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MODEL QUESTION PAPER SUBJECT: MATHEMATICS 1.(a) What is one‐one function and onto function? (1) (b) Verify the following function is one‐one and onto 1,
1,
f:N→N given by f(x)= (2) (c) Consider the functions f(x) =cosx and g(x) = 3x2 . Find fog(x) and gof(x). Also verify fog(x) = gof(x). (2) 2.(a) Prove that sin
(b) Solve tan
3. (a) If A =
2
1
3
4
2
sin
tan
cos
(2) 3
and B =
5
4
(3) 1
,then 2
i) Find AB (1) ii) Find AI , BI and (AB)I (1) iii) Verify (AB)I = BI AI (1) 3
(b) Express A = 4
0
2
1
2
4
5 as a sum of a Symmetric Matrix and a 4
Skew Symmetric Matrix (2) (c) Using elementary row operations, find the inverse of the matrix 4. (a)
1
2
2
(2) 1
1
1
1
(2) (b) Find the area of the triangle with vertices (3,4), (‐4,5) and (7,0) (2) 5. (a) If f(x)=
3
1
5
5
5
i) Find k if f is continous at x=5 (1) ii) Check whether f is continous at x=1 (1) (b) If y = tan
i) Express y in simple form (1) ii) Find (c) If y=
(1) Prove that (1‐x2) y2 ‐ xy1 – y = 0 (2) OR (a) Find if x = a ( t + sint ) and y = ( 1 – cost ) (2) (b) Verify Mean Value Theorem for the function f(x)=x2 in the Interval [2,4] (4) 6. Let AP and BQ be two vertical poles at points A and B respectively with AP=16cm, BQ=22cm and AB=20cm. i)
Let R be a point on AB such that AR=x. Then what is BR. (1) ii)
Find RP and RQ using Pythagoras Theorem. (2) iii) For what value of x, RP2 + RQ2 is minimum (3) 7. Integrate the following i)
(2) ii)
(3) iii)
Sinx+sin2x (1) 8. (a)
2
3
01
(b) (2) (1) ( (c) (3) OR Match the following (6) A B (i) (ii)
(iii)
(iii) (iv) (iv) (v) (vi) 1
(i) sin
(v) Secx+c (vi)
9. Consider the region
,
;0
1, 0
1,0
2 (a) Sketch the graph of the above region (1) (b) Find the intersections of y=x2+1 and y=x+1 (2) (c) Find the area of the region (3) 10. (a) Form the differential equation of the circle with centre at the X‐axis and passing through the origin. (2) (b) Cos2x + y = tanx (0 ≤ x < ) (i) Find the Integrating Factor (1) (ii) Find the solution (2) 11. If = 5i‐j‐3k and =i+3j‐5k a) Find + (1) b) Verify + are perpendicular (2) c) Find the angle between & (2) 12. a) Write the direction cosines of the line =
(1) b) A line passing through the point (5,2,‐4) and which is parallel to the vector 5i‐3j+4k, then write the equation of the line (2) 13. (A) A plane is passing through the intersection of the planes 5x‐3y+7z‐8=0 and 7x‐4y+6z‐2=0 i) Write the equation of the plane which is passing through the point (1,4,‐2) (2) ii) Suppose the plane lies in XZ‐plane, what is its equation. (2) OR (B) A plane passing through two points (6,1,5) and (7,2,‐3) i) write the equation of the plane when it is parallel to the line =(i‐j+k)+λ(3i+4j‐k) (2) ii) write the equation of the plane when it is perpendicular to the plane .(7i‐3j+2k)=5 (2) 14. Consider the linear programming problem Minimise Z = 200 x + 500 y Subject to the constraints x + 2y ≥ 10 3x + 4y ≤ 24 X ≥ 0 , y ≥ 0 i) Give graph of the feasible region (1) ii) Identify the corner points (2) iii) Find the value of Z at each of its corner point (2) iv) Find the minimum Z (1) 15. There are 3 bags. Bag I contains 4 red and 5black balls. Bag II contains 6 red and 8black balls. Bag III contains 7 red and 3black balls. One ball is drawn at random from one of the bags. i)
If E1,E2 and E3 are the events of choosing bagI,bagII and bagIII respectively.Find P(E1),P(E2) and P(E3). (1) ii)
If A is the event of drawing a red ball. Find P(
and P(
) , P(
) ). (2) iii)
Find the probability that the ball was drawn from bagII using the Bayes’ Formula P(
) = ∑
; i=1,2,3,………..,n (3) .
OR (B) Two cards are drawn simultaneously from az well shuffled pack of 52 cards. i) If X denote the number of Kings, what are possible values of X. (1) ii) For each value of X, fine P(X) and write the probability distribution of X. (3) iii) Find Mean and Variance using the formula. Mean = E(X) and Variance = E(X2)‐[E(X)]2. Where E(Xr)=∑
(2)