Let S be non-empty subset of R. Consider the following statement P: There is a rational number \[x\in S\] such that \[x>0\] Which of the following statement is the negation of the statement P?
A)
There is no rational number \[x\in S\] such that\[x\le 0\].
doneclear
B)
Every rational number \[x\in S\] satisfies \[x\le 0\].
doneclear
C)
\[x\in S\] and \[x\le 0\Rightarrow x\] is not rational.
doneclear
D)
There is a rational number \[x\in S\] such that\[x\le 0\].
If the area enclosed between \[f(x)=\] Min. \[({{\cos }^{-1}}(\cos x),{{\cot }^{-1}}(\cot x))\] and \[x\]-axis in \[x\in (\pi ,2\pi )\] is \[\frac{{{\pi }^{2}}}{k}\] where \[k\in N\], then k is equal to
For \[x\in (-1,\,\,\,1)\], the number of solutions of the equation \[{{\tan }^{-1}}(x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+.....\,\infty )+\] \[{{\cot }^{-1}}(-6+6x-6{{x}^{2}}+......\infty )=\frac{\pi }{2}\] is
If the smallest radius of a circle passing through the intersection of \[{{x}^{2}}+{{y}^{2}}+2x=0\] and \[x-y=0\], is r then the value of \[(10\,{{r}^{2}})\] is equal to
If distance of the point (2, 5) from the line \[3x+y=4\] measured parallel to the line \[3x-4y+8=0\], is \[\frac{p}{q}(p,q\in N)\], then least value of \[(p+q)\] equals
The area of the rectangle formed by the perpendiculars from the centre of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a>b>0)\] to the tangent and normal at its point whose eccentric angle is \[\frac{\pi }{4}\] is
If three points A(6, 2), B(4, 0) and \[C(\alpha ,\beta )\] are such that \[\left( \left| AC+CB \right|+\left| AC-CB \right| \right)\] is minimum, then the value of \[\left( \alpha +\beta \right)\] is equal to
If the circle \[{{x}^{2}}+{{y}^{2}}-2x-4y+k=0\] and director circle of ellipse \[\frac{{{x}^{4}}}{4}+{{y}^{2}}=1\] intersects orthogonally then k equals
Let ABC be a variable triangle such that A is (1,2), B and C lie on the line \[y=x+\lambda \] (where \[\lambda \] is a variable). The locus of the orthocentre of triangle ABC is a straight line whose y-intercept is equal to
In triangle ABC, if \[\left| \begin{matrix} 1 & a & b \\ 1 & c & b \\ 1 & b & c \\ \end{matrix} \right|=0\] then the value of \[{{\sin }^{2}}A+{{\cos }^{2}}B+{{\tan }^{2}}C\] is equal to
A straight line passes through a fixed point\[({{x}_{0}},{{y}_{0}})\]. If the equation of the locus of the middle point of it intercepted between the coordinate axes is \[y{{x}_{0}}x{{y}_{0}}=\lambda xy\] (where \[\lambda \in N\]), then \[\lambda \] equals
A circle \[S=0\] passes through points of intersection of circles \[{{x}^{2}}+{{y}^{2}}-2x+4y=1\] and \[{{x}^{2}}+{{y}^{2}}-4x-2y-5=0\] and cuts the circle \[{{x}^{2}}+{{y}^{2}}-4=0\] orthogonally. Then the length of tangent from origin on circle \[S=0\], is
Let \[f(x)=\left\{ \begin{matrix} \frac{\int\limits_{0}^{{{x}^{2}}}{\sin \sqrt{x}dx}}{{{x}^{3}}} & x>0 \\ k, & x=0 \\ \end{matrix} \right.\] If \[f(x)\] is continuous at \[x=0\] then k equals
A function f is continuous and differentiable on Ro and satisfies the condition \[x\,\,f'(x)+f(x)=1\] throughout its domain, with \[f(1)=2\]. Then the range of the function is
Let \[g:[-2,2]\to R\] where\[g(x)={{x}^{2015}}+\sgn (x)+\left[ \frac{{{x}^{2}}+1}{p} \right]\] be an odd function for all \[x\in [-2,2]\] then the smallest integral value of p is equal to [Note: [k] denote the greatest integer less than or equal to k.]
From the point (4, 6), a pair of tangent lines are drawn to the parabola \[{{y}^{2}}=8x\]. The area of the triangle formed by these pair of tangent hues and the chord of contact of the point (4, 6) is
Let A and B are two square matrices of order 3 each that det. \[(A)=3\] and det. \[(B)=2\], then the value or det \[\left( {{\left( adj.\left( {{B}^{-1}}{{A}^{-1}} \right) \right)}^{-1}} \right)\]is equal to [Note: adj M denotes the adjoint of a square matrix M.]
To find the distance d over which a signal can be seen clearly in foggy conditions, a railway engineer uses dimensional analysis and assumes that the distance depends on the mass density \[\rho \] of the fog, intensity (power / area) S of the light from the signal and its frequency f. The engineer finds that S is proportional to \[{{S}^{1/n}}\]. The value of n.
A body is projected vertically upwards. The times corresponding to height h while ascending and while descending are \[{{t}_{1}}\] and \[{{t}_{2}}\] respectively. Then, the velocity of projection is (g is acceleration due to a, gravity)
The front wheel on an ancient bicycle has radius \[0.5\,m\] It moves with an angular velocity given by the function \[\omega \,(t)=2+4\,{{t}^{2}}\], where \[t\] is in sec. About how far does the bicycle move between \[t=2\,s\] and \[t=3\,s\].
The work done on particle of mass m by a force, \[K\left[ \frac{x}{{{({{x}^{2}}+{{y}^{2}})}^{3/2}}}\hat{i}+\frac{y}{{{({{x}^{2}}+{{y}^{2}})}^{3/2}}}\hat{j} \right]\] (K being a constant of appropriate dimension) when the particle is taken from the point (a, 0) to the point (3a,4a) along a circular path of radius a about the origin in the XY-plane is:
A body of mass 5 kg starts from the origin with an initial velocity \[u=(30\,\hat{i}+40\hat{j})m{{s}^{-1}}\] If a constant force \[(-6\hat{i}+5\hat{j})N\] acts on the body, the time in which the y component of the velocity becomes zero is:
A bob of mass m attached to an inextensible string of length \[l\] is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed co rad/s about the vertical line passing through support. About the point of suspension,
A)
angular momentum is conserved
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B)
angular momentum changes in magnitude but not in direction.
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C)
angular momentum changes in direction but not in magnitude.
doneclear
D)
angular momentum changes both in direction and magnitude.
A bus moving on a level road with a velocity v can be stopped at a distance of \[x\], by the application of a retarding force F. The load on the bus is increased by \[25%\] by boarding the passengers. Now, if the bus is moving with the same speed and if the same retarding force is applied, the distance travelled by the bus before it stops is :
A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency \[\omega \]. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform:
A)
for an amplitude of \[2g/3{{\omega }^{2}}\]
doneclear
B)
for an amplitude of \[g/{{\omega }^{2}}\]
doneclear
C)
for an amplitude of \[{{g}^{2}}/{{\omega }^{2}}\]
doneclear
D)
for an amplitude of \[2{{g}^{2}}/3{{\omega }^{2}}\]
A material has poisson's ratio \[0.50\]. If a uniform rod of it suffers a longitudinal strain of \[2\times {{10}^{-3}}\], then the percentage change in volume is:
By sucking through a straw, a student can reduce the pressure in his lungs to 750 mm of Hg (density \[=13.6\,\,gc{{m}^{-3}}\]) Using the straw, he can drink water from a glass upto a maximum depth of
Weight of a body of mass m decreases by \[1%\] when it is raised to height h above the earth's surface. If the body is taken to a depth h in a mine, change in its weight is:
The apparent frequency of the whistle of an engine changes in the ratio \[9:8\] as the engine passes a stationary observer. If the velocity of the sound is \[340\,m{{s}^{-1}}\], then the velocity of the engine is :
Two identical positive charges are fixed on the y-axis at equal distances from the origin O. A negatively charged particle starts on the x-axis, at a large distance from O, moves along the x-axis, passes through O and moves far away from O. Its acceleration 'a? is taken as positive along its direction of motion. The best graph between the particle's acceleration and its x-coordinate is represented by:
A 400 pF capacitor is charged by a 100 V supply. How much electrostatic energy is lost in the process of disconnecting from the supply and connecting another uncharged 400 pF capacitor:
A galvanometer of resistance \[50\,\,\Omega \] is connected to a battery of 3V along with a resistance of \[2950\,\,\Omega \] in series. A full scale deflection of 30 divisions is obtained in the galvanometer. In order to reduce this deflection to 20 divisions, the resitance in series should be:
A solenoid consists of 100 turns of wire and has a length of \[10.0\] cm. The magnetic field inside the solenoid when it carries a current of \[0.500\] A will be
A paramagnetic substance of susceptibility \[3\times {{10}^{-4}}\] is placed in a magnetic field of\[4\times {{10}^{-4}}A{{m}^{-1}}\]. Then, the intensity of magnetization in the unit of \[A{{m}^{-1}}\] is:
Two parallel rails of a railway track insulated from each other and with the ground are connected to a milli voltmeter. The distance between the rails is 1 m. A train is travelling with a velocity of 72 \[km-{{h}^{-1}}\] along the track. The reading of the milli voltmeter (in mV) is (vertical component of the earth's magnetic induction is \[2\times {{10}^{-5}}T\])
The inductance of the oscillatory circuit of a radio station is 10 mh and its capacitance is\[0.25\,\,\mu F\]. Taking the effect of resistance negligible, wavelength of the broadcasted waves will be (velocity of light \[=3.0\times {{10}^{8}}m{{s}^{-1}},\pi =3.14\])
The position of final image formed by the given lens combination from the third lens will be at a distance of \[[{{f}_{1}}=+10\,cm,\,{{f}_{2}}=-10\,cm,{{f}_{3}}=30\,cm]\] 30cm 5cm 10cm
In double slit interference experiment, the fringe width obtained with a light of wavelength \[5900\overset{0}{\mathop{A}}\,\] was \[1.2\] mm for parallel narrow slits placed 2 mm apart. In this arrangement, if the slit separation is increased by one - and - half times the previous value, then the fringe width is :
Different voltages are applied across a p-n junction and the current are measured from each value. Which of the following graphs is obtained between voltage and current?
In process of amplitude modulation of signal to be transmitted. Signal to be modulated is given by \[m(t)={{A}_{m}}\sin {{\omega }_{m}}t\], carrier wave is given by \[c(t)={{A}_{c}}\sin {{\omega }_{c}}t\], modulated signal \[{{c}_{m}}(t)\] is given by
Usually a disilicate share only one oxygen of silicate unit But if in the disilicate, two O atoms are shared, then formula of its salt with potassium is :
For the reaction\[{{N}_{2}}(g)+3{{H}_{2}}(g)2N{{H}_{3}}(g)=93.6\,kJ\]the amount (in moles) of \[{{H}_{2}}\] in the system at equilibrium will increase if -
The first of ionisation potential for Li is \[5.41\] eV and electron affinity of \[C\,l\] is \[3.61\] eV. Enthalpy of following reaction in kJ/mole is\[L{{i}_{(g)}}+C\,{{l}_{(g)}}\xrightarrow{{}}L{{i}_{(g)}}+C{{\ell }^{-}}_{(g)}(1\,eV\]\[=1.6\times {{10}^{-22}}kJ)\]
The specific conductivity of an aqueous solution of a weak monoprotic acid is \[0.00033\,oh{{m}^{-1}}c{{m}^{-1}}\] at a concentration, \[0.02\] M. If at this concentration the degree of dissociation is \[0.043\], calculate the value of \[{{\wedge }_{0}}(oh{{m}^{-1}}c{{m}^{2}}/eqt)\].
An ideal gas expands from \[1.5\] to \[6.5\] litre against as constant pressure of \[0.5\] atm and during this process the gas also absorbs 100J of heat. The change in the internal energy of the gas is-
Which of the following statement is not correct when a mixture of NaCI and \[{{K}_{2}}C{{r}_{2}}{{O}_{7}}\] is gently warmed with concentrated \[{{H}_{2}}S{{O}_{4}}\]?
A)
A deep red vapour is evolved
doneclear
B)
The vapour when passed into NaOH solution gives a yellow solution of \[N{{a}_{2}}Cr{{O}_{4}}\]
The mass of \[{{C}_{6}}{{H}_{12}}{{O}_{6}}\] that should be dissolved in 100 gram of water in order to produce same lowering of vapour pressure as is produced by dissolving 1 gram of urea in same quantity of water is
How many diastereomeric pairs are possible in the given compound when COOH group in the given structure is replaced by \[C{{H}_{3}}\] group\[C{{H}_{3}}-\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,H-\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,H-COOH\]