Board Pattern Mathematics Paper Ii

Mathematics and Statistics SAFE HANDS Paper II Time 2 hrs Q-1

Marks – 40

(A) Attempt any two of the following:(3 + 3)

x + x 2 + x 3 − 14 lim x−2 x →2 2 3( cos θ − 1) 3 (2) Evaluate lim π 6θ − π θ→ (1) Evaluate

6

(3) Find f(π) if the function f(x) =

2 + cos x − 1 ( π − x) 2

is continuous at x = π

(B) Attempt any one of the following: (2)

5x + 2

∫ x 2 − 3x + 2 dx (2) Evaluate ∫ sin x. log(cos x) dx (1) Evaluate

Q2

(A) Attempt any two of the following( 3 + 3) (1) Find the approximate value of f(1.99) if f(x) = x3 -3x + 5.

dy  5 sin x + 4 cos x   find 41 dx  

(2) If y = sin-1 

(3) Find derivative of f(x) = 72x by using first principle. (B) Attempt any one of the following ( 2)

 x + x2 + a2  1  show that dy = (1) If y = log  − x + x2 + a2  dx x2 + a2    1 −1  find dy (2) If y = cos ec   2  dx  2x 1 − x 

Q3.

(A)

(a) Attempt any one of the following (3)

1 + i − i  2 -1 (1)If A =   where i = − 1 then show that A -2A + I = 0 Hence Find A . i 1 − i   (2)Solve the given equation by matrix reduction method- x + y+ z = 2, 2x – y + 3z = 9, x + 5y + z = -2 (b) Attempt any one of the following (3) (1)Prove that the acute angle between the pair of straight lines ax2 + 2hxy + by2 = 0 is given by

 2 h2 − ab   tan −1  a +b    (2)Find separate equation of two lines whose joint equation is x2 + 2xycosec2α + y2 = 0

Q4.

Q5.

(B) Attempt any one of the following (2) (1) Find k if equation 3x2 + 10xy + 3y2 + 16y + k = 0 represents pair of straight lines. (2) Find equation of circle with centre at (3,2) and touching the line 4x + 3y -8 = 0. (a) Attempt any one of the following (3) (1)Show that the equation x2 -16xy -11y2 = 0represents pair of straight lines through origin inclined at an angle 30o with the line x + 2y -1 = 0 (2)Find k if the sum of the slopes of lines given by 2x2 + kxy – 9y2 = 0 is five times their product. (b) Attempt any one of the following (3) (1)A & B are independent events of a sample space if P(A∪B) = 0.7 & P(A∩B) = 0.2 then find P(A) and P(B) (2)A bag contains 4 white, 5 red and 6 black balls. Two balls are drawn at random. Find the probability that both the balls are black or white. (B) Attempt any one of the following (2) (1) Find the equation of tangent to 4x2 9y2 = 36 making equal intercepts on the co ordinate axes. (2) Find equation of tangent to hyperbola 2x2 – 3y2 = 5 at ( -2, -1) (a) Attempt any one of the following (3) (1)Find k if y = x + k touches the ellipse 2x2 + 3y2 = 1 (2)Find equation of parabola whose vertex is at origin and passing through (25,-10) (b) Attempt any one of the following (3) (1)Find the direction cosines of a line equally inclined to coordinate axes (2)Find the angle between the planes 2x – y + 3z + 4 = 0 and 3x + 2y – 4y -1 = 0 (B) Attempt any one of the following (2)

(1)

Find the eccentricity and foci of the ellipse 2x2 5y2 = 10.

(2)

If e1 and e2 are the eccentricities of two conjugate hyperbolas, prove that

1 1 + 2 =1 2 e1 e2