**ARNIKO AWASIYA HIGHER SECONDARY SCHOOL**

**Class: XI Mathematics [116] F.M.:50**

**Time: 1:30 Hour P.M.:20**

*Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. *

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*Group “A”*

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*a) If A = { l, o, v, e } and B = {r, o, s , e}, find the symmetric difference of set A and set B. [2]*

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*b) A, B and C are the subsets of a universal set U. Prove that . [2]*

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*a) If the equation (1 + m*^{2}) x^{2}+ 2mcx + (c^{2}– a^{2}) = 0 has equal roots, show that c^{2}= a^{2}(1+m^{2}). [2]

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*b) If and be in H.P., show that x, y, z are in G.P. [2] *

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*c) a) By mathematical induction, prove that *

* . [2]*

*a) What are the standard forms of equation of a straight line? Find the slope of the line . [2]*

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*b) What do you mean by the left hand limit and right hand limit of a function? What is the condition for the limit of a function to exist at a point? [2]*

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*Group “B”*

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*a) In a group of students, 30 read mathematics, 24 read Economics, 22 read statistics, 14 read mathematics only, 8 read Economics only, 6 read mathematics and statistics only, 2 read mathematics and Economics only and 8 read none of these subjects.*

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*i) **How many students are there in the group? *

*ii) **How many read all the three subjects? [4]*

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*b) Prove that a quadratic equation ax^{2} + bx + c = 0 can not have more than two roots. *

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*a) If a, b, c are rational and***a + b + c = 0**, show that the roots of**(b + c – a) x**are rational.^{2 }+ (c + a – b) x + (a + b – c) = 0

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*b) Evaluate: *

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*a) A function f*(x)*is defined as*

*f(x) = *

*Find the value k so that f(x) is continuous at x=2*

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*b) From first principle, find the derivative of y =3x ^{2} – 2x *

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*Group “C”*

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*The three sides of a triangle have the equations***7x – y + 11 = 0,**

*x + y – 15 = 0 and 7x + 17y + 65 = 0. **Find the equation of the straight line through a vertex and parallel to the opposite side with the equation x + y – 15 = 0. [6]*

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*Find the n*^{th}term and the sum of the first n terms of the series.

* [6]*

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*“HAPPY BIJAYA DASHAMI 2068”*

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