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Properties of inverse trigonometric function part-4

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Inverse Functions (प्रतिलोम फलन)|Inverse Trigonometric Functions (प्रतिलोम त्रिकोणमितीय फलन)|Questions|Properties of Inverse Trigonometric Functions (प्रतिलोम त्रिकोणमितीय फलन के गुणधर्म)|Questions|OMR

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Properties of Inverse Trigonometric Functions (प्रतिलोम त्रिकोणमितीय फलन के गुणधर्म)|Questions|OMR

Maximum & Minimum Value of Trigonometric Function (त्रिकोणमिति फलन के उच्चतम एवं निम्नतम मान)|Trigonometrical Equations with their General Solution (त्रिकोणमितीय समीकरण उनके सामान्य हल)|Practice Questions (अभ्यास प्रश्न)|OMR

Let f_1: RrarrR ,\ \ f_2:(-pi/2,pi/2)->R\ \ f_3:(-1,\ e^(pi/2)-2)rarrR and f_4: RrarrR be functions defined by f_1(x)=sin(sqrt(1-e^-x^2)) , (ii) f_2(x)={(|sinx|)/(tan^(-1)x)\ \ \ \ \ if\ x!=0 1\ \ \ \ \ \ \ \ \ \ \ \ if\ x=0,\ where the inverse trigonometric function tan^(-1)x assumes values in (pi/2,pi/2) , (iii) f_3(x)=[sin((log)_e(x+2))] , where, for t in R , [t] denotes the greatest integer less than or equal to t , (iv) f_4(x)={x^2sin(1/x)\ \ \ \ \ \ if\ x!=0 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ x=0 . LIST-I LIST-II P. The function f_1 is 1. NOT continuous at x=0 Q. The function f_2 is 2. continuous at x=0 and NOT R. The function f_2 is differentiable at x=0 S. The function f_2 is 3. differentiable at x=0 and its is NOT continuous at x=0 4. differentiable at x=0 and its derivative is continuous at x=0 The correct option is Prarr2;\ \ Qrarr3;\ \ Rrarr1;\ \ Srarr4 (b) Prarr4;\ \ Qrarr1;\ \ Rrarr2;\ \ Srarr3 (c) Prarr4;\ \ Qrarr2;\ \ Rrarr1;\ \ Srarr3 (d) Prarr2;\ \ Qrarr1;\ \ Rrarr4;\ \ Srarr3

For any positive integer n , define f_n :(0,oo)rarrR as f_n(x)=sum_(j=1)^ntan^(-1)(1/(1+(x+j)(x+j-1))) for all x in (0, oo) . Here, the inverse trigonometric function tan^(-1)x assumes values in (-pi/2,pi/2)dot Then, which of the following statement(s) is (are) TRUE? sum_(j=1)^5tan^2(f_j(0))=55 (b) sum_(j=1)^(10)(1+fj '(0))sec^2(f_j(0))=10 (c) For any fixed positive integer n , (lim)_(xrarroo)tan(f_n(x))=1/n (d) For any fixed positive integer n , (lim)_(xrarroo)sec^2(f_n(x))=1

cos^(-1) (cos (-5)) + sin^(-1) (sin(6)) - tan^(-1)(tan (12)) is equal to : (The inverse trigonometric functions take the principal values)