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
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 a 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
Solve the differential equation `dy/dx = (x + y+2)/(2(x+y)-1)`
Concept: Introduction of Relations and Functions
Class 12 Mathematics can be very challenging and complicated. But with the right ideas and support, you will be able to make it work. That’s why we have the class 12 Mathematics important questions with answers pdf ready for you. All you need is to acquire the PDF with all the content and then start preparing for the exam. It really helps and it can bring in front an amazing experience every time if you're tackling it at the highest possible level.
Mathematics exams made easy
We make sure that you have access to the important questions for Class 12 Mathematics I.S.C. CISCE. This way you can be fully prepared for any specific question without a problem. There are a plethora of different questions that you need to prepare. And that's why we are covering the most important ones. They are the questions that will end up being more and more interesting and the results themselves can be staggering every time thanks to that. The quality itself will be quite amazing every time, and you can check the important questions for Class 12 Mathematics CISCE whenever you see fit.
The important questions for Class 12 Mathematics 2021 we provide on this page are very accurate and to the point. You will have all the information already prepared and that will make it a lot simpler to ace the exam. Plus, you get to browse through these important questions for Class 12 Mathematics 2021 I.S.C. and really see what works, what you need to adapt or adjust and what still needs some adjustment in the long run. It's all a matter of figuring out these things and once you do that it will be a very good experience.
All you have to do is to browse the important questions for upcoming CISCE Class 12 2021 on our website. Once you do, you will know what needs to be handled, what approach works for you and what you need to do better. That will certainly be an incredible approach and the results themselves can be nothing short of staggering every time. We encourage you to browse all these amazing questions and solutions multiple times and master them. Once you do that the best way you can, you will have a much higher chance of passing the exam, and that's what really matters the most!