The det `Delta=|[d^2+r, de, df],[ de, e^2+r, ef],[ df, ef, f^2+r]|` is divisible by
(a)`r^2`
(b) `(d+e^2+f^2+r)`
(c)`(d^2+e^2+f^2+r)`
(d) `(d^2+e+f^2+r^2)`

Let a , b , c , d , e ,f in R such that a d+b e+cf=sqrt((a^2+b^2+c^2)(d^2+e^2+f^2)) Use vector to prove that (a+b+c)/(sqrt(b^2+b^2+c^2))=(d+e+f)/(sqrt(d^2+e^2+f^2))

f: R->R is defined by f(x)=(e^(x^2)-e^(-x^2))/(e^(x^2)+e^(-x^2)) is :

(a)/(b) = (c)/(d) = (e)/(f) = k(a, b, c, d, e, f gt 0) then (A) k = (a + c +e)/(b + d + f) (B) k = (a^(2) + c^(2) +e^(2))/(b^(2) + d^(2) + f^(2)) (C) k = (c^(2)e^(2))/(d^(2)f^(2)) (D) k = ((ace)/(bdf))^((1)/(3))

Angle of intersection of two circle having distance between their centres d is given by : (A) cos theta = (r_1^2 + r_2^2 - d)/(2r_1^2 + r_2^2) (B) sec theta = (r_1^2 + r_2^2 + d^2)/(2r_1 r_2) (C) sec theta = (2r_1 r_2)/(r_1^2 + r_2^2 - d^2 (D) none of these

For all def ne 0, (4d^(3)e^(4)f^(2))/((2de^(3)f)^(2))=

In A B C , A D and B E are altitudes. Prove that (a r( D E C))/(a r( A B C))=(D C^2)/(A C^2)

From a point O in the interior of a A B C , perpendiculars O D ,\ O E and O F are drawn to the sides B C ,\ C A and A B respectively. Prove that: (i) A F^2+B D^2+C E^2=O A^2+O B^2+O C^2-O D^2-O E^2-O F^2 (ii) A F^2+B D^2+C E^2=A E^2+C D^2+B F^2

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

Let R be a relation on NxxN defined by (a , b)\ R(c , d)hArra+d=b+c\ if\ (a , b),(c , d) in NxxN then show that: (i)\ (a , b)R\ (a , b)\ for\ a l l\ (a , b) in NxxN (ii)\ (a , b)R(c , d)=>(c , d)R(a , b)for\ a l l\ (a , b),\ (c , d) in NxxN (iii)\ (a , b)R\ (c , d)a n d\ (c , d)R(e ,f)=>(a , b)R(e ,f)\ for all \ (a , b),\ (c , d),\ (e ,f) in NxxN

A B C D E F , a r( A B C)=9c m^2 , a r( D E F)=16 c m^2 . If B C=2. 1 c m , then the measure of E F is 2. 8 c m (b) 4. 2 c m (c) 2. 5 c m (d) 4. 1 c m