The eccentricity of the hyperbola 4x2 – 9y2 = 36 is
The length of the latus rectum of the ellipse 16x2 + 25y2 = 400 is
The vertex of the parabola y2 + 6x – 2y + 13 = 0 is
( y −1)2= −6x − 12
( y −1)2= −6(x +12) = 4(-6/4)(x+2)
Vertex →(−2, 1)
The coordinates of a moving point p are (2t2 + 4, 4t + 6). Then its locus will be a
The equation 8x2 + 12y2 – 4x + 4y – 1 = 0 represents
ax2 + by2+ 2hxy + 2gx + 2fy + c = 0
represents ellipse if h2 −ab< 0
3x2 + 12y2− 4x + 4y − 1 = 0
h =0, a= 3, b = 12
h2 −ab< 0
If the straight line y = mx lies outside of the circle x2 + y2 – 20y + 90 = 0, then the value of m will satisfy
x2 +m2 x2− 20mx + 90
x2 (1 + m2)− 20mx + 90 = 0
400m2 − 4× 90 (1 + m2) < 0
40m2 < 360
m2 < 9 ; |m |< 3
The locus of the centre of a circle which passes through two variable points (a, 0), (–a, 0) is
Centre lies on y-axis locus x = 0
The coordinates of the two points lying on x + y = 4 and at a unit distance from the straight line 4x + 3y = 10 are
Let p (h, 4 − h)
|h + 2|= 5
h =3,−7 ; p =1, 1
The intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. Equation of the circle with AB as diameter is
x2 + y2 = 0
( 0, 0 ) , (1,1) as diametric ends (x − 0)(x−1) + (y + 0)(y −1) = 0
x2 +y2− x − y = 0
If the coordinates of one end of a diameter of the circle x2+y2+4x–8y+5=0, is (2,1), the coordinates of the other end is
x2 + y2+ 9x − 8y + 5 = 0
Centre circle (–2, 4)
If the three points A(1,6), B(3, –4) and C(x, y) are collinear then the equation satisfying by x and y is
⇒ 1(3y+ 4x) − (y − 6x) +1(−4 −18) = 0
⇒ 3y+ 4x − y + 6x −12 = 0
⇒ 2y+ 10x − 22 = 0
y +5x= 11
and θ lies in the second quadrant, then cosθ is equal to
θ in 2nd quad Cosθ < 0
The solutions set of inequation cos–1x < sin–1x is
cos–1x < sin–1x
The number of solutions of 2sinx + cos x = 3 is
√5 <3 No solution
If θ+ φ = π/4 then (1 + tanθ)(1 + tanφ) is equal to
If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation
sinθ + cosθ = b/a
sinθ . cosθ = c/a
If A and B are two matrices such that A+B and AB are both defined, then
Addition is defined if order of A is equal to order of B
AB is defined if m = n
⇒ A, B are square matrices of same order
is a symmetric matrix, then the value of x is
A = AT
The equation of the locus of the point of intersection of the straight lines x sin θ + (1 – cos θ) y = a sin θ and x sin θ – (1 + cos θ) y + a sin θ = 0 is
y = a sin θ
x = a cos θ.
x2 + y2= a2
If sinθ + cosθ = 0 and 0 < θ < π, then θ
The value of cos 15o – sin 15o is
he period of the function f(x) = cos 4x + tan 3x is
If y = 2x3 – 2x2 + 3x – 5, then for x = 2 and Δ x = 0.1 value of Δ y is
The approximate value of 5√33 correct to 4 decimal places is
The value of 2∫-2 xcos + xsinx + 1
For the function f(x) = ecos x , Rolle’s theorem is
The general solution of the differential equation
The degree and order of the differential equation are repectively
The function f(x) = ax + b is strictly increasing for all real x if
f′ (x) = a f′(x) > 0 ⇒ a > 0
The general solution of the differential equation
C2 → C2 – C3
C3 → C3 + C2
C3 → C3 + ωC1
C2 → C2 – C1
4 boys and 2 girls occupy seats in a row at random. Then the probability that the two girls occupy seats side by side is
A coin is tossed again and again. If tail appears on first three tosses, then the chance that head appears on fourth toss is
p = 18.104.22.168/2 = 1/2
The coefficient of xn in the expansion of
The sum of the series
The number (101)100– 1 is divisible by
If A and B are coefficients of xn in the expansions of (1+ x)2n and (1+x)2n – 1 respectively, then A/B is equal to
A = 2nCn
B = 2n – 1Cn
If n > 1 is an integer and x ≠0, then (1 + x)n – nx – 1is divisible by
(1 + x)n = nC0 + nC1x + nC2x² + nC3x3 + .......
