# HSC Science (General) 12th Board Exam - Maharashtra State Board Important Questions for 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 ЁЭСб = 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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