Answers created by Ratnaker Mehta

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• How do you write the partial fraction decomposition of the rational expression #x^2 / (x² + x + 2)#?
  
• How do you solve #lim_x->pi/4 (tanx-cotx)/(x-pi/4)# ?
  
• The ordered pair (1, –3) is represented by which number in the complex plane?
  
• If #a^2 - 6b^2 - ab = 0#, what is value of #b/a# ?
  
• How do you evaluate the limit #(sqrt(x^2-5)+2)/(x-3)# as x approaches #3#?
  
• How to prove the following identities cos(30°)-cos(60°)\cos(30°)+sin(60°)=tan(15°)?
  
• #forall u, v#, #((2,3,5,7), (13,17,19,23)) * ((64,28,-18), (-64,-27,18), (15,5,-5), (0,0,1)) * ((1), (u), (v)) = ((11), (29))# ?
  
• How do you solve #csc(x)+cot(x)=1 #?
  
• Show It With Explanation ?
  
• Find the equation of the straight line with positive gradient and inclined at an angle of 45° to the line 3y-x+1=0 and passing through the point (2,0)?
  
• At least how many terms in the GP 20+28+36+.... is greater than 1000 ?
  
• Evaluate the Integral ?
  
• A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/2#. If side C has a length of #28 # and the angle between sides B and C is #pi/12#, what is the length of side A?
  
• How do you factor #4x^2+12x=9#?
  
• How do you find the exact value of cos pi/6 cos pi/3 - sin pi/6 sin pi/3?
  
• A triangle with sides 7cm, 8cm, and 9cm is formed by connecting the centers of the three circles that are tangent to each other. Which of the following is the circumference of the largest circle?
  
• Find the x that makes <2,x> and <4,-6> parallel?
  
• Prove that? tanx+tan2x+tan3x=tanx.tan2x.tan3x
  
• How can #(sin^2(-x))/(tan^2(x))# equal to #cos^2x#?
  
• The sum of two positive integers 'a' and 'b' is 528 and their HCF is 33. Which of the following cannot be the value of (a - b)? (A) 396 (B) 264 (C) 6 (D) 33
  
• Show that The following series is in GP ( Geometric Progression ) ?
  
• Please explain this method to solve series?
  
• How do you factor given that #f(-6)=0# and #f(x)=2x^3+7x^2-33x-18#?
  
• If w is a complex cube root of unity then show that #(2-w)(2-w^2)(2-w^10)(2-w^11)=49#?
  
• If n is a natural number that is not a multiple of 3, prove that #1+w^n+w^(2n)=0#?
  
• Anybody know the answer?
  
• Prove that? : #P(AuuBuuC)=P(A)+P(B)+P(C)-P(AnnB)-P(BnnC)-P(AnnC)+P(AnnBnnC)#
  
• Prove #|(1,a,a^2,a^3+bcd),(1,b,b^2,b^3+cda),(1,c,c^2,c^3+dab),(1,d,d^2,d^3+abc)|=0#?
  
• Given the family of lines, #a(3x+4y+6) + b(x+y+2)=0#. The line of the family situated at the greatest distance from the point #P(2,3)# has the equation?
  
• The position vectors of the points A, B, C of a parallelogram ABCD are a, b, and c respectively. How do I express, in terms of a, b and, the position vector of D?
  
• Let a, b, c be three numbers in G. P . And we take theirs log (eg. Log a, Log b,Log c) is it always true that their log will be in A. P?
  
• How do you find the instantaneous rate of change of #f (x)= x ^2 +2 x ^4# at #x=1#?
  
• Given that #7 cos x + 24 sin x = R cos (x - theta)# where R > 0 and theta is acute, how do you find the minimum and maximum values of #7 cos x + 24 sin x - 12# and the corresponding values of x?
  
• For what values of #x# does #f(x) = 4x + 8 sin x# have a horizontal tangent line?
  
• What is equation of hyperbola given Transverse axis on the y-axis, (2, √2) and (0, -1) are points on the curve?
  
• What is the degree measure of an angle of #(-17pi)/10# radians?
  
• Prove that ( vector a + vector b) * ( vector a - vector b) = a ^2 - b ^2?
  
• How do you find the 7th term of the geometric sequence with the given terms a4 = -4, a6 = -100?
  
• A1,a2,a3,...,a40 are the terms of am AP and a1+a5+a15+a26+a36+a40=105 then find a1+a2+a3+....+a40 ?
  
• Answer The Following Question With Explanation ?
  
• The line through the points (-2,3) and (-1,k) has slope k. What is k?
  
• If #f_1(x)=-((2x+7)/(x+3)), f_(n+1)(x)=f_1(f_n(x))# then #f_2001(2002)# is?
  
• A circle contains a sector with an area of #75cm^2# and a central angle of #270°#. What is the radius of the circle to two decimal places?
  
• How do you differentiate #y=ln(3xe^(1-x))#?
  
• A point moves in a manner that the sum of the squares of its distance from thr origin and the point (2,-3) is always 19.show that the locus of moving point is a circle.find the equation to the circle?
  
• A, B and C form the vertices of a triangle. #BC = 5.7cm# and #angle CBA = 137 degrees#. Given that the area of the triangle is # 10.4 cm^2#, what is the perimeter of the triangle?
  
• Differentiate ³√tanx from first principle ?
  
• The base of a triangle is increased by 66.6% and the altitude is decreased by 40%, then the change in the area of the triangle is ?
  
• X,y,z,m real numbers are so positive in which. Xyz=1 & 2xm/xy+x+1=2ym/yz+y+1=2zm/xz+z+1=1. Prove that m=1/2 ?
  
• The sum of first four terms of a GP is #30# and that of last four terms is #960#. If the first and the last term of the GP is 2 and 512 respectively, find the common ratio.?
  
• In the table below, fill in as many y values as you can if you know that f is an even function, and g is an odd function?
  
• How do you implicitly differentiate #x+sqrty=7#?
  
• How do you find the integral of # (x+3) *sqrt(4-x^2)#?
  
• How do you find the two square roots of -1 + irad3?
  
• How do you solve the equation 𝑤̅×<2,0,−2>=<4,0,4> where 𝑤̅=<𝑤1,𝑤2,𝑤3> is nonzero with a magnitude of 2?
  
• How do you differentiate #e^(x/y)=x-y#?
  
• What is the length of the major axis of the conic section #(x+2)^2/49 + (y-1)^2/25 =1#?
  
• Integrate e^1+cosx .sinx dx =?
  
• How do you find the inverse of #f(x)= absx + 1#?
  
• How do you find the exact value of #sin690#?
  
• What are the unit vectors that make a certain angle with two other vectors?
  
• The product of 4 distinct +ve integers #a,b,c,d# is #8!#. The numbers also satisfy #ab+a+b+1=323 and bc+b+c+1=399# then d=?
  
• The numbers #x,y z# satisfy #abs(x+2)+abs(y+3)+abs(z-5)=1# then prove that #abs(x+y+z)<=1#?
  
• If #cos(pi/4)cos(pi/8)cos(pi/16)...cos(pi/(2^2006))=2^(-a)cos(pi/b)# thsn find a and b?
  
• If #loga/(b-c)=logb/(c-a)=logc/(a-b)# then the numerical value of #a^a*b^b*c^c=?#
  
• Find a unit vector w in the direction of u?
  
• Verifying a Trigonometric Identify: cot(pi/2-x)=tanx I am stuck, I used the difference identity for tan, but tan is undefined at pi/2, so would I just substitute the value for cot pi/2 fir tanA in the tan difference formula? Thanks in advance.
  
• How do you find f'(x) using the definition of a derivative # f(x) = x^2 + x#?
  
• To prove cos3theta +2cos5theta +cos7theta/costheta +2cos3theta +cos5theta=cos2theta-sin2theta×tan3theta ?
  
• How do you find #int5/(16+9cos^2x)dx#?
  
• How do you find#intarcsinx/sqrt(1-x^2)dx#?
  
• How do you use the chain rule to differentiate #y=sqrt(1/(x+1))#?
  
• How do you find formula for the exponential function in the form of f(x)= Ca^x given f(0)=3 and f(1)=15?
  
• How do you differentiate #y=1/(arcsin(2x))#?
  
• How do you solve for x in #cos(-100)= cos(55)cos x+sin(55)sin x#?
  
• How do you simplify # (3+3*sqrt3 i)^4#?
  
• If #u_n=sin^n(theta)+cos^n(theta) and (u_3-u_5)/(u_1)=K((u_5-u_7)/(u_3))# then find K?
  
• Can you explain why as well? There's a picture.
  
• Find the general term for #(2x-5y)^7#?
  
• In PQR triangle , PQ=PR=13. If the length of the altitude drawn from Q perpendicular to side PR is 12 what is the perimeter of PQR?
  
• Which relation is true for A ?
  
• Why is AM=MB=MC in this right triangle?
  
• Show that the propositions p and p v (p ^ q) are logically equivalent? Must do a truth table.
  
• What is the derivative of #y = ln(t^2 + 4) - 1/2 arctan(t/2)#?
  
• How do you find the determinant of #((-2, 1), (3, 8))#?
  
• How can you proof #int dx/(a^2-x^2)=1/(2a) log |(a+x)/(a-x)| + c# using #x = a sintheta# ?
  
• Find #a_1# if the given are #S_n=195, a_n=33 and d=3 ?#
  
• Find two functions f and g such that h(x) can be expressed as the function indicated. h(x) = 4x + x2; f − g?
  
• How do you find the equations of the two tangents to the circle #x^2 + y^2 - 2x - 6y + 6 = 0# which pass through the point P(-1,2)?
  
• How to find the derivative of f(x) = √x − x from first principles.?
  
• Find sin 2theta if theta is in the first quadrant and tan theta = 40/9 ??
  
• How do you simplify this equation 1 into equation 2?
  
• Value of cos(-3888)?
  
• In a #DeltaABC#, prove that the value of #cosA/(sinBsinC)+cosB/(sinCsinA)+cosC/(sinAsinB)# is a prime?
  
• How to evaluate each limit below. (a) lim x+2/x^2+x+1 x→∞ (b) lim (1/√x−2 − 4/x−4)? x->4
  
• If (a+2b 2a-b) (4 -2) (2c+d c-2d)=(4 -3) Find a, b, c, and d?
  
• How do you integrate #int (1-x^2)/((x+1)(x-6)(x-3)) # using partial fractions?
  
• Consider equation ax^2+bx+c=0 whose roots are x and y.Then find quadratic equation whose roots are x/y and y/x?
  
• Linear/Seperable Differential Equations Question (Engineering First Year Calculus)?
  
• How do you integrate #int -x^2/sqrt(144+x^2)dx# using trigonometric substitution?
  
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