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Question 1:

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

1

2

3

4

Question 2:

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

A, B, C, D cannot be coplanar

A, B, C, D are necessarily coplanar

A, B, C, D may or may not be coplanar

no such points A, B, C, D exist

Question 3:

Given that \(b\) 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

\(bc\) is always greater than \(hk\)

\(bc\) is always less than \(hk\)

\(bc\) may be equal to \(hk\)

none of these

Question 4:

If P be a point on ellipse \(4 x^{2}+y^{2}=8\) with eccentric angle \(\frac{\pi}{4}\). Tangent and normal at P intersects the axes at \(A, B, A^{\prime}\) and \(B^{\prime}\) respectively. Then the ratio of area of \(\triangle A P A^{\prime}\) and area of \(\triangle B P B^{\prime}\) is

1

2

3

4

Question 5:

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

0

1

3

4

Question 6:

If \(2 \sin^{2}\bigl(\frac{\pi}{2} \cos^{2} x\bigr)=1-\cos\bigl(\pi \sin 2x\bigr),\; x \neq (2n+1)\tfrac{\pi}{2},\; n \in I\), then

\(\cos 2x\) is \(\tfrac{3}{5}\)

\(\cos 2x\) is \(\tfrac{1}{2}\)

\(\tan x\) is \(\tfrac{1}{2}\)

\(\tan x\) is \(\tfrac{1}{3}\)

Question 7:

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

\(\tfrac{1}{2}\)

1

2

-1

Question 8:

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

I \(>\frac{2}{3}\) and J \(>2\)

I \(<\frac{2}{3}\) and J \(<2\)

I \(<\frac{2}{3}\) and J \(>2\)

I \(>\frac{2}{3}\) and J \(<2\)

Question 9:

Tangent to hyperbola \(x y=c^{2}\) at point \(P\) intersects the x-axis at \(T\) and the y-axis at \(T^{\prime}\). Normal to hyperbola at \(P\) intersects the x-axis at \(N\) and the y-axis at \(N^{\prime}\). If the area of the triangles \(PNT\) and \(PN^{\prime}T^{\prime}\) are \(\Delta\) and \(\Delta^{\prime}\) respectively then \(\frac{1}{\Delta}+\frac{1}{\Delta^{\prime}}\) is equal to

\(c^{2}\)

\(\frac{2}{c^{2}}\)

\(\frac{1}{c^{2}}\)

\(\frac{c^{2}}{2}\)

Question 10:

Consider the function \( y = g(x) \) where \( g^{\prime}(x) < 0 \). Which of the following statements is true regarding the behavior of the function?

The function is increasing.

The function is decreasing.

The function is constant.

The function has a local maximum.

Question 11:

Given a function \( h(x) \) with \( h^{\prime}(x) < 0 \), what can be inferred about the tangent line to the curve at any point \( x \)?

The tangent line has a positive slope.

The tangent line is horizontal.

The tangent line has a negative slope.

The tangent line is vertical.

Question 12:

If a function \( k(x) \) satisfies \( k^{\prime}(x) < 0 \), how does the graph of \( k(x) \) behave as \( x \) increases?

The graph of \( k(x) \) moves upwards.

The graph of \( k(x) \) remains unchanged.

The graph of \( k(x) \) moves downwards.

The graph of \( k(x) \) oscillates.

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