AP EAPCET 2022 4TH JULY EVENING SHIFT

A metal tape is calibrated at $$25^{\circ} \mathrm{C}$$. On a cold day when the temperature is $$-15^{\circ} \mathrm{C}$$, the percentage error in the measurement of length is

(Coefficient of linear expansion of metal $$=1 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$$)

A solution containing 6.0 g of urea is isotonic with a solution containing 10 g of a non-electrolytic solute $$X$$. The molar mass of X (in g $$\mathrm{mol}^{-1}$$) is

The formal charges of atoms (1), (2) and (3) in the ion is

$$\int_{\pi / 4}^{5 \pi / 4}(|\cos t| \sin t+|\sin t| \cos t) d t=$$

In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked randomly. The probability that it is neither red nor green is

$$D, E, F$$ are respectively the points on the sides $$B C, C A$$ and $$A B$$ of a $$\triangle A B C$$ dividing them in the ratio $$2: 3,1: 2,3: 1$$ internally. The lines $$\mathbf{B E}$$ and $$\mathbf{C F}$$ intersect on the line $$\mathbf{A D}$$ at $$P$$. If $$\mathbf{A P}=x_1 \cdot \mathbf{A} \mathbf{B}+y_1 \cdot \mathbf{A C}$$, then $$x_1+y_1=$$

If the equation of the plane passing through the point $$A(-2,1,3)$$ and perpendicular to the vector $$3 \hat{i}+\hat{j}+5 \hat{k}$$ is $$a x+b y+c z+d=0$$, then $$\frac{a+b}{c+d}=$$

In SI units, $$\mathrm{kg}-\mathrm{m}^2 \mathrm{~s}^{-2}$$ is equivalent to which of the following?

Which of the following is not true about vectors $$\mathbf{A}, \mathbf{B}$$ and $$\mathbf{C}$$ ?

An object moving along $$X$$-axis with a uniform acceleration has velocity $$\mathbf{v}=\left(12 \mathrm{cms}^{-1}\right) \hat{\mathbf{i}}$$ at $$x=3 \mathrm{~cm}$$. After 2 s if it is at $$x=-5 \mathrm{~cm}$$, then its acceleration is

$$y=\left(P t^2-Q t^3\right) \mathrm{~m}$$ is the vertical displacement of a ball which is moving in vertical plane. Then the maximum height that the ball can reach is

A force $$\mathbf{F}_1$$ of magnitude 4 N acts on an object of mass 1 kg , at origin in a direction $$30^{\circ}$$ above the positive $$X$$-axis. A second $$F_2$$ of magnitude 4 N acts on the same object in the direction of the positive $$Y$$-axis. The magnitude of the acceleration of the object is nearly.

A cricket ball of mass 50 g having velocity $$50 \mathrm{~cm} \mathrm{~s}^{-1}$$ to stopped in 0.5 s. The force applied to stop the ball is

Two masses $$M_1$$ and $$M_2$$ are arranged as shown in the figure. Let $$a$$ be the magnitude of the acceleration of the mass $$M_1$$. If the mass of $$M_1$$ is doubled and that of $$M_2$$ is halved, then the acceleration of the system is (Treat all surfaces as smooth; masses of pulley and rope are negligible)

A ball of mass 300 g is dropped from a height 10 m above a sandy ground. On reaching the ground, it penetrates through a distance 1.5 m in sand and finally stops. The average resistance offered by the sand to oppose the motion is (acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

Two balls $$A$$ and $$B$$, of masses $$M$$ and $$2 M$$ respectively collide each other. If the ball $$A$$ moves with a speed of $$150 \mathrm{~ms}^{-1}$$ and collides with ball $$B$$, moving with speed $$v$$ in the opposite direction. After collision if ball $$A$$ comes to rest and the coefficient of restitution is 1 (one), then the speed of the ball $$B$$ before it collides with ball $$A$$ is

As shown in the figure, an iron block $$A$$ of volume $$0.25 \mathrm{~m}^3$$ is attached to a spring $$S$$ of unstretched length 1.0 m and hanging to the ceiling of a roof. The spring gets stretched by 0.2 m . This block is removed and another block $$B$$ of iron of volume $$0.75 \mathrm{~m}^3$$ is now attached to the same spring and kept on a frictionless incline plane of $$30^{\circ}$$ inclination. The distance of the block from the top along the incline at equilibrium is

A solid sphere of radius $$R$$ has its outer half removed, so that its radius becomes $$(R / 2)$$. Then its moment of inertia about the diameter is

In a hydraulic lift, compressed air exerts a force $$F$$ on a small piston of radius 3 cm . Due to this pressure the second piston of radius 5 cm lifts a load of 1875 kg . The value of $$F$$ is (Acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

In a $$U$$-shaped tube the radius of one limb is 2 mm and that of other limb is 4 mm . A liquid of surface tension $$0.03 \mathrm{~Nm}^{-1}$$, density $$1500 \mathrm{~kg} \mathrm{~m}^{-3}$$ and angle of contact zero is taken in the tube. The difference in the heights of the levels of the liquid in the two limbs is (Acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

A steady flow of a liquid of density $$\rho$$ is shown in figure. At point 1, the area of cross-section is $$2 A$$ and the speed of flow of liquid is $$\sqrt{2} \mathrm{~ms}^{-1}$$. At point 2 , the area of cross-section is $$A$$. Between the points 1 and 2, the pressure difference is $$100 \mathrm{~Nm}^{-2}$$ and the height difference is 10 cm . The value of $$\rho$$ is (Acceleration due to gravity $$=10 \mathrm{~ms}^{-2}$$)

A gas is expanded from an initial state to a final state along a path on a $$p$$-$$V$$ diagram. The path consists of (i) an isothermal expansion of work 50 J , (ii) an adiabatic expansion and (iii) an isothermal expansion of work 20 J . If the internal energy of gas is changed by $$-$$30 J , then the work done by gas during adiabatic expansion is

In non-rigid diatomic molecule with an additional vibrational mode

The temperature of the sink of a Carnot engine is 250 K . In order to increase the efficiency of the Carnot engine from $$25 \%$$ to $$50 \%$$, the temperature of the sink should be increased by

A body is executing S.H.M. At a displacement $$x$$ its potential energy is 9 J and at a displacement $$y$$ its potential energy is 16 J . The potential energy at displacement $$(x+y)$$ is

Speed of sound in air near room temperature is approximately

A uniform solid sphere of radius $$R$$ produces a gravitational acceleration of $$a_0$$ on its surface. The distance of the point from the centre of the sphere where the gravitational acceleration becomes $$\frac{a_0}{4}$$ is

When monochromatic light of wavelength 600 nm is used in Young's double slit experiment, the fifth order bright fringe is formed at 6 mm from the central bright fringe on the screen. If the experiment is conducted with light of wavelength 400 nm from the central bright fringe, the third order bright fringe will be located at

The radii of curvature of a double convex lens are 4 cm and 8 cm . If the refractive index of the material of the lens is 1.5 , the focal length of the lens is nearly.

A solid sphere of radius $$R$$ carries a positive charge $$Q$$ distributed uniformly throughout its volume. A very thin hole is drilled through it's centre. A particle of mass $$m$$ and charge $$-$$q performs simple harmonic motion about the centre of the sphere in this hole. The frequency of oscillation is

Assertion (A) In a region of constant potential, the electric field is zero and there can be no charge inside the region.

Reason (R) According to Gauss law, charge inside the region should be zero if electric field is zero.

Statement (A) Inside a charged hollow metal sphere, $$E=0, V \neq 0$$, (where, $$E=$$ electric field, $$V=$$ electric potential).

Statement (B) The work done in moving a positive charge on an equipotential surface is zero.

Statement (C) When two like charges are brought closer, their mutual electrostatic potential energy will increase.

