Learn 80 questions from K-TET III - September 2025 - Part III (Mathematics)

K-TET III - September 2025 - Part III (Mathematics) Questions

surface area of a cube= 6a² =  216 sq centimetres
6a²= 216
a² = 216/6 = 36
a = √36 = 6
volume = a³ = 6³ = 216 cu. centimetres

3a=24
a=24/3=8
area= √3/4 a² = √3/4 x 8 x 8 = 16√3

<h3 style="color: rgb(255,255,255);">Polynomial Factorization Techniques</h3>
To find the factors of a cubic polynomial like $P(x) = x^3+2x^2-5x-6$, several methods can be employed, which are commonly tested in competitive exams.

<h4 style="color: rgb(255,255,255);">1. Rational Root Theorem</h4>
<ul style="color: rgb(255,255,255);">
<li>The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root, $rac{p}{q}$, must have 'p' as a factor of the constant term and 'q' as a factor of the leading coefficient.</li>
<li>In $x^3+2x^2-5x-6$, the constant term is -6 and the leading coefficient is 1.</li>
<li>Factors of -6 are: $\pm 1, \pm 2, \pm 3, \pm 6$.</li>
<li>Factors of 1 are: $\pm 1$.</li>
<li>Possible rational roots are: $\pm 1, \pm 2, \pm 3, \pm 6$.</li>
</ul>

<h4 style="color: rgb(255,255,255);">2. Testing Potential Roots</h4>
<ul style="color: rgb(255,255,255);">
<li>We substitute these possible roots into the polynomial to find which ones make $P(x) = 0$.</li>
<li>Test $x = -1$: $P(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0$. Thus, $(x+1)$ is a factor.</li>
<li>Test $x = 2$: $P(2) = (2)^3 + 2(2)^2 - 5(2) - 6 = 8 + 8 - 10 - 6 = 0$. Thus, $(x-2)$ is a factor.</li>
<li>Test $x = -3$: $P(-3) = (-3)^3 + 2(-3)^2 - 5(-3) - 6 = -27 + 18 + 15 - 6 = 0$. Thus, $(x+3)$ is a factor.</li>
</ul>

<h4 style="color: rgb(255,255,255);">3. Polynomial Division or Synthetic Division</h4>
<ul style="color: rgb(255,255,255);">
<li>Once a root (say $x = -1$) is found, we know $(x+1)$ is a factor. We can then divide the original polynomial by $(x+1)$ to obtain a quadratic polynomial.</li>
<li>Using synthetic division with root -1:</li>
<pre> -1 | 1 2 -5 -6
 | -1 -1 6
 -----------------
 1 1 -6 0</pre>
<li>The resulting quadratic is $x^2 + x - 6$.</li>
<li>Factor the quadratic: $x^2 + x - 6 = (x+3)(x-2)$.</li>
<li>Combining the factors, we get $(x+1)(x+3)(x-2)$.</li>
</ul>

<h4 style="color: rgb(255,255,255);">4. Alternative Factoring (Grouping)</h4>
<ul style="color: rgb(255,255,255);">
<li>Sometimes, direct grouping can be applied if the terms align properly, though this is less common for general cubic polynomials.</li>
</ul>

<h4 style="color: rgb(255,255,255);">Key Concepts for Exams</h4>
<ul style="color: rgb(255,255,255);">
<li>Understanding the relationship between roots and factors of a polynomial.</li>
<li>Proficiency in applying the Rational Root Theorem.</li>
<li>Skill in performing polynomial division or synthetic division accurately.</li>
<li>Ability to factor quadratic expressions.</li>
</ul>

1/2 + 1/4 + 1/8 + 1/16

= (8 + 4 + 2 + 1)/16

= 15/16
.
= 0.9375

Factors of 12 = 1, 2, 3, 4, 6, 12
sum = 1+2+3+4+6+12

Interior angle of a regular pentagon = 108

Interior angle of a regular hexagon = 120

Angle of the square = 90

The sum of the angles at a point where three or more lines intersect is always 360°

so,

∠BAC = 360 - [120 + 108 + 90]

= 360 - 318

= 42

In a parallelogram, opposite angles are equal, and the sum of all interior angles is 360°. Given that the sum of two opposite angles is 110°, we can find the sum of the other two angles.

Solving a large number of mathematical problems

participating in the construction of geometrical charts

To help the learners to understand their progress in learning

To compare the learner with other children and advise them study well

Mathematisation of child's thought process

Make a table or a list to organize the information

False, as there is no scope of errors in mathematics

True, as they give ideas about how children construct mathematics concepts

True, identify the weaks students in the class

Helping gifted students in completing project work

To providing opportunities for guessing and doing mental mathematics

To find suitable situation for introducing certain topic

To understand the nature and growth of mathematics

To deveop motivation in children for learning mathematics

Completing the topics within a period of time

A broad planning of instruction for the year

encourage children to search for knowledge

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