If the system of equations
\( \begin{array}{l} x+2 y+3 z=10 \\ -x+y+\lambda z=20 \\ 2 x+3 y+\lambda z=0 \end{array} \)
does not possess a unique solution, then \(\lambda\) is equal to

If \(\begin{vmatrix} 2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1 \end{vmatrix} + 3 = 0\) then the value of x is:

\(\int \sin^3 x dx\) is equal to:

Let \(X\) denote the number of hours a student studies on a selected day. The probability distribution of \(X\) is given by (where \(k\) is some unknown constant)
\(P\left(X=x_i\right)=\left\{\begin{array}{ll}0.5, & \text { if } x_i=0 \\k x_i, & \text { if } x_i=1 \\k\left(4-x_i\right), & \text { if } x_i=2 \text { or } 3 \\0, & \text { otherwise }\end{array}\right.\)
Then the value of \(k\) is

If \(c_{i j}\) denotes the cofactor of element \(c_{i j}\) of the matrix \(A=\left[\begin{array}{ccc}1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3\end{array}\right]\), then the value of \(c_{21} \cdot c_{33}\) is

The following data are from a random sample : \(5,8,10,7,10,14\), then the point estimate of the population standard deviation is

A vector of magnitude 9, which is perpendicular to both the vectors \((4 \hat{i}-\hat{j}+8 \hat{k})\) and \((-\hat{j}+\hat{k})\) is

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 ?

Read the following passage carefully and answer the given questions
The earthworm is referred to as the farmer's friend. This is because they help in the breakdown of complex organic matter as well as in loosening of soil. Similarly, decomposers break down complex organic matter into inorganic substances like carbon dioxide, water and nutrients and the process is called decomposition. Dead plant remains such as leaves, bark, flowers and dead remains of animals, including fecal matter, constitute detritus, which is the raw material for decomposition. Some of the steps in the process of decomposition are fragmentation, catabolism, leaching, humification etc.
Process of accumulation of dark coloured amorphous substance is called:

\(\int\left(e^{x \log a}+e^{a \log x}\right) d x\) is equal to (where \(\left.a>1\right)\)

A uniformly charged conducting sphere of radius 1.3 m has a surface charge density of \( 70 \, \mu\text{C m}^{-2} \). What is the total electric flux leaving the surface of the sphere ?

The value of \(\tan \left(\cos ^{-1} x\right)\) is:

Gabriel synthesis is used for the synthesis of: