BLUE PRINT: CLASS XII MATHS CHAPTER S NAME MARK 4 MARKS 6 MARKS TOTAL. RELATIONS AND FUNCTIONS 5. INVERSE TRIGONOMETRIC 5 FUNCTIONS 3. MATRICES 7 4. DETERMINANTS 6 5. 8 DIFFERENTIATION 6 APPLICATION OF 0 DERIVATIVES 7. INTEGRALS 8. APPLICATION OF 6 INTEGRALAS 9. DIFFERENTIALS 8 EQUATION 0. VECTS 3 7 THREE DIMENSAIONAL 0 GEOMETRY. LINEAR PROGRAMMING 6 3. PROBABILITY 0 TOTAL 0 ( 0) ( 48 ) 7 ( 4 ) 00 Model question paper
Time: 3hrs Mathematics Class: Max.marks:00 General Instructions:. All questions are compulsory. The question paper consists of 9 questions divided into three sections A,B and C. Section A comprises of 0 questions of mark each, Section B comprises of questions of 4 marks each and Section C comprises of 7 questions of 6 marks each. 3. Use of calculators is not permitted. SECTION A. If A is square matrix of order 3 such that = 64, find. If A, find ; 0<. when A + = I. 3. If x = 6.Find x 8 x 8 6 4. Let * be a binary operation on R given by a * b =. Write the value of 3 * 4. 5. Evaluate : Sin {π/3 - }. dx 6. Evaluate x x. 7. If + bx ) dx = +. find the values of a and b. 8. If = i + j, = j + k, find the unit vector in the direction of +. 9. If =, = and angle between and is 60. Find 0. What is the cosine of the angle which the vector i + j + k makes with y-axis. SECTION B. Prove that 5 7 53 3 5 35 sin sin cos. Consider
f : R [ 5, ) given by f x 9x 6x 5. Showthat. f is invertible. Find inverse of f? 3. Show that a b abc a b c c d y 4. If x=a sin pt and y=b cos pt, find the value of 0 at t. dx 5. If y = sin tan x Pr dy x ove that x dx x 6 Find the intervals in which the function f(x) = sinx cosx ; 0 x π is (i) increasing (ii) decreasing x 4 7. Evaluate dx. 4 x x 6 Evaluate 0 x sin x dx. cos x 8. Form the differential equation representing the family of ellipse having foci on x-axis and centre at the origin. Form the differential equation of the family of circles having radii 3. 9. Find the equation of the plane determined by the points A(3,-,), B(5,,4) and C(-,-,6) Also find the distance of the point P( 6,5,9) from the plane dy 0. Solve the differential equation ysec x tan x.sec x ; y(0). dx. If a and b are unit vectors and θ is the angle between them, then prove that cos θ/ = ½ a + b. Find the probability distribution of the number of heads in a single throw of three coins. Three balls are drawn one by one without replacement from a bag containing 5 white and 4 green balls. Find the probability distribution of number of green balls drawn. SECTION C 3. Find the matrix A satisfying the matrix equation
3 0 A 3 5 3 0 4. Evaluate 5. Show that the semi vertical angle of right circular cone of given surface area and maximum volume is sin. 3 6. Using method of integration find the area of the region bounded by the lines x + y =4, 3x - y = 6 and x - 3y + 5 = 0.. 7. Find the distance of the point (3,4,5) from the plane x y z measured parallel to the line x= y =z. Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point (,3,4) from the plane x y + z + 3 = 0. And also find the image of the point in the plane. 8. A dealer in rural area wishes to purchase a number of sewing machines. He has only rupees 5760.00 to invest and has space for at most 0 items. Electronic sewing machines cost him rupees 360 and manually operated sewing machine rs.40. He can sell an electronic sewing machine at a profit of rupees and a manually operated sewing machine at a profit of Rs.8. Assuming that he can sell all the items he can buy, how should he invest his money in order to maximize his profit. Make it as a Linear Programming Problem and solve it graphically. Justify the values promoted for the selection of the manually operated machines. 9. A student is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. To get the probability as, Which value to be promoted among students.. n MARKING SCHEME CLASS XII AdjA A A 8 (). /3
3. x = +6 or -6 4. 5/3 5. 6. log x +c. 7. a = 4 and b = 3 8. i+ j + k / 6 9. 3 0. ½. Let, 5 7 s i n, s i n, 3 5 () t h e n c o s 4, c o s 3 5 ( ) 5 3 c o s 3 5 ( ) 5 7 5 3 s i n s i n c o s 3 5 3 5. let x,x R+ s.t f (x ) = f (x ) prove that x = x () f is -
9 6 5 x x y (3x ) 6 y 3x y 6 ( y 6 ) x f ( y) 3 Therefore for every y ), ( -) / 3 R+ s.t f ( -) / 3 ) = y. f is onto. ( ) since f is - and onto, f is invertible ( ) Therfore (x): from ) to R+ defined as (x) = ( -)/3 () 3. a b c a b c a b c a b c C C C 3 b c abc () a b c b c b c 0 abc( ) () a b c 0 a b c ( ) () a b c 4.
