Let\[\left\{ \begin{matrix} -{{x}^{3}}+{{\log }_{2}}b, & 0\le x<1 \\ 3x, & 1\le x\le 3 \\ \end{matrix} \right.\] . The set of real values of b for which \[f(x)\] has smallest value at \[x=1\], is
The value of 'c' in Rolle's theorem for the function\[f(x)=\left\{ \begin{matrix} {{x}^{2}}\cos \left( \frac{1}{x} \right), & x\ne 0 \\ 0, & x=0 \\ \end{matrix} \right.\]in the interval \[[-1,1]\] is
Let \[\omega =\frac{-1}{2}+i\frac{\sqrt{3}}{2}\]. Then the value of the determinant\[\left| \begin{matrix} 1 & 1 & 1 \\ 1 & -1-{{\omega }^{2}} & {{\omega }^{2}} \\ 1 & {{\omega }^{2}} & {{\omega }^{4}} \\ \end{matrix} \right|\] is
Let \[{{T}_{r}}\] be the \[{{r}^{th}}\] term of a sequence, for\[r=1,2,3,.....\] . If \[3{{T}_{r+1}}={{T}_{r}}\] and \[{{T}_{7}}=\frac{1}{243}\], then the value of \[\sum\limits_{r=1}^{\infty }{({{T}_{r}}.T{{}_{r+1}})}\] is
Let \[A-(-3,2)\] and \[B-(-2,1)\] be the vertices of a triangle ABC. If the centroid of triangle ABC lies on the line \[3x+4y+2=0\], then the locus of vertex C is
Let ABC be an equilateral triangle and suppose KLMN be a rectangle with K, L on BC, M on AC and AN N on AB. If \[\frac{AN}{NB}=2\] and area of triangle BKN is 6, then area of triangle ABC is equal to
If the lines \[\vec{r}=-\hat{i}+\hat{j}-\hat{k}+\lambda (2\hat{i}+\hat{j}+3\hat{k})\] and \[\vec{r}=-2\hat{i}+\alpha \hat{j}+\hat{k}+,\mu (2\hat{i}+3\hat{j}+4\hat{k})\] \[(\lambda ,\mu \in R)\] are coplanar, then the value of \[\alpha \] is
If the projections of a line segment on the \[x,y\] and \[z\] axes in 3-dimensional space are 2, 3 anA5 respectively, then the length of line segment is
STATEMENT-1: The line \[x-2y=2\] meets the parabola \[{{y}^{2}}+2x=0\] only at one point \[(-2,-2)\].
STATEMENT-2: The line \[y=\operatorname{m}x-\frac{1}{2m}(m\ne 0)\] is 2m tangent to the parabola \[{{y}^{2}}=-2x\] at the point\[\left( \frac{-1}{2{{m}^{2}}},\frac{-1}{m} \right)\].
A)
Statement-1 is true, statement-2 is true and Statement-2 is correct explanation for statement-1.
doneclear
B)
Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
If \[P=\left| \begin{matrix} 1 & c & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \\ \end{matrix} \right|\] is the adjoint of a \[3\times 3\] matrix Q and det. \[(Q)=4\], then c is equal to
The area of the rectangle formed by the perpendicular from centre of \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\], to the tangent and normal at the point whose eccentric angle is \[\frac{\pi }{4}\] equals
STATEMENT-1: The system of linear equations\[\begin{align} & x+(\sin \theta )y+(\cos \theta )z=0 \\ & x+(\cos \theta )y+(\sin \theta )z=0 \\ & x-(\sin \theta )y-(\cos \theta )z=0 \\ \end{align}\]has a non-trivial solution for only one value of \[\theta \] lying in the interval\[\left( 0,\frac{\pi }{2} \right)\].
STATEMENT-2: The equation in \[\theta \] \[\left| \begin{matrix} \cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & -\sin \theta & -\cos \theta \\ \end{matrix} \right|=0\] has only one solution lying in the interval\[\left( 0,\frac{\pi }{2} \right)\].
A)
Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for statement- 1.
doneclear
B)
Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for statement-1.
The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is (marks all allotted are integers)
If a curve passes through the point \[(1,0)\] and has slope \[\left( 1+\frac{1}{{{x}^{2}}} \right)\] at any point \[(x,y)\] on it, then the ordinate of point on the curve whose abscissa is \[-3\], is
If \[{{\tan }^{-1}}({{x}^{2}}+3\left| x \right|-4)+{{\cot }^{-1}}(4\pi +{{\sin }^{-1}}(\sin \,14))\]\[=\frac{\pi }{2}\], then the value of \[{{\sin }^{-1}}(\sin 2\left| x \right|)\] is equal to
A bag contains \[(2n+1)\] coins. It is known that n of these coin have a head on both sides, whereas the remaining \[(n+1)\] coins are fair. A coin is selected at random from the bag and tossed once. If the probability the toss results in a head is \[31/42\], then n is equal to
A particle changes its velocity from \[6\,m{{s}^{-1}}\] towards east to \[6\,m{{s}^{-1}}\] towards north in 3 second. Then magnitude of its acceleration will be:
Two bodies of mass 'm' and 9 m are placed at distance r. The gravitational potential at a point on the line joining them where the gravitational field is zero is:
A particle of mass \[2\,kg\] has potential energy\[U=8{{x}^{2}}-4{{x}^{4}}\]. Then its angular frequency for small oscillation about its equilibrium position will be:
A bullet of mass m and charge q is fired towards a fixed solid uniformly charged sphere of radius R and total charge\[+q\]. If it strikes the surface of sphere with speed u. Find the minimum speed u so that it can penetrate through the sphere, (neglect all= resistance or friction acting on bullet except electrostatic force)
A charged particle having some mass is resting in equilibrium at a height H above the centre of a uniformly charged non-conducting horizontal ring of radius R. The force of gravity acts downwards. The nature of equilibrium of the particle will be stable
In the shown circuit, all three capacitor are identical and have capacitance C each. Each resistor has resistance of R. An ideal cell of emf V is connected as shown. Then the potential difference across capacitor \[{{C}_{3}}\] in steady state is:
A bar of diamagnetic substance is placed in a magnetic field with its length making angle \[{{30}^{o}}\] with the direction of the magnetic field. How will the bar behave?
A)
It will align itself parallel to the magnetic field
doneclear
B)
It will align itself perpendicular to the magnetic field.
The magnetic field inside a current carrying toroidal solenoid is B. if its radius is doubled and the current through it is also doubled keeping the total number of turns same, the magnetic field inside the solenoid will be:
As shown in figure, a convergent lens is placed inside a cell filled with liquid. The lens has focal length \[+20\,cm\] when in air and its material has refractive index \[1.50\]. If the liquid has refractive index \[1.60\], the focal length of the system is:
In figure S is a monochromatic source emitting light of wavelength \[\lambda =500\,\,nm\] and a thin lens of focal length \[0.10\,\,m\] is cut into identical halves. The two halves are placed symmetrically about the central axis SO with a gap of\[d=0.5\,mm\]. The distance of the lens from S is \[0.150\,m\] and that from O is \[1.30\,m\]. If the third intensity maximum occurs at a point A on the screen (perpendicular to SO), find the distance OA.
In X-ray tube, when the accelerating voltage V is halved, the difference between the wavelengths of \[{{K}_{\alpha }}\] line and minimum wavelength of continuous X- ray spectrum.
The work function of a metallic surface is \[5.01\,\,eV\] photoelectrons are emitted when light of wavelength \[2000\,\overset{0}{\mathop{A}}\,\]falls on it. The potential difference required to stop the fastest photoelectrons is : \[(h=4.14\times {{10}^{-15}}eVs)\]
If the experiment of Newton's law of cooling, the graph drawn between logarithm of excess of temperature with the surroundings and the time is obtained as:
A)
a straight line passing through origin of positive gradient.
doneclear
B)
a straight line of positive gradient, not passing through origin.
doneclear
C)
a straight line of negative gradient, not passing through origin.
A hydrogen atom and \[L{{i}^{++}}\] ion are both in the second excited state. If \[{{L}_{H}}\] and \[{{L}_{Li}}\] are their respective electronic angular momenta, and \[{{E}_{H}}\] and \[{{E}_{Li}}\] their respectvie energies, then :
A)
\[{{L}_{H}}>{{L}_{Li}}\] and \[\left| {{E}_{H}} \right|>\left| {{E}_{Li}} \right|\]
doneclear
B)
\[{{L}_{H}}={{L}_{Li}}\] and \[\left| {{E}_{H}} \right|<\left| {{E}_{Li}} \right|\]
A neutral metallic sphere is present infront of a uniformly charged conducting plane force between sphere and plane is F and torque on sphere is \['\tau '\] then
The resistance and inductance of a coil are 1 ohm and 2 Henry respectively. A capacitor of \[2\mu F\] capacitor is discharged through the above coil. The frequency (n) and quality factor (Q) of L-C circuit will be:
\[20\,ml\] of gas A at \[1\,atm\] is added to \[40\,ml\]l flask and \[40\,ml\] of gas B at \[2\,\,atm\] is added to \[40\,ml\] flask. Find K if \[10\,ml\] of C is formed and equilibrium pressure is \[2.25\,\,atm\]. (Assume A, B and C are ideal gases) \[A(g)+B(g)\overset{{}}{leftrightarrows}C(g)\]
A certain volume V ml of a gaseous hydrocarbon was exploded with an excess oxygen \[3\,V\] ml of \[C{{O}_{2}}\] is produced and for that combustion \[4\,V\] ml \[{{O}_{2}}\] is required. All volume measurements were under the same conditions of temperature and pressure. What is the molecular formula of the hydrocarbon?
The emf of the cell \[\underset{(0.1M)}{\mathop{Fe\left| F{{e}^{2+}} \right|}}\,\]\[\underset{(2\,atm)}{\mathop{\left| {{H}^{+}} \right|{{H}_{2}}}}\,\], Pt is \[0.22\,V\] at \[{{25}^{o}}C\], pH of the solution at hydrogen electrode is
2 mole of ferrous sulphide are oxidised by \[x\] moles of \[KMn{{O}_{4}}\] in acidic medium into ferric sulphate. 3 moles of ferrous oxalate are oxidised by y moles of \[{{K}_{2}}C{{r}_{2}}{{O}_{7}}\] in acidic medium. The value of \[\frac{x}{y}\]