A circular coil has 100 turns, radius 3 cm and resistance $$4 \Omega$$. This coil is co-axial with a solenoid of 200 turns/$$\mathrm{cm}$$ and diameter 4 cm . If the solenoid current is decreased from 2 A to zero in 0.04 s , then the current induced in the coil is

A toroid has a non ferromagnetic core of inner radius 24 cm and outer radius 25 cm , around which 4900 turns of a wire are wound. If the current in the wire is 12 A , the magnetic field inside the core of the toroid is

A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at $$30^{\circ}$$ with the horizontal. The horizontal component of the earth's magnetic field at the place is 0.3 G . Then the magnitude of the earth's magnetic field at the location is

A steel wire of length $$l$$ and magnetic moment $$M$$ is bent into a semicircular arc of radius $$R$$. The new magnetic moment is

The current 'I' in the circuit shown in the figure is

The electrons take $$40 \times 10^3$$ s to dirift from one end of a metal wire of length 2 m to its other end. The area of cross-section of the wire is $$4 \mathrm{~mm}^2$$ and it is carrying a current of 1.6 A. The number density of free electrons in the metal wire is

A carrier wave is used to transmit a message signal. If the peak voltage of modulating signal and carrier signal are increased by $$1 \%$$ and $$3 \%$$ respectively, the modulation index is changed by

The electric field $$(E)$$ and magnetic field $$(B)$$ of an electromagnetic wave passing through vacuum are given by

$$\begin{aligned} & E=E_0 \sin (k x-\omega t) \\ & B=B_0 \sin (k x-\omega t) \end{aligned}$$

Then the correct statement among the following is

A light of intensity $$12 \mathrm{Wm}^{-2}$$ incidents on a black surface of area $$4 \mathrm{~cm}^2$$. The radiation pressure on the surface is

Capacitive reactance of a capacitor in an AC circuit is $$3 \mathrm{k} \Omega$$. If this capacitor is connected to a new AC source of double frequency, the capacitive reactance will become.

A hydrogen atom at the ground level absorbs a photon and is excited n = 4 level. The potential energy of the electron in the excited state is

The radius of an atomic nucleus of mass number 64 is 4.8 fermi. Then the mass number of another atomic nucleus of radius 6 fermi is

In a photoelectric experiment light of wavelength 800 nm produces photoelectrons with the smallest de-Broglie wavelength of 1 nm . Light of 400 nm produces photoelectrons with smallest de-Broglie wavelength of 0.5 nm. Then the work function of the metal used in the experiment is nearly.

Consider the statements In a semiconductor

(A) There are no free electrons at 0 K.

(B) There are no free electrons at any temperature.

(C) The number of free electrons increases with temperature.

(D) The number of free electrons is less than that in a conductor.

The empirical formula of calgon is

The maximum number of electrons present in an orbital with $$n = 4, l = 3$$ is

Which quantum number provides information about the shape of an orbital?

From the following plots, find the correct option.

Observe the following properties : Volume, enthalpy, density, temperature, heat capacity, pressure and internal energy. The number of extensive properties in the above list is

Match the following.

Which of the following expression is correct?

The pH of 0.01 N lime water is

$$x \%(w / V)$$ solution of urea is isotonic with $$4 \%$$ $$(w / V)$$ solution of a non-volatile solute of molar mass $$120 \mathrm{~g} \mathrm{~mol}^{-1}$$. The value of $$x$$ is

How many grams of Mg is required to completely reduce $$100 \mathrm{~mL}, 0.1 \mathrm{~M} \mathrm{~NO}_3^{-}$$ solution using the following reaction?

$$\mathrm{NO}_3^{-}+\mathrm{Mg} \longrightarrow \mathrm{Mg}^{2+}+\mathrm{NH}_3$$

What is the oxidation state of S in the sulphur containing product of the following reaction?

$$\mathrm{SO}_3^{2-}(a q)+\mathrm{Br}_2(l)+\mathrm{H}_2 \mathrm{O} \longrightarrow$$

In the reaction of phosphorus with conc. $$\mathrm{HNO}_3$$, the oxidised and reduced products respectively are

The macromolecular colloids of the following are

I. Starch solution

II. Sulphur sol

III. Synthetic detergent

IV. Synthetic rubber

Assertion (A) Animal skins are colloidal in nature.

