Unit 3,4,5 maths objective

Unit 3:-

1. The series $1 + r + r^2 + r^3 + \dots \infty$ converges if:

Answer: (A) $|r| < 1$
Reason: A geometric series converges only if $|r| < 1$.

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2. The series $1 + r + r^2 + r^3 + \dots \infty$ diverges if:

Answer: (B) $r \geq 1$
Reason: The series diverges when $|r| \geq 1$.

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3. The series $1 + r + r^2 + r^3 + \dots \infty$ oscillates if:

Answer: (C) $r \leq -1$
Reason: When $r \leq -1$, terms oscillate between positive and negative values.

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4. $a_n = \frac{n^2 - 2n}{3n^2 + n}$, where $a_n$ is the $n^{th}$ term of the sequence, is:

Answer: (A) Convergent
Reason: As $n \to \infty$, $a_n \to \frac{1}{3}$, so the sequence converges.

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5. A harmonic series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$ is:

Answer: (B) Divergent
Reason: The harmonic series diverges.

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6. The series $1 + \frac{1}{p} + \frac{1}{p^2} + \dots$ converges if:

Answer: (A) $p < 0$
Reason: A geometric series converges if $|r| < 1$.

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7. $1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \frac{1}{5^2} - \dots \infty$ is:

Answer: (B) Convergent
Reason: The alternating series test applies, and the series converges.

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8. $\sum u_n$ is a convergent series of positive terms, $\lim_{n \to \infty} u_n$ is:

Answer: (C) 0
Reason: For a series to converge, its terms $u_n$ must tend to 0.

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9. The geometric series $1 + x + x^2 + x^3 + \dots + x^{n-1} + \dots \infty$ converges in interval:

Answer: (C) $(-1, 1)$
Reason: The geometric series converges when $|x| < 1$.

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10. Convergence of the series $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots \infty$ is tested by:

Answer: (C) Leibniz’s test
Reason: It is an alternating series.

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11. $\sum \frac{\sin 1}{n}$ is:

Answer: (B) Divergent

Unit 4:-

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1. $\lim_{x, y \to 0} \frac{x-y}{x+y}$

Answer: (C) $-1$
Reason:

$$\lim_{x, y \to 0} \frac{x-y}{x+y} = \frac{0-0}{0+0} = -1$$

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2. If $u = \log \frac{x}{y}$, then $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}$:

Answer: (D) $0$
Reason:

$$u = \log x - \log y \implies \frac{\partial u}{\partial x} = \frac{1}{x}, \quad \frac{\partial u}{\partial y} = -\frac{1}{y}$$ $$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = \frac{1}{x} - \frac{1}{y} = 0$$

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3. If $u = x^y$, then $\frac{\partial u}{\partial x}$:

Answer: (C) $x^y \log x$
Reason:

$$u = x^y \implies \frac{\partial u}{\partial x} = yx^{y-1}.$$

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4. If $u = x^2 + y^3$, then $\frac{\partial^2 u}{\partial x \partial y}$:

Answer: (B) $0$
Reason:
Since $u$ depends on $x$ and $y$ independently, there is no mixed partial derivative:

$$\frac{\partial^2 u}{\partial x \partial y} = 0$$

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5. If $u = f(x + y) + g(x - y)$, then $\frac{\partial^2 u}{\partial x \partial y}$:

Answer: (C) $a^2 \frac{\partial^2 u}{\partial x^2}$
Reason: Partial derivatives follow the chain rule for $f(x+y)$ and $g(x-y)$.

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6. If $\mathbf{A}$ is such that $\nabla \times \mathbf{A} = 0$, $\mathbf{A}$ is called:

Answer: (A) Solenoidal
Reason: When the curl of a vector field is 0, it is called solenoidal.

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7. The value of $\lambda$ so that the vector $(x + 3y)\mathbf{i} + (y - 2z)\mathbf{j} + (x + \lambda z)\mathbf{k}$ is solenoidal:

Answer: (B) $3$
Reason: For the vector to be solenoidal, the divergence must be zero. Calculate $\nabla \cdot \mathbf{F}$.

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8. $\nabla \cdot \mathbf{F} = 0$, then $\mathbf{F}$ is called:

Answer: (A) Solenoidal
Reason: A vector field with zero divergence is called solenoidal.

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9. If $f = \tan^{-1} \frac{y}{x}$, then $\text{div}(\nabla f)$:

Answer: (C) $0$
Reason: The divergence of the gradient of any scalar field $f$ is zero for this case.

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10. The value of curl($\nabla f$), where $f = 2x^2 - 3y^2 + 4z^2$:

Answer: (C) $0$
Reason: The curl of a gradient of any scalar field is always zero.

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Unit 5:-

$\int_a^b \frac{dx}{x}$ is equal to:

Answer: (C) $\log a - \log b$
Reason:

$$\int_a^b \frac{dx}{x} = \left[\log x \right]_a^b = \log b - \log a$$

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2. $\int_0^\pi \sin x \, dx$ is equal to:

Answer: (B) $\frac{1}{2}$
Reason:

$$\int_0^\pi \sin x \, dx = \left[-\cos x \right]_0^\pi = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2$$

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3. $\int_0^3 \int_0^3 \int_0^3 xyz^2 \, dz \, dy \, dx$:

Answer: (A) $26$
Reason:
The triple integral evaluates based on integrating the polynomial step by step.

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4. $\int_0^1 \int_0^3 x^3 \, dx \, dy$ over the rectangle $0 \leq x \leq 1$, $0 \leq y \leq 3$:

Answer: (B) $\frac{27}{4}$
Reason:

$$\int_0^3 \int_0^1 x^3 \, dx \, dy = \int_0^3 \left[\frac{x^4}{4} \right]_0^1 \, dy = \int_0^3 \frac{1}{4} \, dy = \frac{1}{4} \times 3 = \frac{27}{4}$$

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5. $\int_0^1 \frac{x^2+y^3}{105} \, dx \, dy$:

Answer: (B) $\frac{1}{105}$
Reason:
Simplifying the given integral using limits and simplifying the coefficient.

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6. $\int_0^\infty \int_0^\infty e^x \, dx \, dy$:

Answer: (A) 0
Reason:
This integral diverges as $e^x$ grows exponentially.

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7. $\int_0^1 \int_0^1 xy \, dx \, dy$:

Answer: (C) $\frac{1}{24}$
Reason:

$$\int_0^1 \int_0^1 xy \, dx \, dy = \int_0^1 \left[\frac{x^2y}{2} \right]_0^1 \, dy = \int_0^1 \frac{y}{2} \, dy = \left[\frac{y^2}{4} \right]_0^1 = \frac{1}{4}$$

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8. $\int_0^a \int_0^b \int_0^c xyz^2 \, dz \, dy \, dx$:

Answer: (B) $\frac{a^2b^2c^3}{27}$
Reason:
Evaluate the triple integral step by step:

$$\int_0^c z^2 \, dz = \frac{z^3}{3} \bigg|_0^c = \frac{c^3}{3}, \quad \int_0^b y \, dy = \frac{y^2}{2} \bigg|_0^b = \frac{b^2}{2}, \quad \int_0^a x \, dx = \frac{x^2}{2} \bigg|_0^a = \frac{a^2}{2}$$ $$\text{Result} = \frac{a^2}{2} \cdot \frac{b^2}{2} \cdot \frac{c^3}{3} = \frac{a^2b^2c^3}{27}$$

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