VITEEE 2009 Question Paper is available for download here. VITEEE 2009 Question Paper includes 40 questions from Mathematics/Biology, 35 questions from Physics, 35 questions from Chemistry, 5 questions from English, 10 questions from Aptitude to be attempted in 150 minutes. Candidates can download the VITEEE 2009 Question Paper with Solution PDF using the link below.
VITEEE 2009 Question Paper with Solution PDF
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When a wave traverses a medium the displacement of a particle located at \(x\) at a time \(t\) is given by \(y = a \sin (bt - cx)\). Where \(a\), \(b\) and \(c\) are constants of the wave. Which of the following is a quantity with dimensions?
A body is projected vertically upwards at time \(t = 0\) and it is seen at a height \(H\) at time \(t_1\) and \(t_2\) second during its flight. The maximum height attained is (acceleration due to gravity = \(g\))
A particle is projected up from a point at an angle \(\theta\) with the horizontal direction. At any time \(t\), if \(p\) is the linear momentum, \(y\) is the vertical displacement, \(x\) is horizontal displacement, the graph among the following which does not represent the variation of kinetic energy \(KE\) of the particle is
A motor of power \(P_0\) is used to deliver water at a certain rate through a given horizontal pipe. To increase the rate of flow of water through the same pipe \(n\) times, the power of the motor is increased to \(P_1\). The ratio of \(P_1\) to \(P_0\) is
A body of mass \(5\,kg\) makes an elastic collision with another body at rest and continues to move in the original direction after collision with a velocity equal to \(\dfrac{1}{10}\)th of its original velocity. Then the mass of the second body is
A particle of mass \(4m\) explodes into three pieces of masses \(m\), \(m\), and \(2m\). The equal masses move along X-axis and Y-axis with velocities \(4\,m\,s^{-1}\) and \(6\,m\,s^{-1}\) respectively. The magnitude of velocity of the heavier mass is
A body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If \(R\) is the radius of the earth, the maximum height attained by the body from the surface of the earth is
The displacement of a particle executing SHM is given by \(y = 5\sin\left(4t + \frac{\pi}{3}\right)\). If \(T\) is the time period and the mass of the particle is \(2g\), the kinetic energy of the particle when \(t = \frac{T}{4}\) is given by
If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are \(a\), \(b\) and \(c\) respectively, then the ratio between the increase in lengths of brass and steel wires would be
A soap bubble of radius \(r\) is blown up to form a bubble of radius \(2r\) under isothermal conditions. If \(T\) is the surface tension of soap solution, then energy spent in blowing the bubble is
Eight spherical raindrops of same mass and radius are falling down with a terminal speed of \(6\,cm\,s^{-1}\). If they coalesce to form one big drop, what will be the terminal speed of bigger drop? (Neglect buoyancy of air)
A clock pendulum made of invar has a period of \(0.5\,s\) at \(20^\circ C\). If the clock is used in a place where temperature averages to \(30^\circ C\), how much time does the clock lose in each oscillation? (For invar, \(\alpha = 9 \times 10^{-7}/^\circ C\), \(g =\) constant)
A piece of metal weighs \(45\,g\) in air and \(25\,g\) in a liquid of density \(1.5 \times 10^3\,kg\,m^{-3}\) kept at \(30^\circ C\). When the temperature of the liquid is raised to \(40^\circ C\), the metal piece weighs \(27\,g\) in the density of liquid at \(40^\circ C\) is \(1.25 \times 10^3\,kg\,m^{-3}\). The coefficient of linear expansion of metal is
An ideal gas is subjected to a cyclic process ABCD as depicted in the \(P-V\) diagram given below.
