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IGNOU MTE 1 SOLVED ASSIGNMENT HINDI

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IGNOU MTE 1 SOLVED ASSIGNMENT HINDI

~~Rs. 80~~
Rs. 41

Last Date of Submission of IGNOU MTE-01 (BSC) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MTE 1 SOLVED ASSIGNMENT HINDI
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 1
Subject Name	Calculus
Year	2026
Session
Language	English Medium
Assignment Code	MTE-01/Assignmentt-1//2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 01 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-01 (BSC) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 80~~
Rs. 41

Questions Included in this Help Book

Ques 1.

. निम्नलिखित कथनों में से कौन-से कथन सत्य और कौन-से असत्य हैं? अपने उत्तर के पक्ष में एक संक्षिप्त उपपत्ति या प्रति-उदाहरण दीजिए।
a) समुच्चय $S = \{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ एक अपरिमित समुच्चय है।
b) अधिकतम पूर्णांक फलन, $\mathbf{R}$ पर सतत् होता है।
c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .
d) प्रत्येक समाकलनीय फलन एकदिष्ट होता है।
e) $a \oplus b = \sqrt{a + b}$ परिमेय संख्याओं के समुच्चय $\mathbf{Q}$ , पर एक द्विआधारी संक्रिया है।

Ques 2.

2. a) $f(x) = \sqrt{\frac{2 - x}{x^2 + 1}}$ द्वारा परिभाषित फलन f का प्रांत ज्ञात कीजिए।

Ques 3.

वास्तविक संख्याओं का समुच्चय $\mathbf{R}$ और उस पर सामान्य जोड़ (+) तथा सामान्य गुणनफल $(\cdot)$ दिए गये हैं। $(), \mathbf{R}$ पर निम्नलिखित से परिभाषित है :
$a b = \frac{a + b}{2}, \forall a, b \in \mathbf{R}.$
क्या $(), \mathbf{R}$ सहयोगी है? क्या $(\cdot), \mathbf{R}$ में () पर वितरित है? जाँच कीजिए।

Ques 4.

यदि $|z - 1 + 2i| = 4$ है, तो दर्शाइए कि बिन्दु z + i एक वृत्त निरूपित करता है। इस वृत्त को खींचिए।

Ques 5.

$\frac{x - 1}{x^3 - x^2 - 2x}$ को आंशिक भिन्नों के योग में व्यक्त कीजिए।

Ques 6.

a) $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , जहाँ a > 0, b > 0 हैं, का न्यूनतम मान ज्ञात कीजिए।

Ques 7.

$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} \, dx$ का मान ज्ञात कीजिए।

Ques 8.

दो समुच्चयों S और T के लिए दर्शाइए कि:

$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Ques 9.

$\mathbf{R}$ पर $f(x) = x^3 - x^2 - 8x + 12$ और
और $g(x) = \begin{cases} \frac{f(x)}{x + 3}, & \text{जब } x \neq -3 \\ \alpha, & \text{जब } x = -3 \end{cases}$

द्वारा परिभाषित दो फलन f और g लीजिए।

Ques 10.

$\alpha$ का वह मान ज्ञात कीजिए जिसके लिए $g, x = -3$ पर सतत् है।

Ques 11.

$f(x) = 0$ के सभी मूल ज्ञात कीजिए।

Ques 12.

वक्र $y^2(4 - x) = x^3$ और इसकी y-अक्ष के समांतर अनंतस्पर्शी के बीच का क्षेत्रफल ज्ञात कीजिए

Ques 13.

यदि एक आय फलन $\frac{dR}{dx} = 15 + 2x - x^2$ द्वारा दिया गया है, जहाँ x निवेश है, तो अधिकतम आय ज्ञात कीजिए। यदि प्रारम्भिक आय 0 है, तो आय फलन R भी ज्ञात कीजिए।

Ques 14.

वक्र $y^2(x + 1) = x^2(3 - x)$ का आरेखण कीजिए और ऐसा करने के लिए प्रयोग किए गये गुणधर्म भी लिखिए।

Ques 15.

चक्रज $x = \alpha(\theta - \sin \theta), y = \alpha(1 - \cos \theta)$ की लम्बाई ज्ञात कीजिए और दर्शाइए कि रेखा $\theta = \frac{2\pi}{3}$ इसे 1 : 3 के अनुपात में विभक्त करती है।

Ques 16.

