Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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Important Questions for PUC Karnataka Science Class 12 - Department of Pre-University Education, Karnataka - Mathematics

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Mathematics
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Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`

Appears in 7 question papers
Chapter: [0.11] Three - Dimensional Geometry
Concept: Three - Dimensional Geometry Examples and Solutions

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.

Appears in 5 question papers
Chapter: [0.08] Applications of the Integrals
Concept: Area Under Simple Curves
 

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

 
Appears in 4 question papers
Chapter: [0.06] 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: [0.06] Applications of Derivatives
Concept: Maxima and Minima

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Appears in 4 question papers
Chapter: [0.07] Integrals
Concept: Methods of Integration: Integration by Substitution

Find: `I=intdx/(sinx+sin2x)`

Appears in 4 question papers
Chapter: [0.07] Integrals
Concept: Methods of Integration: Integration Using Partial Fractions

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: [0.11] Three - Dimensional Geometry
Concept: Coplanarity of Two Lines

Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].

Appears in 3 question papers
Chapter: [0.01] Relations and Functions
Concept: Types of Relations

Let f : N→N be a function defined as f(x)=`9x^2`+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find `f^-1`(43) and` f^−1`(163).

Appears in 3 question papers
Chapter: [0.01] Relations and Functions
Concept: Inverse of a Function

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

Appears in 3 question papers
Chapter: [0.02] Inverse Trigonometric Functions
Concept: Properties of Inverse Trigonometric Functions

if `2[[3,4],[5,x]]+[[1,y],[0,1]]=[[7,0],[10,5]]` , find (xy).

Appears in 3 question papers
Chapter: [0.03] Matrices
Concept: Equality of Matrices

Solve the following matrix equation for x: `[x 1] [[1,0],[−2,0]]=0`

Appears in 3 question papers
Chapter: [0.03] Matrices
Concept: Operations on Matrices > Addition of Matrices

Prove that  `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.

Appears in 3 question papers
Chapter: [0.03] Matrices
Concept: Elementary Transformations

Write the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by `a_(ij)=∣(i−j)/2∣`

Appears in 3 question papers
Chapter: [0.03] Matrices
Concept: Introduction of Operations on Matrices

If `A=([2,0,1],[2,1,3],[1,-1,0])` find A2 - 5A + 4I and hence find a matrix X such that  A2 - 5A + 4I + X = 0

Appears in 3 question papers
Chapter: [0.03] Matrices
Concept: Operations on Matrices > Addition of Matrices

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: [0.03] Matrices
Concept: Multiplication of Two Matrices

If `[[3x,7],[-2,4]]=[[8,7],[6,4]]`, find the value of x

Appears in 3 question papers
Chapter: [0.03] Matrices
Concept: Introduction of Operations on Matrices

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: [0.04] Determinants
Concept: Inverse of Matrix > Inverse of a Square Matrix by the Adjoint Method

Prove that  `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.

Appears in 3 question papers
Chapter: [0.04] Determinants
Concept: Elementary Transformations
 

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: [0.04] Determinants
Concept: Properties of Determinants
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