Let \(y=138.86+7.64(x-2021)\) be a straight line of best fit by using least square method.

\(YEAR(X)\)	2018	2019	2020	2021	2022	2023	2024
\(PROFIT(y)\)	144	130	126	144	138	156	164

Then the trend value for the year 2024 is :

The derivative of \(\sec (\tan \sqrt{x})\) with respect to \(x\) is :

If \(A =\left(\begin{array}{cc}-a & b \\ c & a\end{array}\right)\) and \(A ^2= I\), where \(I\) is an Identity matrix of order 2, then which of the following is correct?

Match List I with List II

LIST I	LIST II
A. If \(A^{-1}=\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]\) then \(A\) is	I. \(\left[\begin{array}{ll}5 & 2 \\ 7 & 3\end{array}\right]\)
B. If \(A=\left[\begin{array}{cc}2 & -1 \\ 3 & 1\end{array}\right]\) then \(A A^T\) is	II. \(\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\)
C. If \(A^{-1}=\left[\begin{array}{ll}5 & 7 \\ 2 & 3\end{array}\right]\) then \(\left(A^T\right)^{-1}\) is	III. \(\left[\begin{array}{cc}-3 & 2 \\ \frac{5}{2} & -\frac{3}{2}\end{array}\right]\)
D. If \(A=\left[a_{i j}\right]_{2 \times 2}\) where \(a_{i j}=(i-j)^2\), then \(A\) is	IV. \(\left[\begin{array}{cc}5 & 5 \\ 5 & 10\end{array}\right]\)

The function \(f\) is given by
\(f(x)=\left\{\begin{array}{ll}x^3+3, & \text { if } x \neq 0 \\ 4, & \text { if } x=0\end{array}\right.\)
Then number of points of discontinuity for this function is :

The direction cosines of the line
\(x-y+z-5=0\) and \(x-3 y-6=0\) are:

In a game of 100 points, \(A\) can give 20 points to \(B\) and 28 points to \(C\). How many points can \(B\) give \(C\) ?

Read the passage carefully and give the answers to the question :
Does the number of species in a community really matter to the functioning of the ecosystem? This is a question for which ecologists have not been able to give a definitive answer. For many decades, ecologists believed that communities with more species, generally, tend to be more stable than those with less species. According to the International Union for Conservation of Nature and Natural Resources (IUCN) (2004), the total number of plant and animal species described so far is more than 1.5 million.
The scientist who proved that species richness directly correlates with the stability of a community, was

If \(\left[\begin{array}{ll}3 & 1 \\ 2 & 1\end{array}\right] A\left[\begin{array}{ll}2 & 1 \\ 1 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\), then matrix ' \(A\) ' is

Ethyl bromide is converted to succinic acid by using the following reagents in steps:
 A. \(KCN\)
B. \(Br_2\)
C. \(KOH\ (alc.)\)
D. \(H_2O/H^+\)
Choose the correct sequence of conversion:

The relation \(R\) on the set \(N \times N\) defined by \((a, b) R(c, d) \Longleftrightarrow a+d=b+c\), \(\forall(a, b),(c, d) \in N \times N\), where N is the set of natural numbers, is:

The area (in sq. units) bounded by the parabola \(y^2=16 x\) and its latus rectum is

The maximum torque acting on a coil having an effective area \( 0.04 \text{ m}^2 \) and carrying current of \( 100 \mu A \) is \( 4 \times 10^{-8} \text{ N m} \), when placed in a magnetic field. The magnitude of the magnetic field is