Question

41. If 2x = $$\begin{gathered} \sqrt{\mathrm{a}}+\frac{1}{\sqrt{a}},\mathrm{then}\frac{\sqrt{{\mathrm{x}}^2-}}{\mathrm{x}-\sqrt{{\mathrm{x}}^2-1}}\mathrm{is}\mathrm{equal}\mathrm{to} \\ \left(\mathrm{A}\right)\frac{1}{2}\left(\mathrm{a}-\frac{1}{\mathrm{a}}\right)\left(\mathrm{B}\right)\frac{1}{2}\left(\mathrm{a}+1\right)\left(\mathrm{C}\right)\frac{1}{2}\left(\sqrt{\mathrm{a}}-1\right)\left(\mathrm{D}\right)\frac{1}{2}\left(\mathrm{a}-1\right) \end{gathered}$$

42. A man saves Rs. 200 in each of the first three months of his service. In each of the subsequences months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
(A) 20 months (B) 21 months (C) 18 months (D) 19 months

43. A string of length 12 is bent first into a square PQRS and then into aright angled triangle PQT by keeping the side PQ of the square fixed. Then the area of PQRS is equal to
(A) area of $∆$PQT ​(B) $\frac{3}{2}$(area of $∆$PQT) (C) 2 (area of $∆$PQT) (D) $\frac{1}{3}$(area of $∆$PQT)

44. If cos x + cos² x = 1 then sin¹² x + 3 sin¹⁰ x + 3 sin⁸ x + sin⁶ x =

(A) 0 (B) $\sqrt{2}$ (C) 1 (D) 2

45. If P(x) is a polynomial in x, and x²³+23x¹⁷–18x¹⁶–24x¹⁵+108x¹⁴=(x⁴–3x²–2x+9). P(x) for all values of x, then the sum of the coefficients of P(x) is

(A) 12 (B) 14 (C) 16 (D) 18

Answer 1

Dear student
$$\begin{gathered} \mathrm{Side}\mathrm{of}\mathrm{square}=\frac{12}{4}=3\mathrm{unit} \\ \Rightarrow \mathrm{ar}\left(\mathrm{PQRS}\right)=3^2=9\mathrm{square}\mathrm{units} \\ \mathrm{One}\mathrm{side}\mathrm{of}△\mathrm{PQT}\mathrm{is}\mathrm{the}\mathrm{side}\mathrm{PQ}\mathrm{of}\mathrm{the}\mathrm{square}\mathrm{i}.\mathrm{e}3\mathrm{units} \\ \mathrm{The}\mathrm{other}\mathrm{is}\mathrm{one}\mathrm{more}\mathrm{than}\mathrm{its}\mathrm{side},\mathrm{i}.\mathrm{e}(3+1)=4\mathrm{units} \\ \Rightarrow \mathrm{ar}(△\mathrm{PQT})=\frac{1}{2}\left(3\right)\left(4\right)=6\mathrm{square}\mathrm{units} \\ \mathrm{Therefore},\mathrm{ar}\left(\mathrm{PQRS}\right)=\frac{3}{2}\left\{\mathrm{ar}\left(△\mathrm{PQT}\right)\right\} \end{gathered}$$

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