If the corner points of the bounded feasible region of an LPP are (0,2), (3,0), (6,0), (6,8) and (0,5) then the minimum value of the objective function F = 4x+6y occurs at

The corner points of the bounded feasible region for an LPP are (0,4), (4, 4), (6, 6), (0, 12).
If the objective function is \(Z=p x+q y, p>0, q>0\) then the condition on p and q so that the maximum of Z occurs at (6, 6) and (0, 12) is

The window of the house is in the form of a rectangle surmounted by a semi-circular portion, having a perimeter of 10 m as shown in figure.

If \(x\) and \(y\) represent the length and breadth of rectangular region, then relation between \(x\) and \(y\) is:

Which of the following terms are associated with a linear programming problem?
(A) Constraints
(B) Independent events
(C) Feasible region
(D) Objective function
Choose the correct answer from the options given below :

Number of solutions of the equation \(\left|\begin{array}{ccc}-1 & 0 & \sin \theta \\ \sin \theta & -1 & 0 \\ 0 & \sin \theta & -1\end{array}\right|=0\) in \((0, \pi)\) is:

Vector in the direction of \(\hat{i}+\hat{j}+\hat{k}\) with magnitude 5 units is :

\(A\) and \(B\) take turns in throwing a pair of dice, the first to throw a sum of 9 wins the prize.
If \(A\) throws first, then the ratio of the probabilities of \(A\) and \(B\) winning is :

Which of the following statements are true with reference to homology or homologous organs?
(A) Homology indicates common ancestry.
(B) Whale and Cheetah share similarities in the pattern of the bones of the forelimbs.
(C) Vertebrate heart is an example of homologous organs.
(D) Thorn of Bougainvillea and tendrils of cucurbita represent homology.
Choose the correct answer from the options given below:

Given \(X=\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]\), then value of K if \(X^2=8 X+K I\) is :

Read the passage carefully and answer the following questions
Aldehydes, carboxylic acids and ketones are widespread in the animal and plant kingdoms. They play a pivotal role in various biochemical processes of life. Their presence in nature adds fragrance and flavor. These compounds are widely used in food products, pharmaceuticals, paints, resins and other important product industries. These compounds are prepared by various laboratory methods, which mainly include oxidation, formylation, acylation and reduction. Due to the polar nature of the carbonyl group in aldehydes and ketones, they can exhibit different reactions like nucleophilic addition. They do exhibit redox and various condensation reactions which lead to the formation of various important compounds. On the other hand, the carboxylic acids are mainly prepared by oxidation and hydrolysis of different compounds. The carboxylic acid consists of a carbonyl group and the hydroxy group (attached to the carbonyl carbon atom). This makes it possible for the carboxylic acid to participate in various chemical reactions which involve cleavage of the C-OH bond and the O-H bond along with the reactions involving the complete - COOH group.
Identify the incorrect statement for Cannizaro reaction.

p-block elements are placed in groups 13 to 18 of the periodic table. Group 15 is nitrogen family.
Nitrogen makes various compounds like \(NH _3, HNO _3, N_2 O , NO _2\) etc. Phosphorus exists in the form of white and red phosphorus. White phosphorus glows in dark. \(PH _3\) gas is used as smoke screen. Group 16 is called oxygen family. Dry air contains \(21\% O_2\) gas by volume. Combined sulphur exists as sulphates such as gypsum, epsome salt, baryte etc. The allotropic forms of sulphur are Rhombic and monoclinic sulphur. \(S_2\) is paramagnetic like \(O_2\) and it stable at high temperature.
Group 17 is Halogen family. \(F_2\) is highly reactive, \(Cl _2\) is greenish yellow gas. Both \(SO _2\) and \(Cl _2\) are used for bleaching. Group 18 is Noble gas family but some noble gases also form compounds.
Complete hydrolysis of \(XeF _6\) gives HF and \(XeO _3\) as main products.
At elevated temperature \((\sim 1000 K)\) the dominant form of sulphur is :

The following solutions were prepared by dissolving 1 g of solute in 1 L of the solution. Arrange the following solutions in decreasing order of their molarity
(A) Glucose (molar mass = 180 g mol^-1)
(B) NaOH (molar mass = 40 g mol^-1)
(C) NaCl (molar mass = 58.5 g mol^-1)
(D) KCl (molar mass = 74.5 g mol^-1)
Choose the correct answer from the options given below :

Which of the following statement(s) is/are incorrect?
(A) The Kirchhoff's laws are equally applicable in alternating current circuit and direct current circuits.
(B) It is not necessary that the wire stretched on the meter bridge must have a uniform cross-sectional area.
(C) The internal resistance of the cell does not affect the reading of the null point in metre bridge.
(D) Electrical potential has no direction, the electrical potential gradient has a direction.