In zone-refining method the molten zone

Four newly  married couples are dancing at a function the selection of the partner is random. The number of ways that exactly one husband  is not dancing with his own wife is

There are 12 intermediate stations on a railway line between 2 stations. The number  number of ways that a train can be made to stop at 4 of these intermediate stations no two of these halting stations being consecutive is

The number of ways of forming an arrangement of 4 letters from the letters of the word "IITJEE" is

 equal to

is equal to

If then

For such questions, we should know how to find the dot product and cross product. Dot product of two vectors is always a scalar quantity. The cross product of two vectors is a vector quantity.

If are non coplanar vectors then the roots of equation

The point of intersection of the plane will the stright line passing through the origin and perpendicular to the plane is

The position vector of the centre of the circle

The locus of midpoints of chords of  which subtend a right angle  at the centre is

𝐓𝐡𝐞𝐫𝐞 𝐚𝐫𝐞 𝐭𝐰𝐨 𝐔𝐫𝐧𝐬. 𝐔𝐫𝐧 𝐀 𝐡𝐚𝐬 𝟑 𝐝𝐢𝐬𝐭𝐢𝐧𝐜𝐭 𝐫𝐞𝐝 𝐛𝐚𝐥𝐥𝐬 𝐚𝐧𝐝 𝐮𝐫𝐧 𝐁 𝐡𝐚𝐬 𝟗 𝐝𝐢𝐬𝐭𝐢𝐧𝐜𝐭 𝐛𝐥𝐮𝐞 𝐛𝐚𝐥𝐥𝐬. 𝐅𝐫𝐨𝐦 𝐞𝐚𝐜𝐡 𝐮𝐫𝐧 𝐭𝐰𝐨 𝐛𝐚𝐥𝐥𝐬 𝐚𝐫𝐞 𝐭𝐚𝐤𝐞𝐧 𝐨𝐮𝐭 𝐚𝐭𝐫𝐚𝐧𝐝𝐨𝐦 𝐚𝐧𝐝 𝐭𝐡𝐞𝐧 𝐭𝐫𝐚𝐧𝐬𝐟𝐞𝐫𝐫𝐞𝐝 𝐭𝐨 𝐭𝐡𝐞 𝐨𝐭𝐡𝐞𝐫. 𝐓𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐰𝐚𝐲𝐬 𝐢𝐧 𝐰𝐡𝐢𝐜𝐡 𝐭𝐡𝐢𝐬 𝐜𝐚𝐧 𝐛𝐞 𝐝𝐨𝐧𝐞 𝐢𝐬

𝐓𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐞𝐫𝐦𝐮𝐭𝐚𝐭𝐢𝐨𝐧𝐬 𝐟𝐫𝐨𝐦 𝐭𝐡𝐞 𝐥𝐞𝐭𝐭𝐞𝐫𝐬 𝐀 𝐭𝐨 𝐆 𝐬𝐨 𝐭𝐡𝐚𝐭 𝐧𝐞𝐢𝐭𝐡𝐞𝐫 𝐭𝐡𝐞 𝐬𝐞𝐭 𝐁𝐄𝐆 𝐧𝐨𝐫 𝐭𝐡𝐞 𝐬𝐞𝐭 𝐂𝐀𝐃 𝐚𝐩𝐩𝐞𝐚𝐫𝐬 𝐢𝐬