वह प्रतिबंध ज्ञात कीजिए कि वक्र $ax^2 + by^2 = 1$ और $a'x^2 + b'y^2 = 1$ एक-दूसरे को लम्बवत् प्रतिच्छेद करते हैं।

Ques 17.

) यदि $y = e^{m \sin^{-1} x}$ है, तो दर्शाइए कि $(1 - x^2)y_2 - xy_1 - m^2y = 0$ है। इस प्रकार लाइब्निट्ज के सूत्र का प्रयोग करके (1 - x²)y_n+2 - (2n + 1)xy_n+1 का मान निकालिए।

Ques 18.

$f(x) = \begin{cases} 2x & , x > 5 \\ x + 5 & , 1 \leq x \leq 5 \\ |x| & , x < 1 \end{cases}$ द्वारा परिभाषित $\mathbf{R}$ फलन $f : \mathbf{R} \to \mathbf{R}$ के जिस भी सबसे बड़े समुच्चय पर सतत् है वह निकालिए।

Ques 19.

समीकरण $x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$ हल कीजिए, जिसके सभी मूल G.P. में हैं।

Ques 20.

) $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$ ज्ञात कीजिए।
10. a) यदि $I_{m,n} = \int x^3 (\log x)^n \, dx$ है, तो दर्शाइए कि :

Ques 21.

) यदि $I_{m,n} = \int x^3 (\log x)^n \, dx$ है, तो दर्शाइए कि :

$(m + 1) I_{m,n} = x^{m+1}(\log x)^n - n I_{m,n-1}$ .
है। इस प्रकार $\int x^4 (\log x)^3 \, dx$ ज्ञात कीजिए।

Ques 22.

$f(x) = 2x^2 - 7x - 10$ द्वारा परिभाषित फलन f के लिए अंतराल [2, 5] पर लैग्रांज माध्यमान प्रमेय सत्यापित कीजिए।

Ques 23.

Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.
a) The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.
b) The greatest interger function is continuous on $\mathbf{R}$ .
c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .
d) Every integrable function is monotonic.
e) $a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

Ques 24.

Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$

Ques 25.

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is () associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

Ques 26.

Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$ .

Ques 27.

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is () associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

Ques 28.

If $|z - 1 + 2i| = 4$ , show that the point z + i describes a circle. Also draw this circle.
b) Express $\frac{x-1}{x^3 - x^2 - 2x}$ as a sum of partial fractions.

Ques 29.

Find the least value of $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , where a > 0, b > 0.
b) Evaluate:

$$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} dx$

Ques 30.

For any two sets S and T, show that:

$$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Depict this situation in the Venn diagram.

Ques 31.

Let f and g be two functions defined on R by:

$f(x) = x^3 - x^2 - 8x + 12$

and $g(x) = \begin{cases} \frac{f(x)}{x+3}, & \text{when } x \neq -3 \\ \alpha, & \text{when } x = -3 \end{cases}$
i) Find the value of $\alpha$ for which f is continuous at $x = -3$ .

ii) Find all the roots of $f(x) = 0$ .

Ques 32.

) Find the area between the curve $y^2(4 - x) = x^3$ and its asymptote parallel to y-axis.

Ques 33.

If the revenue function is given by $\frac{dR}{dx} = 15 + 2x - x^2$ , x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.
b) Trace the curve $y^2(x + 1) = x^2(3 - x)$ , clearly stating all the properties used for tracing it.

Ques 34.

Find the length of the cycloid $x = \alpha(\theta - \sin \theta), y = \alpha(1 - \cos \theta)$ and show that the line $\theta = \frac{2\pi}{3}$ divides it in the ratio 1 : 3.
b) Find the condition for the curves, $ax^2 + by^2 = 1$ and $a'x^2 + b'y^2 = 1$ intersecting orthogonally.

Ques 35.

If $y = e^{m \sin^{-1} x}$ , then show that $(1 - x^2)y_2 - xy_1 - m^2y = 0$ . Hence using Leibnitz's formula, find the value of (1 - x²)y_n+2 - (2n + 1)xy_n+1.
b) Find the largest subset of R on which the function $f : \mathbf{R} \to \mathbf{R}$ defined as:
$$f(x) = \begin{cases} 2x, & x > 5 \\ x + 5, & 1 \le x \le 5 \\ |x|, & x < 1 \end{cases}$
is continuous.

Ques 36.

Solve the equation:
$$x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$
given that its roots are in G.P.
b) Evaluate:
$$\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$

Ques 37.

