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# Important Questions for HSC Arts 12th Board Exam - Maharashtra State Board - Mathematics and Statistics

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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: [4] 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: [10] 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: [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: [15] Integration
Concept: Methods of Integration - Integration by Parts

Find : int(x+3)sqrt(3-4x-x^2dx)

Appears in 4 question papers
Chapter: [15] Integration
Concept: Methods of Integration - Integration by Substitution

Find: I=intdx/(sinx+sin2x)

Appears in 4 question papers
Chapter: [15] Integration
Concept: Methods of Integration - Integration Using Partial Fractions

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

Appears in 3 question papers
Chapter: [4] Pair of Straight Lines
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.

Appears in 3 question papers
Chapter: [7] Vectors
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

Appears in 3 question papers
Chapter: [9] Line
Concept: 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.

Appears in 3 question papers
Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems

If x = a sin 2t (1 + cos 2t) and y = b cos 2t (1 – cos 2t), find the values of  dy/dx .

Appears in 3 question papers
Chapter: [13] Differentiation
Concept: Derivatives of Functions in Parametric Forms

If y = f(x) is a differentiable function of x such that inverse function x = f–1 (y) exists, then prove that x is a differentiable function of y and dx/dy=1/((dy/dx)) " where " dy/dx≠0

Appears in 3 question papers
Chapter: [13] Differentiation
Concept: Derivative > Derivative of Inverse Function

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: [15] Integration
Concept: Methods of Integration - Integration by Parts

Find intsqrtx/sqrt(a^3-x^3)dx

Appears in 3 question papers
Chapter: [15] Integration
Concept: Methods of Integration - Integration by Substitution

Evaluate :

∫_(-pi)^pi (cos ax−sin bx)^2 dx

Appears in 3 question papers
Chapter: [15] Integration
Concept: Evaluation of Definite Integrals by Substitution

If y = P eax + Q ebx, show that

(d^y)/(dx^2)=(a+b)dy/dx+aby=0

Appears in 3 question papers
Chapter: [17] Differential Equation
Concept: General and Particular Solutions of a Differential Equation

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.

Appears in 3 question papers
Chapter: [19] Probability Distribution
Concept: Random Variables and Its Probability Distributions

find the symbolic fom of the following switching circuit, construct its switching table and interpret it.

Appears in 2 question papers
Chapter: [1] Mathematical Logic
Concept: Mathematical Logic > Application - Introduction to Switching Circuits

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Appears in 2 question papers
Chapter: [2] Matrices
Concept: Elementary Operation (Transformation) of a Matrix

Show that:

cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)

Appears in 2 question papers
Chapter: [3] Trigonometric Functions
Concept: Basic Concepts of Trigonometric Functions
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