The partition p' is a refinement of P if:

1. P' ⊃ P
2. P' = P
3. P' < P
4. P' ⊃ P

1. Sup L(p,f,α)_p∈[a,b]
2. inf U(p,f,α)_p∈[a,b]
3. ∑_n=1^∞MΔα
4. ∑_n=1^∞mΔα

Let f∈R(α) on [a,b], m ≤ f ≤ M, φ be continuous [m, M] and h(x) = φ(f(x)) on [a,b]. Then-

In rectifiable curve Λ_(p) is given by-

1. ∑_i=1ⁿ|Δx_i|
2. ∑_i=1ⁿ|Δx_i|-4(x_i-1)
3. ∑_i=1ⁿ|Δx_i|-4(x_i-1)|
4. ∑_i=1ⁿ|Δx_i|+4(x_i-1)|

If the set of numbers Λ_(p) is unbounded then 4 is called-

1. rectifiable
2. vector-valued function
3. non rectifiable
4. None of these

Let ∑_n=1^∞u_n be a series of real numbers then ∑_n=1^∞u_n is said to be conditional convergent if the series ∑_n=1^∞u_n converges, but :

1. ∑_n=1^∞u_n oscillates
2. ∑_n=1^∞u_n diverges
3. ∑_n=1^∞u_n converges and diverges both
4. None of these

If a sequence is uniformly convergent then it is necessarily -

1. divergent
2. oscillates
3. (a) & (b) both
4. pointwise convergent

If f_n(x) = n²x / (1+n²x²) defined in [0, 1] then by M_n test it is-

1. non-uniformly convergent
2. uniformly convergent
3. divergent
4. None of these

If {f_n} is a sequence of continuous functions on E and if f_n → f uniformly on E, then-

1. f is discontinuous on E
2. f is continuous on E
3. f is complete metric space on E
4. None of these

Every span is a-

1. vector space
2. dimension
3. (a) & (b) both
4. None of these

||A|| is given by :

1. inf_{x∈Rⁿ, |x|=1} |AX|
2. sup_{x∈Rⁿ, |x|=1} |AX|
3. inf_{x∈Rⁿ, |x|≠0} |X|
4. sup_{x∈Rⁿ, |x|≠0} |AX|

A contraction map is always:

1. discontinuous
2. continuous
3. reimann integrable
4. (a) & (b) both

Radius of convergence R of the power series ∑c_nzⁿ, where c_n = nⁿ is:

1. ⅓
2. 3

A mapping f of E into Rⁿ is said to be of class C^II if each component fⁱ is of:

1. Class C^II
2. Class C^I
3. Class f
4. None of these

If A_x ∈ L(Rⁿ), A_y ∈ L(Rⁿ) then A_(h,k) is given by:

1. A_x h + A_y k
2. A_x h - A_y k
3. -A_x h
4. -A_y k

If P* is a refinement of p, then prove that L (p, f, α) ≤ L (p*, f, α).

If f ∈ R(α) and g ∈ R(α) on [a,b], then prove that :

1. fg ∈ R(α)
2. |f| ∈ R(α)and ∫_a^b|f|dα ≤ |∫_a^bf dα|

Define Integration of vector-valued functions.

Explain Rectifiable Curves.

Explain the difference of pointwise convergence and uniform convergence.

Prove that the sequence of functions {f_n} defined on E, converges uniformly on E if and only if for every ε > 0 there exists an integer N such that m ≥ N, n ≥ N, x ∈ E implies |f_m(x) - f_n(x)| ≤ ε.

If A ∈ L(Rⁿ), Rⁿ then prove that ||A|| < ∞ and A is uniformly continuous mapping of Rⁿ into Rⁿ.

Suppose E be an open set in Rⁿ, f maps E into Rⁿ and x ∈ E and Lim_h→0 &frac{|f(x+h)-f(x)-Ah|}{|h|} = 0 holds with A=A₁ and with A=A_x. Then prove that A₁ = A_x.

Suppose f is defined in an open set E ¹ R² and D₁f and D₂f exist at every point of E. Suppose Q ¹ E is a closed rectangle with sides parallel to the coordinate axes, having (a, b) and (a + h, b + k) as opposite vertices (h ≠ 0, k ≠ 0). put Δ(f,Q) = f(a + h, b + k) - f(a + h, b) - f(a, b + k) + f(a, b) Then prove that there is a point (x, y) in the interior of Q such that Δ(f, Q) = hk, D₁D₂f(x, y).

If A ∈ L(Rⁿ, R^m) and if A₁ is invertible, then there corresponds to every k ∈ R^m a unique h ∈ Rⁿ such that A(h, k) = 0. Then prove that this h can be computed from k by the formula h = -(A₁)^-1 A₂k.

Let f be a bounded function and α be a monotonically increasing function on [a, b]. Then prove that f ∈ R (α) on [a, b] if and only if for every ε > 0 there exists a partition p such that U (p, f, α) - L (p, f, α) < ε.

Let f ∈ R on [a, b]. For a ≤ x ≤ b put F(x) = ∫_a^xf(t)dt. Then prove that F is continuous on [a, b]; furthermore if f is continuous at a point x₀ of [a, b], then F is differentiable at x₀ and F'(x₀) = f(x₀).

Let γ be a continuously differentiable curve on [a, b], then prove that γ is rectifiable and Λ_γ(a, b) = ∫_a^b|γ'(t)|dt.

Let ∑a_n be a given series of real number which converges but not absolutely. Suppose -∞ ≤ α ≤ β ≤ ∞ Then prove that there exists a rearrangement ∑a'_n with partial sums s'_n such that lim inf s'_n = α, lim sup s'_n = β.

Suppose f_n → f uniformly on a set E in a metric space. Let x be a Limit point of E, and Suppose that lim_t→x f_n(t) = A_n (n=1, 2, 3, ...) Then prove that {A_n} converges and lim_n→∞ A_n = lim_t→x f(t).

Let α be monotonically increasing on [a, b] suppose f_n ∈ R(α) on [a, b], for n=1, 2, 3, ....., and suppose f_n → f uniformly on [a, b]. Then prove that f ∈ R (α) on [a, b] and ∫_a^bfdα = lim_n→∞ ∫_a^bf_n dα.

Suppose E is an open set in Rⁿ, f maps E into R^m, f is differentiable at x₀ ∈ E. g maps an open set containing f(E) into R^k, and g is differentiable at f(x₀). Then prove that the mapping F of E into R^k defined by F(x) = g(f(x)) is differentiable at x₀ and F'(x₀) = g'(f(x₀)) f'(x₀).

Suppose f maps a convex open set E ¹ Rⁿ into Rⁿ, f is differentiable. in E and there is a real number M such that ||f'(x)|| ≤ M for every x ∈ E. Then prove that ||f(b) - f(a)|| ≤ M|b-a| ∀ a ∈ E, b ∈ E.

Let ∑a_nxⁿ be a power series with unit radius of convergence and let f(x) = ∑a_nxⁿ. If the series ∑a_n converges then prove that lim f(x) = ∑a_n.

Let f be a c¹-mapping of an open set E ¹ Rⁿ into Rⁿ, such that f(a, b) = 0 for some point (a, b) ∈ E. Put A = f¹ (a, b) and

assume that A₁ is invertible. Then prove that there exist open set U ¹ R^nm and W ¹ Rⁿ with (a, b) ∈ U and b ∈ W, having the following property: To every Y ∈ W corresponds a unique x such that (x, y) = 0 and f(x,y)=0. If this x is defined to be g(y), then prove that g is a c¹-mapping of W into Rⁿ. g (b) = a, f (g (y), y) = 0 (y ∈ w).