`f(x)=(x-1)|(x-2)(x-3)|dot` Then `f` decreases in (a)`(2-1/(sqrt(3)),2)` (b) `(2,2+1/(sqrt(3)))` (c)`(2+1/(sqrt(3)),3)` (d) `(3,oo)`

If |cos^(-1)((1-x^2)/(1+x^2))| < pi/3,t h e n (a) x in [-1/3,1/(sqrt(3))] (b) x in [-1/(sqrt(3)),1/(sqrt(3))] x in [0,1/(sqrt(3))] (d) none of these

4 cos^(2)x + sqrt(3) = 2(sqrt(3)+1)

int (dx)/(sqrt((2)/(3)x^(2)-x+(1)/(3)))

Factorise : (sqrt(3)-1)x^(2)-2x-(sqrt(3)+1)

If tan^(-1)x+2cot^(-1)x=(2pi)/3, then x , is equal to (a) (sqrt(3)-1)/(sqrt(3)+1) (b) 3 (c) sqrt(3) (d) sqrt(2)

d/(dx)[tan^(-1)((sqrt(x)(3-x))/(1-3x))] is (a) 1/(2(1+x)sqrt(x)) (b) 3/((1+x)sqrt(x)) (c) 2/((1+x)sqrt(x)) (d) 3/(2(1+x)sqrt(x))

If 3sin^(-1)((2x)/(1+x^2))-4cos^(-1)((1-x^2)/(1+x^2))+2tan^(-1)((2x)/(1-x^2))=pi/3, w h e r e|x|<1, then x is equal to 1/(sqrt(3)) (b) -1/(sqrt(3)) (c) sqrt(3) (d) -(sqrt(3))/4

The straight line 3x+4y-12=0 meets the coordinate axes at Aa n dB . An equilateral triangle A B C is constructed. The possible coordinates of vertex C (a) (2(1-(3sqrt(3))/4),3/2(1-4/(sqrt(3)))) (b) (-2(1+sqrt(3)),3/2(1-sqrt(3))) (c) (2(1+sqrt(3)),3/2(1+sqrt(3))) (d) (2(1+(3sqrt(3))/4),3/2(1+4/(sqrt(3))))

The values of parameter a for which the point of minimum of the function f(x)=1+a^2x-x^3 satisfies the inequality (x^2+x+2)/(x^2+5x+6)<0a r e (a) (2sqrt(3),3sqrt(3)) (b) -3sqrt(3),-2sqrt(3)) (c) (-2sqrt(3),3sqrt(3)) (d) (-2sqrt(2),2sqrt(3))

If I=int(sinx+sin^3x)/(cos2x)dx=Pcosx+Qlog|f(x)|+R , then (a) P=1/2,Q=-3/(4sqrt(2)) (b) P=1/4, Q=1/(sqrt(2)) (c) f(x)=(sqrt(2)cosx+1)/(sqrt(2)cosx-1) (d) f(x)=(sqrt(2)cosx-1)/(sqrt(2)cosx+1)