Questions asked by sahil

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• If # tan(alpha+beta)=a+b# and # tan(alpha-beta)=a-b# then show that #atanalpha-btanbeta=a^2-b^2#?
  
• (at^2,2at) is a point on the line 2x-5y+12a=0,from this find the co-ordinates of two points on the line?
  
• If #a+b+c !=0# and #a^3+b^3+c^3-3abc=0#, then what will be the relation between #a#, #b#. and #c#?
  
• Find the equatioon of straight line which are at a distance of 1 unit from the orgin and which passes through the point (3,1)?
  
• Find the ratio in which the staight line 5x+4y=4 divides the join of (4,5) and (7,-1)?
  
• The variable coefficient p,q,r in the equation of the straight line px+qy+r=0 are connected by the relation pa+qb+rc=0 where a,b and c are fixed contants.show that the variable line always passes through a fixed point?
  
• If ab+bc+ca=0,show that the lines x/a+y/b=1/c,x/b+y/c=1/a and x/c+y/a=1/b are concurrent?
  
• A variable straight line throgh the point of intersction of the two lines x/3+y/2=1 and x/2+y/3=1 meets the co-ordinate axes at A and B .find the locus of the middle point of AB?
  
• A straight line passes through the point A(1,2) and makes an angle theta with the positive direction of x axes. find theta,given that the distance of A from the point of intersection of this line with the line x+y=4 is root5/3 unit?
  
• Find the ratio in which the staight line 5x+4y=4 divides the join of (4,5) and (7,-1)?
  
• In triangleRST,angle S is right angle.The mid points of two sides RS ans ST are X and Y respectively.Let us prove that, RY^2+XT^2=5XY^2?
  
• The area of a rectangular land is 500 sq.m.If the length is decreased by 3m and breadth is increased by 2m.then land formed a square.let us write by calculating the length and breadth of land?
  
• If an AP consists of n terms and sum of first three terms is x and sum of last three terms is y then show that sum of all terms is equal to (n/6)(x+y)?
  
• If n is a positive integers,show that (n+1)^2+(n+2)^2+....+4n^2=n/6(2n+1)(7n+1)?
  
• Find the sum to n terms 1/(1.4)+1/(4.7)+1/(7.10)+......to n terms?
  
• Let a_n denotse the nth term of an A.P.and p and q be two positive integers with p<q .if a_p+1+a_p+2+a_p+3+...+a_q=0 find the sum of first p+q terms of the A.P?
  
• Between the roots of the quadratic equation 3px^2-10px+5q=0(p>0,q/p<5/3),an odd number of A.M.S are insertred and their sum exceeds their number by 10.find the number of A.M.S inserted?
  
• Find the sum of 3+7+14+27+52 +...... to n terms?
  
• If A is the A.M and G is the G.M of two unequal positive real numbers p and q,prove that A>G and |G|>G^2/A?
  
• If A is the A.M and G is the G.M of two unequal positive real numbers p and q ,prove that A>G and |G|>G^2/A?
  
• IF Sn=1+1/2+1/2^2+.......+1/(2^n-1),find the least value of n for which 2-Sn<1/100?
  
• If #u_1,u_2,u_3,...# form a G.P with common ratio #k#, find the sum of #u_1u_2+u_2u_3+...+u_n u_(n+1)# in terms of #u_1# and #k#?
  
• If the first term ,common ratio and the sum of first n terms of a G.P be a,r and Sn respectively, find the value of S1+S2+S3+...+Sn?
  
• The first term of an A.P is same as that of G.P .the common difference of one and the common ration of other are both 4.if the sum of the first three terms of both series are same ,find the fourteenth terms of each series?
  
• Find the sum to n terms of the series 1/2+3/2^2+5/2^3+...+(2n-1)/2^n?
  
• Find the sum to n terms of the series 2+3.3+4.3^2+5.3^3+.....?
  
• If #xcos alpha+ysin alpha=xcos beta+ysin beta=k# then show that #x/cos ((alpha+beta)/2)=y/sin ((alpha+beta)/2)=k/cos ((alpha-beta)/2)#?
  
• Prove that CotA/2-3cot3A/2=4sinA/1+2cosA?
  
• If # tan (beta/2)=4tan (alpha/2)#,prove that #tan ((beta-alpha)/2)=(3sin alpha)/(5-3cos alpha)#?
  
• Prove that #cot (A/2)- 3cot ((3A)/2) = (4sinA)/(1+2cosA)#?
  
• If #sec(theta+alpha)+sec(theta-alpha)=2sec(theta)#,prove that #cos(theta)=sqrt2 cos(alpha/2)#?
  
• If A=170 degrees,prove that,tan A/2=(-1-root(1+tan^2A))/tanA?
  
• Prove that #2sin(A/2)=+-sqrt(1+sinA) +-sqrt(1-sinA)#?
  
• Prove that #cot(A/2)-3cot((3A)/2)=(4sinA)/(1+2cosA)#?
  
• If cosA=x/(y+z),cosB=y/(z+x),cosC=z/(x+y),show that tan^2A/2=tan^2B/2=tan^2C/2=1?
  
• If# cosx/a_1=(cos2x)/a_2=(cos3x)/a_3# then show that #sin^2(x/2)=(2a_2-a_1-a_3)/(4a_2)#?
  
• #cos alpha=x/(y+z), cos beta=y/(z+x), cos gamma=z/(x+y),#show that #tan^2 (alpha/2)+tan^2 (beta/2)+tan^2 (gamma/2)=1#?
  
• Find the solutio of 4sin 4theta+1=root5?
  
• Solve it 4sin4theta+1=root5?
  
• Solve #2(sec^2x+sin^2x)=5#?
  
• IF #P={x:sinx-cosx=sqrt2 cosx} and Q={x:sinx+cosx=sqrt2 sinx} # then show that P=Q?
  
• IF #(1+x)^n=c_0+c_1x+c_2x^2+cdots+c_nx^n# then show that #c_1+2c_2+3c_3+cdots+nc_n=n.2^(n-1)#?
  
• If #(1+x)^n=c_0+c_1x+c_2x^2+cdots+c_nx^n# then show that #c_0.c_n+c_1.c_(n-1)+c_2.c_(n-2)+cdots+c_n.c_0=((2n)!)/(n!)^2#?
  
• If #(1+x)^n=c_0+c_1x+c_2x^2+cdots+c_nx^n# then show that #c_0+3c_1+5c_2+cdots+(2n+1)c_n=(n+1)2^n#?
  
• If ( 1 + x ) ^n = c 0 + c 1 x + c 2 x 2 + ⋯ + c n x n then show that C0/1+C2/3+C4/5+........=2^n/(n+1)?
  
• If ( 1 + x ) ^n = C 0 + C 1 x + C 2 x 2 + ⋯ + C n x n then show that (C0+C1+C2+...+Cn)^2=2nC0+2nC1+2nC2+...+2nC2n?
  
• If ( 1 + x ) ^n = c 0 + c 1 x + c 2 x 2 + ⋯ + c n x n then show that C0+4C1+8C2+12C3+...+4nCn=1+n.2^(n+1)?
  
• Show that 2^n-n/1!.2^(n-1)+n(n-1)/2!.2^(n-2)-.....+(-1)^n=1?
  
• If # (1+x)^n = c_0 + c_1x + c_2x^2 + ... + c_nx^n # then show that # c_0c_1 + c_1c_2 + ... + c_(n-1)c_n = (2n)!(n+1)!(n-1)! #?
  
• If #( 1 + x )^n = C_0 + C_1 x_1 + C_2 x_2 + ⋯ + C_n x_n# then show that #C_0C_r+C_1C_(r+1)+C_2C_(r+2)+....C_nC_(r+n)=((2n)!)/((n+r)!(n-r)!) ?#
  
• If ( 1 + x ) n = c 0 + c 1 x + c 2 x 2 + ⋯ + c n x n then show that 3 . n C 0 − 8 . n C 1 + 13 . n C 2 − 18 . n C 3 + ... . upto ( n + 1 ) terms = 0 ?
  
• Show #lim_ (x->2) {(1+x)^n-3^n}/(x-2)=n.3^(n-1)#?
  
• Show that lim x->a (x^3/8-a^3/8)/(x^5/3-a^5/3)?
  
• Show that #lim_( x->a) (x^(3/8)-a^(3/8))/(x^(5/3)-a^(5/3))=9/40 a^(-31/24)#?
  
• Find the value of limit #{sqrt(1+x^2)-sqrt(1+x)}/{sqrt(1+x^3)-sqrt(1+x)}# as x approaches to 0?
  
• Find the value of #lim ((cosx)^(1/2)-(cosx)^(1/3))/sin^2x# as #x# approaches #0#?
  
• Find the value of limit (cosah-cosbh)/h^2 as h approaches to 0?
  
• Find the value of limit sin(x^2+4x)/(x^3-5x^2+2x) as x approaches to 0?
  
• Show that limit of log(1+2x)/sin3x=2/3 as x approaches to 0?
  
• Show that limit of #sqrt(x){sqrt(x+2)-sqrt(x)}=1# as x approaches to infinity?
  
• Show that limit of #{(n+1)^(1/3)-n^(1/3)}=0# as x approaches to infinity?
  
• Putting x=theta-pi/4 show that limit of (sintheta-costheta)/(theta-pi/4)=sqrt2 as theta approaches to pi/4?
  
• Putting z=x-pi/4 show that limit of (1-tanx)/(1-sqrt2 sinx)=2 as x approaches to pi/4?
  
• If #lim_(x rarr 2) (ax^2-b)/(x-2)=4# then find the value of a and b?
  
• Show that limit of (4^x-4)/(x-1)=8log2 as x approaches to 1?
  
• Limit of (ax^2-b)/(x-2)=4 as x approaches to 2 find the value of a and b?
  
• Putting z=x-pi/4 show that limit of (1-tanx)/(1-sqrt2 sinx)=2 as x approaches to pi/4?
  
• Putting x=theta-pi/4 show that limit of (sintheta-costheta)/(theta-pi/4)=sqrt2 as theta approaches to pi/4?
  
• Find the value of limit of {1+2+3+......to (3n+2)terms}/(n+1)^2 as n approaches to infinity?
  
• Limit of (ax^2-b)/(x-2)=4 as x approaches to 2 find the value of a and b?
  
• If #f(x)=lim_(n->oo)1/(1+nsin^2pix)# , find the value of #f(x)# for all real values of #x#?
  
• The height of a house subtends a right angle at the window of another house situated just on the oposite side of the road.The angle of elevation of the top of the first house from the window is 30deg.If the road is 4sqrt3 meter broad,find hgt of 1st hous?
  
• Show that limit of log(2x-3)/2(x-2)=1 as x approaches to 2?
  
• Find the limit of {4sqrt3-(cosx+sinx)^5}/(1-sin2x) as x approaches to pi/4?
  
• If f(x)=lim of 1/(1+nsin^2pix) as n approaches to infinity,find the value of f(x) for all real values of x?
  
• Find the volume of the gas at NTP by treating 200gm of CaCO3 with excess of dil Hcl?
  
• Find the volume of ammonia required for the preparation of 66gm of ammonium sulphate according to the reaction? 2NH3+H2SO4=(NH4)2SO4?
  
• Find derivative of (tanx/x) log(e^x/x^x)?
  
• Let f(x)=5x-4,when 0<x<1 and f(x)=4x^2-3x, when x>1,discuss the differentiability of f(x) at x=1?
  
• Find the #lim_(x to pi/4) {4sqrt3-(cos(x)+sin(x))^5}/(1-sin(2x))#?
  
• If 1,w,w^2 denotes the cube root of unity ,find the roots of (x+5)^3+27=0?
  
• Find all roots of x^3-1, show that if w is a complex root of this equation, the other complex root is w^2 and 1+w+w^2=0?
  
• If a,b,p,q are real and #(a-ib)^(1/3)=p-iq# prove that #(a+ib)^(1/3)=p+iq#?
  
• Find the square roots of a^2+1/a^2+4i(a+1/a)-2?
  
• If #1,omega,omega^2# denotes the cube root of unity ,find the roots of # (x+5)^3+27=0#?
  
• Show that one value of #(1+i)^(1/2)-(1-i)^(1/2) = i sqrt{2(sqrt2-1)}#?
  
• Show that?
  
• If z=x+iy and |z-1|^2+|z+1|^2=4, determine the position of the points z in the complex plane?
  
• If |z1|=|z2| and argz1~argz2=pi,show that z1+z2=0?
  
• If x,y,b are real,z=x+iy and (z-i)/(z-1)=ib,show that (x-1/2)^2+(y-1/2)^2=1/2?
  
• If x+iy=2/(3+costheta+isintheta),show that 2x^2+2y^2=3x-1?
  
• If #y=sqrt(x^2+6x+8)#,show that one value of #sqrt(1+iy)+sqrt(1-iy)=sqrt(2x+8)#?
  
• Factorize #x^2+y^2+z^2-xy-yz-zx?#
  
• Express (1+x^2)(1+y^2) as the sum of two squares?
  
• If a,b are real and a^2+b^2=1 then show that the equation {sqrt(1+x)-isqrt(1-x)}/{sqrt(1+x)+isqrt(1-x)}=a-ib is satisfy by a real value of x?
  
• Factorize x ^2 + y ^2 + z ^2 − x y − y z − z x ? by using complex number
  
• If a,b are real and a^2+b^2=1 then show that the equation {sqrt(1+x)-isqrt(1-x)}/{sqrt(1+x)+isqrt(1-x)}=a-ib is satisfy by a real value of x?
  
• In the complex plane ,the vertices of an equilateral triangle are represented by the complex numbers #z_1,z_2# and #z_3# ,prove that #1/(z_1-z_2)+1/(z_2-z_3)+1/(z_3-z_1)=0#?
  
• If #(z_1)^2+(z_2)^2+(z_3)^2-z_1z_3-z_3z_2-z_1z_2=0#, prove that,#|z_2-z_3|=|z_3-z_1|=|z_1-z_2|#, where #z_1,z_2,z_3# are complex numbers?
  
• The volume of a sphere varies directly as the cube of its radius.the diameter of a sphere made of lead is 14cm.prove by method of variationthat by melting the sphere 8 sphere can be formed each of diameter 7cm(suppose that after melting the volume cnstnt?
  
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