Chapter 2

$A$ line $L$ with slope 2 cuts parabola $y^2=8x$ to form a chord. If mid- point of chord lies on $x=4$ then equation of line $L$ is (1) $y=2x-6$ (2) $x=2y-6$ (3) $2x+y=6$ (4) $x-2y=6$

The order of differential equation of family of circles passing through intersection of $L\equiv 3x+4y-7=0$ and $S\equiv x^2+y^2-2x-2y+1=0$ is $(1)1$ (2) 2 (3) 3 (4) 4

Let $ABCD$ be a square with $A(0,0),C(2,2)$. If $\mathrm{M}$ is mid- point of $\mathrm{AB}$ and $\mathrm{P}$ is a variable point on $\mathrm{BC}$. The smallest value of $\mathrm{DP}+\mathrm{PM}$ is (1) $\sqrt{13}$ (2) $\sqrt{2}+\sqrt{5}$ (3) $2\sqrt{3}$ (4) $1+2\sqrt{2}$

The equation of plane through intersection of the planes $x+2y+z-1=0$ and $2x+y+3z-2=0$ and perpendicular to plane $x+y+z=1$ is (1) $x+4y-3z+1=0$ (2) $x-4y+3z-1=0$ (3) $x-4y+3z=0$ (4) $x+4y-3z=0$

A man running round a race course notes that the sum of the distance of two flag - posts from him is always 10 meters and distance between the flag - posts is 8 meters. The area of the path he encloses in square meters is (1) $15\pi$ (2) $12\pi$ (3) $18\pi$ (4) $8\pi$

Let $\mathrm{H}$ be a set of hyperbolas. If a relation $\mathrm{R}$ on $\mathrm{H}$ be defined by $\text{Missing left or extra right}$, then the relation $R$ is (1) reflexive and symmetric but not transitive (2) symmetric and transitive but not reflexive (3) reflexive and transitive but not symmetric (4) Equivalence relation

Let $\omega =\frac{-1+i\sqrt{3}}{2}$, and $\omega ,\omega ,{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_8$ be roots of equation $x^{10}+ax+b=0$, where ${\alpha }_1\ne {\alpha }_i,i\ne j$ value of $(\omega -{\alpha }_1)(\omega -{\alpha }_2)\dots .(\omega -{\alpha }_8)$ $=$ (1) $90{\omega }^2$ (2) ${}^{10}{\mathrm{C}}_2{\omega }^2$ (3) $10\omega +a$ (4) 0

If $x_1$ and $x_2$ are the real and distinct roots of $a{x}^2+bx+c=0$, then ${\mathrm{Lim}}_{x\to x_1}{(1+\sin (a{x}^2+bx+c))}^{\frac{1}{x-x_1}}$ is equal to (1) $e^{x_1-x_2}$ (2) $e^{x_2-x_1}$ (3) $e^{a(x_1-x_2)}$ (4) $e^{a(x_2-x_1)}$

Let $p,q$ be real numbers such that $\frac{1}{p}-\frac{1}{q}=1$ and \(0

${\int }_0^{\pi /2}\frac{xdx}{1+\sin x}=$ (1) $\mathrm{n}2+\pi /4$ (2) $\ell \mathrm{n}2-\pi /4$ (3) $1+\pi /2+\ell n2$ (4) $\ell \mathrm{n}2$