The total number of diprotic acids among the following is:

H₃PO₄, H₂SO₄, H₃PO₃, H₂CO₃, H₂S₂O₇, H₃BO₃, H₃PO₂, H₂CrO₄ and H₂SO₃

In the reaction,

The structure of the product T is :

One mole of an ideal gas is taken from $\mathbf{a}$ to $\mathbf{b}$ along two paths denoted by the solid and the dashed lines as shown in the graph below. If the work done along the solid line path is $W_{\text {s }}$ and that dotted line path is $W_{\mathrm{d}}$, then the integer closest to the ratio $W_{\mathrm{d}} / W_{\mathrm{s}}$ is

All the compounds listed in Column I react with water. Match the result of the respective reactions with the appropriate options listed in Column II.

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$
(A)$$\,\,\,\,$$ A line from the origin meets the lines $$\,{{x - 2} \over 1} = {{y - 1} \over { - 2}} = {{z + 1} \over 1}$$
and $${{x - {8 \over 3}} \over 2} = {{y + 3} \over { - 1}} = {{z - 1} \over 1}$$ at $$P$$ and $$Q$$ respectively. If length $$PQ=d,$$ then $${d^2}$$ is
(B)$$\,\,\,\,$$ The values of $$x$$ satisfying $${\tan ^{ - 1}}\left( {x + 3} \right) - {\tan ^{ - 1}}\left( {x - 3} \right) = {\sin ^{ - 1}}\left( {{3 \over 5}} \right)$$ are
(C)$$\,\,\,\,$$ Non-zero vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c \,\,$$ satisfy $$\overrightarrow a \,.\,\overrightarrow b \, = 0.$$
$$\left( {\overrightarrow b - \overrightarrow a } \right).\left( {\overrightarrow b + \overrightarrow c } \right) = 0$$ and $$2\left| {\overrightarrow b + \overrightarrow c } \right| = \left| {\overrightarrow b - \overrightarrow a } \right|.$$
If $$\overrightarrow a = \mu \overrightarrow b + 4\overrightarrow c \,\,,$$ then the possible values of $$\mu $$ are
(D)$$\,\,\,\,$$ Let $$f$$ be the function on $$\left[ { - \pi ,\pi } \right]$$ given by $$f(0)=9$$
and $$f\left( x \right) = \sin \left( {{{9x} \over 2}} \right)/\sin \left( {{x \over 2}} \right)$$ for $$x \ne 0$$
The value of $${2 \over \pi }\int_{ - \pi }^\pi {f\left( x \right)dx} $$ is

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$Column-$$II$$
(p)$$\,\,\,\,$$ $$-4$$
(q)$$\,\,\,\,$$ $$0$$
(r)$$\,\,\,\,$$ $$4$$
(s)$$\,\,\,\,$$ $$5$$
(t)$$\,\,\,\,$$ $$6$$

[Note : Here z takes value in the complex plane and Im z and Re z denotes, respectively, the imaginary part and the real part of z.]

Column I

Column II

If $$g$$ is a function defined on $$R$$ with values in the interval $$\left( {0,\infty } \right)$$ such that $$$f\left( x \right) = ln\,\left( {g\left( x \right)} \right),\,\,for\,\,all\,\,x \in R$$$
then the number of points in $$R$$ at which $$g$$ has a local maximum is ___________.

[Note :[k] denotes the largest integer less than or equal to k ]

Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$B$$.

The coordinates of $$A$$ and $$B$$ are

The orthocentre of the triangle $$PAB$$ is

Tangents are drawn from the point $$P(3, 4)$$ to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ touching the ellipse at points $$A$$ and $$B$$.

The equation of the locus of the point whose distances from the point $$P$$ and the line $$AB$$ are equal, is

Let $k$ be a positive real number and let

$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{array}{ccc} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{array}\right] . \end{aligned} $$

If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$

is equal to _________.

[ Note : adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$ ].

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The area bounded by the curve $$y=f(x)$$ and the lines $$x=0,$$ $$y=0$$ and $$x=t,$$ lies in the interval

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The real numbers lies in the interval

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The function$$f'(x)$$ is

If the radius of the opening of the dropper is $$r$$, the vertical force due to the surface tension on the drop of radius R (assuming $$r$$ << R) is

If r = 5 $$ \times $$ 10⁻⁴ m, $$\rho $$ = 10³ kg m⁻³ , g = 10 m/s² , T = 0.11 Nm⁻¹ , the radius of the drop when it detaches from the dropper is approximately

After the drop detaches, its surface energy is

A block of mass 2 kg is free to move along the x-axis. It is at rest and from t = 0 onwards, it is subjected to a time-dependent force F(t) in the x-direction. The force F(t) varies with t as shown in the figure. The kinetic energy of the block after 4.5 s is

A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is

A large glass slab ($$\mu$$ = 5/3) of thickness 8 cm is placed over a point source of light on a plane surface. It is seen that light emerges out of the top surface of the slab from a circular area of radius R cm. What is the value of R?

Image of an object approaching a convex mirror of radius of curvature 20 m along its optical axis is observed to move from $${{25} \over 3}$$ m to $${{50} \over 7}$$ m in 30 s. What is the speed of the object in km per hour?

To determine the half-life of a radioactive element, a student plots a graph of $$\ln \left| {{{dN(t)} \over {dt}}} \right|$$ versus t. Here, $${{dN(t)} \over {dt}}$$ is the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is __________.

At time t = 0, a battery of 10 V is connected across points A and B in the given circuit. If the capacitors have no charge initially, at what time (in seconds) does the voltage across them becomes 4 V? (Take ln5 = 1.6, ln3 = 1.1)

A diatomic molecule has moment of inertia I. By Bohr's quantization condition, its rotational energy in the nth level (n = 0 is not allowed) is

In a CO molecule, the distance between C (mass = 12 amu) and O (mass = 16 amu), where 1 amu $$ = {5 \over 3} \times {10^{ - 27}}$$ kg, is close to :

Two transparent media of refractive indices $\mu_1$ and $\mu_3$ have a solid lens shaped transparent material of refractive index $\mu_2$ between them as shown in figures in Column II. A ray traversing these media is also shown in the figures. In Column I different relationships between $\mu_1, \mu_2$ and $\mu_3$ are given. Match them to the ray diagram shown in Column II :

You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of 50 Hz frequency (the next three circuits) in different ways as shown in Column II. When a current I (steady state for DC or rms for AC) flows through the circuit, the corresponding voltage $V_1$ and $V_2$ (indicated in circuits) are related as shown in Column I. Match the two :