The sum of the following series \[1+6+\frac{9\,({{1}^{2}}+{{2}^{2}}+{{3}^{3}})}{7}+\frac{12\,({{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}})}{9}\] \[+\frac{15\,({{1}^{2}}+{{2}^{2}}+...+{{5}^{2}})}{11}+...\] up to 15 terms, is:
For each \[x\in R\] let \[[x]\] be the greatest integer less than or equal to\[x\]. then Then \[\underset{x\to {{0}^{+}}}{\mathop{\text{lim}}}\,\frac{x([x])+\left| x \right|\sin [x]}{\left| x \right|}\] is equal to:
Let f : \[[0,1]\to \text{R}\]be such that\[f(xy)=f(x)f(y)\] for all \[x,y\in [0,1]\] and\[f(0)\ne 0.\] If \[y=y(x)\] satisfies the differential equation, \[\frac{dy}{dx}=f(x)\] with \[y(0)=1,\]then \[y\left( \frac{1}{4} \right)+y\left( \frac{3}{4} \right)\] is equal to:
If both the roots of the quadratic equation \[{{x}^{2}}-5x+4=0\]are real and distinct and they lie in the interval [1, 5], then m lies in the interval.
Let S be the set of all triangle in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is:
Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P. then \[\frac{a}{c}\]is equal to:
The equation of the plane containing the straight line \[\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\]and perpendicular to the plane containing the straight lines \[\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\]and \[\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\]is:
A data consists of n observations: \[{{x}_{1}},\]\[{{x}_{2}},\]?\[{{x}_{n}}.\] If\[\sum\limits_{i=1}^{n}{{{({{x}_{i}}+1)}^{2}}}=9n\] and \[\sum\limits_{i=l}^{n}{{{({{x}_{i}}-1)}^{2}}=5n,}\] then the standard deviation of this data is:
Let f be a differentiable function R to R such that \[\left| f(x)-f(y) \right|\le 2{{\left| x-y \right|}^{\frac{3}{2}}},\]for all \[x,y\in R.\] If \[f(0)=1\]then \[\int\limits_{0}^{1}{{{f}^{2}}}(x)dx\]is equal to:
Figure shows a student, sitting on a stool that can rotate freely about a vertical axis. The Student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose moment of inertia is I about its central axis. The wheel is rotating at an angular speed en; from an overhead perspective, the rotation is counter clockwise. The axis of the wheel point?s vertical, and the angular momentum l, of the wheel points vertically upward. The student now inverts the wheel; as a result, the student and stool rotate about the stool axis. With what angular speed and direction does the student then rotate? (The moment of inertia of the student + stool + wheel system about the stool axis is in).
A light cylindrical vessel is kept on a horizontal surface. Its base area is A.A hole of cross sectional area a is made just at its bottom side. The minimum coefficient of friction necessary for sliding of the vessel due to the impact force of the emerging liquid is \[~\left( a<<A \right):\]
A solid sphere of radius \[{{R}_{1}}\] and volume charge density \[\rho =\frac{{{\rho }_{0}}}{r}\] is enclosed bt a hollow sphere of radius \[{{R}_{2}}\] with negative surface charge density \[\sigma ,\]such that the total charge in the system is zero. \[{{\rho }_{0}}\] is a positive constant and \[r\]is the distance from the Centre of the sphere. The ratio \[{{R}_{2}}/{{R}_{1}}\] is
A boy of height \[h\] is walking away from a street lamp with a constant speed v. the height of the street lamp is \[3h\]. The rate at which the length of the boy's shadow is increasing when he is at a distance of \[10h\] from the base of the street lamp is:
An ac source of angular frequency \[\omega \] is fed 3 across a resistor r and a capacitor C in series. The current registered is \[I\]. If now the frequency Of source is changed to \[\omega /3\] (but maintaining the same voltage), the current in the circuit is found to be halved. Calculate the ratio of reactance to resistance at the original frequency \[\omega \]
The tension in a string holding a solid block below the surface of a liquid of density greater than that of solid as shown in figure is Ty, when the system is at rest. Tension in the string if the system has upward acceleration 'a' will be:
A thin spherical shell of radius R lying on a rough horizontal surface is hit sharply and horizontally by a cue. Where should it be hit so that the shell does not slip on the surface?
