If \(\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}3 & 0 \\ 4 & -8\end{array}\right|\), then value(s) of \(x\) is/are

\(\int\left(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\right) d x=\)
(Given that c is an arbitrary constant )

The value of the integral \(\int\left(x^6+x^4\right) d\left(x^2\right)\) is equal to:

Which one of the following inequalities is redundant for the shaded feasible region (ABCDA) shown below?

Consider the LPP, with objective function optimize \(Z=2 x-y+5\) subject to constraints \(3 x+4 y \leq 60 ; x+3 y \leq 30 ; x, y \geq 0\). If the corner points of feasible region are \(A(0,10), B(12,6), C(20,0)\) and \(O(0,0)\), then match List I with List II:

LIST 1	LIST 2
Minimum value of Z	45
Maximum value of Z	50
The sum of minimum and maximum value of Z	-5
Maximum of Z - Minimum of Z	40

Anshu takes a personal loan of ₹\(10,00,000\) at the rate of \(12\%\) per annum for 2 years, then the EMI by using flat rate method is

\(\int \frac{1}{(x+1)(x+2)} d x=\)

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdom.
Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon stom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidizing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
The starting compound Y in the following reaction is
\(\text{Y}\) \(\xrightarrow{\text{oxidation}}\) \(\text{aldehyde}\) \(\xrightarrow{\text{oxidation}}\) \(\text{carboxylic acids}\)

The area of the region \(\left\{(x, y): x^2+y^2 \leq 1 \leq x+y\right\}\) is

The products for the reaction of KMNO₄ with potassium iodide in acidic solutions are

Match List I with List II

LIST I	LIST II
A. \(\int \sin x \cos x d x\)	I. \(\log _e\left(1+\sin ^2 x\right)+C\)
B. \(\int \frac{d x}{\sin ^2 x \cos ^2 x}\)	II. \(\tan \frac{x}{2}+C\)
C. \(\int \frac{\sin 2 x}{1+\sin ^2 x} d x\)	III. \(-\frac{1}{4} \cos 2 x+C\)
D. \(\int(1-\cos x)\operatorname{cosec}^2 x d x\)	IV. \(\tan x-\cot x+C\)

Which of the following is true ?
(A) The equilibrium of a charged particle under the action of coulomb force alone can never be stable.
(B) A hollow metal sphere placed on an insulating stand is charged positively. The electric potential inside is same as on surface
(C) A free electron and a free proton are placed in a uniform field, both will experience same magnitude of the force
(D) A free electron and a free proton are placed in a uniform field, both will experience same acceleration
Choose the correct answer from the options given below :

If a function \(f: R _{+} \rightarrow[-5, \infty)\) given by \(f(x)=9 x^2+6 x-5\) is one-one and onto, then \(f^{-1}(y)\) is