If $I_{m,n} = \int x^m (\log x)^n dx$ , show that:

$$(m+1)I_{m,n} = x^{m+1} (\log x)^n - n I_{m, n-1}.$
Hence find the value of $\int x^4 (\log x)^3 dx$ .

Ques 38.

Verify Lagrange's mean value theorem for the function f defined by $f(x) = 2x^2 - 7x - 10$ over

Ques 39.

Ques 40.

2. a) $f(x) = \sqrt{\frac{2 - x}{x^2 + 1}}$ द्वारा परिभाषित फलन f का प्रांत ज्ञात कीजिए।

Ques 41.

Ques 42.

Ques 43.

$\frac{x - 1}{x^3 - x^2 - 2x}$ को आंशिक भिन्नों के योग में व्यक्त कीजिए।

Ques 44.

a) $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , जहाँ a > 0, b > 0 हैं, का न्यूनतम मान ज्ञात कीजिए।

Ques 45.

$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} \, dx$ का मान ज्ञात कीजिए।

Ques 46.

दो समुच्चयों S और T के लिए दर्शाइए कि:

$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Ques 47.

$\mathbf{R}$ पर $f(x) = x^3 - x^2 - 8x + 12$ और
और $g(x) = \begin{cases} \frac{f(x)}{x + 3}, & \text{जब } x \neq -3 \\ \alpha, & \text{जब } x = -3 \end{cases}$

द्वारा परिभाषित दो फलन f और g लीजिए।

Ques 48.

$\alpha$ का वह मान ज्ञात कीजिए जिसके लिए $g, x = -3$ पर सतत् है।

Ques 49.

$f(x) = 0$ के सभी मूल ज्ञात कीजिए।

Ques 50.

Ques 51.

Ques 52.

Ques 53.

Ques 54.

Ques 55.

Ques 56.

Ques 57.

समीकरण $x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$ हल कीजिए, जिसके सभी मूल G.P. में हैं।

Ques 58.

) $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$ ज्ञात कीजिए।
10. a) यदि $I_{m,n} = \int x^3 (\log x)^n \, dx$ है, तो दर्शाइए कि :

Ques 59.

) यदि $I_{m,n} = \int x^3 (\log x)^n \, dx$ है, तो दर्शाइए कि :

$(m + 1) I_{m,n} = x^{m+1}(\log x)^n - n I_{m,n-1}$ .
है। इस प्रकार $\int x^4 (\log x)^3 \, dx$ ज्ञात कीजिए।

Ques 60.

Ques 61.

Ques 62.

Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$

Ques 63.

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is () associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

Ques 64.

Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$ .

Ques 65.

The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is () associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

Ques 66.

If $|z - 1 + 2i| = 4$ , show that the point z + i describes a circle. Also draw this circle.
b) Express $\frac{x-1}{x^3 - x^2 - 2x}$ as a sum of partial fractions.

Ques 67.

Find the least value of $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , where a > 0, b > 0.
b) Evaluate:

$$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} dx$

Ques 68.

For any two sets S and T, show that:

$$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Depict this situation in the Venn diagram.

Ques 69.

Let f and g be two functions defined on R by:

$f(x) = x^3 - x^2 - 8x + 12$

and $g(x) = \begin{cases} \frac{f(x)}{x+3}, & \text{when } x \neq -3 \\ \alpha, & \text{when } x = -3 \end{cases}$
i) Find the value of $\alpha$ for which f is continuous at $x = -3$ .

ii) Find all the roots of $f(x) = 0$ .

Ques 70.

) Find the area between the curve $y^2(4 - x) = x^3$ and its asymptote parallel to y-axis.

Ques 71.

Ques 72.

Ques 73.

Ques 74.

Solve the equation:
$$x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$
given that its roots are in G.P.
b) Evaluate:
$$\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$

Ques 75.

If $I_{m,n} = \int x^m (\log x)^n dx$ , show that:

$$(m+1)I_{m,n} = x^{m+1} (\log x)^n - n I_{m, n-1}.$
Hence find the value of $\int x^4 (\log x)^3 dx$ .

Ques 76.

Verify Lagrange's mean value theorem for the function f defined by $f(x) = 2x^2 - 7x - 10$ over

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IGNOU MTE 1 SOLVED ASSIGNMENT HINDI

IGNOU MTE 1 SOLVED ASSIGNMENT HINDI

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