Column - I Column - II (A) If a, b, c, d are four real numbers such that a2 + b2 = c2 + d2 = 4 and ac + bd = 0, then the least value of (ab – cd + 6) is (P) 1 (B) If x1, x2, x3 ........ x10 are the smallest positive values of x satisfying the equation 1 + 2{x} + 22{x}2 + 23{x}3 +...+ \infty = Min. (sin2(2πx) + cosec2(2πx)) then \left[\right. \frac{\sum_{r = 1}^{10} x_{r}}{10} \left]\right. equal [Note: [k] and {k} denote greatest integer and fractional part functions of k respectively.] (Q) 2 (C) The least positive integral value of p for which px2 + (p – 4x) \int_{0}^{\pi / 2} \left(\right. xsinx \left.\right) dx > 0 is true for all x is (R) 3 (D) If A = \left[\right. 3 & - 4 \\ 1 & - 1 \left]\right. \textrm{ }\textrm{ } then \textrm{ }\textrm{ } \left|\right. \textrm{ } A^{10076} - 2 A^{10075} \textrm{ } \left|\right. is equal to (S) 4 (T) 5

Engineering

Mathematics

Properties of Definite Integral

Question

Column - I	Column - II
(A) If a, b, c, d are four real numbers such that a² + b² = c² + d² = 4 and ac + bd = 0, then the least value of (ab – cd + 6) is	(P) 1
(B) If x₁, x₂, x₃ ........ x₁₀ are the smallest positive values of x satisfying the equation 1 + 2{x} + 2²{x}² + 2³{x}³ +...+ $\infty$ = Min. (sin²(2πx) + cosec²(2πx)) then $[\frac{\sum_{r = 1}^{10} x_{r}}{10}]$ equal [Note: [k] and {k} denote greatest integer and fractional part functions of k respectively.]	(Q) 2
(C) The least positive integral value of p for which px² + (p – 4x) $\int_{0}^{π / 2} (xsinx) dx > 0$ is true for all x is	(R) 3
(D) If A = $[\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}] then \| A^{10076} - 2 A^{10075} \|$ is equal to	(S) 4
	(T) 5

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This is a matching question with four parts (A, B, C, D) and five possible answers (P, Q, R, S, T). You must match each problem statement to its correct numerical answer from the second column.

For example, part (A) involves real numbers a, b, c, d with constraints $a^{2} + b^{2} = 4$ and $c^{2} + d^{2} = 4$ , and $a c + b d = 0$ . You must find the least value of the expression (ab - cd + 6).

Final Answer: The task is to find the correct pairings: A-?, B-?, C-?, D-?.