The area (in sq. units) of the region bounded by the curve \(y=2 x^3, x\)-axis and ordinates \(x=-1\) and \(x=1\) is.

If \(A=\left[\begin{array}{ccc}0 & 2 & c \\ a & b & -1 \\ -5 & 1 & 0\end{array}\right]\) is a skew-symmetric matrix, then \((a+b+c)^3=\)

Which of the following statements are true?
(A) If \(\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}\), then \(x, y, z\) are called direction ratios of \(\vec{r}\).
(B) For any two vectors \(\vec{a}\) and \(\vec{b}, \vec{a}+\vec{b}=\vec{b}+\vec{a}\).
(C) \(\vec{a} \perp \vec{b}\) if and only if \(\vec{a} \times \vec{b}=\overrightarrow{0}\).
(D) Projection of \(\vec{b}\) on \(\vec{a}\) is \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}\).
Choose the correct answer from the options given below :

Match List - I with List - II

List - I (Matrix/Equations)	List - II (Values)
(A) \(\left[\begin{array}{cc}2 x+1 & 3 y \\ 0 & y^2-5 y\end{array}\right]=\left[\begin{array}{cc}x+3 & y^2+2 \\ 0 & -6\end{array}\right]\)	(I) \(x = 2, y = - 1\)
(B) \(\left[\begin{array}{ccc}1 & 2 & -1 \\ x & 0 & 3 \\ y & 3 & 4\end{array}\right]\) is symmetric	(II) \(x = 2, y = 2\)
(C) \(\left[\begin{array}{ll}x & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -2 & -3\end{array}\right]\left[\begin{array}{ll}5 & 2 \\ 0 & y\end{array}\right]=0\)	(III) \(x = - 2, y = 2\)
(D) \(\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}4 & 0 \\ -1 & y / 2\end{array}\right]\)	(IV) \(x = 2, y = 0\)

Choose the correct answer from the options given below:

Arrange the following as per statistical inferences :
(A) Data Analysis
(B) Population
(C) Making Inferences
(D) Data Collection
Choose the correct answer from the options given below :

\(\int e^x\left(\tan ^{-1} x+\frac{1}{1+x^2}\right) d x\) is equal to :

In the following matrix equation, the value of \(x\) is :
\(\left[\begin{array}{cc}x+y & 4 \\ -5 & 3 y\end{array}\right]=\left[\begin{array}{cc}3 & 4 \\ -5 & 6\end{array}\right]\)

Answer the question on the basis of passage given below :
Formaldehyde is the first member of the aldehydic series. It is present in green leaves of plants where its presence is supposed to be due to the reaction of \(CO_2\) with water in the presence of sunlight and chlorophyll. Commercially, formaldehyde is obtained by the controlled oxidation of methane in the presence of molybdenum oxide (\(MoO\)) acting as a catalyst. Formaldehyde is a constituent of a number of polymers also called resins. These are non volatile solids or semi solids which are obtained directly from certain plants and can be synthesized in the laboratory.
Question: Formaldehyde on reaction with benzaldehyde in basic medium gets converted to:

Two bulbs marked 220 V, 200 W and 220 V, 1000 W, respectively, are connected in series to 220 V mains supply. The ratio of heat generated in them, respectively will be

The Abingdon tortoise became extinct in Galapagos islands due to introduction of which of the following animal?

The sunlight is allowed to fall on a pinhole of diameter \( 10^{-4} \text{ mm} \) and thus a diffraction pattern is obtained. How will the size and intensity of the diffraction pattern be affected if the diameter of the pinhole is slightly increased?

The type of pollination in which genetically different pollen grains fall on the stigma is known as:

In Young's double slit experiment, the light has a frequency of \(6 \times 10^{14}\) Hz and the distance between the centers of adjacent fringes is 0.75 mm. If the screen is 1.5 m away, what is the distance between the slits?