A to Z Exams

The equation of the tangent to the curve 2x2-xy+3y2=18 at (3, 1) is

If [cos(θ1-θ2)/ cos(θ1+θ2)]+ [cos(θ3+θ4)/ cos(θ3-θ4)] =0, then tan θ3tan θ4=

The roots of 6x4-5x3-38x2-5x+6=0 are

The equation of the auxiliary circle of x2/16-y2/25=1 is

The distances between the objective and the eye lens of an astronomical telescope when adjusted for parallel light is 100cm.The measured value of the magnification is 19.The focal length of objective and eye piece are

In Young’s double slit experiment,the intensity at appoint P on the screen is half the maximum intensity in the interference pattern.If the wavelength of light used is λ and d is the distance between the slits, the angular separation between point P and the center of the screen is

If the vectors 2i+3j, 5i+6j, 8i+ λj have their initial point at (1, 1) then the value of λ so that the vectors terminated on one line is

If n positive integers are taken at random and multiplied together , the probability that the last digit of the product is 2,4,6 or 8 is

The intercept made by the circle x2+y2-2hx sin θ-2ky sinθ=h2 cos2θ on the x-axis is

A body is sliding down on a rough inclined plane of inclination 45. The time taken by it to slide down on the rough inclined plane is 3 times the time taken by the body to slide down the smooth inclined plane of same inclination and length. The co-efficient of friction between the rough inclined plane and the body is

If h is the height of the maximum cone inscribed in a sphere of radius r then h:r=

If y= ax2+b/x then x2y2=

x= cos 550 y= cos 650, z= cos 1750 then xy+yz+zx=

If α,β,γ are the roots of x3+2x2-5x+2=0 then the equation whose roots α-1/βγ,β-1/γα,γ–1/αβ is

If y= x log|x+√(1+x2)|-√(1+x2) then dy/dx=

In a ∆ABC ,  ∑(b+c) tan a/2 tan(b-c)/2  is equal to

Which of the following has the highest first IP

If the points 3i-2j-k, 2i+3j-4k, i+j+2k, 4i+5j+λk are coplanar then λ=

cos π/11 cos 2π/11 cos 3π/11 cos 4π/11 cos 5π/11=

The solution of excot y dx+(1-ex)cosec2ydy=0 is

If the foot of the perpendicular from (0,0,0) to a plane is (1,2,3), then the equation of the plane is

The general term of (2a-3b)-1/2 is

A man slides down on a telegraphic pole with  an acceleration equal to one-fourth of acceleration due to gravity. The frictional force between man and pole is equal to in terms of man's weight w

A certain current liberated 0.504g of hydrogen in 2hrs. How many grams of copper can be librated by the same current flowing for the same time in a copper sulphate solution

If f(x)= x(√x-√(x+1)) then

cos 240 cos 480 cos 960 cos 1680 =

One atom of 3919K contains:

The centre of the circle passing through the points (0, 0), (1, 0) and touching the circle x2+y2=9 is

If u=(x-y) (y-z) (z-x) then ux+uy+uz=

The equation of the line dividing the line segment joining the points (2, -3), (1, 2) in the ratio 2:3 and perpendicular to   2x+5y-1=0. Is

If the line 4x+3y+k=0 is a normal to the circle 2x2+2y2+7x+4y+8=0 then k=

A bag contains 5 red, 3 black balls and another bag contains 4 red and 5 black balls. One of the bags is chosen at random and a draw of two balls is made from it. The chance that one id red and the other is black is

The angle between the pair of lines 2x2+5xy+2y2+3x+3y+1=0,is:

The condition that a root of ax2+bx+c=0 may be the reciprocal of a root of a1x2+b1x+c1=0 is

The orthocenter of the triangle formed by (0, 0), (3, 1), (1, 3) is

Two vertices of a triangle are (5, -1) and (-2, 3). If the orthocenter of the triangle is the origin, then the third vertex is

The hydrolysis constant Kh of a salt of NaOH and a weak acid (HX) If the Ka of the acid is Ka=2x10-6

I: The locus of poles of chords of the ellipse x2/a2+y2/b2=1  which touch the parabola y2= 4px is p a2y2+b4x=0 II: The locus of poles of chords of the ellipse x2/a2+y2/b2=1  which touch the x2/α2+y2/β2=1 is α 2x2/a4+β2y2/b4=1

75 ml of 0.2 M HCl is mixed with 25 ml of 1 M HCl. To this solution, 300 ml of distilled water is added. What is the pH of the resultant solution

The solution of sin 2θ+cos 2θ+sin θ+ cos θ+1=0 in the first quadrant is

If the pairs of lines 3x2-5xy+py2=0 and 6x2-xy-5y2=0 have one line in common, then p=

If the locus of the centre of the circle which cuts the circles x2+y2+4x-6y+9=0 and x2+y2-4x+6y+4=0 orthogonally is ax+by+c=0,(a>0) then the ascending order of a,b,c is

The equation of the circle passing the origin having its centre on the line x+y=4 and cutting the circle x2+y2-4x+2y+4=0  orthogonally is

The equations of the tangents to the circle x2+y2+8x-4y-5=0 and perpendicular to 2x+3y+5=0 are

If x2+xy+y2=a2, then y2=

138 g of ethyl alcohol is mixed with 72 g of water. The ratio of mole fraction of alcohol to water is

The derivative of cos hx/2w.r.to x is

If y=2 Tan-1(x√2/1-x2)+log(1+x√2+x2/1-x√2+x2), then dy/dx=

Through the vertex O of the parabola y2 = 4ax a perpendicular is drawn to any tangent meeting it at P and the parabola at Q.Then OP.OQ =

Thomson coefficient of a conductor is 10μV/K.The two ends of it are kept at 500C and 600C respectively.Amount of heat absorbed by the conductor when a charge of 10C flows through it is

The ratio of the coefficient of (r + 1)th term in the expansion of (1 + x)n+1 to the sum of the coefficients of rth and (r + 1) terms in the expansion of ( 1 +x )n is:

If 2,3 are roots of 2x3+mx2-13x+n=0 then the other root is

An open pipe resonates to a frequency ‘f1’ and a closed pipe resonates to a frequency ‘f2’.If they are joined together to form a longer tube,then it will resonate to a frequency of(neglect end corrections)

The set of all values of a for which the function f(x)=(a2-3a+2)(cos2x/4-sin2x/4)+(a-1)x+sin 1 does not possess critical points is

The derivative of Sin-12x/1+x2 w.r.to Tan-12x/1-x2is

The extreme values of  4 cos(x2) cos(π/3  + x2 ) cos(π/3 - x2) over IR are

The shortest distance from (-2, 14) to the circles x2+y2-6x-4y-12=0 is

Two identical sringed instruments have frequency 100Hz. If tension in one of them is increased by 4% and they are sounded together, then number of beats in one second is

If xr occurs in the expansion (x+1/x2)2n, then its coefficient is:

If (1, 2),(4, 3),(6, 4) are the midpoints of the sides BC,CA,AB of Δ ABC, then the equation of AB is

The domain of Cosh-1 3x is

The vant Hoff factor for 0.1M Barium nitrate is 2.74. The percentage of dissociation of barium nitrate is

A box is made from a piece of sheet of metal 12 inch square by cutting equal small squares from each corner and tuning up the edge. The dimensions of the box of largest volume which can be made in this way are

If α,β,γ are the roots of x3+3px+q=0 then the equation whose roots are α+1/β+γ–α,β+1/γ+α–β and γ+1/α+β–γ is

If a is any vector then (axi)2+(axj)2+(axk)2=

(sin 650+sin 250)/(cos 650+ cos 250)=

If A=(1,1) ,B=(4,5) and C=(6,13) then cos A=

The equation of the circle passing  through (2, 1) and touching the coordinate axes is