An ideal gas is subjected to a cyclic process involving four thermodynamic states, among these state \(Q\) and work \(W\) involved in each of these stages are: \(Q_1 = 6000\,J,\; Q_2 = -5500\,J,\; Q_3 = -3000\,J,\; Q_4 = 3500\,J\) \(W_1 = 2500\,J,\; W_2 = -1000\,J,\; W_3 = -1200\,J,\; W_4 = x\,J\)
The ratio of the net work done by the gas to the total heat absorbed by the gas is \(n\). The values of \(x\) and \(n\) respectively are
Two cylinders \(A\) and \(B\) fitted with pistons contain equal number of moles of an ideal monatomic gas at \(400K\). The piston of \(A\) is free to move while that of \(B\) is held fixed. Same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in \(A\) is \(42K\), the rise in temperature of the gas in \(B\) is
Three rods of same dimensions have thermal conductivities \(3K\), \(2K\) and \(K\). They are arranged as shown in the figure. The ends are maintained at \(100^\circ C\), \(50^\circ C\) and \(0^\circ C\). Then, the temperature of the junction in steady state is
Two sources \(A\) and \(B\) are sending notes of frequency \(680\,Hz\). A listener moves from \(A\) and \(B\) with a constant velocity \(u\). If the speed of sound in air is \(340\,m\,s^{-1}\), what must be the value of \(u\) so that he hears \(10\) beats per second?
Two identical piano wires have a fundamental frequency of \(600\) cycles per second when kept under the same tension. What fractional increase in the tension of one wire will lead to the occurrence of \(6\) beats per second when both wires vibrate simultaneously?
In the Young's double slit experiment, the intensities at two points \(P_1\) and \(P_2\) on the screen are respectively \(I_1\) and \(I_2\). If \(P_1\) is located at the centre of a bright fringe and \(P_2\) is located at a distance equal to a quarter of fringe width from \(P_1\), then \(\dfrac{I_1}{I_2}\) is
In Young's double slit experiment, the \(10^{th}\) maximum of wavelength \(\lambda_1\) is at a distance \(y_1\) from the central maximum. When the wavelength of the source is changed to \(\lambda_2\), the \(5^{th}\) maximum is at a distance of \(y_2\) from the central maximum. The ratio \(\left(\dfrac{y_1}{y_2}\right)\) is
Four light sources produce the following four waves:
(i) \(y_1 = a\sin(\omega t + \phi_1)\)
(ii) \(y_2 = a\sin 2\omega t\)
(iii) \(y_3 = a\sin(\omega t + \phi_2)\)
(iv) \(y_4 = a\sin(3\omega t + \phi_1)\)
Superposition of which two waves give rise to interference?
The two lenses of an achromatic doublet should have
Two bar magnets \(A\) and \(B\) are placed one over the other and are allowed to vibrate in a vibration magnetometer. They make \(20\) oscillations per minute when the similar poles of \(A\) and \(B\) are on the same side, while they make \(15\) oscillations per minute when their opposite poles lie on the same side. If \(M_A\) and \(M_B\) are the magnetic moments of \(A\) and \(B\), and \(M_A > M_B\), the ratio \(M_A : M_B\) is
A bar magnet is \(10\,cm\) long and is kept with its north pole pointing north. A neutral point is formed at a distance of \(15\,cm\) from each pole. Given the horizontal component of earth's field is \(0.4\) Gauss, the pole strength of the magnet is
An infinitely long straight wire has uniform linear charge density of \(\dfrac{1}{3}\,cm^{-1}\). Then, the magnitude of the electric intensity at a point \(18\,cm\) away is (given \(\varepsilon_0 = 8.8 \times 10^{-12}\,C^2N^{-1}m^{-2}\))
Two point charges \(-q\) and \(+q\) are located at points \((0,0,-a)\) and \((0,0,a)\), respectively. The electric potential at a point \((0,0,z)\), where \(z > a\) is
In the adjacent shown circuit, a voltmeter of internal resistance \(R_v\), when connected across \(B\) and \(C\) reads \(\dfrac{100}{3}\,V\). Neglecting the internal resistance of the battery, the value of \(R_v\) is
A cell in secondary circuit gives null deflection for \(2.5\,m\) length of wire for a potentialmeter having \(10\,m\) length of wire. If the length of the potentiometer wire is increased by \(1\,m\) without changing the cell in the primary, the position of the null point now is
The following series L-C-R circuit, when driven by an emf source of angular frequency \(70\) kilo-radians per second, the circuit effectively behaves like
