For the given linear programming problem \(z=a x+b y, a, b>0\) subject to the constraint
\(2 x+y \leq 10, x+3 y \leq 15, x, y \geq 0\). If the corner points are \((0,0),(5,0),(3,4)\) and \((0,5)\) and \(z\) is maximum at both \((3,4)\) and \((0,5)\), then the relationship between \(a\) and \(b\) is

If \(m\) and \(n\) are respectively the order and degree of the differential equation :
\(\left(\frac{d^2 y}{d x^2}\right)^5+6 \frac{\left(\frac{d^2 y}{d x^2}\right)^3}{\frac{d^3 y}{d x^3}}+\frac{d^3 y}{d x^3}=x^2+5\), then \(:\)

Which of the following are NOT correct regarding the equation of tangent and normal to the curve \(y=\frac{x-11}{(x-2)(x-3)}\) at the point, where it cuts the \(x\)-aris?
(A) The point of contact is \((11,0)\).
(B) The equation of tangent is \(x-72 y-11=0\).
(C) The equation of normal is \(7 2 x + y - 1 1 = 0\).
(D) The slope of the tangent at the given point of contact is \(\frac{1}{88}\).
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A function \(f: R \rightarrow R\) defined by \(f(x)=\frac{x}{x^2+1}\) is (where \(R\) is the set of real numbers) :

If \(x \sqrt{1+y}+y \sqrt{1+x}=0\), where \(|x|<1,|y|<1\) and \(x \neq y\), then

Let two independent random samples of sizes \(n_1\) and \(n_2\) respectively have been drawn from the same normal population. Let \(\overline{X_1}\) and \(\overline{X_2}\) be the means and let \(s_1\) and \(s_2\) be their standard deviations. In order to test whether the the two sample means \(\overline{X_1}\) and \(\overline{X_2}\) differ significantly or not, the \(t\)-test statistic is given by

The value of the determinant \(\left|\begin{array}{lll}a^2 & a & 1 \\ b^2 & b & 1 \\ c^2 & c & 1\end{array}\right|\) is:

Match List - I with List - II.

List - I	List - II
(A) Resolving power of an astronomical telescope	(I) \( 1.22\lambda/D \)
(B) Limit of resolution of an astronomical telescope	(II) \( \mu \sin \theta \)
(C) Numerical aperture	(III) \( D/1.22\lambda \)
(D) Fresnel distance	(IV) \( a^2/\lambda \)

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Read the passage carefully and give the answers to the question :
Does the number of species in a community really matter to the functioning of the ecosystem? This is a question for which ecologists have not been able to give a definitive answer. For many decades, ecologists believed that communities with more species, generally, tend to be more stable than those with less species.
According to the International Union for Conservation of Nature and Natural Resources (IUCN) (2004), the total number of plant and animal species described so far is more than 1.5 million.
In 'rivet popper hypothesis' the 'rivet' signifies :

In a pure crystal, the lattice point cannot be occupied by:

Select scalar quantities from the following:
A. electric flux
B. Electric current
C. Current density
D. Electromagnetic force
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Identify curative method (s) used to treat ADA deficiency disease in human.
A. Gene Therapy
B. Bone Marrow Transplantation
C. Complete Blood Replacement
D. Enzyme Replacement Therapy
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Progestasert and LNG-20 are the examples of: