Question.13. If A and B are focus of hyperbola 25[(x-1)^2+(y+1)^2]=k^2(3x+4y+5)^2,k>1 and P is a variable point on hyperbola such that triangle APB is right angled isosceles triangle. If K=tanα, α∈[0,2π] and sum of all values of α is (aπ/2b) (where a and b are co-prime numbers), then (7a+2b) is

Question.12. If Sr=A^r+B^r+C^r then prove the following ( Matrix and Determinant )

Question.11. Prove that : tanA/(1-cotA) + cotA/(1-tanA) = 1 + cosecA*secA ( Trigonometry )

Question.10. Find lim (n->infinity ) e^2n(n!)^2/(2n^(2n+1)) . ( Limit )

Here following concept is used : -

Question.9. If tanx+tan2x+tan3x=1, then the value of 2cos6x-2cos4x+cos2x is equal to ( Trigonometry )

(A) 1/2          (B) 2             (C) 1          (D) 3

Question.8. Value of k so that equation (x2-2x)2-3(x2-2x)+(k+2)=0 has 2 real solution- ( Quadratic Equation )

(A) (-infinity,-6)                             (B){1/4}

(C) (-infinity,-6)U{1/4}                  (D)none of these

Question.7. If -3sin2A+2sin2B=1 and 3sin2A-2sin2B=0, where A,B are positive acute angles, then A-2B = ? ( Trigonometry )

(A) pi/2       (B) pi/3         (C) pi/4         (D) pi/6

NOTE : - Here A=Alpha and B= Beta

Question.6. If -5+iβ,-5+iγ, β2!=γ2,β,γ∈R are roots of x3+15x2+cx+860=0, c∈R then ( Roots of Equation )

(A) c=222

(B) all the three roots are imaginary

(C) two roots are imaginary but not complex conjugate of each other.

(D) -5+7sqrt(3)i, -5-7sqrt(3)i are imaginary roots.

Question.5. Show that the straight lines represented by 3x2+48xy+23y2=0 and 3x-2y+13=0 form an equilateral triangles of area 13/sqrt(3) sq. units. ( Straight Line )

                 

Question.4. Suppose f:[0,1]->R is Differentiable function. If f(0)=0, f(x)=0 for all x belongs (0,1), then which of the following is necessarily true - ( Differentiability )

(A) f'(c)/f(c)=f'(1-2c)/f(1-2c) for at least one c belongs (0,1/2)

(B) f'(c)/f(c)=f'(1-3c)/f(1-3c) for at least one c belongs (0,1/3)

(C) f'(c)/f(c)=3f'(1-3c)/2f(1-3c)+1/(1-2c) for at least one c belongs (0,1/3)

(D) f'(c)/f(c)=3f'(1-3c)/2f(1-3c)+1/(1-2c) for at least one c belongs (1/3,1/2)