PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Mark the correct alternative in the following question:

The relation S defined on the set R of all real number by the rule aSb if a ≥ b is _______________ .

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The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .

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Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then, R is _____________ .

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Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m ∈ L. Then, R is ______________ .

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Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b for all a, b ∈ T. Then, R is ____________ .

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Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is _____________ .

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For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .

A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
 

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.

Determine the order and degree (if defined) of the following differential equation:-

\[\left( \frac{ds}{dt} \right)^4 + 3s\frac{d^2 s}{d t^2} = 0\]

Determine the order and degree (if defined) of the following differential equation:-

y"' + 2y" + y' = 0

Determine the order and degree (if defined) of the following differential equation:-

(y"')² + (y")³ + (y')⁴ + y⁵ = 0

Determine the order and degree (if defined) of the following differential equation:-

y"' + 2y" + y' = 0

Determine the order and degree (if defined) of the following differential equation:-

y" + (y')² + 2y = 0

Determine the order and degree (if defined) of the following differential equation:-

y" + 2y' + sin y = 0

Determine the order and degree (if defined) of the following differential equation:-

y"' + y² + e^y' = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = x² + 2x + C            y' − 2x − 2 = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = cos x + C            y' + sin x = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(1+x^2)`                     `y'=(xy)/(1+x^2)`

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = x sin x              `xy'=y+xsqrt(x^2-y^2)`

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.