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Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r = (mbarb + nbara)/(m + n) `

Concept: Section Formula

Show that four points A, B, C and D whose position vectors are

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Concept: Coplanarity of Two Lines

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius *r* is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

Concept: Maxima and Minima

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

Concept: Methods of Integration: Integration by Parts

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Concept: Methods of Integration: Integration by Substitution

Find: `I=intdx/(sinx+sin2x)`

Concept: Methods of Integration: Integration Using Partial Fractions

**Prove that: **

`{:(int_(-a)^a f(x) dx = 2 int_0^a f(x) dx",", "If" f(x) "is an even function"),( = 0",", "if" f(x) "is an odd function"):}`

Concept: Fundamental Theorem of Integral Calculus

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours `("Given" sqrt(2) = 1.414)`

Concept: Application of Differential Equations

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Concept: Variance of Binomial Distribution (P.M.F.)

Show that every homogeneous equation of degree two in x and y, i.e., ax^{2} + 2hxy + by^{2} = 0 represents a pair of lines passing through origin if h^{2}−ab≥0.

Concept: Pair of Straight Lines > Pair of Lines Passing Through Origin - Homogenous Equation

Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r = (mbarb + nbara)/(m + n) `

Concept: Section Formula

Show that four points A, B, C and D whose position vectors are

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Concept: Coplanarity of Two Lines

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius *r* is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

Concept: Maxima and Minima

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

Concept: Methods of Integration: Integration by Parts

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Concept: Methods of Integration: Integration by Substitution

Find: `I=intdx/(sinx+sin2x)`

Concept: Methods of Integration: Integration Using Partial Fractions

**Solve the differential equation: **

(1 + y^{2}) dx = (tan^{−1 }y − x) dy

Concept: General and Particular Solutions of a Differential Equation

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Concept: Variance of Binomial Distribution (P.M.F.)

Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk` are coplanar.

Concept: Scalar Triple Product of Vectors

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Concept: Distance of a Point from a Line