Column I Column II (A) If f(x) = \int_{0}^{x^{2}} \left(\right. cos \frac{1}{\sqrt{t}} + 1 \left.\right) dt , x \geq 1 then the value of f "(x) is greater than or equal to (P) 1 (B) If H'(x) = (2x + 3) (x3 + x – 2)2 (x2 – 1) (x2 + 3x + 2) and ln g(x) = H(x) then number of local minimum of g(x) is less than or equal to (Q) 2 (C) The value of \underset{n \textrm{ } \rightarrow \textrm{ } \infty}{Lim} \frac{1}{\sqrt{4 n}} \left(\right. 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + . .... + \frac{1}{\sqrt{9 n}} \left.\right) is greater than (R) 3 (D) If a conic satisfying the differential equation x2dy – dx = 2xy dx which passing through the point \left(\right. 1 , \frac{- 1}{3} \left.\right) has the eccentricity e then e2, is less than (S) 4 (T) 6

Engineering

Mathematics

Linear Differential Equation

Question

Column I	Column II
(A) If f(x) = $\int_{0}^{x^{2}} (\cos \frac{1}{\sqrt{t}} + 1) dt, x \geq 1$ then the value of f "(x) is greater than or equal to	(P) 1
(B) If H'(x) = (2x + 3) (x³ + x – 2)² (x² – 1) (x² + 3x + 2) and ln g(x) = H(x) then number of local minimum of g(x) is less than or equal to	(Q) 2
(C) The value of $\underset{n \to \infty}{Lim} \frac{1}{\sqrt{4 n}} (1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + . .... + \frac{1}{\sqrt{9 n}})$ is greater than	(R) 3
(D) If a conic satisfying the differential equation x²dy – dx = 2xy dx which passing through the point $(1, \frac{- 1}{3})$ has the eccentricity e then e², is less than	(S) 4
	(T) 6

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This is a matching question with four parts (A-D) and options (P-T). Each part requires solving a calculus or differential equation problem to find a numerical bound, then matching it to the correct option.

(A) Find f"(x) for f(x)= $\int_{0}^{x^{2}} (\cos \frac{1}{\sqrt{t}} + 1) d t$ . Use the Fundamental Theorem of Calculus and chain rule. f'(x)=2x(cos(1/|x|)+1). f"(x)≥2, so answer is (Q).

(B) Given H'(x) and ln g(x)=H(x), g'(x)/g(x)=H'(x). Find critical points from H'(x)=0. Number of local minima ≤3, so answer is (R).

(C) The limit is a Riemann sum: $\lim_{n \to \infty} \frac{1}{\sqrt{n}} \sum_{k = 1}^{9 n} \frac{1}{\sqrt{k}}$ . This approximates $\int_{0}^{9} \frac{1}{\sqrt{x}} d x = 6$ . Value >4, so answer is (S).

(D) Solve the differential equation x²dy - dx = 2xydx. Rewrite as dy/dx = (2xy+1)/x². This is linear; integrating factor is 1/x². Solution is a hyperbola y = x - 1/(3x). Eccentricity e²=1+(b²/a²). For hyperbola y=x-1/(3x), e²<4, so answer is (R).

Final Answer: A→Q, B→R, C→S, D→R