XII HSC MATHS - II SAMPLE TEST SERIES Time : 2 hrs. M.M.: 40 GENERAL INSTRUCTIONS : 1. All questions are compulsory. 2. Figure to right indicates full marks. 3. Graph paper is not necessary for LPP. 4. Answer to every question must be written on a new page. Q1. (A) Attempt any two of the following : [8] (i) Evaluate : lim 1 − sin x (π − 2 x) 2 (3) (ii) Evaluate : lim log (5 + x) − log (5 − x) x (3) x→ π/2 x→0 (5sin x − 1) 2 (iii) If f (x) = , x ≠ 0 is continuous at x = 0 , find f (0). x log (1 + 2 x) (3) (B) Attempt any one of the following : 1 − tan x (i) Evaluate : ∫ 1 + tan x (ii) Evaluate : ∫x 2 dx (2) sin x dx (2) Q2. (A) Attempt any two of the following : [8] (i) If y = [ x + x 2 + a 2 ]n , then prove that (ii) If y = (iii) If y = dy = dx ny + a2 x2 sin −1 x x sin −1 x dy 2 , then prove that + log = 1 − x (1 − x 2 )3/2 dx 1 − x2 1− x dy , prove that (1 − x2) +y=0 1+ x dx (3) (3) (3) (B) Attempt any one of the following : (i) If y = xey, prove that y dy = dx x (1 − y) (ii) If sin y = x sin (a + y), show that sin 2 (a + y) dy = dx sin a Q3. (A) (a) Attempt any one of the following : 1 dx x {6 (log x) 2 + 7 log x + 2} (i) Evaluate : ∫ (ii) Evaluate : ∫ ( x + 2) ( x 8 2 + 4) dx (2) (2) [8] (3) (3) (b) Attempt any one of the following : π/2 (i) Evaluate : ∫ 0 dx 5 + 4 cos x (3) 1/2 (ii) Evaluate : ∫ 0 dx (3) (1 − 2 x ) 1 − x 2 2 (B) Attempt any one of the following : (i) Solve : (x + y + 1)2 dy = dx (2) (ii) Form the differential equations by eliminating the arbitrary constants from the following equations : Ax2 + By2 = 1 (2) Q4. (A) (a) Attempt any one of the following : [8] (i) Find the 20th term of the sequence 2, 6, 12, 20, 30, ....... (3) (ii) f (x) is a polynomial of degree 2 in x. If f (0) = 8, f (1) = 12 , f (2) = 18, find f (x). (3) (b) Attempt any one of the following : (i) Solve the following differential equations : x2y dx − (x3 + y3) dy = 0 (ii) Find the particular solution of the differential equation : (x + 1) dy + 1 = 2e−y , when x = 1, y = 0 dx (3) (3) (B) Attempt any one of the following : (i) Find correct to 4 decimal places, the approximate value of : (3.07)4 (2) (ii) The radius of a sphere is measured as 12 cm with an error of 0.06 cm. Find approximately the relative error and percentage error in calculating its surface area. (2) Q5. (A) (a) Attempt any one of the following : [8] (i) (Derivative of a Difference). If u and v are differentiable functions of x and y = u − v, then prove that dy du dv − = . dx dx dx (3) (ii) (Derivative of a Composite Function). If y is a differentiable function of u and u is a differentiable function of x, then prove that dy dy du × = . dx du dx (3) (b) Attempt any one of the following : π/2 (i) Evaluate : ∫ cos2x log sinx dx (3) sin 2 x . log (tan x) dx (3) π/4 π/2 (ii) Evaluate : ∫ 0 (B) Attempt any one of the following : (i) A rectangle has an area equal to to 50. Show that its perimeter is the least when it is square. (ii) Find the approximate value of log10 103 given that log10 e = 0.4343. x x (2) (2)

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