The corner points of the bounded feasible region determined by the system of linear constraints are \((0,0), (5,0), (6,5)\), \((6,8),(4,10), (0,8)\). Let \(Z=3 x-4 y\) be the objective function. The minimum value of \(Z\) occurs at

By using the determinants the area of the triangle with vertices \((0,0),(0,1)\) and \((1,0)\) is :

The point on the curve \(\frac{x^2}{4}+\frac{y^2}{9}=1\) at which the tangent to the curve is parallel to the \(x\)-aris is

If \(y=\sin ^{-1} \sqrt{\frac{x}{x+1}}+\sec ^{-1} \sqrt{\frac{x+1}{x}}\), then \(\frac{d y}{d x}\) is

Corner points of the feasible region determined by the system of linear constraints are \((0, 3), (1, 1)\) and \((3, 0)\). Let \(Z = px + qy\) where \(p, q > 0\). Condition on \(p\) and \(q\) so that minimum of \(Z\) occurs at \((3, 0)\) and \((1, 1)\) is:

"The probability distribution of a random variable \(x\) is :
\(P(X=x)=\left\{\begin{array}{ll}k x^2, & \text { for } x=1,2,3 \\2 k x, & \text { for } x=4,5,6 \text { where } k \text { is a constant. } \\0, & \text { otherwise }\end{array}\right.\)
Match List-I with List-II

List I	List II
(A) k	(I) 7/22
(B) \(P(X \geq 4)\)	(II) 1/44
(C) \(P(X<4)\)	(III) 95/22
(D) E[X]	(IV) 15/22

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Match List - I with List - II

List - I	List - II
(A) The degree of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^2+\left(\frac{d y}{d x}\right)=x \sin \left(\frac{d y}{d x}\right)\)	(I) 4
(B) The degree of differential equation \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^{1 / 4}+x^{1 / 5}=0\)	(II) 1
(C) The degree of differential equation \(\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^3+6 y^5=0\)	(III) Not defined
(D) The degree of differential equation \(1+\left(\frac{d y}{d x}\right)^4=7\left(\frac{d^2 y}{d x^2}\right)^3\)	(IV) 3

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Read the following paragraph carefully and answer the given questions.
Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. For determining the cause of allergy, the patient is exposed to or injected with very small doses of possible allergens, and the reactions studied. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. This is obtained by acetylation of morphine. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal.
Which of the following technique is not very useful to detect the cancers of internal organs?

Which of the following statements is incorrect with respect to the plot of molar conductivity (\(\Lambda_m\)) versus square root of concentration (\(c^{1/2}\)) for electrolytes in solution?

If \(M=\left[\begin{array}{c}2 \\ -1 \\ 3\end{array}\right]\) and \(N=\left[\begin{array}{lll}7 & 1 & -4\end{array}\right]\), then \((M N)^T\) will be equal to:

Consider the following hypothesis \(H_0: \mu=315\) and \(H_a: \mu \neq 315\).
A sample of 60 provided a sample mean of 324.6 . The standard deviation ( \(\sigma\) ) is 14 and level of significance \(\alpha=0.05\). Then the confidence interval is :
[Given : \(Z_{\alpha / 2} \frac{14}{\sqrt{60}}=3.54\)]

The Young's double slit experiment is performed with monochromatic blue, green, yellow and red lights, respectively. Arrange the distance (x) of 4th maxima from the central maximum for all the colors in decreasing order:
(A) \( X_{red} \)
(B) \( X_{blue} \)
(C) \( X_{green} \)
(D) \( X_{yellow} \)
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Select the correct statement/s about sporopollenin found in pollen grain.
(A) The hard outer layer of pollen grains is made up of sporopollenin.
(B) It is one of the most resistant organic material known.
(C) It does not withstand high temperature and strong acids.
(D) No enzyme can degrades it.
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