Let the line intersect the plane containing the lines and at the point . Then the value of equals _____________. ...

Equation of plane containing the line 4ax - y + 5z - 7a = 0 = 2x - 5y - z - 3 can be written as 4ax - y + 5z - 7a + (2x - 5y - z - 3) = 0 (4a + 2 )x - (1 + 5 )y \\\\ + (5 - )z - (7z + 3 ) = 0 Which is coplanar with the line x - 4 1 = y + 1 - 2 = z 1 4(4a + 2 ) +...

Let the line frac x37frac y21frac z34 intersect the plane co: null (Mathematics)

Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$ and $4 a x-y+5 z-7 a=0=2 x-5 y-z-3, a \in \mathbb{R}$ at the point $P(\alpha, \beta, \gamma)$ . Then the value of $\alpha+\beta+\gamma$ equals _____________.

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated, is

The area of the region enclosed by the parabolas y=4x-x^2 and 3y=(x-4)^2 is equal to

Area of the region (x,y):x^2+(y-2)^2≤4,x^2≥2y is

For (2 leq r leq n,binom{n}{r} + 2binom{n}{r - 1} + binom{n}{r - 2} =)

The integral int_0^1 \frac{1}{7^{\left[\frac{1}{x}\right]}} {dx}, where [.] denotes the greatest integer function is equal to

Let A = { 1,3,4,6,9} and B = { 2,4,5,8,10}. Let R be a relation defined on A X B such that R = {( ( a_{1},b_{1} ), ( a_{2},b_{2} )):a_{1} ? b_{2} and b_{1} ? a_{2}}. Then the number of elements in the set R is

For integers n and r let n r nCr ifn ge r ge 0 0 otherwise

Let Z be the set of integers. If A = { x in Z:2^{(x + 2) ( x^{2} - 5x + 6 )} = 1 } and = { x in Z : - 3 < 2x - 1 < 9}, then the . number of subsets of the set A X B is

The minimum number of elements that must be added to the relation R = {(a,b),(b,c),(b,d)} on the set { a,b,c,d} so that it is an equivalence relation is

The area enclosed by the curves xy+4y=16 and x+y=6 is equal to :

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