Reason (R) Animal skin has positively charged particles.

Match List I with List II.

Which of the following solids is not a molecular solid?

38.6 amperes of current is passed for 100 seconds through an aqueous $$\mathrm{CuSO}_4$$ solution using platinum electrodes. The mass of copper consumed from the solution and volume of gas liberated at STP are respectively (molar mass of $$\mathrm{Cu}=63.54 \mathrm{~g} \mathrm{~mol}^{-1}$$).

The time required for completion of $$93.75 \%$$ of a first order reaction is $$x$$ minutes. The half-life of it (in minutes) is

In which of the following, elements are arranged in the correct order of their electron gain enthalpies?

The set of molecules in which the central atom is not obeying the octet rule is

The hybridisations of carbon in graphite, diamond and $$\mathrm{C}_{60}$$ are respectively

The pair of elements that form both oxides and nitrides, when burnt in air are

Which of the following is formed when $$\mathrm{SO}_3$$ is absorbed by concentrated $$\mathrm{H}_2 \mathrm{SO}_4$$ ?

Identify the correct statements about the anomalous behaviour of boron.

I. Boron trihalides can form dimeric structures.

II. Boron shows +1 as stable oxidation state.

III. Maximum covalency of boron is four.

IV. Boron does not form $$\mathrm{BF}_6^{6-}$$ ion.

Among $$\mathrm{P}_4, \mathrm{~S}_8$$ and $$\mathrm{N}_2$$ the elements which undergo disproportionation when heated with NaOH solution.

Assertion (A) Transition metals and their complexes show catalytic activity.

Reason (R) The activation energy of a reaction is lowered by the catalyst.

The crystal field theory is successful in explaining which of the following?

I. Ligands as point charges.

II. Formation and structures of complexes.

III. Colour.

IV. Magnetic properties.

V. Covalent character of metal-ligand bonding.

Identify the ortho and para-directing groups towards aromatic electrophilic substitution reactions from the following list

$$\mathrm{\mathop { - OH}\limits_I \mathop { - CN}\limits_{II} \mathop { - C{O_2}H}\limits_{III} \mathop { - OC{H_3}}\limits_{IV} \mathop { - NHCOC{H_3}}\limits_V \mathop { - CHO}\limits_{VI}}$$

The major product of the following reaction is

Identify the major product (Y) from the following reaction,

Finkelstein reaction is used for the synthesis of

Identify the major product of the following reaction.

Which among the following will have highest density?

Identify the major product of the following reaction,

An aryl carboxylic acid on treatment with sodium hydrogen carbonate liberates a gaseous molecule. Identify the gas molecule liberated.

Pernicious anemia is caused due to deficiency of which vitamin ?

Which of the following vitamins cannot be stored in the body?

$$\left\{x \in R / \frac{\sqrt{|x|^2-2|x|-8}}{\log \left(2-x-x^2\right)}\right.$$ is a real number $$\}=$$

If $$4+6\left(e^{2 x}+1\right) \tanh x=11 \cosh x+11 \sinh x$$ then $$x=$$

The sum of the real roots of the equation $$|x-2|^2+|x-2|-2=0$$ is

If $$\frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6}$$, then $$A^2+B^2=$$

In how many ways can the letters of the word "MULTIPLE" be arranged keeping the position of the vowels fixed?

A natural number $$n$$ such that $$n!$$ ends in exactly 1000 zeroes is

A true statement among the following identities is

If $$A+B+C=\pi, \cos B=\cos A \cos C$$, then $$\tan A \tan C=$$

The value of $$\tan \left(\frac{7 \pi}{8}\right)$$ is

$$1+\sec ^2 x \sin ^2 x=$$

In a $$\triangle A B C, 2(b c \cos A+a c \cos B+a b \cos C)=$$

In a $$\triangle A B C, \frac{a}{b}=2+\sqrt{3}$$ and $$\angle C=60^{\circ}$$. Then, the measure of $$\angle A$$ is

In a $$\triangle A B C,\left(\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}\right)^2 \leq$$

In a $$\triangle A B C$$ $$(b+c) \cos A+(c+a) \cos B+(a+b) \cos C=$$

Suppose $$\triangle A B C$$ is an isosceles triangle with $$\angle C=90^{\circ}, A=(2,3)$$ and $$B=(4,5)$$. Then, the centroid of the triangle is

If $$a=2, b=3, c=4$$ in a $$\triangle A B C$$, then $$\cos C=$$

Condition that 2 curves $$y^2=4 a x, x y=c^2$$ cut orthogonally is

The number of those tangents to the curve $$y^2-2 x^3-4 y+8=0$$ which pass through the point $$(1,2)$$ is

If the straight line $$x \cos \alpha+y \sin \alpha=p$$ touches the curve $$\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$$ at the point $$(a, b)$$ on it and $$\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}$$, then $$k=$$

A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is

Two particles $$P$$ and $$Q$$ located at the points $$P\left(t, t^3-16 t-3\right), Q\left(t+1, t^3-6 t-6\right)$$ are moving in a plane, the minimum distance between the points in their motion is

By eliminating the arbitrary constants from $$y=(a+b) \sin (x+c)-d e^{x+e+f}$$, then differential equation has order of

If $$2 x-y+C \log (|x-2 y-4|)=k$$ is the general solution of $$\frac{d y}{d x}=\frac{2 x-4 y-5}{x-2 y+2}$$, then $$C=$$

If $$l$$ and $$m$$ are order and degree of a differential equation of all the straight lines at constant distance of $$P$$ units from the origin, then $$l m^2+l^2 m=$$

If $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x}) d x=e^{\sqrt{x}}[A x+B \sqrt{x}+C]+K$$ then $$A+B+C=$$

If $$f(x)=\int x^2 \cos ^2 x\left(2 x \tan ^2 x-2 x-6 \tan x\right) d x$$ and $$f(0)=\pi$$, then $$f(x)=$$

If $$\int \frac{1+\sqrt{\tan x}}{\sin 2 x} d x=A \log \tan x+B \tan x+C$$, then $$4 A-2 B=$$

$$\int \frac{1+\tan x \tan (x+a)}{\tan x \tan (x+a)} d x=$$

The domain of the real valued function $$f(x)=\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right.$$ is

If $$[\cdot]$$ denotes greatest integer function, then $$\lim _\limits{x \rightarrow \frac{-3}{5}} \frac{1}{\dot{x}}\left[\frac{-1}{x}\right]=$$

Let $$f(x)=\left\{\begin{array}{cl}\frac{1}{|x|}, & \text { for }|x|>1 \\ a x^2+b, & \text { for }|x| \leq 1\end{array}\right.$$. If $$\lim _\limits{x \rightarrow 1^{+}} f(x)$$ and $$\lim _\limits{x \rightarrow 1^{-}} f(x)$$ exist, then the possible values for $$a$$ and $$b$$ are

$$\frac{d}{d x}\left(\lim _{x \rightarrow 2} \frac{1}{y-2}\left(\frac{1}{x}-\frac{1}{x+y-2}\right)\right)=$$

If $$f(x)=\left\{\begin{array}{cc}\frac{x^2 \log (\cos x)}{\log (1+x)} & , \quad x \neq 0 \\ 0 & , x=0\end{array}\right.$$, then at $$x=0, f(x)$$ is

If $$l, m(l< m)$$ are roots of $$a x^2+b x+c=0$$, then $$\lim _\limits{x \rightarrow \alpha} \frac{\left|a x^2+b x+c\right|}{a x^2+b x+c}=$$

$$\int_0^{\pi / 4} e^{\tan ^2 \theta} \sin ^2 \theta \tan \theta d \theta=$$

If $$f(x)=\max \{\sin x, \cos x\}$$ and $$g(x)=\min \{\sin x, \cos x\}$$, then $$\int_0^\pi f(x) d x+\int_0^\pi g(x) d x=$$

If $$x \neq 0$$ and $$f(x)$$ satisfies $$8 f(x)+6 f(1 / x) =x+5$$, then $$\frac{d}{d x}\left(x^2 f(x)\right)$$ at $$x=1$$ is

The area of the circle passing through the points $$(5, \pm 2),(1,2)$$ is

For any real number $$t$$, the point $$\left(\frac{8 t}{1+t^2}, \frac{4\left(1-t^2\right)}{1+t^2}\right)$$ lies on a / an

The ratio of the largest and shortest distances from the point $$(2,-7)$$ to the circle $$x^2+y^2-14 x-10 y-151=0$$ is

A circle has its centre in the first quadrant and passes through $$(2,3)$$. If this circle makes intercepts of length 3 and 4 respectively on $$x=2$$ and $$y=3$$, its equation is

The image of the point $$(3,4)$$ with respect to the radical axis of the circles $$x^2+y^2+8 x+2 y+10=0$$ and $$x^2+y^2+7 x+3 y+10=0$$ is

If the normal to the rectangular hyperbola $$x^2-y^2=1$$ at the point $$P(\pi / 4)$$ meets the curve again at $$Q(\theta)$$, then $$\sec ^2 \theta+\tan \theta=$$

If the vertices and foci of a hyperbola are respectively $$( \pm 3,0)$$ and $$( \pm 4,0)$$, then the parametric equations of that hyperbola are

Suppose a parabola passes through $$(0,4),(1,9)$$ and $$(4,5)$$ and has its axis parallel to the $$Y$$-axis. Then, the equation of the parabola is

Suppose $$P$$ and $$Q$$ are the mid-points of the sides $$A B$$ and $$B C$$ of a triangle where $$A(1,3), B(3,7)$$ and $$C(7,15)$$ are vertices. Then, the locus of $$R$$ satisfying $$A C^2+Q R^2=P R^2$$ is

If the points of intersection of the coordinate axes and $$|x+y|=2$$ form a rhombus, then its area is

Suppose, in $$\triangle A B C, x-y+5=0, x+2 y=0$$ are respectively the equations of the perpendicular bisectors of the sides $$A B$$ and $$A C$$. If $$A$$ is $$(1,-2)$$, the equation of the line joining $$B$$ and $$C$$ is

If the pair of straight lines $$9 x^2+a x y+4 y^2+6 x+b y-3=0$$ represents two parallel lines, then

A line passing through $$P(2,3)$$ and making an angle of $$30^{\circ}$$ with the positive direction of $$X$$-axis meets $$x^2-2 x y-y^2=0$$ at $$A$$ and $$B$$. Then the value of $$P A: P B$$ is

The focal distances of the point $$\left(\frac{4}{\sqrt{5}}, \frac{3}{\sqrt{5}}\right)$$ on the ellipse $$\frac{x^2}{4}+\frac{y^2}{9}=1$$ are

In second period of the long form of the periodic table an element $$X$$ has second lowest first ionisation enthalpy and element $$Y$$ has second highest first ionisation enthalpy values. What are $$X$$ and $$Y$$ ?

If $$f(f(0))=0$$, where $$f(x)=x^2+a x+b, b \neq 0$$, then $$a+b=$$

If the difference between the roots of $$x^2+a x+b=0$$ and that of the roots of $$x^2+b x+a=0$$ is same and $$a \neq b$$, then

For what values of $$a \in Z$$, the quadratic expression $$(x+a)(x+1991)+1$$ can be factorised as $$(x+b)(x+c)$$, where $$b, c \in Z$$ ?

