Text Explanation&nbsp;:∵ 3 is a root of f(x) = 0,∴ 9a + 3b + c = 0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;…(i)Also,f(5) = −3f(2)∴ 25a + 5b + c = −3(4a + 2b + c)∴ 37a + 11b + 4c = 0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;…(ii)On solving equations (i) and (ii), we get,a – b = 0∴ a = bWe know that sum of the roots of a quadratic equation (ax2&nbsp;+ bx + c = 0) is –b/a∴ 3 + other root = −1∴ Other root = −4Hence, option (b).

Text Explanation&nbsp;:ax2&nbsp;+&nbsp;bx&nbsp;+ 1 = 0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;… (i)For equation (i) to have real roots,&nbsp;b2&nbsp;− 4a&nbsp;≥ 0i.e.&nbsp;a&nbsp;≤&nbsp;b2/4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;… (ii)For&nbsp;b&nbsp;= 4: to satisfy (ii),&nbsp;a&nbsp;= 1, 2, 3, 4∴ 4 equations are possible.For&nbsp;b&nbsp;= 3: to satisfy (ii),&nbsp;a&nbsp;= 1, 2∴ 2 equations are possible.For&nbsp;b&nbsp;= 2: to satisfy (ii),&nbsp;a&nbsp;= 1∴ 1 equation is possible.Thus, total number of possible equations = 7Hence, option (b).

Text Explanation&nbsp;:The roots at f(x) = 0 are 3 and −4∴ The equation can be written as (x – 3)(x + 4) = 0Or, x2&nbsp;− x + 12 = 0The co-efficient of x2&nbsp;is 1 here, but all equations which are multiple of this equation will also have same roots.For example,&nbsp;10(x2&nbsp;− x + 12) = 0 will also have same roots∴ (a + b + c) cannot be determined uniquely.Hence, option (e).

Test Name	No. of Questions	Marks (of each)	Time	Take Test
Test-1	1	3	40 Minutes
Test-2	2	3	40 Minutes

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Quadrilaterals And Polygons

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<div><p>Let f(x) = ax<sup>2</sup> + bx + c, where, a, b and c are certain constants and a &ne; 0. It is known that f(5) = &minus;3f(2) and that 3 is a root of f(x) = 0.</p></div>

Question 1.

<div>What is the other root of f(x) = 0?</div>

Question 2.

<div>If both \( a \) and \( b \) belong to the set \(\{1, 2, 3, 4\}\), then the number of equations of the form $$ ax^2 + bx + 1 = 0 $$</div>

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<div>What is the value of a + b + c?</div>

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