If \(x=a \cos ^3 \theta, y=a \sin ^3 \theta\). The value of \(\frac{d^2 y}{d x^2}\) at \(\theta=\frac{\pi}{6}\) is :

In an AC circuit, the current leads the voltage by \(\pi / 2\). The circuit is :

The following data shows the percentage of rural and urban Indian households who have a high speed internet connection.

Year(x)	Rural Household \((y \%)\)	Urban Household\((y \%)\)
2020	2	6
2021	3	7
2022	3	9
2023	4	15
2024	5	20

Based on the above data, which of the following statements are correct?
(A) A straight line trend by the method of least squares for rural Indians \(y_t=3.4+0.7(x-2022)\)
(B) A straight line trend by the method of least squares for urban Indians is \(y_t=11.4+4.3(x-2022)\)
(C) The forecast for the year 2025 for urban group using the trend equation is \(22.2 \%\)
(D) The forecast for the year 2025 for rural group using the trend equation is \(5.5 \%\)
Choose the correct answer from the options given below :

If A and B are square matrices of order n × n, then \((A-B)^2\) is equal to:
(A) \(A^2-A B-B A+B^2\)
(B) \(A^2-2 A B+B^2\)
(C) \(A^2-2 B A+B A\)
(D) \(A^2-B^2\)
(E) \(A^2+2 A B+B^2\)

The interval in which where the function \(f(x)=x^3-3 x^2+4 x+1, x \in R\) is increasing, is:

If \(A=\left[\begin{array}{cc}2 & -3 \\ -4 & 7\end{array}\right]\) and \(2 A^{-1}=K I-A\), where \(K\) is a real number and \(I\) is the identity matrix of order 2 , then the value of \(K\) is :

\(\tan ^{-1} \frac{1}{\sqrt{3}}-\sin ^{-1} \frac{1}{2}\) is equal to :

The maximum value of the function f(x) = \(x^2\) (60 - x) in [20, 80] is:

Match List-I with List-II

List-I	List-II
(A) Light	(I) 1 mm to \(10^{-3} nm\)
(B) Infra-red	(II) 700 nm to 400 nm
(C) X-rays	(III) 400 nm to 1 nm
(D) UV rays	(IV) 1 mm to 700 nm

Choose the correct answer from the options given below :

An electron is moving at a velocity of 100 m/s along the X-axis in the presence of an electric field of 30 V/m directed along the Y-axis and a magnetic field of 300 G directed along the Z axis. The magnitude and direction of force acting on the electron is (Given: the charge on the electron is "e")

Which of the following metals cannot be used to adsorb H2 gas on its surface?

Which of the following properties are desired in materials for permanent magnets ?
(A) High retentivity
(B) High coercivity
(C) High permeability
(D) low permeability
Choose the correct answer from the options given below:

If \(A=\left[\begin{array}{ll}3 & -5 \\ 4 & -7\end{array}\right]\), then \(A^{-1}\) is: