MCQ
The remainder when $19^{200}+23^{200}$ is divided by $49$ , is $.........$.
- A$28$
- B$27$
- ✓$29$
- D$26$
$\Rightarrow 2\left[{ }^{100} C _0 21^{200}+200 C _2 21^{198} \cdot 2^2+\ldots . .+{ }^{200} C _{198} 21^2\right.$
$\left.2^{198}+2^{200}\right]$
$\Rightarrow 2\left[49 I _1+2^{200}\right]=49 I _1+2^{201}$
Now, $2^{201}=(8)^{67}=(1+7)^{67}=49 I _2+{ }^{67} C _0{ }^{67} C _1 \cdot 7=$ $49 I _2+470=49 I _2+49 \times 9+29$
$\therefore$ Remainder is $29$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| $column 1$ | $column 2$ | $column 3$ |
| $(I)$ $x^2+y^2=a^2$ | $(i)$ $m y=m^2 x+a$ | $(P)$ $\left(\frac{a}{m^2}, \frac{2 a}{m}\right)$ |
| $(II)$ $x^2+a^2 y^2=a^2$ | $(ii)$ $y=m x+a \sqrt{m^2+1}$ | $(Q)$ $\quad\left(\frac{-m a}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$ |
| $(III)$ $y^2=4 a x$ | $(iii)$ $y=m x+\sqrt{a^2 m^2-1}$ | $(R)$ $\quad\left(\frac{-a^2 m}{\sqrt{a^2 m^2+1}}, \frac{1}{\sqrt{a^2 m^2+1}}\right)$ |
| $(IV)$ $x^2-a^2 y^2=a^2$ | $(iv)$ $y=m x+\sqrt{a^2 m^2+1}$ | $(S)$ $\quad\left(\frac{-a^2 m}{\sqrt{a^2 m^2-1}}, \frac{-1}{\sqrt{a^2 m^2-1}}\right)$ |
($1$) The tangent to a suitable conic (Column $1$) at $\left(\sqrt{3}, \frac{1}{2}\right)$ is found to be $\sqrt{3} x+2 y=4$, then which of the following options is the only CORRECT combination?
$[A] (II) (iii) (R)$ $[B] (IV) (iv) (S)$ $[C] (IV) (iii) (S)$ $[D] (II) (iv) (R)$
($2$) If a tangent to a suitable conic (Column $1$) is found to be $y=x+8$ and its point of contact is $(8,16$ ), then which of the following options is the only CORRECT combination?
$[A] (III) (i) (P)$ $[B] (III) (ii) (Q)$ $[C] (II) (iv) (R)$ $[D] (I) (ii) (Q)$
($3$) For $a=\sqrt{2}$, if a tangent is drawn to a suitable conic (Column $1$ ) at the point of contact $(-1,1)$, then which of the following options is the only CORRECT combination for obtaining its equation?
$[A] (II) (ii) (Q)$ $[B] (III) (i) (P)$ $[\mathrm{C}]$ $(I) (1) (P)$ $[D] (I) (ii) (Q)$