A wire of length \(l\) is bent into a circular loop of radius \(R\) and carries a current \(I\). The magnetic field at the centre of the loop is \(B\). The same wire is now bent into a double loop of equal radii. If both loops carry the same current \(I\) and it is in the same direction, the magnetic field at the centre of the double loop will be
An infinitely long straight conductor is bent into the shape as shown below. It carries a current of \(1\) ampere and the radius of the circular loop is \(R\) metre. Then, the magnitude of magnetic induction at the centre of the circular loop is
The work function of a certain metal is \(3.31 \times 10^{-19}\,J\). Then, the maximum kinetic energy of photoelectrons emitted by incident radiation of wavelength \(5000\,AA\) is (given \(h = 6.62 \times 10^{-34}\,Js\), \(c = 3 \times 10^8\,m\,s^{-1}\), \(e = 1.6 \times 10^{-19}\,C\))
A photon of energy \(E\) ejects a photoelectron from a metal surface whose work function is \(W_0\). If this electron enters into a uniform magnetic field of induction \(B\) in a direction perpendicular to the field and describes a circular path of radius \(r\), then the radius \(r\) is given by (in the usual notation)
Two radioactive materials \(x_1\) and \(x_2\) have decay constants \(10\lambda\) and \(\lambda\) respectively. Initially they have the same number of nuclei, then the ratio of the number of nuclei of \(x_1\) to that of \(x_2\) after a time \(t\) will be \(1/e\). The value of \(t\) is
Current flow in each of the following circuit A and B respectively are
A bullet of mass \(0.02\,kg\) travelling horizontally with velocity \(250\,m\,s^{-1}\) strikes a block of wood of mass \(0.23\,kg\) which rests on a rough horizontal surface. After the impact, the block and bullet move together and come to rest after travelling a distance of \(40\,m\). The coefficient of kinetic friction on the rough surface is \((g = 9.8\,m\,s^{-2})\)
Two persons \(A\) and \(B\) are located in X-Y plane at points \((0,0)\) and \((0,10)\) respectively. (The distances are measured in MKS unit). At a time \(t = 0\), they start moving simultaneously with velocities \(\vec{v_A} = 2\hat{i}\,m\,s^{-1}\) and \(\vec{v_B} = 2\hat{i}\,m\,s^{-1}\) respectively. Determine time after which \(A\) and \(B\) are at their closest distance.
A rod of length \(l\) is held vertically stationary with its lower end located at a point \(P\) on the horizontal plane. When the rod is released to topple about \(P\), the velocity of the upper end of the rod with which it hits the ground is
A wheel of radius \(0.4\,m\) can rotate freely about its axis as shown in the figure. A string is wrapped over its rim and an mass of \(4\,kg\) is hung. An angular acceleration of \(8\,rad\,s^{-2}\) is produced in it due to the torque. (Take \(g = 10\,m\,s^{-2}\)) The moment of inertia of the wheel is
Given that \(\Delta H_f(H) = 218\,kJ/mol\), express the \(H-H\) bond energy in \(kcal/mol\).
Identify the alkyne in the following sequence of reactions:
Alkyne \(\xrightarrow[Lindlar's catalyst]{H_2}\) A \(\xrightarrow[only]{Ozonolysis}\) \(\longrightarrow\) \(Wacker Process \longrightarrow CH_2=CH_2\)
Fluorine reacts with dilute NaOH and forms a gaseous product \(A\). The bond angle in molecule of \(A\) is
One mole of alkene on ozonolysis gave one mole of acetate aldehyde and one mole of acetone. IUPAC name of \(X\) is
The number of \(\pi\) and \(\pi^\ast\) \(\pi_{z}\) bonds present in \(XeO_3\) and \(XeO_4\) molecules, respectively are
The wave velocities of electron waves in two orbits is \(a:5\). The ratio of kinetic energy of electrons is
Which one of the following sets correctly represents the increase in the paramagnetic property of the ions?
Electrons with a kinetic energy of \(6.023 \times 10^{-19}\,J\) are evolved from the surface of a metal, when it is exposed to a radiation of wavelength of \(600\,nm\). The minimum amount of energy required to remove an electron from the metal atom is
The chemical entities present in thermosphere of the atmosphere are
The type of bonds present in sulphuric anhydride are
In Gattermann reaction, a diazonium group is replaced by \(X\) using \(Y\). \(X\) and \(Y\) are
Which pair of oxyacids of phosphorus contains \(P-P\) bonds?
Dipole moment of HCl = \(1.03\,D\), HI = \(0.38\,D\). Bond length of HCl = \(1.3\,AA\) and HI = \(1.6\,AA\). The ratio of fraction of electric charge \(\delta\) existing on each atom in HCl and HI is
SiCl\(_4\) on hydrolysis forms \(X\) and HCl. Compound \(X\) loses water at \(1000^\circ C\) and gives \(Y\). Compounds \(X\) and \(Y\) respectively are
\(1.5g\) of \(CdCl_2\) was found to contain \(0.9g\) of Cd. Calculate the atomic weight of Cd.
Aluminium reacts with NaOH and forms compound \(X\). If the coordination number of aluminium in \(X\) is \(6\), the correct formula of \(X\) is
The average kinetic energy of one molecule of an ideal gas at \(27^\circ C\) and \(1\,atm\) pressure is
Assertion (A): \(K\), \(Rb\) and \(Cs\) form superoxides.
Reason (R): The stability of superoxides increases from \(K\) to \(Cs\) due to decrease in lattice energy.
How many mL of perhydrol is required to produce sufficient oxygen which can be used to completely convert \(2L\) of \(SO_2\) gas to \(SO_3\) gas?
pH of a buffer solution decreases by \(0.02\) units when \(0.12\,g\) of acetic acid is added to \(250\,mL\) of a buffer solution of acetic acid and potassium acetate at \(27^\circ C\). The buffer capacity of the solution is
Match the following:
List I \(\quad\) List II
(A) Felspar \(\quad\) (I) \([Ag_3Sb_3]\)
(B) Asbestos \(\quad\) (II) \(Al_2O_3\cdot H_2O\)
(C) Pyrargyrite \(\quad\) (III) \(MgSO_4\cdot H_2O\)
(D) Diaspore \(\quad\) (IV) \(KAlSi_3O_8\)
\(\quad\) (V) \(CaMg_3(SiO_3)_4\)
Which one of the following order is correct for the first ionisation energies of the elements?
What are \(X\) and \(Y\) in the following reaction sequence?
\[ C_2H_5OH \xrightarrow{Cl_2} X \xrightarrow{Cl_2} Y \]
What are \(A\), \(B\), \(C\) in the following reactions?
\[ (i)\; (CH_3CO)_2Ca \xrightarrow{\Delta} A \]
\[ (ii)\; CH_3CO_2H \xrightarrow{HI,\;Red\,P} B \]
\[ (iii)\; 2CH_3CO_2H \xrightarrow{P_4O_{10}} C \]
One percent composition of an organic compound \(A\) is carbon: \(85.71%\) and hydrogen \(14.29%\). Its vapour density is \(14\). Consider the following reaction sequence:
\[ A \xrightarrow{Cl_2/H_2O} B \xrightarrow{(i)KCN/EtOH\;(ii)H_3O^+} C \]
Identify \(C\).
How many tripeptides can be prepared by linking the amino acids glycine, alanine and phenyl alanine?
A codon has a sequence of \(A\) and specifies a particular \(B\) that is to be incorporated into a \(C\). What are \(A\), \(B\), \(C\)?
Parkinson's disease is linked to abnormalities in the levels of dopamine in the body. The structure of dopamine is
During the depression in freezing point experiment, an equilibrium is established between the molecules of
Consider the following reaction:
\[ C_2H_5Cl + AgCN \xrightarrow[EtOH/H_2O]{} X \;(major) \]
Which one of the following statements is true for \(X\)?
(I) It gives propionic acid on hydrolysis
(II) It has an ester functional group
(III) It has nitrogen linked to ethyl carbon
(IV) It has a cyanide group
For the following cell reaction:
\[ Ag|Ag^+||AgCl|Cl^-|Cl_2,Pt \]
\[ \Delta G_f^\circ(AgCl) = -109\,kJ/mol \]
\[ \Delta G_f^\circ(Cl^-) = -129\,kJ/mol \]
\[ \Delta G_f^\circ(Ag^+) = 78\,kJ/mol \]
\(E^\circ\) of the cell is
The synthesis of crotonaldehyde from acetaldehyde is an example of ...... reaction.
At \(25^\circ C\), the molar conductances at infinite dilution for the strong electrolytes NaOH, NaCl and BaCl\(_2\) are \(248\times 10^{-4}\), \(126\times 10^{-4}\) and \(280\times 10^{-4}\,S\,m^2\,mol^{-1}\) respectively. \(\lambda_m^\circ\) of \(Ba(OH)_2\) in \(S\,m^2\,mol^{-1}\) is
The cubic unit cell of a metal (molar mass = \(63.55\,g\,mol^{-1}\)) has an edge length of \(362\,pm\). Its density is \(8.92\,g\,cm^{-3}\). The type of unit cell is
The equilibrium constant for the given reaction is \(100\).
\[ N_2(g) + 2O_2(g) \rightleftharpoons 2NO_2(g) \]
What is the equilibrium constant for the reaction given below?
\[ NO_2(g) \rightleftharpoons \frac{1}{2}N_2(g) + O_2(g) \]
For a first order reaction at \(27^\circ C\), ratio of time required for \(75%\) completion to \(25%\) completion of reaction is
The concentration of an organic compound in chloroform is \(6.15\,g\) per \(100\,mL\) of solution. A portion of this solution in a 5 cm polarimeter tube causes an observed rotation of \(-1.2^\circ\). What is the specific rotation of the compound?
\(20\,mL\) of \(0.1\,M\) acetic acid is mixed with \(50\,mL\) of potassium acetate. \(K_a\) of acetic acid \(= 1.8\times 10^{-5}\). At \(27^\circ C\), calculate the concentration of potassium acetate if pH of the mixture is \(4.8\).
Calculate \(\Delta H_f^\circ\) for the reaction:
\[ Na_2O(s) + SO_3(g) \rightarrow Na_2SO_4(g) \]
given the following:
(A) \(Na(s)+H_2O(l)\rightarrow NaOH(s)+\frac{1}{2}H_2(g)\), \(\Delta H^\circ=-146\,kJ\)
(B) \(Na_2SO_4(s)+H_2O(l)\rightarrow 2NaOH(s)+SO_3(g)\), \(\Delta H^\circ=+418\,kJ\)
(C) \(2Na_2O(s)+2H_2(g)\rightarrow 4Na(s)+2H_2O(l)\), \(\Delta H^\circ=+259\,kJ\)
Which one of the following is the most effective in the coagulation of an \(As_2S_3\) sol?
If \(f:[2,3]\rightarrow \mathbb{R}\) is defined by \(f(x)=x^3+3x-2\), then the range \(f(x)\) is contained in the interval
The number of subsets of \(\{1,2,3,\ldots,9\}\) containing at least one odd number is
A binary sequence is an array of 0's and 1's. The number of \(n\)-digit binary sequences which contain even number of 0's is
If \(x\) is numerically so small so that \(x^2\) and higher powers of \(x\) can be neglected, then
\[ \left(1+\frac{2x}{3}\right)^{3/2}\left(32+5x\right)^{-1/5} \]
is approximately equal to
The roots of
\[ (x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0 \]
where \(a\in \mathbb{R}\) are always
Let \(f(x)=x^2+ax+b\), where \(a,b\in \mathbb{R}\). If \(f(x)=0\) has all its roots imaginary, then the roots of \(f(x)+f'(x)+f''(x)=0\) are
If \(f(x)=2x^4-13x^2+ax+b\) is divisible by \(x^2-3x+2\), then \((a,b)\) is equal to
If \(p,q,r\) are all positive and are the \(p^{th}\), \(q^{th}\) and \(r^{th}\) terms of a geometric progression respectively, then the value of the determinant
\[ \left|\begin{matrix} \log x & p & 1
\log y & q & 1
\log z & r & 1 \end{matrix}\right| \]
equals
The locus of \(z\) satisfying the inequality
\[ \left|\frac{z+2i}{2z+i}\right|<1,\; where z=x+iy, \]
is
If \(n\) is an integer which leaves remainder one when divided by three, then
\[ (1+\sqrt{3}i)^{n}+(1-\sqrt{3}i)^{n} \]
equals
The period of \(\sin^4x+\cos^4x\) is
If \(3\cos x \neq 2\sin x\), then the general solution of
\[ \sin^2x-\cos2x=2-\sin2x \]
is