Two long cylindrical metal tubes stand on insulating floor. A dielectric oil is filled between plates. Two tubes are maintained with potential difference \[V.\text{ }A\]small hole is opened at bottom then
A)
Reading of ammeter decreases
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B)
Capacitance of system increases
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C)
Current in circuit is dependent on area of hole
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D)
Current in circuit is inversely proportional to dielectric constant
A thin movable plate is separated from two fixed plates \[{{P}_{1}}\]and \[{{P}_{2}}\] by two highly viscous liquids of coefficients of viscosity \[{{n}_{1}}\] and \[{{n}_{2\,}}\] as shown where \[{{n}_{2}}=9{{n}_{1}}.\]Area of contact of movable plate with each fluid is same. If the distance between two fixed plates is h, then the distance \[{{h}_{1}}\] movable plate form upper plate such that movable plate can be moved with a finite velocity by applying the minimum possible force on movable plate is (assume only linear velocity distribution in each liquid)
A single slit of width a is illuminated by violet light of wavelength 400 nm and the width of the diffraction pattern is measured as' y. When half of the slit width is covered and illuminated by yellow light of wavelength 600 nm, the width of the diffraction pattern is
If x is horizontal and y is vertical direction and magnetic field in the space is\[{{b}_{0}}j,\] the jumper can remain in equilibrium when y coordinate of its ends is (\[i\]= current in jumper)
A torsional pendulum consists of a solid disc connected to a thin wire \[\alpha =2.4\times 10-5/{}^\circ C\]at its centre. Find the percentage change in the time period between peak winter \[\left( 5{}^\circ C \right)\]and peak summer\[\left( 45{}^\circ C \right)\].
A bar magnet suspended at a place P where Dip angle is \[60{}^\circ \] gives 10 oscillations per minute. The same bar magnet suspended at another place Q where dip angle is \[30{}^\circ \] gives 20 osciilations per minute. The ratio of magnetic Field at \[P\] and \[Q\],\[\frac{{{B}_{P}}}{{{B}_{Q}}}\] is
A chain of length t is placed on a smooth Spherical surface of radius \[R\] with one of its ends fixed at the top of the sphere. What will be the acceleration of the each element of the chain when its upper end is released? It is assumed that the length of the chain \[\ell <\left( \frac{\pi R}{2} \right)\]
A rod of length \[\ell \] is pivoted about a horizontal, frictionless pin through one end. The rod is released from rest in a vertical position shown in Figure. Find velocity of the C.M. of the rod, when rod is inclined at an angle \[\theta \] from the vertical.
A given object takes \[n\]times as much time to slide down a \[45{}^\circ \] rough incline as it takes to slide down a perfectly smooth \[45{}^\circ \] incline. The coefficient of friction between the object and the incline is
A 'thermacole' icebox is a cheap and efficient method for storing small quantities of cooked food in summer in particular. A cubical icebox of side 30 cm has a thickness of \[5.0\] cm. If \[4.0\] kg of ice is put in the box, estimate the amount of ice remaining after 6h. The outside temperature is \[45{}^\circ \]C and coefficient of thermal conductivity of thermacole is 0.01 J/s-m-K. (Heat of fusion of water =\[335\times {{10}^{3}}\]J/kg)
Two \[{{l}^{st}}\] order reactions have half-lives in the ratio 3 : 2. Then the ratio of time intervals \[{{t}_{1}}:{{t}_{2}},\] will be? Where \[{{t}_{1}}\] is the time period for 25% completion of the first reaction and \[{{t}_{2}}\] is time required for 75% completion of the second reaction, \[[\log 2=0.3,\,\,\log 3=0.48]\]
Decomposition of A follows first order kinetics by the following equation.
\[4A\,(g)\xrightarrow{{}}B\,(g)+2C\,(g)\]
If initially, total pressure was 800 mm of Hg and after 10 minutes it is found to be 650 mm of Hg. What is half-life of A? (Assume only A is present initially)
Electrolysis of a solution of \[HSO_{4}^{\,-}\] ions produces \[{{S}_{2}}{{O}_{8}}^{-\,-}.\] Assuming 75% current efficiency, what current should be employed to achieve a production rate of 1 mole of \[{{S}_{2}}{{O}_{8}}^{-\,-}\] per hour?
In the reaction sequence \[Cr{{O}_{2}}C{{l}_{2}}\xrightarrow{NaOH}A\xrightarrow{\text{dil}\,\,{{H}_{2}}S{{O}_{4}}}B\xrightarrow{NaOH}C\xrightarrow{AgN{{O}_{3}}}D\]
A solution of 0.2 mole Kl \[(\alpha =100%)\] in 1000 g water freezes at \[{{T}_{1}}^{0}C.\] Now to this solution 0.1 mole \[Hg{{I}_{2}}\] added and the resulting solution freezes at \[{{T}_{2}}^{0}C.\] Which of the following is correct:
Consider the following statements in respect of the reaction \[B{{r}^{-}}+R-C{{H}_{2}}-\overset{+}{\mathop{O}}\,{{H}_{2}}\xrightarrow{{}}Br-C{{H}_{2}}-R+\text{ }{{H}_{2}}O\] 1. \[B{{r}^{-}}\] is a nucleophile and protonated alcohol is an electrophile. 2. It is nucleophilic displacement of water from protonated alcohol by \[B{{r}^{-}}\]nucleophile. Which of the statements given above is /are correct?
A hyperbola has its centre at the origin, passes through the point (4. 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity of the hyperbola is:
Let\[A\,(4,-\,\,4)\] and B(9, 6) be points on the parabola \[{{y}^{2}}=4x.\] be chosen on the arc AOB of the parabola, where O is the origin, such the area of \[\Delta ACB\] is the area (in sq. units) of \[\Delta ACB\] is:
Let the equation of two sides of a triangle be \[3x-2y+6=0\] and \[4x+5y-20=0\] the orthocentre of this triangle is at (1, 1), then the equation of its third side is:
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is:
Let \[\vec{a}=\hat{i}+\sqrt{2}\hat{k},\] \[\vec{b}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+\sqrt{2}\hat{k}\] and \[\vec{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k},\] be three vectors such that the projection vector of \[\vec{b}\] on \[\vec{a}\] is \[\vec{a}\]. If \[\vec{a}+\vec{b}\] is perpendicular to \[\vec{c},\] then \[\left| {\vec{b}} \right|\]is equal to:
A bulb of capacity 500 \[c{{m}^{3}}\]is joined by a narrow tube to a B of capacity 250 \[c{{m}^{3}}\]and they are filled with hydrogen at S.T.P. if the temperature of bulb A is raised to \[100{}^\circ C,\]find the new pressure of the system and the mass of air which is transferred effectively from one blub to another after a long time. Density of air at S.T.P. is \[1.28g/litre.\]
An artificial satellite of the moon revolves in a circular orbit whose radius exceeds the radius of the moon \[\eta \] times. The process of motion the satellite experiences a slight resistance due to cosmic dust. Assuming the resistance force to depend on the velocity of the satellite as \[F=\alpha {{\nu }^{2,}}\] where \[\alpha \]is a constant, find how long the satellite will stay in orbit until it falls onto the moon?s surface
A body falling freely from a give height ?H? hits an inclined plane in its path at a height ?h? As a result of this impact the direction of the velocity of the body becomes horizontal. For what value of \[(h/H)\] the body will take maximum time to reach the ground?
In the arrangements shown, a block (of mass\[m\]) is being moved up against gravity, by two identical balloons, with constant speed\[v\]. The balloons carry \[+Q\]charge each and the connecting strings are massless. \[T\] and \[B\] respectively represent tension in each of the connecting strings and buoyant force on each of the balloons. Choose the incorrect alternative.
A 4\[\mu F\] capacitor, a resistance of \[2.5M\Omega \] is in series with 12 V battery. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. \[\left[ Given\,\ell n(2)=0.693 \right]\]
Rishabh of Raxaul skated the 10,000 m race in Salt Lake City in 12 min, 58.92 seconds. The oval track is made up of two straight 112.00m sections and two essentially identical semicircular curves. There are two lanes, each 5.00m wide. The 400m Lap starts at \[A\] on the inner straightway, rounds the inner curve, crosses over in the next straight section in the shortest diagonal path to the outside lane (the other skater crosses over the other way), and rounds the outer curve, ending up on the adjacent lane at \[B\] (see dotted line). The measurement is made 5cm out from the inner edge of the lane, and is exactly 400m for one lap. What is the radius of the inner curve, R, in m?
In given figure, a wire loop has been bent so that it has three segments: segment \[ab\] (a quarter circle), \[bc\] (a square corner), and \[ca\] (straight). Here are three choices for a magnetic field through the loop:
Where \[\overset{\to }{\mathop{B}}\,\] is in millitesla and t is in seconds. If the induced current in the loop due to \[{{\overset{\to }{\mathop{B}}\,}_{1,}}{{\overset{\to }{\mathop{B}}\,}_{2}}\] and \[{{\overset{\to }{\mathop{B}}\,}_{3}}\] are \[{{i}_{1,}}\,\,{{i}_{2}}\,\,\text{and}\,\,\,{{i}_{3}}\]respectively then
For the two parallel rays \[AB\] and \[DE\] shown here, \[BD\] is the wave front. For what value of wavelength of rays destructive interference takes place between ray \[DE\] and reflected ray \[CD\]?
Water is boiled in a rectangular steel tank of thickness 2 cm by a constant temperature furnance. Due to vaporization water level falls at a steady rate of 1 cm in 9 min. calculate temperature of the furnance. Given, K for steel \[=0.2\,cal{{s}^{-1}}{{m}^{-1}}{}^\circ {{C}^{-1.}}\]
You are given the following cell at 298 K, \[Zn\,\,\left| \begin{matrix} Z{{n}^{++}}_{(aq.)} \\ 0.01\,\,M \\ \end{matrix} \right|\,\,\left| \begin{matrix} HC{{l}_{(aq.)}} \\ 1.0\,\,\text{lit} \\ \end{matrix} \right|\,\,\left| \begin{matrix} {{H}_{2}}\,(g) \\ 1.0\,\,\,\text{atm} \\ \end{matrix} \right|\,\,Pt\] with \[{{E}_{\text{cell}}}=0.701\] and \[E_{Z{{n}^{2\,+}}/Zn}^{0}=-\,0.76\,V.\] Which of the following amounts of NaOH \[(\text{equivalent }\,\text{weight}=\text{40})\] will just make the pH of cathodic compartment to be equal to 7.0:
The maximum radius of an atom which can occupy empty spaces (voids) in a body centred structure, of an element having atomic radius R, without causing any distortion, can be:
How many moles of sucrose should be dissolved in 500 gms of water so as to get a solution which has a difference of \[104{}^\circ C\]between boiling point and freezing point. \[({{K}_{f}}=1.86K\,\,\text{Kg}\,\,\text{mo}{{\text{l}}^{-1}},\,\,{{K}_{b}}=0.52K\,\,\text{Kg}\,\,\text{mo}{{\text{l}}^{-1}})\]
For the decomposition of \[{{H}_{2}}{{O}_{2}}\,(aq)\] it was found that \[{{V}_{{{O}_{2}}}}\,(t=15\min .)\] was 100 mL (at \[0{}^\circ C\]and 1 atm) while \[{{V}_{{{O}_{2}}}}\](maximum) was 200 mL (at \[0{}^\circ C\] and 2 atm). If the same reaction had been followed by the titration method and if \[V_{KMn{{O}_{4}}}^{(c\,M)}\,(t=0)\] had been 40 mL, what would \[V_{KMn{{O}_{4}}}^{(c\,M)}\,(t=15\min )\] have been?
If dominant C and P genes are essential for the development of purple colour in sweet pea flowers, what would be the ratio of white and purple colour in a cross between \[CcPp\times Ccpp\text{ }-\]
In a cross between individuals homozygous for (a, b) and wild type (++) 700 out of 1000 individuals were of parental type. Then the distance between a and b is -
Given below is an incomplete table about certain hormones, their source glands and one major effect of each on the body in humans, identify the correct option for the three blanks A, B and C.
Gland
Secretion
Effect on Body
Ovary
Maintenance of secondary sexual characters in female