NA VRACHANA SCHOOL, SAMA- VADODARAPRACTISE PAPER

MATHEMATICSMARKS: 100TIME: 3Hrs

I Section 'A'-One Mark Questions1. If A is any matrix such that IA I = 14, find A.(adj A).

2. Let l :N ~ N defined by fix) = x2. Is it injective and surjective?3. If A = [a ..] is a square matrix such that a·· = i2 -]2, then write whether A is symmetric

LJ LJor skew symmetric.

2cosS -2sinS4. Evaluate:

sinS cosS

. . ( 1)5. Using the principal values, evaluate the following: tan-1 1 + sin-1 -2 .

fSin]; d6. Evaluate: ]; x.

%7. Evaluate: f cosec x dx.

~

8. Find A if a = 4i - J + k and b = Ai - 2J + 2h are perpendicular to each other.

9. Find the cosine of an acute angle between the vectors 2£- 3J + h and i + J - 2h .

10. Find the direction cosines of a line which makes equal angles with the coordinate axes.

Section 'B'-Four Marks Questions11. Let Y = Ix2 : x E N} c N. Consider f: N ~ Y as fix) = x2. Shov/ that f is invertible.

Find the inverse of f.OR

Let l: {l, 2, 31 ~ la, b, cl be one-one and onto function given by fi1) = a, fi2) = bandf(3) = c. Show that there exists a function g : la, b, c} -7 {l, 2, 3} such that gof = Ixand fog = II" where, X = II, 2, 3) and Y = la, q, cl.

-1 12 . -1 3 . -1 56cos -+SIn - = SIn -13 5 65

-be b2 + be e2 + ben ? = (ab + be + eo.)3a'" + ac -o.c e~ + o.e

0.2 + ab b2 + ab -ab

r1-sin3 x.

2 '3 cos x

14. Let fix) = 1 a;

b(l- sin x)2(7( - 2x)

values of 'a' and 'b'.

7(X<-,-

27( TI .

X = 2 . If fix) is a continuous function at x = 2' find the

7(x>-

2

ORLet fix) is continuous at x = 0, find the values of a, b, and c.

sin (a + l)x + sin xx<O

x=O.Jx+bx2 -.fX.

bx3/2 '

Verif\r L.M.V. theorem for the function lex) = x3 - 5x2 - 3x on [1, 3].

16. Differentiate: Sin-1( 2X

+1

]w.r.t. x.1 + 4'\

17. A person plays a game of tossing acoin thrice. For each head, he is given ~ 2 by theorganiser of the games and for each tail, he has to give ~ 1.50 to the organiser. LetX denote the amount gained or lost by the person. Show that X is a random variableand exhibit it as a function a sample space of the experiment. Find its range also.

ORA random variable X has the following probability distribution:

-

X 0 1 2 3 4 5 (:i ,...I-+--

\,

P(X) ]{ 'l 2[{2 7K2 + K'0 2K 2[( 3K IC'

o18. Evaluate: f f(x) dx where f(x) = Ixl + Ix + 21 + Ix + 51·

-5

19. Show that the differential equation 2ye-t/y dx + (y - 2xe-tly) dy = 0 is homogeneous andfind its particular solution, given that, x = 0 when y = 1.

ORFor the following differential equation, find a particular solution satisfying the givencondition:

dy 3 . 2 h 1t- - Y cot x = sm x; y = 2 w en x .= -dx 2

20. Form the differential equation of the family of circles in the second quadrant andtouching the coordinate axes.

21. Show that each of the given these vectors is a unit vector:

~ (2£ + 3J + 6k), ~ (3i - 6J + 2k), ~ (6£ + 2J - 3k).Also, show that they are mutually perpendicular to each other.

22. A vari?ble plane is at a constant distance 'p' from the origin and meets the co-or~inateaxes in A, B, C. Show that focus of the centroid of the tetrahedron OABC isx-2 + y-2 + z-2 = 16p-2.

I Section 'C'-Six .Marks Questions

[0 a -tan ~]

23. If A = and 1is the identity matrix of order 2,tan- 0

2

(1_A) [co.s a - sin aJ.show that 1+ A == sma cosa24. Find the points on the curve y2 = 2x, which is at a minimum distance from the point

(1, 4).OR

~ 3 4 4 3 2 36x .Find intervals in which the function given by fix) = -x - -x - 3x +- + IllS10 5 5 .

(a) strictly increasing (b) strictly decreasing.etan-1 x

25. Evaluate: f 2 dx.(l+x2

) ,\

26. A letter is known to have come from either TATANAGAR or CALCUTTA. On theenvelope just two consecutive letters TA are visible. What is the probability that theletter has come from:

OR. Find the probability distribution of the number of white balls drawn in a random draw

of 3 balls without replacement from a bag containing 4 white and 6 red balls. Also, findthe mean and variance of the distribution.

27. Find the area intercepted between the line 2y = 3x + 12 and the parabola 4y = 3x2

(using integration).28. Find the equation of the plane through the point (-2, 1, 3) and passing through the

intersection of ;. (2i -7J + 4k) = 3 and;· (3i - 5J + 4k) + 11 = O.

29. Coloured balls are distributed in 4 boxes as follows:

Colour

Box Black White Red Blue-"- --

A 3 4 5 6B 2 2 2 2C 1 2 3 1D 4 3 1 5

A box is selected at random and a ball is drawn. If the colour of the ball is black, what. . is the probability that the ball drawn is from box C?