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If y = x^{x}, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Concept: Simple Problems on Applications of Derivatives
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
Find : `int(x+3)sqrt(3-4x-x^2dx)`
Concept: Methods of Integration: Integration by Substitution
Solve the differential equation:
(1 + y^{2}) dx = (tan^{−1}y − x) dy
Concept: General and Particular Solutions of a Differential Equation
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Concept: Inverse of Matrix > Inverse of a Square Matrix by the Adjoint Method
If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).
Concept: Properties of Determinants
Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
SchoolArticle | |||
A | B | C | |
Hand-fans | 40 | 25 | 35 |
Mats | 50 | 40 | 50 |
Plates | 20 | 30 | 40 |
Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation.
Concept: Multiplication of Two Matrices
Using properties of determinants, prove that
`|[b+c , a ,a ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc
Concept: Properties of Determinants
If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of `dy/dx `at t = `pi/4`
Concept: Derivatives of Functions in Parametric Forms
Find `intsqrtx/sqrt(a^3-x^3)dx`
Concept: Methods of Integration: Integration by Substitution
Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`
Concept: Evaluation of Simple Integrals of the Following Types and Problems
Evaluate :
`∫_(-pi)^pi (cos ax−sin bx)^2 dx`
Concept: Evaluation of Definite Integrals by Substitution
Find the integrating factor of the differential equation.
`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`
Concept: Solutions of Linear Differential Equation
If y = P e^{ax} + Q e^{bx}, show that
`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`
Concept: General and Particular Solutions of a Differential Equation
Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`
Concept: Solutions of Linear Differential Equation
A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.
Concept: Independent Events
The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A
Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.
Concept: Product of Two Vectors > Scalar (Or Dot) Product of Two Vectors
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
`x^2+y^2=4 at (1, sqrt3)`
Concept: Area Under Simple Curves
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