Board Pattern Mathematics Paper I

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

Marks – 40

(A) Attempt any two of the following:(3 + 3) (1) Prepare the truth table for the statement pattern [ p ∧ (p ⇒q)] ⇒ q. Interpret the result. (2) In the statement p and q are true and statement r is false then find the truth value of the following(a) p ∨ (q ⇒ r); (b) ¬q ∧ (p⇒q) (c) (∼r↔p) → ∼q (3) Find the logical form of the following switch circuit and write its behavior C A

Q.2.

B

B’ (B) Attempt any one of the following: (2) (1) Draw the graph and state only the vertices of possible region of the following inequalities 5x + 3y ≥ 15 , x + 2y ≥ 6; 0 ≤ x ≤ 6 ; 0 ≤ y ≤ 5 (2) Two different kinds of food A and B are being considered to form a weekly diet. The minimum weekly requirements for fats, carbohydrates and proteins are 18, 24 and 16 units respectively. One kg of food A has 4,16 and 8 units respectively of these ingredients and one kg food B has 12, 4 and 6 units respectively. The cost of A and B are 4Rs/kg and 3 Rs/kg formulate the problem as LPP in order to minimize the cost. (A) Attempt any two of the following( 3 + 3) (1) By vector method prove that the altitudes of a triangle are concurrent.

(2) Show that ( b + c ) • [ ( (3) If

a

,

b

and

c

+a ) X (

a

+

b

) = 2.

a

•(

X

b

c

)

are three non zero non coplanar vectors then prove that any vector

c

be uniquely expressed as linear combination of

a

,

b

and

c

r in the space can

.

(B) Attempt any one of the following ( 2) (1) If A(2,3,-1); B ( -2,-3,-3); C(1,7,2) and D(-6, 2, 2) then find [ AB, AC, AD ] and interpret the result. (2) If Q3.

a

,

b

and

c

are the position vectors of A, B and C respectively such that 2

a

+3

b

-5

c

=0. Find

ratio in which C divides the segment AB Show that A, B, C are collinear. (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 – 4z -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