If 4y=x+7 is diameter of the circumscribing circle of the rectangle ABCD and A(-3, 4), B(5, 4), then the area of the rectangle is

A large tank filled with water to a height h is to be emptied through a small hole at the bottom The ratio of times taken for the level of water to fall from h to h/2 and h/2 to zero is

(1-ω+ω2)(1-ω2+ω4) (1-ω4+ω8) (1-ω8+ω16)=

Consider the following statements A and B and identify the correct answers given below :A : Peltier coefficient is numerically equal to the potential difference across the junctions of the thermocouple through which current is flowing.B : According to Thomson, energy is neither absorbed nor evolved at the junction of a thermocouple but is observed or evolved only along the lengths of both the conductors.

P(-2, -1) and (0, -3) are the limiting points of a coaxal system of which C= x2+y2+5x+y+4=0 is a member. The circle S= x2+y2-4x-2y-15=0 is orthogonal to the circle C. The point where the polar P cuts the circle S is

Number of coulombs of current required to convert completely 1 mole of MnO4-ions in acid medium to one mole of Mn2+ions electrolytically is

A circular coil of radius 11cm,carries and current of 2.5 amperes.If it has 700 turns,the flux density at the centre of the coil is

The equation of the sphere which passes through the four points (0,0,0), (1,0,0), (0,1,0) and (0,0,1) is

n moles of an ideal gas at temperature T (in kelvin) occupy VL of volume, exerting a pressure of P atmospheres.What is the concentration (in mol / L)?

The equilibrium constant for the reactionSO2(g) + (1/2) O2(g) <====> SO3(g) is  5 x 10-2 atm  The equilibrium constant of the reaction2SO3(g)  <====> 2SO2(g) + O2(g) would be

Which of the following is not correct

d/dx{1-cos 2x/3+2 sin 2x}=

If y= log sin x then y2=

If the tangents at (3, -4) to the circle x2+y2-4x+2y-5=0 w.r.t the circle x2+y2+16x+2y+10=0 in A and B, then the midpoint of AB is

If the line 2x+3y+1=0 and 3x-y-4=0 lie along diameters of a circle of circumference 10π, then the equation of the circle is

The locus of the point of intersection of tangents to the  hyperbola  x2-y2=a2 which includes an angle of 450 is

The radius of the circle r =12cosθ+5sinθ is

Radius of the director circle of the hyperbola (x2/81) - (y2/36) = 1 is

A boats crew consists of 16 men, 6 of whom can only row on one side and 4 only on the other. The number of ways in which the crew can be arranged 8 on each side is

A sonometre wire vibrating in 4 segments is in unison with the tuning fork. Keeping the same length, if the tension made 4 times the tuning fork can still be in resonance with the wire provided the wire now vibrates in

The centres of similitude of the circles x2+y2-2x-6y+9=0, x2+y2=1 is

The minimum value of x3-9x2+24x-12 is

If x2+y2-2x+3y+k=0 and  x2+y2+8x-6y-7=0 cut each orthogonally, the value of k must be

The sum of the series 1+3x+5x2+7x3+....+(2n-1)xn-1+....is:

The angle made by the line joining (5,2),(6,-15) at (0 ,0) is

The multiples roots of x5-3x4-5x3+27x2-32x+12=0 are

The points (k,2-2k), (1-k,2k) and (-4-k,6-2k)are collinear. Then k =

The weight in grams of a non-volatile solute (M.wt:60) to be dissolved in 90 g of water to produce a relative lowering of vapour pressure of 0.02 is

The derivative of cosec-1(ax)w.r.to x is

The point of intersection of the diagonals of the quadrilateral with vertices (1, 2), (3, 4), (2, 1), (-1, -2) is

x grams of calcium carbonate was completely burnt in air. The weight of the solid residue formed is 28g. What is the value of x (in grams)

The condition that the pairs of lines ax2+2pxy-ay2=0, bx2-2qxy-by2=0 are such that each pair bisects the angle between the other pair is