Question 41. The value of k, for which 
(cos x + sin x)${}^2$ + k sin x cos x - 1 = 0 is an identity, is

Question 42. If for all real values of $x,\frac{4x^2+1}{64x^2-96xsin\alpha +5}<\frac{1}{32}$, then $\alpha$ lies in the interval.

Question 43. If $\theta$ is an acute angle and $sin\theta =\frac{p-6}{8-p},$ then p must satisfy

Question 44.   $IfA,B,C$ be the angle of a triangle, the $\sum \frac{cotA+cotB}{tanA+tanB}$
 

Question 45. If $A=si{n}^8\theta +co{s}^{14}\theta$, then for all real value of $\theta$

Question 46.

Question 47. Answer the questions on the basis of the information given below:
In the adjoining figure, I and II are circles with centres P and Q respectively. The two circles touch each other and have a common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the ratio 4 : 3. It is also known that the length of PO is 28 cm.

What is the radius of the circle II?                                

Question 48. Answer the questions on the basis of the information given below:
In the adjoining figure, I and II are circles with centres P and Q respectively. The two circles touch each other and have a common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the ratio 4 : 3. It is also known that the length of PO is 28 cm.

The length of SO is:  

Question 49. In the adjoining figure, points A, B, C and D lie on the circle, AD = 24 and BC = 12. What is the ratio of the area of triangle $\Delta$CBE to that of the triangle $\Delta$ADE ? (CAT 1997)

Question 50. AB > AF > BF; CD > DE > CE; and BF = 6$\sqrt{5}$ cm. which is the closest pair of points among all the six given points?            (CAT 1999)