Gurdeep Singh Arora
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Mechanical Engineer having experience of Mechanical Maintenance, Inspection, Construction, Commissioning of Ammonia/Urea Oil and Gas
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Large numbers can be expressed in standard form, also known as scientific form, by using positive integral exponents.A number is in standard form when it is written as `a*10^n` , where a is a... 
Answered a Question in Math
Given parametric equations are: `x=3t^2,y=t^3t` We need to find the second derivative, to determine the concavity of the curve. `dy/dx=(dy/dt)/(dx/dt)` Let's take the derivative of x and y with... 
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Parametric curve (x(t),y(t)) has a horizontal tangent if its slope `dy/dx` is zero i.e when `dy/dt=0` and `dx/dt!=0` It has a vertical tangent if its slope approaches infinity i.e it is... 
Answered a Question in Math
To find a tangent line to a polar curve,`r=f(theta)` we regard `theta` as parameter and write it's parametric equations as, `x=rcos(theta)=f(theta)cos(theta)` `y=rsin(theta)=f(theta)sin(theta)` We... 
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One of the vertex form of the parabola is, `(yk)^2=4p(xh)` where (h,k) is the vertex and p is the distance between vertex and focus and also the same distance between the vertex... 
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Draw a table for different values of t and plot the corresponding points (x,y) obtained from the table.Connect the points to a smooth curve. ( Refer the attached image). The direction in which the... 
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Draw a table for different values of t and plot the corresponding points (x,y) obtained from the table. Connect the points to a smooth curve.(Refer the attached image). The direction in which the... 
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Given parametric equations are: `x=1+1/t ,y=t1` Draw a table for different values of t and plot the points obtained from the table. Connect the points to a smooth curve.( Refer the attached... 
Answered a Question in Math
`(1/(3x^23))/(5/(x+1)(x+4)/(x^23x4))` Let's factorize the denominators of the terms in the complex fraction, `=(1/(3(x^21)))/(5/(x+1)(x+4)/(x^2+x4x4))`... 
Answered a Question in Math
`(x+2)/(x4)+2/x+(5x)/(3x1)` LCD of the expression is `x(x4)(3x1)` `=(x(3x1)(x+2)+2(x4)(3x1)+5x(x)(x4))/(x(x4)(3x1))` Expand the terms in the numerator and combine the like terms,... 
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`(x+3)/(x^225)(x1)/(x5)+3/(x+3)` Factorize the denominator of the first term by using the identity: `a^2b^2=(a+b)(ab)` `=(x+3)/((x+5)(x5))(x1)/(x5)+3/(x+3)` LCD of the above... 
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`x/(x^29)+(x+1)/(x^2+6x+9)` Apply the following identities to factorize the denominators of the above rational functions: `a^2b^2=(a+b)(ab)` and `a^2+2ab+b^2=(a+b)^2`... 
Answered a Question in Math
`(1/(2x5)7/(8x20))/(x/(2x5))` `=(1/(2x5)7/(4(2x5)))/(x/(2x5))` LCD of all the denominators in the complex fraction is `(2x5)` Multiply both the numerator and denominator of the... 
Answered a Question in Math
`(3/(x2)6/(x^24))/(3/(x+2)+1/(x2))` `=(3/(x2)6/((x+2)(x2)))/(3/(x+2)+1/(x2))` `=((3(x+2)6)/((x+2)(x2)))/((3(x2)+1(x+2))/((x+2)(x2)))` Divide the... 
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`18/(x^23x)6/(x3)=5/x` LCD is `x^23x=x(x3)` Multiply all the terms of the equation by LCD and simplify, `x(x3)(18/(x^23x))x(x3)(6/(x3))=x(x3)(5/x)` `186x=(x3)5` `186x=5x15`... 
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`(x+1)/(x+6)+1/x=(2x+1)/(x+6)` LCD is `x(x+6)` Multiply all the terms of the equation by LCD and simplify, `x(x+6)((x+1)/(x+6))+x(x+6)(1/x)=x(x+6)((2x+1)/(x+6))` `x(x+1)+(x+6)=x(2x+1)`... 
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`(6x)/(x+4)+4=(2x+2)/(x1)` LCD is `(x+4)(x1)` Multiply each term of the equation by LCD and simplify, `(x+4)(x1)((6x)/(x+4))+4(x+4)(x1)=(x+4)(x1)((2x+2)/(x1))`... 
Answered a Question in Math
`x/(x^22)=(1)/x` Cross multiply, `x*x=1(x^22)` `x^2=x^2+2` `x^2+x^22=0` `2x^22=0` Factorize, `2(x^21)=0` `2(x+1)(x1)=0` Use the zero product property, `x+1=0` or `x1=0` `x=1`... 
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`x/(x+1)=3/(x+1)` cross multiply, `x(x+1)=3(x+1)` `x*x+x*1=3*x+3*1` `x^2+x=3x+3` `x^2+x3x3=0` Factorize, `x(x+1)3(x+1)=0` `(x+1)(x3)=0` use zero product property, `x+1=0` or... 
Answered a Question in Math
`6/(x1)=9/(x+1)` Cross multiply, `6(x+1)=9(x1)` `6x+6=9x9` `6x9x=96` `3x=15` `x=(15)/(3)` `x=5` Check the solution: `6/(51)=9/(5+1)` `6/4=9/6` `(3*2)/(2*2)=(3*3)/(3*2)` `3/2=3/2` It's... 
Answered a Question in Math
`4/(2x)=5/(x+6)` Cross multiply, `4(x+6)=5(2x)` `4x+4*6=10x` `4x+24=10x` `4x10x=24` `6x=24` `x=(24)/(6)` `x=4` Check the solution: `4/(2*4)=5/(4+6)` `1/2=5/10` `1/2=1/2` It's true. So, the... 
Answered a Question in Math
`log(12x11)=log(3x+13)` Using the property of logarithmic equality, `12x11=3x+13` `=>12x3x=13+11` `=>9x=24` `=>x=24/9` `=>x=8/3` Let's plug back the solution in the equation to check... 
Answered a Question in Math
`ln(4x7)=ln(x+11)` Using one to one property of logarithms, `4x7=x+11` `=>4xx=11+7` `=>3x=18` `=>x=18/3` `=>x=6` Plug back the solution in the equation to check the solution,... 
Answered a Question in Math
`log_5(2x7)=log_5(3x9)` Using one to one property of logarithms, `2x7=3x9` `=>2x3x=9+7` `=>x=2` `=>x=2` Let's plug back the solution in the equation, `log_5(2*27)=log_5(3*29)`... 
Answered a Question in Math
` f(x)=(4x)/(x^2+2x3), c=0` Let's first factorize the denominator of the function, `x^2+2x3=x^2+3xx3` `=x(x+3)1(x+3)` `=(x+3)(x1)` Now let, `(4x)/(x^2+2x3)=A/(x+3)+B/(x1)`... 
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If x is a variable, then an infinite series of the form `sum_(n=0)^ooa_n(xc)^n=a_0+a_1(xc)+a_2(xc)^2+.......+a_n(xc)^n+......` is a power series centered at x=c, where c is a constant.... 
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If x is a variable, then an infinite series of the form `sum_(n=0)^ooa_n(xc)^n=a_0+a_1(xc)+a_2(xc)^2+......+a_n(xc)^n+.....` is a power series centered at x=c, where c is a constant.... 
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The given parametric equations are , `x=2picos(t), y=2tpisin(t)` The curve crosses itself for different values of t , which give the same x and y value. So, to get the point where the curve... 
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Given parametric equations are : `x=t3` (1) `y=t/(t3)` (2) Draw a table for different values of t and plot the points obtained from... 
Answered a Question in Math
Parametric curve (x(t),y(t)) has a horizontal tangent if its slope `dy/dx` is zero i.e. when `dy/dt=0` and `dx/dt!=0` It has a vertical tangent, if its slope approaches infinity i.e.... 
Answered a Question in Math
Parametric curve (x(t),y(t)) has a horizontal tangent if its slope `dy/dx` is zero, i.e when `dy/dt=0` and `dx/dt!=0` Curve has a vertical tangent line, if its slope approaches infinity... 
Answered a Question in Math
Parametric curve has a horizontal tangent if its slope `dy/dx` is zero i.e. when `dy/dt=0` and `dx/dt!=0` Tangent line is vertical if its slope approaches infinity i.e. `dx/dt=0`... 
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Parametric curve (x(t),y(t)) has a horizontal tangent when its slope `dy/dx` is zero, i.e. `dy/dt=0` and `dx/dt!=0` . Curve has a vertical tangent if its slope approaches infinity i.e.... 
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Given parametric equations are: `x=2sin(2t)` `y=3sin(t)` Let's make a table of x and y values for different values of t. (Refer the attached image).The point where the curve crosses itself will... 
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The parametric equations are: `x=t^36t` (1) `y=t^2` (2) From equation 2, `t=+sqrt(y)` Substitute `t=sqrt(y)` in equation... 
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Given parametric equations are: `x=t^2t` `y=t^33t1` We have to find the point where the curves cross. Let's draw a table for different values of t, and find different values of t which give the... 
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Arc length of a curve C described by the parametric equations x=f(t) and y=g(t), `a<=t<=b` where f' and g' are continuous on [a,b] and C is traversed exactly once as t increases from a to b,... 
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Arc length of a curve C described by the parametric equations x=f(t) and y=g(t),`a<=t<=b` , where f'(t) and g'(t) are continuous on [a,b] and C is traversed exactly once as t increases from a... 
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`x=sqrt(t)` (1) `y=t5` (2) Draw a table for different values of t and plot the points obtained from the table.(Refer the attached image).... 
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`x=t^3` (1) `y=t^2/2` (2) Draw a table for different values of t and plot the points obtained from the table.Connect the points to a... 
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`x=1+t` (1) `y=t^2` (2) Draw a table for different values of t and plot the points obtained from the table (Refer the attached... 
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`x=2t3` (1) `y=3t+1` (2) From equation 1, `x+3=2t` `=>t=(x+3)/2` Plug in the value of t in equation (2) `y=3((x+3)/2)+1` `y=(3x+9)/2+1`... 
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`sum_(n=0)^oo3(x4)^n` Given series is a geometric series with ratio `r=(x4)` , so the series converges if `r<1` `x4<1` `=>1<(x4)<1` `=>3<x<5` Given series... 
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`sum_(n=0)^oo2(x/3)^n` Given series is a geometric series with ratio `r=x/3` , so the series converges if `r<1` `x/3<1` `=>1<x/3<1` `=>3<x<3` So the... 
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`sum_(n=0)^oo((x3)/5)^n` It is a geometric series with common ratio (r)`=(x3)/5` , so the series converges if `r<1` `(x3)/5<1` `=>1<(x3)/5<1` `=>5<(x3)<5`... 
Answered a Question in Math
`a_n=nsin(1/n)` Apply n'th term test for divergence, which states that, If `lim_(n>oo) a_n!=0` , then `sum_(n=1)^ooa_n` diverges `lim_(n>oo)nsin(1/n)=lim_(n>oo)sin(1/n)/(1/n)`... 
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Integral test is applicable if f is positive, continuous and decreasing function on infinite interval `[k,oo)` where `k>=1` and `f(n)=a_n` . Then the series `sum_(n=k)^ooa_n` converges or... 
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Consider an irregularly shaped planar lamina of uniform density `rho` , bounded by graphs `x=f(y)` , `x=g(y)` and `c<=y<=d` . The mass `m` of this region is given by:... 
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Consider an irregularly shaped planar lamina of uniform density `rho` , bounded by graphs `y=f(x),y=g(x)` , and `a<x<b` .The mass `m` of this region is given by:... 
Answered a Question in Math
`sum_(n=0)^oo(1)^nx^(2n)` `=sum_(n=0)^oo(x^2)^n` It's a geometric series with ratio`r=x^2` A geometric series `sum_(n=0)^ooar^n` with ratio r diverges if `r>=1` . If `0<r<1` , then...
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