If a set $$A$$ has $$m$$-elements and the set $$B$$ has $$n$$-elements, then the number of injections from $$A$$ to $$B$$ is

Which of the following is not a property of a Binomial distribution?

Five digit numbers are formed by using digits $$1,2,3,4$$ and 5 without repetition. Then, the probability that the randomly chosen number is divisible by 4 is

A manager decides to distribute ₹ 20000 between two employees $$X$$ and $$Y$$. He knows $$X$$ deserves more than $$Y$$, but does not know how much more. So, he decides to arbitrarily break ₹ 20000 into two parts and give $$X$$ the bigger part. Then, the chance that $$X$$ gets twice as much as $$Y$$ or more is

For two events $$A$$ and $$B$$, a true statement among the following is

In a Binomial distribution $$B(n, p)$$, if the mean and variance are 15 and 10 respectively, then the value of the parameter $$n$$ is

The least value of $$n$$ so that $${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$$

$$O A B C$$ is a tetrahedron. If $$D, E$$ are the mid-points of $$O A$$ and $$B C$$ respectively, then $$\mathbf{D E}=$$

If $$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$$ and $$|\mathbf{a}|=7,|\mathbf{b}|=5,|\mathbf{c}|=3$$ then the angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is

If $$P$$ and $$Q$$ are two points on the curve $$y=2^{x+2}$$ in the rectangular cartesian coordinate system such that $$\mathbf{O P} \cdot \hat{i}=-1, \mathrm{OQ} \cdot \hat{i}=2$$, then $$\mathrm{OQ}-4 \mathrm{OP}=$$

If the equation of the plane which is at a distance of $$1 / 3$$ units from the origin and perpendicular to a line whose directional ratios are $$(1,2,2)$$ is $$x+p y+q z+r=0$$, then $$\sqrt{p^2+q^2+r^2}=$$

If $$(2,3, c)$$ are the direction ratios of a ray passing through the point $$C(5, q, 1)$$ and also the mid-point of the line segment joining the points $$A(p,-4,2)$$ and $$B(3,2,-4)$$, then $$c \cdot(p+7 q)=$$

If $$x$$-coordinate of a point $$P$$ on the line joining the points $$Q(2,2,1)$$ and $$R(5,2,-2)$$ is 4, then the $$y$$-coordinate of $$P=$$

If $$A=\left[\begin{array}{ccc}1 & 0 & 0 \\ a & -1 & 0 \\ b & c & 1\end{array}\right]$$ is such that $$A^2=I$$, then

If $$a, b, c$$ are respectively the 5 th, 8 th, 13 th terms of an arithmetic progression, then $$\left|\begin{array}{ccc}a & 5 & 1 \\ b & 8 & 1 \\ c & 13 & 1\end{array}\right|=$$

If $$A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]$$, then $$\left(A^T\right)^2+(12 A)^T=$$

Let $$A=\left[\begin{array}{ccc}-2 & x & 1 \\ x & 1 & 1 \\ 2 & 3 & -1\end{array}\right]$$. If the roots of the equation $$\operatorname{det} A=0$$ are $$l, m$$ then $$l^3-m^3=$$

If the mean of the data $$p, 6,6,7,8,11,15,16$$, is 3 times $$p$$, then the mean deviation of the data from its mean is

Multiplicative inverse of the complex number $$(\sin \theta, \cos \theta)$$ is

$$\sum_\limits{k=0}^{440} i^k=x+i y \Rightarrow x^{100}+x^{99} y+x^{242} y^2+x^{97} y^3=$$

If $$e^{i \theta}=\operatorname{cis} \theta$$, then $$\sum_\limits{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}=$$

If $$I_n=\int_0^{\pi / 4} \tan ^n x d x$$, then $$\frac{1}{I_2+I_4}+\frac{1}{I_3+I_5}+\frac{1}{I_4+I_6}=$$