dx dy ap cos pt bp sin pt dt dt dy b tan pt dx a () d y b 3 sec pt dx d y dx a b a at t0 5..Let x = cos t Y=sin =sin ( ) Y=sin t= () = ( ) 6. f (x)=cos(x) + sin(x) f (x)=0 tan(x) = - () x=3π/4 and 7π/4 Intervals are [0,3π/4),(3π/4,7π/4), (7π/4,π] () [0,3π/4)- f is positive, so f(x) is increasing (3π/4,7π/4)- f is negative, so f(x) is decreasing (7π/4,π]-f is positive, so f(x) is increasing () 7. Consider 4 x 4 x dx x x 6 6 x x () dt 4 =, wheret x t 3 x () = tan t c 3 3 () x 4 = tan c 3 3x () dx 4
consider 0 x x x x dx dx cos cos x sin cos sin () x 0 x By solving further we get given = = x x sec tan () x dx dx 0 0 tan () 4 8. The equation of the ellipse is + = () x/a + y/b dy/dx = 0 (y/x)dy/dx=-b /a () Differentiating again and getting the differential equation as (xy)d y/dx + x(dy/dx) ydy/dx = 0 ()
9. Equation of the plane x 3 y z 3 0 4 0 4 i. e., 3x 4y 3z 9 0 () Now the perp. Distance from (6,5,9)to this plane is 3.6 4.5 3.9 9 9 6 9 6 34 units ------------------- () 0. Which is in linear differential equation For finding I.F.= x dx tan x e e () Solution y.i.f. = () tan x tan x =tan x. e e c () When x =0, y = c and writing the completed solution (). Consider a b cos () =(+cos ) -----------------------------() =4 cos cos () a b. Let X be the number of heads,x=0,,,3 P(having head)=p=/ q=/, n=3.- --------------------------() Now P(X=0)= 3 p q q C 0 8 Probability distribution 0 3 3 X 0 3 P(X) /8 3/8 3/8 /8 -------------------------------------()
3. 4. I= SECTION C 3 Let B and C 3 5 3 B 0 and c 0 3 adjb (3) 3 adjc 5 3 3 B and C 3 5 3 3. (3) 3 5 3 0 A B C I= () I=π () I=π I=π () For getting the answer as (3)
5. Let r, h, l,s and V be the radius, height, slant height. surface area and the volume of the cone. h l r S= rl r S r l r ----------------------------------------- () 4 and V r h V r h 3 9 ----------------------------------------- () dv 0 dr ( 3 rs 8 S r ) 0 9 For max or min r l 3 ---------------------------() d V now ( at S 4 r ) o dr V is max imum. r sin Now l 3 sin ( ). 3 V is max imum. ----------------------------------- () ----------------------------------------- () 6. Let AB x + y =4 BC 3x-y=6 and AC x-3y+5=0 Solving and B(,0), C(4,3) and A(,) ( ) For the correct figure () Area of triangle = dx - dx ( )
For integrating and getting area=7/ sq.units () 7. Line is x/=y/=z/ -------------------- () Line PQ through P(3,4,5)and II to the given line is -------------------- () (x-3)/=(y-4)/ =(z-5)/= General point on the line is Q( 3, 4, 5) If this point lies on the plane x+y+z=, 3 4 5 Q(, 0,) PQ (3) (4 0) (5 ) 6. -------------------- () -------------------- ()
8. Suppose number of electronic operated machine = x and number of manually operated sewing machines = y x+y < 0 - - - (i) and, 360 x + 40y < 5760 or 3x+y < 48 - - - (ii) x>0, y>0 To maximise Z=x + 8y Corners of feasible region are A (0,0), P(8,), B(6,0) ZA = 8x0 = 360, ZP = x8+8x = 39, ZB = 35 ½
Z is maximum at x=8 and y= The dealer should invest in 8 electric and manually operated machines (½) Keeping the save environment factor in mind the manually operated machine should be promoted so that energy could be saved. () 9. Let E be the event that the student reports that 6 occurs in the throwing of die and let S be the event that 6 occurs and S be the event that 6 does not occur. P(S )=/6, P(S )=5/6, () P(E/S )=3/4, P(E/S )=/4 () P(S /E)= =3/8 () To get the probability as one, the student should always speak truth. The value to be promoted among students is truth value. ()