A to Z Exams

The inverse of f(x)=10x-10-x/ 10x+10-x is

If θ= Sin-1 x+ Cos-1 x+ Tan-1 x, 0≤x≤1, then the smallest interval in which θ lies is given by

If cos 2x = (√2 + 1)(cos x - 1/√2) , cos x ≠ 1/2 then x belongs to

cos4 π/8+ cos4 3π/8+ cos4 5π/8+ cos4 7π/8=

The number of ways that all the letters of the word SWORD can be arranged such that no letter is in its original position is

Let f(x)=-2sinx, if x≤-π/2; f(x)=a sinx+b,if –π/2

If f(x,y)=xy+(1/x)+(1/y) then fxx?fyy-fxy2 at (1,1) is

cos(π/4+A) cos(π/4-B)+ sin(π/4+A) sin(π/4-B)=

Mr. A is called for 3 interviews .There are 5 candidates at the first interview, 4 at the second and 6 at the third .If the selection of each candidates is equally likely then the probability that A will be selected for at least  one post is

If (1,2) is the midpoint of a chord of the circle (x-2)2+(y-4)2=10 and the equation of the chord is ax+by+c=0(a>0) then a-b+c=

If 1,-1,2 are the roots of x3+Ax2+Bx+C=0 then the ascending order of A,B,C is

A bag A contains 2 white and 3 red balls and a bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. The probability that it was drawn from bag B is

The polar of given point with respect to any one of the circles x2+y2-2kx+c2=0, (k is a variable) always passes through a fixed point whatever be the value of k is

If y=4x-5 is a tangent to the curve y2=px3+q at (2, 3), then

If ΔABC is right angled at A, then r2 + r3 is equal to

If tan-13 + tan-1n= tan-18, then n is equal to

The area of the parallelogram whose diagonals are i-3j+2k, -i+2j is

Equation of the circle passing through (0,0),(a,b) and (b,a) is

If (1,2,3), (2,3,1) are two vertices of an equilateral triangle then its third vertex is

sin 850-sin 350- cos 650

The equations of the tangents to the hyperbola 2x2-3y2=6 which are perpendicular to the line x-2y+5 =0 are

The locus of midpoints of the chord of the circle x2+y2 = 25 which pass through a fixed point (4, 6) is a circle. The radius of that circle is

If A,B,C are collinear points such that A=(3,4), B=(7,7) and AC=10 then C=

If b + c = 3a, then cot B/2 cot C/2  is equal to :

The equation of the circle whose center lies on the X- axis and which passes through the points (0, 1) (1, 1) is

If a= sin θ+ cos θ, b= sin3 θ+ cos3θ then

The area (in square units) bounded by the curves y2 = 4x and x2 = 4y in the plane is :

If f(x)= (a-xn)1/n, where a>0 and n?N, then (fof)(x)=

If tan A,tan B are the roots of x2-px+q=0,the value of sin2(A+B) is

If the equation x2-2mx+7m-12=0 has equal roots then m=

If the circles x2+y2+2x-2y+4=0 cuts the circle x2+y2+4x-2fy+2=0 orthogonally, then f=

The height of a hill is 3300 mt. From a point P on the ground the angle of elevation of the top of elevation of the top of the hill is 600. A balloon is moving with constant speed vertically upwards from P. After 5 minutes of its movement, a person sitting in it observes the angle of elevation of the top of the hill is 300. What is speed of the balloon?

The vector r satisfying the conditions that i) it is perpendicular to 3i+2i+2k and 18i-22j-5k ii) it makes an obtuse angle with y-axis, iii)|r|=14 is

If the 3rd, 4th and 5th terms of (x+a)n are 720, 1080 and 810 respectively then (x,a,n)=

If x2+y2-4x+6y+c=0 represents a circle radius 5 then c=

(l1,m1,n1) and (l2,m2,n2) are D’rs of two lines inclined at an angle 1200 then D.C’s of the line bisecting the angle between them are

A(-1, 1) B(5, 3) are opposite vertices of a square. The equation of the other diagonal (not passing through A, B) of the square is

If A+B+C+D= 2π, then -4 cos (A+B/2) sin (A+C/2) cos (A-D/2)=

If the roots of the equation ax2+bx+c=0 is of the form k+1/k and k+2/k+1(k≠0),then (a+b+c)2 is equal to

The tangent and normal to the ellipse 4x2+9y2 =36 at a point P on it meets the major axis in Q nd R respectively. If QR=4, then the eccentric angle of P is

(tan 230+ tan220)/(1- tan 230 .tan220)=

The quadrilateral formed by the pairs of lines 6x2-5xy-6y2=0, 6x2-5xy-6y2+x+5y-1=0 is

An observer finds that the angular elevation of a tower is θ. On advancing ‘a’ metres towards the tower, the elevation is 450 and on advancing b metres the elevation is 900-θ. The height of the tower is

If 2,-2,4 are the roots of ax3+bx2+cx+d=0 then the roots of 8ax3+4bx2+2cx+d=0 are

2 cos 540. Sin 660=

Cos23π/5 + Cos24π/5 is equal to

If the circle x2 + y2 + 2x + 3y + 1 = 0 cuts another circle x2+y2 + 4x + 3y + 2 = 0 in A and B, then the equation of the circle with AB as a diameter is :

sin A+sin 5A+sin 9A)/(cos A+ cos 5A+ cos 9A)=

The multiplicative inverse of (4+3i) 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 pole of the straight line x+4y = 4 With respect to the ellipse x2 + 4y2 = 4 is

y-axis divides the line segment joining (3,5), (-4,7) in the ratio

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

If 2,3 are the roots of the equation 2x3+px2-13x+q=0,then (p,q)=

The straight line x + y = k touches the parabola y = x-x2, if  k =

The intersection of the sphere x2+y2+z2-3x+3y+4z=8 is the same as the intersection of one of the sphere and the plane

The cosine of the angle A of the triangle with vertices A(l, -1, 2), B(6,11, 2),C(1, 2, 6)is

The minimum value of sin6 x+ cos6 x is

Angle between the tangents to the curve y=x2-5x+6 at the points (2, 0) and (3, 0) is

The common chord of x2+y2-4x-4y=0 and x2+y2=16 substends at the origin an angle equal to

The vector equation of the plane passing through A and perpendicular to AB where 3i+j+2k, i-2j-4k are the position vectors of A, B respectively

The length of the common chord of the circles x2+y2+2hx=0, x2+y2-2ky=0 is

There are three events A,B and C one of which and only one can happen. The odds are 7 to 3 against A and 6 to 4 against B.The odds against C are

The extremities of a diameter of a circle have coordinates (-4, -3) and (2, -1). The length of the segment cut off by the circle on y-axis is

If the point of intersection of kx+4y+2=0, x-3y+5=0 lies on 2x+7y-3=0, then k=

Let an=10n / n! for n=1,2,3.................. Then the greatest value of n for which an is the greatest is

The extremities of a diagonal of a parallelogram are the points (3,-4) and (-6,5). If the third vertex is (-2,1) then the fourth vertex is

If x2+y2=a2 then dy/dx=

The product of the slopes of the tangents to the ellipse 2x2+3y2=6 draw from the point (1, 2) is

P and Q are two points on the line x-y+1=0. If OP=OQ=6 then length of median of Δ OPQ through O is

A, B, C, D are four points with the position vectors a, b, c, d respectively such that (a-d). (b-c)=(b-d).(c-a)=0. The point D is the ….of ΔABC

The 1st  and  2nd points of trisection of the join of (-2, 11), (-5, 2) are

If A2 = A, B2 = B, AB = BA = O then (A+B)2 =

If P is a point on the parabola y2 = 4ax such that the subtangent and subnormal at P are equal, then the coridinate of P are

The radical centre of the circle x2+y2=1, x2+y2-2x=1, x2+y2-2y=1 is

If ω is a complex cube root of unity then ( 1 - ω + ω2)6 + ( 1- ω2 + ω)6 =

(4/1.3)-(6/2.4)+(12/5.7)-(14/6.8)+…….=

If a denotes the number of permutations of x+2 things taken all at a time, b the number of permutations of x-11 things taken all at a time such that a=182bc, then the value of x is

2+ 5/(2!.3)+5.7/(3!.3)+5.7/(3!.32 )+...........∞=

If α,β are the roots of ax2+bx+c=0 and γ,δ are the roots of lx2+mx+n=0,then the equation whose roots are αγ+βδ and αδ+βγ is

In a ∆ ABC, (a-b)2cos2(C/2)+(a+b)2sin2(C/2) is equal to

The perpendicular distance of radical axis determined by the circles x2 + y2 + 2x + 4y – 7 =0 and x2 + y2 – 6x + 2y – 5 =0 from the origin is:

The polar of the point(2t, t-4) w.r.t the circle x2+y2-4x-6y+1=0 passes through the point

If the quadratic equations ax2+2cx+b=0 and ax2+2bx+c=0, (b ≠ c) have a common root then a+4b+4c =

The circumcentre of the triangle with vertices at A(5, 12),B(12, 5), c (2√(13 ) ,3√(13 )) is

If the circles x2+y2-4x+6y+8=0, x2+y2-10x-6y+14=0 touch each other , then the point of contact is

The solution of (12x+5y-9)dx+(5x+2y-4)dy=0 is

The circle x2+y2-2x+5y-24=0 cuts the x-axis at A and B and Y-axis at C and D then AB+CD=

A stone thrown upwards, has its equation of motion s=490t-4.9t2. Then the maximum height reached by it is

The slope of a stright line passing through A(5, 4) is -5/12.The points on the line that are 13 unit  away from A are

Tangents OA and OB are drawn to the circle x2+y2+gx+fy+c=0 from O(0,0). The equation of the circum circle of the ?OAB is

Two equal sides of an isosceles triangle are given by equation 7x-y+3=0 and x+y-3=0 and its third side passes through the point (1, 0).The equation of the third side is

tan 2030+tan 220+tan 2030 tan 220 =

The solution set of x2>4x-5 is

The differential equation obtained by eliminating the arbitrary constants a and b from xy=aex + be-x is

I: The circum centre of the triangle with vertices (1, √3), (1, √2), (3, -√3) is (2, 0). II: The ortho-centre of the triangle formed by the lines 4x-7y+10=0, x+y=5, 7x+4y=15 is (1, 2)

If tan θ+sin θ=m ,tan θ- sin θ=n then (m2-n2)2

The equation to one asymptote of the hyperbola 14x2+38xy+20y2+x-7y-91=0 is 7x+5y-3=0, then the other asymptote is

The value of a such that x3+3ax2+3a2x+b is increasing on R-{-a} are

If f(x)= x(√x-√(x+1)) then