Let $$A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & -11 & 6\end{pmatrix}$$. What is the trace of $$A^5$$?
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Let $$A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & -11 & 6\end{pmatrix}$$. What is the trace of $$A^5$$?
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What is the value of $$9^7-\binom{9}{1}8^7+\binom{9}{2}7^7-\binom{9}{3}6^7+\binom{9}{4}5^7-\binom{9}{5}4^7+\binom{9}{6}3^7-\binom{9}{7}2^7+\binom{9}{8}1^7$$?
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Let $$y=y(x)$$ satisfy $$\frac{1}{y}\frac{dy}{dx}-\frac{2e^x\cos x}{y}=1$$ and $$y(0)=0$$. Then $$y\left(\frac{\pi}{2}\right)=$$
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$$\overline{r\ }=\left(3,4,1\right)+t\left(0,2,-1\right)$$ and $$P=\left(-4,7,6\right)$$ determine a plane $$\pi$$. What is the perpendicular distance from the point $$Q(1,1,1)$$ to $$\pi$$?
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The $$n^{th}$$ terms of two sequences are $$a_n=\frac{3n+1}{2^{n-1}}$$ and $$b_n=\frac{3n-2}{2^n}$$ $$\forall n\in\mathbb{N}$$. $$A_n$$ and $$B_n$$ are the sums of the first $$n$$ terms, respectively, of these sequences. What is the value of $$A_6-B_6$$?
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How many $$5$$-digit odd natural numbers having distinct digits and whose one of the digits is $$0$$, exist?
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The vectors $$5\hat{i}+\hat{j}+\hat{k}, \hat{i}+5\hat{j}+\hat{k}$$ and $$\hat{i}+\hat{j}+5\hat{k}$$ are the three face diagonals of three faces having a common vertex in a parallelopiped. What is the volume of the parallelopiped?
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The value of $$\sin^{-1}\cot\left[\sin^{-1}\left\{\frac{1}{2}\left(1-\sqrt{\frac{5}{6}}\right)\right\}+\cos^{-1}\sqrt{\frac{2}{3}}+\sec^{-1}\sqrt{\frac{8}{3}}\right]$$ is:
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There are two bags $$B_1$$ and $$B_2$$. $$B_1$$ has two white and three black balls. $$B_2$$ has four white and two black balls. A ball is first drawn from $$B_1$$. Its colour is noted and it is put back into $$B_1$$. If the colour is white, then a ball is picked from $$B_2$$, else a ball is picked from $$B_1$$. What is the probability that the second ball is white?
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The focus of a parabola is at $$S(2,1)$$ and the lines $$y=x$$ and $$x+y=0$$ touch the parabola. What is the equation of the parabola?
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Let $$E:\frac{x^2}{36}+\frac{y^2}{16}=1$$ and $$C$$ be its auxiliary circle. $$AB$$ is a chord of $$E$$. $$A', B'$$ are corresponding points of $$A, B$$ respectively on $$C$$. If $$\angle A'OB'=\frac{\pi}{3}$$ and the slope of $$AB$$ is $$\frac{1}{\sqrt{3}}$$, then what is the value of $$AB^2$$?
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Let $$f:\left(0,\frac{7}{5}\right)\longrightarrow$$ $$\mathbb{R}$$ and $$g:\left(0,\frac{7}{5}\right)\longrightarrow$$ $$\mathbb{R}$$ be functions defined by $$f(x)=2[x^2]$$ and $$g(x)=(2|x-1|+3|x-2|)f(x)$$ (where $$[x]$$ is the greatest integer less than or equal to $$x$$). Let
$$a=$$ number of points of discontinuity of $$f$$,
$$b=$$ number of points of non-differentiability of $$f$$,
$$c=$$ number of points of discontinuity of $$g$$, and
$$d=$$ number of points of non-differentiability of $$g$$.
What is the value of $$a+b+c+d$$?
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$$\int_0^1\left(x^4-9\right)e^xdx=$$
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The set $$\{z\in\mathbb{C}:(2+6i)z+(9-i)\overline{z}+10=0\}$$ on the Argand plane represents:
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For $$a\in\mathbb{R}$$, the solution set of $$x^2-3(a+1)x+a^2+a-1\le0$$ is an interval of length $$4$$. What is the sum of the squares of the different values of $$a$$?
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If $$2x+y>0$$, $$2y>x$$, and $$y<3$$, what is the range of all possible values of $$x+y$$?
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Let $$I\left(m,n\right)=\int_0^m\tan^nx\ dx$$. What is the value of $$2I\left(\frac{\ \pi\ }{4},2\right)+3I\left(\frac{\ \pi\ }{4},3\right)+2I\left(\frac{\ \pi\ }{4},4\right)+3I\left(\frac{\ \pi\ }{4},5\right)$$?
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Let $$L_1:\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-1}{2}$$ and $$L_2:\frac{x-2}{1}=\frac{y-3}{4}=\frac{z}{1}$$ be two lines in space. $$M$$ and $$N$$ are points on $$L_1$$ and $$L_2$$ respectively such that $$MN$$ is the shortest distance between $$L_1$$ and $$L_2$$. What is the sum of the coordinates of $$M$$?
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$$\Sigma\ _{n=1}^{15}\left(\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+2\right)\left(n+3\right)}\right)=$$
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What is the area bounded by $$y=x^3-7x^2+14x-8$$ and the x-axis in the interval $$[0,2]$$?
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$$6\int_{0}^{\pi} |(\sin3x+\sin2x+\sin x)|dx$$ is equal to _________.
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Let $$a_{1}=1$$ and for $$n\geq1,a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^2}$$. Then $$\left|\sum_{ n=1}^{ \infty}\left( a_n - \frac{2}{n^2}\right)\right|$$ is equal to ______.
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Let S= {(m, n) :m, n $$\epsilon$$ {1, 2, 3, .... , 50}}. lf the number of elements (m, n) in S such that $$6^m+9^n$$ is a multiple of 5 is p and the number of elements (m, n) in S such that m + n is a square of a prime number is q, then p +q is equal to ________.
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Let $$f:R\rightarrow R$$ be a twice differentiable function such that the quadratic equation $$f(x)m^{2}-2 f'(x)m+ f''(x)=0$$ in m, has two equal roots for every $$x \epsilon R$$. If $$ f(0)=1,f'(0)=2$$, and $$(\alpha,\beta)$$ is the largest interval in which the function $$f(\log_{e}{x-x})$$ is increasing, then $$\alpha+\beta$$ is equal to ________.
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For some $$\alpha,\beta\epsilon R$$, let $$A=\begin{bmatrix}\alpha & 2 \\ 1 & 2 \end{bmatrix}\text{ and }B=\begin{bmatrix}1 & 1 \\1 & \beta \end{bmatrix}$$ be such that $$A^{2}-4A+2I=B^2-3B+I=O$$. Then $$(det(adj(A^3-B^3)))^2$$ is equal to _______.
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The best representing velocity -time ($$v-t$$) graph corresponding to displacement -time ($$s-t$$ ) graph shown is
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Find the equivalent resistance of the given electrical network across terminals A and B:
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Find the value of current $$i$$ in the given circuit
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A wave pulse is propagating on a long string along the length taken as +x axis. Shape of the string
at $$t=0$$ is given by $$y=\frac{1}{x^2+8x+19}$$ and at $$t=2\ \sec$$ the shape is $$y=\frac{1}{x^2+3}$$. find the speed of wave on string
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$$AB$$ is a quarter of a smooth circular track of radius $$R=2m$$ as shown in figure. A particle $$P$$ of mass $$m=5kg$$ moves along the track from $$A\ to\ B$$ under the action of a force which always directed toward point $$B$$ and has magnitude $$10\ N$$. Find the work done by force in moving the object from $$A\ to\ B$$.
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The amplitude of the vibrating particle due to superposition of two SHMs
$$x_1=\ 2\ Sin\left(\omega t\right)$$ and
$$x_2=\ 2\ Sin\left(\omega t+\ \frac{2\pi}{3}\right)$$ .
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A wooden block is floating in a liquid, $$40\ \%\ $$ of its volume is inside the liquid when the vessel is stationary. Percentage of volume immersed when the vessel moves upwards with an acceleration $$a=5\ m.s^{-2}$$ is
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The activity of a radioactive sample decreases to one tenth of the original activity $$A_{0\ }$$ in a period of one year. After 9 more years its activity would be
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The wall of a house is made of two different materials of same thickness. The temperature of the outer wall is $$T_2$$
and that of inner wall is $$T_{1\ }<\ T_2$$ . the temperature variation inside the wall is as shown in the figure in steady state, then
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Two protons move parallel to each other, keeping distance $$r$$ between them, both moving with same velocity $$v$$. Then the ratio of the electric and magnetic force of interaction between them is
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Two smooth spherical non conducting shells each of radius $$R$$ having uniformly distributed charge $$Q$$ & $$-Q$$ on their surfaces are released on a smooth non-conducting surface when the distance between their centers is $$10R$$. The mass of A is $$m$$ and that of B is $$2m$$. The speed of A just before A and B collide is: [Neglect gravitational interaction]
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From a solid sphere of mass $$M$$ and radius $$R$$, a spherical portion of radius $$\left(\frac{R}{2}\right)$$ is removed as shown in the figure. Taking gravitational potential $$V = 0$$ at $$r = \infty$$, the potential at the centre of the cavity thus formed is ($$G$$ = gravitational constant)
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A narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius R . What should be the refractive index if the beam is to be focused at the surface of the sphere,
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The radiation force experienced by body exposed to radiation of intensity $$I$$ , assuming surface of body to be perfectly absorbing is :
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A cubical volume is bounded by the surfaces $$x = 0$$, $$x = a$$, $$y = 0$$, $$y = a$$, $$z = 0$$, $$z = a$$. The electric field in the region is given by $$\vec{E} = E_0 x \hat{i}$$. Where $$E_0 = 4 \times 10^4$$ NC$$^{-1}$$ m$$^{-1}$$. If $$a = 2$$ cm, the charge contained in the cubical volume is $$Q \times 10^{-14}$$ C. The value of $$Q$$ is ______. (Take $$\epsilon_0 = 9 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$m$$^{-2}$$)
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The angle of polarization for any medium is $$60^{\circ\ }$$, what will be critical angle for this :
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A uniform chain of length $$l\ $$ has one of its end attached to the wall a point A, while $$\frac{3l}{4}$$of the length of the chain is lying on table as shown in figure. Find the minimum co-efficient of friction between table and chain so that chain remains in equilibrium is.
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A rod of length 5L is bent right angle keeping one side length as 2L.
The position of the centre of mass of the system: (Consider L = 10 cm)
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A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is
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A thin circular ring of mass $$m$$ and radius $$R$$ is rotating about its axis with a constant angular velocity $$\omega$$. Two objects each of mass $$M$$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $$\omega' =$$
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Two identical thin rods of mass M kg and length L m are connected as shown in the figure below. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $$\frac{x}{12}ML^{2}\text{kg m}^{2}$$. The value of x is ____ .
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10 mole of oxygen is heated at constant volume from $$30^{\circ}C \text{to} 40^{\circ}C$$. The change in the internal energy of the gas is ____ cal (the molecular specific heat of oxygen at constant pressure, $$C_{p}= 7 \text{cal}/\text{mol}.^{\circ}C \text{and} R = 2 \text{cal}./\text{mol}.^{\circ}C).$$
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A collimated beam of light of diameter 2 mm is propagating along x-axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. lf first lens has focal length 40 mm, then the focal length of second lens is ____ mm.
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The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of 1Ω is connected across these points is ____ J
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In a microscope the objective is having focal length $$f_{0}=2 \text{cm}$$ and eye-piece is having focal length $$f_{e} = 4 \text{cm}$$ The tube length is 32 cm. the magnification produced by this microscope for normal adjustment is _______.
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Which of the following statement is correct?
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The oxidation number of sulphur in Peroxydisulphuric acid is :
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$$\text{Which of the following will NOT undergo } S_\mathrm{N}1 \text{ or } S_\mathrm{N}2 \text{ reactions?}$$
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Which of the following is the correct IUPAC name of the given coordination compound?
$$\mathrm{(NH_4)_2[FeF_5(H_2O)]}$$
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Match column-I with column-II according to their shape.
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Which of the following f-block elements in the +3 oxidation state is colourless?
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Which of the following graphs is correct for a second-order reaction?
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Which of the following statement(s) is incorrect?
(i) The order of electron gain enthalpy for halogens is: Cl > F > Br >I
(ii) The first ionization energy follows the order: C < O < N <F
(iii) Stability of +3 oxidation state in group 15 increases in the order: $$\mathrm{N^{+3} > P^{+3} > As^{+3} > Sb^{+3} > Bi^{+3}}$$
(iv) The order of atomic radii in group 13 is: Tl > In > Ga > Al > B
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Equivalent conductivity of $$\mathrm{BaCl_2}$$, $$\mathrm{H_2SO_4}$$ and $$\mathrm{HCl}$$ are $$x_1,\ x_2$$ and $$x_3\ \mathrm{S\,cm^{-1}\,eq^{-1}}$$ at infinite dilution. If the conductivity of saturated $$\mathrm{BaSO_4}$$ solution is $$x\ \mathrm{S\,cm^{-1}}$$, then $$K_{sp}$$ of $$\mathrm{BaSO_4}$$ is:
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Find the iso-electric point of the given amino acid:
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The orbital having one radial node as well one angular node is:
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Assertion: In case of isomeric dihalobenzenes, the p-isomers have greater boiling point than the ortho and meta-isomers.
Reason: The greater symmetry of p-isomers allows them to pack more efficiently in the crystal lattice than their ortho and meta counterparts.
Choose the correct option.
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Statement I: $$pK_b$$ of imidazole is lower than that of aniline.
Statement II: The order of basicity for ethyl substituted amine in aqueous solution is $$\mathrm{(C_2H_5)_2NH > C_2H_5NH_2 > (C_2H_5)_3N > NH_3}$$.
Choose the correct option.
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The ratio of number of keto groups to the number of carbons in the major product is:
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A first order reaction is 50% completed in 20 minutes at 27 °C and in 5 minutes at 47 °C. The energy of activation of the reaction is:
(Given: ln 4=1.386)
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Benzene diazonium salt reacts with ethanol to form compound B, which on treatment with oleum gives compound C. Compound C, when treated with NaOH at 350∘C, followed by hydrolysis, forms compound D. Compound D is then treated with sodium dichromate in the presence of sulphuric acid. The final product is:
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The final product is:
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For an isothermal free expansion of an ideal gas into vacuum, which one of the following set of values is correct?
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The most stable canonical structure of given molecule is:
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Among the following, the Newmann projections of meso-2,3-butanediol are:
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Consider the following reaction sequence
The percentage of nitrogen in product 'T' formed is ____ %. (Nearest integer)
(Given molar mass in g $$mol^{-1}$$ H : 1, C: 12, N : 14, 0: 16)
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Pre-exponential factors of two different reactions of same order are identical. Let activation energy of first reaction exceeds the activation energy of second reaction by 20 kJ $$mol^{-1}$$. If $$k_{1}\text{ and }k_{2}$$ are the rate constants of first and second reaction respectively at 300 K, then In $$\frac{k_{2}}{k_{1}}$$ will be ___.
(nearest integer) $$[R=8.3JK^{-1}mol^{-1}]$$
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Use the following data :
One mole each of $$A_{2}(g)$$ and $$B_{2}(g)$$ are taken in a 1 L closed flask and allowed to establish the equilibrium at 500K
$$A_{2}(g)+B_{2}(g)\rightleftharpoons2AB(g)$$
The value of x in $$( kJ mol^{-1})$$ is ____ . (Nearest integer)
(Given: log K=2.2 R= 8.3 kJ $$K^{-1} mol^{-1}$$)
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Consider the following reactions:
$$NaCl+K_{2}Cr_{2}O_{7}+H_{2}SO_{4}\rightarrow A+KHSO_{4}+NaHSO_{4}+H_{2}O$$
$$A+NaOH\rightarrow B+NaCl+H_{2}O$$
$$B+H_{2}SO_{4}+H_{2}O_{2}\rightarrow C+Na_{2}SO_{4}+H_{2}O$$
In the product 'C, 'X' is the number of $$O_{2}^{2-}$$ units, 'Y' is the total number oxygen atoms present and 'Z' is the oxidation state of Cr·. The value of X + Y + Z is ______
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The pH and conductance of a weak acid (HX) was found to be 5 and $$4\times10^{-5}S$$. respectively. The conductance was measured under standard condition using a cell where the electrode plates having a surface area of 1 $$cm^{2}$$ were at a distance of 15 cm apart. The value of the limiting molar conductivity is ______ S $$m^{2}mol^{-1}$$ (nearest integer)
(Given : degree of dissociation of the weak acid ($$\alpha$$) < < 1)
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