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If a function \( y=f(x) \) is such that \( f'(x)<0 \), then the number of integral values of ' \( a \) ' for which the major axis of ellipse \( f(a+11) x^{2} + f(a^{2}+2a+5) y^{2} = f(a+11) f(a^{2}+2a+5) \) becomes \( x \)-axis is

In a \( \triangle A B C, A, B, C \) are in \( A P \) and \( a, b, c \) are in GP then value of \( a^{3}+b^{3}+c^{3}-a^{2} b-b^{2} c-c^{2} a \) is

The four points \( A, B, C, D \) in space are such that angle \( A B C, B C D, C D A \) and \( D A B \) are all right angles, then

b and c are arithmetic means between a and d (a>d>0) and h and k are the geometric means between a and d then

If \( \sin x+\sin y \geq \cos \alpha \cos x \forall x \in R \) then \( \sin y+\cos \alpha \) is equal to

Let \(I=\int_{0}^{1} \frac{\sin x}{\sqrt{x}}\,dx\) and \(J=\int_{0}^{1} \frac{\cos x}{\sqrt{x}}\,dx\). Then which one of the following is true?