Solve your IGNOU Doubts

Question:

निम्नलिखित में से कौन-से कथन सत्य है? अपने उत्तरों की पुष्टि कीजिए। (इसका अर्थ है कि यदि आप सोचते हैं कि कोई कथन असत्य है, तो एक संक्षिप्त उपपत्ति या एक उदाहरण ऐसा दीजिए जो यह दर्शाए कि कथन असत्य है। यदि वह सत्य है, तो ऐसा कहने की एक संक्षिप्त उपपत्ति दीजिए। उदाहरणार्थ, यह दर्शाने के लिए कि {1, पदमा, नीला एक समुच्चय है' सत्य है, आपको यह कहने की आवश्यकता है कि यह सत्य है, क्योंकि यह तीन वस्तुओं का एक सुपरिभाषित संग्रह है।)

i) वेन आरेखों का संग्रह एक समुच्चय है।

ii) $(A\setminus B)\cup C=A\setminus(B\cap C)$ , किन्हीं तीन समुच्चयों A, B और C के लिए।

iii) यदि $z\in\mathbb{R}$ , तो | $|z|=z$ है।

iv) $x^m+a_1 x^{m-1}+\dots+a_{m-1}x+a_m=0,\quad a_i\in\mathbb{R}\quad\forall i=1,\dots,m,$ का $\mathbb{R}$ में केवल तभी मूल होता है जब m एक विषम संख्या हो।

v) x+3>1 के लिए x > 0 आवश्यक है।

vi) दो चरों में तीन रैखिक समीकरणें के किसी भी निकाय का कोई हल नहीं होता है।

vii) यदि किसी आव्यूह में n² प्रविष्टियाँ हैं, जहां n ∈ $\mathbb{N}$ , तो वह एक वर्ग आव्यूह है।

viii) n > 4 के लिए, प्रथम n प्राकृत संख्याओं का आम, n+1 से बड़ा होता है।

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Question:

Write an odd natural number as a sum of two integers m, and m₂, in a way that m₁m, is maximum.

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Question:

Let x ∈ R such that 0 < x ≤ x ≤...≤x, n ≥2, and

$\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n}=1.$ Then show that

$\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n}\ge(n-1)\left(\frac{1}{\sqrt{x_1}}+\cdots+\frac{1}{\sqrt{x_n}}\right).$

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Question:

Prove that $\frac{1}{2}(x+y+z)\le\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y},\quad\text{for}x,y,z>0.$

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Question:

Show that $1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}\ge\sqrt{2(n-1)},\text{for}n\in\mathbb{N},n>1.$

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Question:

Is Cramer's Rule applicable for solving the linear system below? If yes, apply it. Otherwise, alter the last equation in the system so that the solution can be obtained by applying the Rule.

x + y + z = π

– πχ + π + √2z = 0

π²x + π²y + 2z = 0.

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Question:

Give examples, with justification, of the following:

i) two non-zero, 3×3 matrices A and B , with A| = 0 |B| = 5/7 i;

ii) two non-singular 2× 2 matrices C and D , with |C| = √2| D |.

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Question:

Solve by substitution, the system you have obtained in 5(a).

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Question:

Apply the Gaussian elimination process to determine values of λ for which the following linear system is consistent:

x-3y+4=0, 3x - 2y = λ, y = 6-2x.

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Question:

The annual bonus given to the employees of a company is 10% of their taxable incomes, after the state and central taxes are deducted. The state tax is 15% of taxable income. The central tax is 15% of taxable income after deducting the state tax. Formulate this situation for determining the bonus, as a linear system.

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Question:

Solve x4-8x3+21x220x+5=0 given that the sum of two of its roots is 4.

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Question:

Find the roots of the equation 2x³-x2-22x-24=0, if two of them are in the ratio 3:4.

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Question:

Obtain the discriminant of the equation 2x³ - 23x²+82x-78=0. Hence provide the nature of its roots.

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Question:

Find the polynomial over R of least degree which has i-2 and √3+5i as its roots.

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Question:

Find the sum of the 5^th roots of unity.

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Question:

Apply De Moivre's theorem to write (√5+i) in the form a + ib, with a, b ∈ R.

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Question:

Obtain the polar and exponential representations of z₁, z₂ and z₁z₂, where z₁ = 1/2 - i and z₂ = 3+i

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Question:

Express z = 2/-4-i in standard (algebraic) form. Further, give an Argand diagram in which z, Z and -z are plotted.

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Question:

Find x and y, given that (5x + y, 3x - y) = (2, 2x)∈QxQ.

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Question:

Draw a Venn diagram of sets A, B and C where A⊆B, A∩C, BOC≠ φ.B, ∩ C = φ. What is the universal set you have chosen? Justify your choice of sets in the diagram.

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