6)$, $C(1,1)$: $CS_1$, $CS_2$ divide the triangle into 3 equal areas ($S_1,S_2$ on $AB$). Lines through origin parallel to $CS_1$, $CS_2$ form $y^2+\lambda xy-\mu x^2=0$. Value of $(\lambda+\mu)$. II) Area enclosed by $y=x$ and $x^2+y-2=0$ is $p/q$ (coprime). Value of $(p-q)$. III) Curve satisfying $dy/dx=(x^2+y^2)/(2xy)$, passing through $(2,1)$ is a hyperbola with eccentricity $e$. Value of $e^4$. IV) $\displaystyle\lim_{n\to\infty}\frac{1}{n^{2m}}[(n^2+1^2)^m(n^2+2^2)^m\cdots(n^2+n^2)^m]^{1/n}=\left(\frac{a}{e^2}\cdot\frac{e^{\pi/2}}{2}\right)^m$. Value of $a$. **Column-II:** P) 2, Q) 4, R) 6, S) 7, T) 8

Match the following: **Column-I:** I) Remainder when $25^{24}\cdot 26$ is divided by $7$. II) $x^8+y^8+6=8xy$ where $x,y\in\mathbb{Z}$. Number of ordered pairs $(x,y)$. III) $y=3x-8$ is tangent at $(7,13)$ to a parabola with focus $(-1,-1)$ and latus rectum length $l$. Value of $[l]$. IV) Hyperbola with centre at origin, one focus at $(6,8)$, two directrices $3x+4y\pm 10=0$, eccentricity $e$. Value of $\dfrac{4e^2}{5}$. **Column-II:** P) 0, Q) 1, R) 2, S) 3, T) 4

Multiple Topics Questions (8)