= 1 + nx + x² (nC2 + nC3 x + .........) (1 + x)n – nx – 1
= x² (nC2+ nC3x + ........)
If nC4, nC5 and nC6 are in A.P., then n is
nC4, nC5 and nC6
The number of diagonals in a polygon is 20. The number of sides of the polygon is
nC2 –n = 20
n = 8
Let a , b, c be three real numbers such that a + 2b + 4c = 0. Then the equation ax2 + bx + c = 0
If the ratio of the roots of the equation px2 + qx + r = 0 is a : b, then ab/(a + b)2 is
If α and β are the roots of the equation x2 + x + 1 = 0, then the equation whose roots are α19 and β7 is
α and β are the roots of x2 + x + 1 = 0
α = ω
α19 = ω
β7 = ω2
x2 – (α19 + β7)x + α19 β7 = 0
Thou, x2 – (ω + ω 2) x + ω . ω 2 = 0
x2 + x + 1 = 0
For the real parameter t, the locus of the complex number z = (1 – t²) + i√(1 + t2) in the complex plane is
Let z = x + iy
x = 1 – t2
y2 = 1 + t2
Thus, x + y2 = 2
y2 = 2 – x
y2 = – (x – 2)
if x + 1/x = 2cosθ, then for any integer n , xn + 1/xn
x + 1/x = 2cosθ
Let x = cos θ + 1 sin θ
1/x = cosθ − 1sinθ
thus, xn + 1/xn = 2 cos nθ
If ω ≠ 1 is a cube root of unity, then the sum of the series S = 1 + 2ω + 3ω ² + .......... + 3nω 3n – 1 is
S = 1 + 2ω + 3ω ² + .......... + 3nω 3n – 1
Sω = ω + ω ² + .......... + (3n-1)ω 3n – 1 + 3nωn
s(1 – ω) = 1 + ω + ω² + ...........+ ω3n–1 – 3nω3h
= 0 – 3n
If log3x + log3y = 2 + log32 and log3(x + y) = 2, then
log3x + log3y = 2 + log32
⇒ x.y = 18 log (x + y) = 2
⇒ x + y = 9
we will get x = 3 and y = 6
If log 7 2 = λ, then the value of log49 (28) is
log4928 = log724 × 7
The sequence log a, log a2/b, loga3/b2, ....... is
log a . (2log a – log b)(3log a – 2 log b)
= T2 – T1 = log a – log b
= T3– T2 = log a – log b
If in a triangle ABC, sin A, sin B, sin C are in A.P., then
c1 → c1 + c2 + c3
The area enclosed between y2 = x and y = x is
Let f(x) = x3e–3x, x > 0. Then the maximum value of f(x) is
f(x) = x3e–3x
= f′(x) = 3x2e–3x + x3 e–3x (–3)
= x23e–3x[1 – x] = 0, x = 1
Maximum at x = 1
f(1) = e–3
The area bounded by y2 = 4x and x2 = 4y is
The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8m/sec2. The time taken by the particle to move the second metre is
The solution of
Integrating Factor (I.F.) of the defferential equation
The differential equation of y = aebx (a & b are parameters) is
y = a.ebx ............ (i)
y1 = abebx
y1 = by............(ii)
y2 = by1 ...........(iii)
The value of
∫ 2x (f ′(x) + f (x) log 2)dx is
I = ∫ 2x f ′(x)dx+∫ 2x f (x) log 2dx
Let f(x) = tan–1x. Then f′(x) + f′′(x) is = 0, when x is equal to
f(x) = tan–1x
If the function
if f(x) is continuous at x = 2, the value of λ will be
The even function of the following is
If f(x + 2y, x – 2y) = xy, then f(x, y) is equal to
The locus of the middle points of all chords of the parabola y2 = 4ax passing through the vertex is
2h = x, 2k = y
y2 = 4ax
k2 = 2ah
y2 = 2ax
The charge on the capacitor of capacitance C shown in the figure below will be
∴ Potential difference across R2 ,
∴ Charge on the capacitor
The resistance across A and B in the figure below will be
Resistance are in parallel = R/3
Five equal resistance, each of resistance R, are connected as shown in figure below. A battery of V volt is connected between A and B. The current flowing in FC will be
Two cells with the same e.m.f. E and different internal resistances r1 and r2 are connected in series to an external resistance R. The value of R so that the potential difference across the first cell be zero is
Current through ABC and A'B'C' is I. What is the magnetic field at P? BP = PB' = r (Here C'B' PBC are collinear)
The magnetic field at the point of intersection of diagonals of a square wire loop of side L carrying a current I is
In an inelastic collision an electron excites as hydrogen atom from its ground state to a M-shell state. A second electron collides instantaneously with the excited hydrogen atom in the M-State and ionizes it. At least how much energy the second electron transfers to the atom in the M-state?
Minimum energy required by electron should be +1.51 eV
A radioactive nucleus of mass number A, initially at rest, emits an α-particle with a speed ν . The recoil speed of the daughter nucleus will be
From conservation of momentum
In the nuclear reaction
Which type of Gate the following truth table represents?
Given A =2iˆ+ 3ˆj and B = ˆi+ˆj. The component of vector A along vector B is
A cubical vessel of height 1 m is full of water. What is the amount of work done in pumping water out of the vessel? (Take g = 10 m s–2)
V = l3=1m3
m = 1 x 1000 = 1000kg
w = mgh = 1000 x 10 x 1/2= 5000J
A stone of relative density K is released from rest on the surface of a lake. If viscous effects are ignored, the stone sinks in water with an acceleration of
If a person can throw a stone to maximum height of h metre vertically, then the maximum distance through which it can be thrown horizontally by the same person is
A body of mass 6 kg is acted upon by a force which causes a displacement in it given x = t2/4 metre where t is the time in second. The work done by the force in 2 seconds is
A box is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to
A particle is moving with a constant speed ν in a circle. What is the magnitude of average velocity after half rotation?
A cricket ball of mass 0.25 kg with speed 10 m/s collides with a bat and returns with same speed within 0.01 S. The force acted on bat is
If the Earth were to suddenly contract to 1/n th of its present radius without any change in its mass, the duration of the new day will be nearly
I1ω1=I 2 ω 2
If g is the acceleration due to gravity on the surface of the earth, the gain in potential energy of an object of mass m raised from the earth’s surface to a height equal to the radius R of the earth is
A material has Poisson’s ratio 0.50. If a uniform rod of it suffers a longitudinal strain of 2 × 10–3, then the percentage change in volume is
Two identical springs are connected to mass m as shown (k = spring constant). If the period of the configuration in (a) is 2S, the period of the configuration (b) is
T = 1S
An object weighs m1 in a liquid of density d1 and that in liquid of density d2 is m2. The density d of the object is
V(d – d1)g = m1g
V(d – d2)g = m2g
A body floats in water with 40% of its volume outside water. When the same body floats in an oil, 60% of its volume remains outside oil. The relative density of oil is
Vσg = 0.6 Vσ1g ...... (1)
Vσg = 0.4 Vσ2g .................. (2)
Dividing (1) and (2)
Two soap bubbles of radii x and y coalesee to constitute a bubble of radius z. Then z is requal to
n = n1 + n2
pv = p1v1 + p2v2
A particle of mass m is located in a one dimensional potential field where potential energy is given by : V(x) = A(1 – cos px), where A and p are constants. The period of small oscillations of the particle is
v x =A(1− cos px )
The period of oscillation of a simple pendulum of length l suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination α, is given by
In Young’s double slit experiment the two slits are d distance apart. Interference pattern is observed on a screen at a distance D from the slits. A dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light is
A plane progressive wave is given by y = 2 cos 6.284 (330 t – x). What is period of the wave ?
y = 2 cos 2π (330 t – x)
ω = 2π × 330
T = 1/330 S
The displacement of a particle in S.H.M. varies according to the relation x = 4(cos πt + sin πt). The amplitude of the particle is
R sin δ = 4
R cos δ = 4
R = 4√2
Two temperature scales A and B are related by At which temperature two scales have the same reading ?
, A = B
2A – 84 = A – 72
A = 12
An ideal gas is compressed isothermally until its pressure is doubled and then allowed to expand adiabatically to regain its original volume (γ = 1.4 and 2-14= 0.38). The ratio of the final to initial pressure is
Air inside a closed container is saturated with water vapour. The air pressure is p and the saturated vapour pressure of water is p- . If the mixture is compressed to one half of its volume by maintaining temperature constant, the pressure becomes
1.56 × 105 J of heat is conducted through a 2 m2 wall of 12 cm thick in one hour. Temperature difference between the two sides of the wall is 200C. The thermal conductivity of the material of the wall is (in W m–1 K–1)
A diver at a depth of 12 m in water ( μ = 4/3) sees the sky in a cone of semivertical angle :
Two thin lenses of focal lengths 20 cm and 25 cm are placed in cotact. The effective power of the combination is
P = P1 + P2
A convex lens of focal length 30 cm produces 5 times magnified real image of an object. What is the object distance ?
If the focal length of the eye piece of a telescope is doubled, its magnifying power (m) will be
m = -f0/fe
m' = m/2
A plano-concave lens is made of glass of refractive index 1.5 and the radius of curvature of its curved face is 100 cm. What is the power of the lens ?
Four charges equal to –Q are placed at the four corners of a square and a charge q is at its centre. If the system is in equilibrium, the value of q is
Two aromatic compounds having formula C7H8O which are easily identifiable by FeCl3 solution test (violet colouration) are
O – cresol contains phenolic group, thus it gives violet coloration with FeCl3 where as benzylalchol donot contains phenolic group, hence no coloration with FeCl3. Hence Identifiable
The ease of dehydrohalogenation of alkyl halide with alcoholic KOH is
Such dehydrohalogenation follows E2 mechanism. The driving force of such reactions is the stability of alkene produced. Since tetriary alkyl halide can give more substituted alkene, it reacts fastest followed by secondary and primary i.e. 3o > 2o > 1o.
The ease of Nitration of the following three hydrocarbons follows the order
The correct order of decreasing acidity of nitrophenols will be
Due to – I and – R influence, NO2 in ortho-postion should have raised the acidity to the maximum extent. But it is due to intramolecular H – bonding, ortho-nitrophenol is less acidic than para – nitrophenol.
Among the alkenes which one produces tertiary bytyl alcohol on acid hydration
Which of the following compounds has maximum volatility?
Due to intramolecular H – bonding
Which one of the following will show optical isomerism?
The central carbon is attached to four different substituents, hence it is chiral carbon, therefore optically active.
The pH of an aqueous solution of CH3COONa of concentrated C(M) is given by
In case of Hydrolysis of salt of weak acid and strong base, the pH is given by
The standard reduction potential Eo for half reations are
Zn = Zn+2 + Ze Eo = +0.76 V
Fe = Fe+2 + Ze Eo = + 0.41 V
The EMF of hte cell reaction
Fe+2 + Zn = Zn+2 + Fe
If the equilibrium constants of the following equilibria
are given by K1 and K2 respectively, which of the following relations is correct
The energy of an electron in first Bohr orbit of H – atom is – 13.6 eV. The possible energy value of electron in the excited state of Li2+ is
For the excited state, n = 2 and for Li++ ion, z = 3
The amount of the heat released when 20 ml 0.5 M NaOH is mixed with 100 ml 0.1 M HCl is x kJ. The heat of neutralization is
During formation of 10 millimole of H2O the heat released is x KJ. Therefore heat of neutralisation is – 100 x KJ/mol (heat released hence negative)
Which one of the following has the lowest ionization energy?
It’s an alkalimetal; hence least I.P
The ozone layer forms naturally by
2 gm of metal carbonate is neutralized completely by 100 ml of 0.1 (N) HCl. The equivalent weight of metal carbonate is
Number of gram equivalents of HCl = 100 x 0.1/1000 = 0.01
Number of gram equivalents of metal carbonate required for neutralisation must also be 0.01. Thus, mass of 1 gram eqivalent of carbonate salt 2/0.01= 200g
Which one of the following is not true at room temperature and pressure
SO3 is a colourless gas, crystalline transparent solid at room temperature.
An electric current is passed through an aqueous solution of a mixture of alanine (isoelectric point 6.0) glutamic acid (3.2) and arginine (10.7) buffered at pH 6. What is the fate of the three acids?
At pH = 6, glutamic acid exists as a dianionic species & migrates to anode while arginine exists as cationic species & moves to cathode. Alanine does not migrate to any electrode at its isoelectric point .
