Un triangolo ha la base di 21 cm e l'altezza di 8 cm . Se si aumenta l'alte \( z z a \) di 2 cm di quan- to si deve aumentare o diminuire la base per mantenere invariata la superficie? [diminuire di \( 4,2 \mathrm{~cm} \) ]

How can you determine the surface area of a cube using its net diagram? Describe the steps involved.

Given the equations of lines \( \boldsymbol{l}_{1} \) and \( \boldsymbol{l}_{2} \) are \( \mathrm{r}=(\mathrm{i}+2 \mathrm{j}+3 \mathrm{k})+\lambda(\mathrm{i}-\mathrm{j}+2 \mathrm{k}) \) and \( \mathrm{r}=(5 \mathrm{i}+\mathrm{j}+5 \mathrm{k})+\mu 2 \mathrm{i}+\mathrm{j}-2 \mathrm{k}) \) respectively, where \( \lambda \) and \( \mu \) are real patameters. (a) Show that lines \( \boldsymbol{\ell}_{1} \) and \( \boldsymbol{\ell}_{2} \) intersect and hence, state the coordinates of the point of intersection. (b) Find the Cartesian equation of the plane \( \Pi_{1} \), which contains lines \( \boldsymbol{\ell}_{1} \) and \( \boldsymbol{\ell}_{2} \). (c) The planes \( \Pi_{2} \) and \( \Pi_{3} \) are with equations \( 2 \mathrm{r}+2 \mathrm{y}+z=3 \) and \( 2 \mathrm{r}+p y+2 z=2 \) respectively, where \( p \) is a constant. (i) Find the vector equation of the line of intersection between planes \( \Pi_{1} \) and \( \Pi_{2} \). (ii) Determine the value of \( p \) such that the planes \( \Pi_{1}, \Pi_{2} \) and \( \Pi_{3} \) do not have any common point.

(c) The planes \( \Pi_{2} \) and \( \Pi_{3} \) are with equations \( \left.2 x+2 y+1\right) \) respectively, where \( p \) is a constant. (i) Find the vector equation of the line of intersection between planes \( \Pi_{1} \) and \( \Pi_{2} \). (ii) Determine the value of \( p \) such that the planes \( \Pi_{1}, \Pi_{2} \) and \( \Pi_{3} \) do not have any common point.

यदि एक समांतर चतुर्भुज \( A B C D \) के बिंदुओं \( A, B \) तथा \( C \) के निर्देशांक क्रमश: \( (1,2),(3,4) \) तथा \( (2,5) \) है, तो विकर्ण \( A D \) का समीकरण है [JEE (Main) 2019] \( \begin{array}{ll}\text { (a) } 5 x+3 y-11=0 & \text { (b) } 3 x-5 y+7=0 \\ \text { (c) } 3 x+5 y-13=0 & \text { (d) } 5 x-3 y+1=0\end{array} \)

A circle whose centre \( (-2,5) \) passes through point \( P(10,10) \). Find the equation of the is the form of \( a x^{2}+b y^{2}+c x+d y+k=0 \). Where ab, c \( d \) and \( k \) are constants.