The vector equals

is a tetrahedron in which is the origin and position vector of points A,B,C  are and respectively. A value of a for which shortest distance between and is

𝐓𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐰𝐚𝐲𝐬 𝐢𝐧 𝐰𝐡𝐢𝐜𝐡 𝐭𝐡𝐞 𝐟𝐨𝐮𝐫 𝐟𝐚𝐜𝐞𝐬 𝐨𝐟 𝐚 𝐫𝐞𝐠𝐮𝐥𝐚𝐫 𝐭𝐞𝐭𝐫𝐚𝐡𝐞𝐝𝐫𝐨𝐧 𝐜𝐚𝐧 𝐛𝐞 𝐩𝐚𝐢𝐧𝐭𝐞𝐝 𝐰𝐢𝐭𝐡 𝐟𝐨𝐮𝐫 𝐝𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐭 𝐜𝐨𝐥𝐨𝐮𝐫𝐬 𝐢𝐬

T𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐰𝐚𝐲𝐬 𝐢𝐧 𝐰𝐡𝐢𝐜𝐡 𝟓 𝐢𝐝𝐞𝐧𝐭𝐢𝐜𝐚𝐥 𝐛𝐚𝐥𝐥𝐬 𝐜𝐚𝐧 𝐛𝐞 𝐤𝐞𝐩𝐭 𝐢𝐧 𝟏𝟎 𝐢𝐝𝐞𝐧𝐭𝐢𝐜𝐚𝐥 𝐛𝐨𝐱𝐞𝐬, 𝐢𝐟 𝐧𝐨𝐭 𝐦𝐨𝐫𝐞 𝐭𝐡𝐚𝐧 𝐨𝐧𝐞 𝐜𝐚𝐧 𝐠𝐨 𝐢𝐧𝐭𝐨 𝐚 𝐛𝐨𝐱, 𝐢𝐬

𝐓𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐰𝐚𝐲𝐬 𝐭𝐨 𝐟𝐢𝐥𝐥 𝐞𝐚𝐜𝐡 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐨𝐮𝐫 𝐜𝐞𝐥𝐥𝐬 𝐨𝐟 𝐭𝐡𝐞 𝐭𝐚𝐛𝐥𝐞 𝐰𝐢𝐭𝐡 𝐚 𝐝𝐢𝐬𝐭𝐢𝐧𝐜𝐭 𝐧𝐚𝐭𝐮𝐫𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 𝐬𝐮𝐜𝐡 𝐭𝐡𝐚𝐭 𝐭𝐡𝐞 𝐬𝐮𝐦 𝐨𝐟 𝐭𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 𝐢𝐬 𝟏𝟎 𝐚𝐧𝐝 𝐭𝐡𝐞 𝐬𝐮𝐦𝐬 𝐨𝐟 𝐭𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 𝐩𝐥𝐚𝐜𝐞𝐝 𝐝𝐢𝐚𝐠𝐨𝐧𝐚𝐥𝐥𝐲 𝐚𝐫𝐞 𝐞𝐪𝐮𝐚𝐥 𝐢𝐬

𝐒𝐢𝐱 𝐱′ 𝐬 𝐡𝐚𝐯𝐞 𝐭𝐨 𝐛𝐞 𝐩𝐥𝐚𝐜𝐞𝐝 𝐢𝐧 𝐭𝐡𝐞 𝐬𝐪𝐮𝐚𝐫𝐞 𝐨𝐟 𝐠𝐢𝐯𝐞𝐧 𝐟𝐢𝐠𝐮𝐫𝐞 𝐬𝐮𝐜𝐡 𝐭𝐡𝐚𝐭 𝐞𝐚𝐜𝐡 𝐫𝐨𝐰 𝐜𝐨𝐧𝐭𝐚𝐢𝐧𝐬 𝐚𝐭𝐥𝐞𝐚𝐬𝐭 𝐨𝐧𝐞 𝐱. 𝐓𝐡𝐞𝐧 𝐭𝐡𝐞 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐰𝐚𝐲𝐬 𝐝𝐨𝐢𝐧𝐠 𝐬𝐨 𝐚𝐫𝐞

The number of positive integral solutions of  where n is positive integer, is

f𝐨𝐮𝐫 𝐛𝐨𝐲𝐬 𝐩𝐢𝐜𝐤𝐞𝐝 𝐮𝐩 𝟑𝟎 𝐦𝐚𝐧𝐠𝐨𝐞𝐬. 𝐈𝐧 𝐡𝐨𝐰 𝐦𝐚𝐧𝐲 𝐰𝐚𝐲𝐬 𝐜𝐚𝐧 𝐭𝐡𝐞𝐲 𝐝𝐢𝐯𝐢𝐝𝐞 𝐭𝐡𝐞𝐦, 𝐢𝐟 𝐚𝐥𝐥 𝐦𝐚𝐧𝐠𝐨𝐞𝐬 𝐛𝐞 𝐢𝐝𝐞𝐧𝐭𝐢𝐜𝐚𝐥

A non zero vector is parallel to the line of intersection of the plane determined by the vectors and the plane determined by the vectors then the angle between and is

For such questions, we should know how to find the normal to the plane. We should know how to find the line of intersection of two planes. And, we should know how to take a cross product.

Let and . Then, the line of intersection of planes one determined by and other determined by is perpendicular to

For such questions, we should know how to find the line intersecting two planes. We should also know how to find the normal to the plane using two vectors.

𝐀 𝐠𝐮𝐚𝐫𝐝 𝐨𝐟 𝟏𝟐 𝐦𝐞𝐧 𝐢𝐬 𝐟𝐨𝐫𝐦𝐞𝐝 𝐟𝐫𝐨𝐦 𝐧 𝐬𝐨𝐥𝐝𝐢𝐞𝐫𝐬.𝐈𝐟 𝐀 𝐚𝐧𝐝 𝐁 𝐚𝐫𝐞 𝐭𝐡𝐫𝐞𝐞 𝐭𝐢𝐦𝐞𝐬 𝐚𝐬 𝐨𝐟𝐭𝐞𝐧 𝐭𝐨𝐠𝐞𝐭𝐡𝐞𝐫 𝐨𝐧 𝐠𝐮𝐚𝐫𝐝 𝐚𝐬 𝐂, 𝐃, 𝐄 𝐭𝐡𝐞𝐧 𝐧 =

Number of ways of selecting 3 consecutive objects out of 15 objects (distinct) placed around a circle is

If , then 

The number of ways in which 6 gentlemen and 3 ladies be seated round a table so that every gentleman may have a lady by his side is