In the given figure, feasible region represented by the constraints \(4 x+y \geq 80, x+5 y \geq 115,3 x+2 y \leq 150, x, y \geq 0\) is

The value of \(\left|\begin{array}{llr}\sin 15^{\circ} & \cos 15^{\circ} & 2 \sin 30^{\circ} \\ \sin 30^{\circ} & \cos 30^{\circ} & \sin 45^{\circ} \\ \cos 75^{\circ} & \sin 75^{\circ} & \sin 90^{\circ}\end{array}\right|\) is

The general solution of the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\) is :
(Where \(C\) is constant of integration)

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 \(\frac{1}{88}\).
Choose the correct answer from the options given below:

The least positive integer \(x\) satisfying \(28=x(\bmod 6)\) is :

If z = 5x + 8y is the objective function of a LPP and (0,0), (3, 1), (2, 4), (0, 3), (5,0) are corner points of the bounded feasible region, then the maximum value of the objective function is

Let \(A=\left[a_{i j}\right]\) be a square matrix, where \(a_{i j}=\left\{\begin{array}{ll}0, & \text { when } i=j \\ 1, & \text { otherwise }\end{array}\right.\)
If \(|\operatorname{adj} A|=|A|^2\), then which of the following statements are correct?
(A) A is a skew-symmetric matrix.
(B) A is a non-singular matrix.
(C) A is a square matrix of order 4.
(D) A is a symmetric matrix.
Choose the correct answer from the options given below :

Read the passage carefully and answer the Questions
Haloalkanes undergo nucelophilic substitution reactions owing to the polarity of C X bond. The nucleophile reacts with the haloalkane on the carbon possessing a partial positive charge holding the halogen atom. The halogen atom \(X\) is replaced by a nucleophile. Depending on the kinetics and mode of bond breaking, the mechanism can be either SN1 or SN2 reaction. The rate of SN1 reaction is governed by the stability of carbocation and in SN2 reaction, the rate of reaction is governed by steric factor. Chirality is the main factor in both SN1and SN2. In SN1 reaction, the chirality of alkyl halide is accompanied by racemization of the product, while in SN2 reaction, the product is characterized by inversion of configuration. The structure of alkyl halide and the nature of the solvent also governs the mechanism of the substitution.
What is chiral carbon ?

\(\int_1^2 \frac{1}{x(x+1)} d x, x>0\) equals

Let the matrix \(A=\left[a_{i j}\right]_{3 \times 3}\) be defined by
\(a_{i j}=\left\{\begin{array}{ll}2 i+3 j, & i<j \\5, & i=j \\3 i-2 j, & i>j\end{array}\right.\)
The number of elements in the matrix \(A\) which are greater than 7 , is:

The maximum value of \(z=2 x+3 y\) subject to the constraints \(3 x-3 y \geq 0,2 x+2 y \leq 12, x \geq 0, y \geq 0\) occurs at the point:

Arrange the following structures involved in embryogenesis in the correct sequence.
(A) Zygote
(B) Blastocyst
(C) Blastomeres
(D) Morula
Choose the correct answer from the options given below:

Number of coulombs required for the reduction of \(1\) mol of \(MnO_4^-\ \text{to}\ Mn^{2+}\) is: