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HSC Science (Computer Science) 12th Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

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Mathematics and Statistics
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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.

Appears in 4 question papers
Chapter: [0.016] Line and Plane
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

Appears in 4 question papers
Chapter: [0.022000000000000002] Applications of Derivatives
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`

Appears in 4 question papers
Chapter: [0.023] Indefinite Integration
Concept: Methods of Integration: Integration by Parts

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Appears in 4 question papers
Chapter: [0.023] Indefinite Integration
Concept: Methods of Integration: Integration by Substitution

Find: `I=intdx/(sinx+sin2x)`

Appears in 4 question papers
Chapter: [0.023] Indefinite Integration
Concept: Methods of Integration: Integration Using Partial Fractions

Show that every homogeneous equation of degree two in x and y, i.e., ax2 + 2hxy + by2 = 0 represents a pair of lines passing through origin if h2ab0.

Appears in 4 question papers
Chapter: [0.04] Pair of Straight Lines
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.

Appears in 4 question papers
Chapter: [0.1] Plane
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

Appears in 4 question papers
Chapter: [0.14] Applications of Derivative
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`

Appears in 4 question papers
Chapter: [0.15] Integration
Concept: Methods of Integration: Integration by Parts

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Appears in 4 question papers
Chapter: [0.15] Integration
Concept: Methods of Integration: Integration by Substitution

Find: `I=intdx/(sinx+sin2x)`

Appears in 4 question papers
Chapter: [0.15] Integration
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.

Appears in 3 question papers
Chapter: [0.015] Vectors
Concept: Scalar Triple Product of Vectors

Let `"A" (bar"a")` and `"B" (bar"b")` 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" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

Appears in 3 question papers
Chapter: [0.015] Vectors
Concept: Section Formula

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

Appears in 3 question papers
Chapter: [0.016] Line and Plane
Concept: 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.

Appears in 3 question papers
Chapter: [0.017] Linear Programming
Concept: Graphical Method of Solving Linear Programming Problems

Find the derivative of the inverse of function y = 2x3 – 6x and calculate its value at x = −2

Appears in 3 question papers
Chapter: [0.021] Differentiation
Concept: Derivatives of Inverse Functions

The displacement of a particle at time t is given by s = 2t3 – 5t2 + 4t – 3. Find the velocity when 𝑡 = 2 sec

Appears in 3 question papers
Chapter: [0.022000000000000002] Applications of Derivatives
Concept: Derivatives as a Rate Measure

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.

Appears in 3 question papers
Chapter: [0.023] Indefinite Integration
Concept: Methods of Integration: Integration by Parts

Find `intsqrtx/sqrt(a^3-x^3)dx`

Appears in 3 question papers
Chapter: [0.023] Indefinite Integration
Concept: Methods of Integration: Integration by Substitution

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`

Appears in 3 question papers
Chapter: [0.026000000000000002] Differential Equations
Concept: Application of Differential Equations
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