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Question:

Check whether the set of integers is countable or not.

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Question:

Show that the equation $x^3 + x^2 - 2x - 2 = 0$ has a real root other than $x = -1$

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Question:

Using the principle of mathematical induction, show that
$$ 1^2 + 3^2 + 5x + ... + (2n - 1)^2 = \frac{1}{3} n(4n^2 - 1) \forall n \in \mathbf{N}.$

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Question:

Show that $5 + \sqrt{2}$ is an algebraic number.

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Question:

Find a and b such that $\lim_{x \to 0} \frac{a \tan x + bx}{x^3}$ exists.

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Question:

Use Cauchy's mean value theorem to prove that:
$$ \frac{\cos \alpha - \cos \beta}{\sin \alpha - \sin \beta} = \tan \theta, 0 < \alpha < \theta < \beta < \frac{\pi}{2}$

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Question:

a) If the partition P₂ is a refinement of the partition P₁ of [a,b], then $L(P_1, f) \le L(P_2, f)$ and $U(P_2, f) \le U(P_1, f)$ . Verify this result for the function $f(x) = 2\cos x$ defined over the interval $\left[0, \frac{\pi}{2}\right]$ and the partitions $P_1 = \left\{0, \frac{\pi}{3}, \frac{\pi}{2}\right\}$ and $P_2 = \left\{0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}\right\}$ .
b) Evaluate: $\displaystyle \lim_{n \to \infty} \sum_{r=1}^{2n} \frac{n^2}{(2n + r)^3}$ .

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Question:

Show that $\mathbf{R_n}(x)$ , the lagrange's form of remainder in the Maclaurin series expansion of e^4x, tends to zero as $n \to \infty$ . Hence obtain the Maclaurin's infinite expansion for e^4x.

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Question:

Determine the local minimum and local maximum values of the function f defined by $f(x) = 3 - 5x^3 + 5x^4 - x^5$ .

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Question:

a) Test the following series for convergence,

  (i)     $\displaystyle \sum_{n=1}^{\infty} n x^{n-1}, x > 0.$
                (ii)    $\displaystyle \sum_{n=1}^{\infty} \left[ \sqrt{n^4 + 9} - \sqrt{n^4 - 9} \right]$
        b)      Show that $\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1} \frac{5}{7n+2}$ is conditionally convergent.

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Question:

Verify Bozano-Weierstrass Theorem for the following sets:
i)      Set of non-negative integers.
ii)     Interval $[-1, \infty]$
d)      Check whether the limit $\lim_{x \to 0} (x \operatorname{cosec} x)^x$ exists or not?

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Question:

Write the inequality $4 \le 2x + 3 \le 6$ in the modulus form.

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Question:

Prove that a strictly decreasing function is always one-one.

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Question:

Check whether the intervals ]5,9] and [6,12[ are equivalent or not.

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Question:

Find the following limit

$$ \lim_{x \to 0} \frac{1 - \cos x^2}{x^2 \sin x^2}$

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Question:

a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

$<div style=$ $f(x)=\begin{cases}-x^2,&\text{when}x\le&space;0\\[4pt]4-5x,&\text{when}0%3Cx\le&space;1\\[4pt]3x-4x^2,&\text{when}1%3Cx\le&space;2\\[4pt]-12x+2x,&\text{when}x%3E2\end{cases}$ ">

Also check whether the function f is derivable at x = 1.

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Question:

Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function $f : [-2, 2] \to \mathbf{R}$ defined by $f(x) = \frac{4x + 3}{x^2 + 1}$ is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function $f : [0, 2] \to \mathbf{R}$ defined by $f(x) = x + [x]$ is not integrable.

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Question:

शिक्षकों के व्यावसायिक विकास के लिए आई.सी.टी. का प्रयोग किस प्रकार किया जासकता है? शिक्षकों के व्यावसायिक विकास के लिए प्रयोग की जा सकने वाले आई.सी.टी. के कुछ उपकरणों और प्लेटफार्मों की परिचर्चा कीजिए।

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Question:

प्रौद्योगिकी मध्यस्थ अधिगम (टेक्नोलॉजी मीडिएटेड लर्निंग) के लिए संज्ञानवाद के निहितार्थों की व्याख्या कीजिए।

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Question:

शिक्षा में प्रौ‌द्योगिकी प्रयोग के नैतिक एवं स्वास्थ्यपरक परिप्रेक्ष्यों की चर्चा कीजिए।

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