\(\cos^{-1}\left(-\frac{1}{2}\right)-2\sin^{-1}\left(\frac{1}{2}\right)+3\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)-4\tan^{-1}(-1)\) equals
In a \(\triangle ABC\)
\[ \frac{(a+b-c)(b+c-a)(c+a-b)(a+b-c)}{4b^2c^2} \]
equals
The angle between the lines whose direction cosines satisfy the equations
\[ l+m+n=0,\quad l^2+m^2-n^2=0 \]
is
If \(m_1,m_2,m_3,m_4\) are respectively the magnitudes of the vectors
\[ \vec{a_1}=2\hat{i}-\hat{j}+\hat{k},\quad \vec{a_2}=3\hat{i}-4\hat{j}-4\hat{k}, \] \[ \vec{a_3}=\hat{i}+\hat{j}-\hat{k},\quad \vec{a_4}=-\hat{i}+3\hat{j}+\hat{k} \]
then
If \(X\) is a binomial variable with the range \(\{0,1,2,3,4,5,6\}\) and \(P(X=2)=4P(X=4)\), then the parameter \(p\) of \(X\) is
The area (in square unit) of the circle which touches the lines \(4x+3y=15\) and \(4x+3y=5\) is
The area (in square unit) of a triangle formed by \(x+y+1=0\) and the pair of straight lines \(x^2-3xy+2y^2=0\) is
The pairs of straight lines \(x-3y+2y^2=0\) and \(x^2-3xy+2y^2-x-2=0\) form a
The equations of the circle which pass through the origin and makes intercepts of lengths \(4\) and \(8\) on the \(x\)-axis and \(y\)-axis respectively are
The point \((3,-4)\) lies on both the circles
\[ x^2+y^2-2x+8y+13=0 \] \[ x^2+y^2-4x+6y+11=0 \]
Then, the angle between the circles is
The equation of the circle which passes through the origin and cuts orthogonally each of the circles
\[ x^2+y^2-6x+8=0 \]
and
\[ x^2+y^2-2x-2y=7 \]
is
The number of normals drawn to the parabola \(y^2=4x\) from the point \((1,0)\) is
If the circle \(x^2+y^2=a^2\) intersects the hyperbola \(xy=c^2\) in four points \((x_1,y_1)\) for \(i=1,2,3,4\), then \(y_1+y_2+y_3+y_4\) equals
The mid point of the chord \(4x-3y=5\) of the hyperbola \(2x^2-3y^2=12\) is
The perimeter of the triangle with vertices at \((1,0,0),(0,1,0)\) and \((0,0,1)\) is
If a line in the space makes angles \(\alpha,\beta,\gamma\) with the coordinate axes, then
\[ \cos2\alpha+\cos2\beta+\cos2\gamma+\sin^2\alpha+\sin^2\beta+\sin^2\gamma \]
equals
The radius of the sphere
\[ x^2+y^2+z^2=12x+4y+3z \]
is
Evaluate
\[ \lim_{x\to 0}\left(\frac{x+5}{x+2}\right)^{x+3} \]
If \(f:\mathbb{R}\rightarrow \mathbb{R}\) is defined by
\[ f(x)= \begin{cases} \dfrac{2\sin x-\sin 2x}{2x\cos x}, & x\neq 0
a, & x=0 \end{cases} \]
then the value of \(a\) so that \(f\) is continuous at \(0\) is
If
\[ x=\cos^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right), \quad y=\sin^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right), \]
then \(\frac{dy}{dx}\) is equal to
If
\[ \frac{d}{dx}\left[a\tan^{-1}x+b\log\left(\frac{x-1}{x+1}\right)\right]=\frac{1}{x^4-1} \]
then \(a-2b\) is equal to
If
\[ y=e^{a\sin^{-1}x}=(1-x^2)y_{n+2}-(2n+1)xy_{n+1} \]
is equal to
The function \(f(x)=x^3+ax^2+bx+c\), \(a^2\leq 3b\) has
If
\[ \int \left(\frac{2-\sin2x}{1-\cos2x}\right)e^x\,dx \]
is equal to
If \(I_n=\int \sin^n x\,dx\), then \(I_n-nI_{n-2}\) equals
The line \(x=\frac{\pi}{4}\) divides the area of the region bounded by \(y=\sin x\), \(y=\cos x\) and x-axis \((0\leq x\leq \frac{\pi}{2})\) into two regions of areas \(A_1\) and \(A_2\). Then \(A_1:A_2\) equals
The solution of the differential equation
\[ \frac{dy}{dx}=\sin(x+y)\tan(x+y)-1 \]
is
If \(p\Rightarrow(\sim p\vee q)\) is false, then the truth value of \(p\) and \(q\) are respectively
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