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ISC (Science) Class 12 - CISCE Important Questions for Mathematics

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Mathematics
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If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`

 
Appears in 4 question papers
Chapter: [0.032] Applications of Derivatives
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

Appears in 4 question papers
Chapter: [0.032] Applications of Derivatives
Concept: Maxima and Minima

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Appears in 4 question papers
Chapter: [0.033] Integrals
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?

Appears in 3 question papers
Chapter: [0.021] Matrices and Determinants
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).

 
Appears in 3 question papers
Chapter: [0.021] Matrices and Determinants
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.

Appears in 3 question papers
Chapter: [0.021] Matrices and Determinants
Concept: Multiplication of Two Matrices

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

Appears in 3 question papers
Chapter: [0.021] Matrices and Determinants
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`

Appears in 3 question papers
Chapter: [0.031] Continuity, Differentiability and Differentiation
Concept: Derivatives of Functions in Parametric Forms

Find `intsqrtx/sqrt(a^3-x^3)dx`

Appears in 3 question papers
Chapter: [0.033] Integrals
Concept: Methods of Integration: Integration by Substitution

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Appears in 3 question papers
Chapter: [0.033] Integrals
Concept: Evaluation of Simple Integrals of the Following Types and Problems

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Appears in 3 question papers
Chapter: [0.033] Integrals
Concept: Evaluation of Definite Integrals by Substitution

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

Appears in 3 question papers
Chapter: [0.034] Differential Equations
Concept: Solutions of Linear Differential Equation

If y = P eax + Q ebx, show that

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

Appears in 3 question papers
Chapter: [0.034] Differential Equations
Concept: General and Particular Solutions of a Differential Equation

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

Appears in 3 question papers
Chapter: [0.034] Differential Equations
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.

Appears in 3 question papers
Chapter: [0.04] Probability (Section A)
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

Appears in 3 question papers
Chapter: [0.05] Vectors (Section B)
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.

Appears in 3 question papers
Chapter: [0.05] Vectors (Section B)
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)`

Appears in 3 question papers
Chapter: [0.07] Application of Integrals (Section B)
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.

Appears in 3 question papers
Chapter: [0.1] Linear Programming (Section C)
Concept: Graphical Method of Solving Linear Programming Problems

If f : R → R, f(x) = x and g: R → R , g(x) =  2x+ 1, and R is the set of real numbers, then find fog(x) and gof (x)

Appears in 2 question papers
Chapter: [0.01] Relations and Functions (Section A)
Concept: Composition of Functions and Invertible Function
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