The sum of order and degree of the differential equation \(\left(x^2 \frac{d^2 y}{d x^2}\right)^{\frac{3}{4}}=5\left(\frac{d y}{d x}\right)^2-3\) is equal to :

The value of \(\left[\begin{array}{lll}1 & b c & b c(b+c) \\ 1 & c a & c a(c+a) \\ 1 & a b & a b(a+b)\end{array}\right]\) is

The integral \(\int \frac{2 x+x^3}{1+x^2} d x\) is equal to :

The sum of the order and degree of the differential equation representing the family of curves \(y=m x+m^4\), where \(m\) is arbitrary constant, is

The vectors \(\lambda \hat{i}+\hat{j}+2 \hat{k}, \hat{i}+\lambda \hat{j}-\hat{k}\) and \(2 \hat{i}-\hat{j}+\lambda \hat{k}\) are coplanar if :

Approximate percentage increase in the volume of a sphere if its radius is increased by \(3 \%\) is:

If \(x_1, x_2, x_3,\)........,\(x_n\) are \(n\) observations in a sample then which of following is/are TRUE?
(A) The mean \(\bar{x}\) has \(n\) degree of freedom
(B) The mean \(\bar{x}\) has ( \(n-1\) ) degree of freedom
(C) The standard deviation of the sample has ( \(n-1\) ) degree of freedom
(D) The standard deviation of the sample has \(n\) degree of freedom
Choose the correct answer from the options given below :

Given that \(\vec{a}=-3 \hat{i}-6 \hat{j}+4 \hat{k}, \quad \vec{b}=9 \hat{i}-\lambda \hat{j}-12 \hat{k}\)
If \(\vec{a} \times \vec{b}=\overrightarrow{0}\), then the value of \(\lambda\) is

Let \(A=\left[\begin{array}{ccc}2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3\end{array}\right]\), and \(a_{i j}\) be any element of \(A, i, j \in\{1,2,3\}\), then which of the following are TRUE?
(A) Minor of \(a_{23}=16\)
(B) Minor of \(a_{23}=-8\)
(C) Cofactor of \(a_{23}=-16\)
(D) Cofactor of \(a_{23}=8\)
(E) Cofactor of \(a_{13}=4\)
Choose the correct answer from the options given below :

If \(f(x)=\frac{1}{1-x}\), then for x > 1, f(x) is:

The area of the region bounded by parabola \(x^2=4 y\), straight line \(x=2\) and \(x\)-axis, is

Read the passage carefully and answer the 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 final product (X) of the chemical reaction obtained from the following transformation :

Which one of the following is not true for a molecule that can act as a genetic material?