If the determiant `|{:(x+a,p+u,l+f),(y+b,q+v,m+g),(z+c,r+w,n+h):}|` splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.

Let f be an injective map with domain {x ,\ y ,\ z} and range {1, 2, 3} such that exactly one of the following statements is correct and the remaining are false. f(x)=1,\ \ f(y)!=1,\ f(z)!=2 . The value of f^(-1)(1) is (a) x (b) y (c) z (d) none of these

A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let r be the distance of the body frm the centre of the star and let its linear velocity be v, angular velocity omega , kinetic energy K, gravitational potential energy U, total energy E and angular momentum p. As the radius r of the orbit increases, determine which of the above quantities increase and which one decrease.

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. The centre of the sphere (x-4)(x+4)+(y-3)(y+3)+z^2=0 is

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius a is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. Equation of the sphere having centre at (3, 6, -4) and touching the plane rcdot(2hat(i)-2hat(j)-hat(k))=10 is (x-3)^2+(y-6)^2+(z+4)^2=k^2 , where k is equal to

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. Radius of the sphere, with (2, -3, 4) and (-5, 6, -7) as xtremities of a diameter, is

Assume X, Y, Z, W and P are matrices of order 2 × n , 3 × k , 2 × p , n × 3 and p × k respectively.If n = p, then the order of the matrix 3X - 6Z is:

Let f(x)=x+2|x+1|+x-1| . If f(x)=k has exactly one real solution, then the value of k is (a) 3 (b) 0 (c) 1 (d) 2

Let L denotes the number of subjective functions f : A -> B , where set A contains 4 elementset B contains 3 elements. M denotes number of elements in the range of the function f(x) = sec^-1(sgmx) + cosec^-1(sgn x) where sg n x denotes signum function of x. And N denotes coefficient of t^5 in (1+t^2)^5(1+t^3)^8 . The value of (LM+N) is lambda , then the value of lambda/19 is

In a three-dimensional coordinate system, P ,Q ,a n dR are images of a point A(a ,b ,c) in the x-y ,y-za n dz-x planes, respectively. If G is the centroid of triangle P Q R , then area of triangle A O G is ( O is the origin) a. 0 b. a^2+b^2+c^2 c. 2/3(a^2+b^2+c^2) d. none of these

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A,B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?