The representation of the ground state electronic configuration of He by box – diagram as ↑↑ is wrong because it violates
According to Pauli Exclusion Principle, In any orbital, maximum two electrons can exist, having opposite spin.
The electronic transitions from n = 2 to n = 1 will produce shortest wavelength in (where n = principal quantum state)
Hence, for shortest λ, z must be maximum, which is for Li+2.
In the following electron – dot structure, calculate the formal charge from left to right nitrogen atom;
Formal chargl = Number of electrons in
Valence shell –(1/2) x numbers of electrons as bond pair + numbers of electrons as lone pair)
If the molecular wt. of Na2S2O3 and I2 are M1 and M2 respectively, then what will be the equivalent wt. of Na2S2O3 and I2 in the following reaction?
A radioactive atom yxM emits two α particles and one β particle successively. The number of neutrons in the nucleus of the product will be
Number of neutrons
= Mass no. – Atomic no.
= X – 8 – Y + 3
= X – Y – 5
An element belongs to Group 15 and third period of the periodic table. Its electonic configuration will be
General valence shell electronic configuration of 15 group elements is ns2np3. where n = period number.
Which one of the following is paramagnetic?
Platinum, Palladium and Iridium are called noble metals because
Which one is not a constituent of nucleic acid?
Guanine is the constituent of nucleic acid and not guanidine.
The sp3d2 hybridization of central atom of a molecule would lead to
In aqueous solution glucose remains as
Which of the following is used to prepare Cl2 gas at room temperature from concentrated HCl?
2MnO4-+ 16 H+ + 10Cl– → 2Mn2+ + 5Cl2 + 8H2O
NO2 is not obtained on heating
The normality of 30 volume H2O2 is
Volume strength = 5.6 × normality
30 = 5.6 × N
N = 30/5.6 = 5.3
Reaction of formaldehyde and ammonia gives
6HCHO + 4NH3 → (CH2)6 N4 + 6H2O
A plot of In k against 1/T (abscissa) is expected to be a straight line with intercept on ordinate axis equal to
ΔG° = – RT InK
or, ΔH° – TΔS° = – RT InK
Which of the following represents the composition of Carnallite mineral?
The solubility of Ca3(PO4)2 in water is y moles / litre. Its solubility product is
Anhydrous ferric chloride is prepared by
Which one of the following is s-butyl phynylvinyl methane?
Hybridization of C2 and C3 of H3C – CH = C = CH – CH3 are
Which of the following compounds is not formed in iodoform reaction of acetone
Glucose and amino acids are reabosorbed in the
Glucose and amino acids are reabsorbed in the proximal tubule of nephron.
The amount of CSF in the cranial cavity
The amount of CSF in the cranial cavity is 140 ml.
Which one is imino acid?
Proline and hydroxyproline are imino acids.
The main difference between Gram positive and Gram negative bacteria is
ACTH is secreted from
ACTH is secreted from anterior pituitary
Which of the following is the correct pathway for propagation of cardiac impulse?
Inner surface of the bronchi, bronchioles and fallopian tubes are lined by
Ciliated epithelium is found in inner surface of bronchi, bronchioles and fallopian tubes
Electric potential of the brain is recorded by
Electrical potential of brain is recorded by EEG
Which of the following is related to humoral immunity?
Humoral immunity is due to B-lymphocyte because it secretes antibody in the blood plasma.
Fertilization occur in
Fertilization occurs in fallopian tube at the junction of ampulla and isthmus.
The Gastrin is secreted from
Gastrin hormone is secreted from “G-cells” of stomach.
The cause of cretinism is
Cretinism is caused by hyposecretion of thyroxine in children.
Which of the following is a minerelocorticoid?
Aldosterone is secreted from adrenal cortex and controls RAAS. mechanism.
The part of the brain where the centre for hunger and thirst is located is
Hypothalamus is the centre for hunger and thirst.
The reflex arc, which is made of two neurones is known as
Monosynaptic reflex are has two neurons sensory and motor, which forms one synapse in CNS.
The lactase hydrolyzes lactose into
Lactose → Glucose + Galactose
In 24 hours, total glomerular filtrate formed in human kidney is
GFR is 120 ml/min, so, approx. 170 litre ultra fitrate is produced in 24 hrs.
When the oxygen supply to the tissue is inadequate, the condition is
Inadequate supply of oxygen to the tissue is called hypoxia
Which one of the following is not a second messenger in hormone action?