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

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

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

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: Concept of Line > Distance of a Point from a Line

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

Concept: Graphical Method of Solving Linear Programming Problems

Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

= 0, if f (x) is an odd function.

Concept: Methods of Integration - Integration by Parts

Find `intsqrtx/sqrt(a^3-x^3)dx`

Concept: Methods of Integration - Integration by Substitution

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.

Concept: Random Variables and Its Probability Distributions

If θ is the measure of acute angle between the pair of lines given by `ax^2+2hxy+by^2=0,` then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|,a+bne0`

Concept: Acute Angle Between the Lines

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: Concept of Line > Distance of a Point from a Line

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