Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation Knowledge Understanding Applicatoin HOTS Evaluation SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 Time allowed : 3 Hours MAX.MARKS Blue Print Marks Marks 4 Marks 6 Marks Sl.N o Chapters / typology Relations and functions Inverse trigonometric functions 3 Matrices 4 Determinants 5 Continuity and differentiability 6 Applications of derivatives 7 Integrals 8 Applications of the Integrals 9 Differential equations Vector Algebra Three dimensional Geometry Linear Programming 3 Probability Total 3 3 4 3
SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 Time allowed : 3 Hours MAX.MARKS GENERAL INSTRUCTIONS: (i). All questions are compulsory. (ii). The question paper consists of 9 questions. (iii). Question 4 in SECTION A are very short-answer type question carrying marks each. (iv). Question 5 in SECTION B are very short-answer type question carrying marks each. (v). Question 3 3 in SECTION C are very short-answer type question carrying 4 marks each. (vi). Question 4 9 in SECTION D are very short-answer type question carrying 6 marks each. Questions to 4 carry mark each. Section A. If a b = 3a + 4b, find the value of 3?. Find the value of Sin (Sin 3 ). 3. Find x such that 3 4 5 = x 3 x 5 4. Find the position vector of R, which divides the line joining P(3a b ) and Q(a + b ) in the ratio : internally. Section B Questions 5 to carry mark each. 5. Find the projection of vector a = i + 3j + 6k on vector b = i + j + 3k. 6. Find the value of λ such that the vectors 3i + λj + 5k, i + j 3k and i j + k are coplanar. 7. Prove that tan + 5 tan 3 = tan. 4 43 8. Obtain the differential equation representing the family of parabolas having vertex at the origin and axis along the positive direction of x-axis. 9. Two cards are drawn at random and without replacement from a pack of 5 playing cards. Find the probability of drawing at least one red card.. Show that the points A(a, b + c), B(b, c + a) and C(c, c + a) are collinear.. Evaluate e x ( x +x ) dx.. If a b = c d, and a c = b d show that (a d ) is parallel to (b c ), provided a d and b c. Questions 3 to 3 carry 4 marks each. Section C 3. Determine the constants a and b so that the function f defined as 3ax + b, if x > f(x) = {, if x = is continuous at x =, find the values of a and b. 5ax b, if x < 4. From a lot of 5 bulbs, which includes 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of the number of defective bulbs. Hence, find the mean of the distribution. 5. Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets,, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw,, 3 or 4 with the die? 6. Solve the differential equation : x dy + (y x 3 ) dx =.
7. The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after second. Solve the differential equation: (x + y)dy + (x y)dx = given that y = when x =. x 8. Evaluate. dx x +3x+ Evaluate Cos x log sin x dx. 9. Prove that Sin x. log tan x dx =. Show that log( + x) x, x > is an increasing function of x throughout its domain. +x. A ladder 5m long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4m away from the wall. Prove that the curves x = y and xy = k cut at right angles, if 8k =.. Find the equations of the perpendicular drawn from the point P(, 4, ) to the line x+5 z 6 9. Also write down the co-ordinates of the foot of the perpendicular from P to the line. = y+3 = 4 3. Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), and newspaper s envelope as carry bags. It is found that the shopkeepers A use polythene bags, 3 handmade bags, 4 newspaper envelopes, B use 3 polythene bags, 4 handmade bags, newspaper envelopes, and C use 4 polythene bags, handmade bags, 3 newspaper envelopes. The shopkeepers A, B, C spent `5, `7 & ` on these carry bags respectively. Find the cost of each carry bags using matrices. Keeping in mind the social & environmental conditions, which shopkeeper is better? Why? Section C Questions 4 to 9 carry 6 marks each 4. Evaluate (e x + 3x ) dx as limit of sum. Evaluate log( + cos x). dx 5. Find the inverse of the matrix given below by using elementary row transformations. A = [ 3 ] 6. Let A = R {3} and B = R {}. Consider the function f: A B defined by f(x) = x x 3 Show that f is one-one and onto and hence find f. 7. Prove that the image of the point (3,, ) in the plane 3x y + 4z = lies on the plane x + y + z + 4 =. Find the equation of the plane through the line of intersection of the planes x + y + z = and x + 3y + 4z = 5, which is perpendicular to the plane x y + z =, also find the distance of the plane obtained above, from the origin. 8. Using integration, find the area of the region bounded by the parabola y = 4x and the circle 4x + 4y = 9. Find the area of the smaller region bounded by the ellipse x 9 + x 4 = and the line x 3 + y =.
9. A medical company has two factories at two places, A and B. From these places, supply is made to each of its three agencies situated at P, Q and R. The monthly requirements of the agencies are respectively 6, 4 and 5 packets of the medicines, while the production capacity of the factories A and B are 6 and 9 packets respectively. The transportation cost per packet, from the factories to the agencies are given below. To From A B P 5 4 Q 4 R 3 5 How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost.
SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 MARKING SCHEME Section A Questions to 4 carry mark each.. If a b = 3a + 4b, the value of 3 = 6+ = 8. (). The value of Sin (Sin )= 3 Sin (Sin ( ))= 3 Sin (Sin 3 )= 3 () 3. Given 3 4 5 = x 3 = 5x 6x x = x 5 () 4. OR = OP +OQ 3 3a b +a +b 3 Questions 5 to carry mark each. 5a 3. () Section B 5. The projection of vector a = i + 3j + 6k on vector b = i + j + 3k is a.b b ( M) a. b = + 6 + 8 = 6, b = + 4 + 9 = 4, Proj of a on b is 6 4. ( M) 6. The vectors 3i + λj + 5k, i + j 3k and i j + k are coplanar. Implies [3i + λj + 5k, i + j 3k, i j + k ] = ( M) 3 λ 5 3 = 3( ) λ(7) + 5( 5) = 7λ = 8, λ = 4 ( M) 7. Prove that tan 5 = tan ( tan 5 + tan 5 = 4 tan ( 5 5 + 4 5 48 ) = tan ( 5 ) ( M) ) = tan 3 43 ( M) 8. Finding equation of family of curve ( M), forming DE ( M) 9. P(A: first card black) = 5 5, P(B: second card black) =, P(A B)= 5 ( M) P(at least one card is red) = 5 5 ( M). The points A(a, b + c), B(b, c + a) and C(c, c + a) are collinear if area of ABC = ( M) a b + c b + c b c + a = (a + b + c) c + a = (a + b + c)() = ( M) c a + b a + b. e x ( x +x ) dx = e x ( x +x (+x ) ) dx = ex +x + c (Split, Integration ). a b = c d, and a c = b d a b a c = c d b d ( M) a (b c ) (c b ) d = (a d ) (b c ) =, so (a d ) (b c ). ( M) Questions 3 to 3 carry 4 marks each. Section C 3. Obtaining equations 3a + b =, 5a b = M Finding the values a=3 and b= M 4. Distribution table 3 M Mean M 5. A: she gets a 5 or 6, B: She gets,, 3 or 4, E: She obtained exactly one head P(A)=, 3 P(B)=, P(E 3 A )=3, P(E 8 B )=. M P( B E ) = P(B).P( E B ) = P(A).P( E 3 A )+P(B).P(E B ) 8 + 3 = 8 M
6. x dy + (y x 3 ) dx =. dy dx x x, Integrating factor = e dx x = x M xy = x 3. dx 4xy = x 4 + c 7. Given that dv = k, dv = K. dt dt v = Kt + C Finding C = 36 Finding K = 84 3 Finding r = 9 dy, Put y = Vx dx y+x ½ M obtaining the form v+ dx dv = v + x obtaining log(x + y ) + tan y = c x ½ M 8. obtaining c = 4 log x x +3x+ x+3 dx 3 x +3x+. dx A(x + 3) + B = x, A=, B= 3 dx (x+ 3 ) 4 log x + 3x + 3 log x+ + C x+ 4 log x + log x + + C Cos x log sin x dx. By ILATE rule, Cos x is second function, Log Sin x is second function. log Sin x. Cos x. dx cos x sin x log sin x. sin x. cos x dx log sin x sin x x I = sin x I= sin x sin x 4 9. Let I = Sin x. log tan x dx log cot x. dx. cos x. dx. dx + C ½ M. (log tan x + log cot x). dx ½M I=. f(x) = log( + x) x, x > f `(x) = f `(x) = 4 +x (+x) +x x (+x)(+x), x > so x + >, hence f `(x) > on (, ). Let the distance between foot of ladder to wall x The height of the top of ladder from the base is y x + y = 5, when x = 4. then y = 3 x dx dy + y = dt dt 8 + 3 dy dx =, dy dt = 8 3 Consider (x, y ) be the point of intersection of both the curves x = y, and x y = k Slope of tangent to first curve at (x, y ) is y M M ½ M ½ M
Slope of tangent to second curve at (x, y ) is y x Curves cut at right angles, so, y = x y x =, y = k = or 8k =. Any point on the given line is (k 5, 4k 3, 9k + 6) ½ M If F is foot of perpendicular from P,dr s of PF: (k 7, 4k 7, 9k + 7) PF is perpendiculat to given line, so, (k 7) + 4(4k 7) 9( 9k + 7) = k = and foot of perpendicular is F( 4,, 3) Equation to PF is x 6 3 ½ M 3. Writing correct matrices + Product of two matrices and finding cost of each type of bag Value Section C Questions 4 to 9 carry 6 marks each 4. nh = and f( + rh) = e +rh + 3( + rh) (e x + 3x ) dx = lim n h e +rh + 3( + rh) lim n h[e (e h ) r + 3h r + ] e. lim [ heh (e ) h e h = e e + 5 I = ] + lim log( + cos x). dx 3h n(n+) n + M = log( cos x). dx ½ M I = log(sin x). dx = log(sin x). dx I = log(sin x). dx ½ M I = log(cos x). dx ½ M I = log ( sin x ) I = log(sin x). dx. dx log I = I log, hence I = log ½M 5. [ 3 ] = [ ] A [ 5 ] = [ ] A M 5 6 3 [ 5 ] = [ ] A M 5 3 6 3 6 [ ] = [ ] A, A = [ ] M 5 5 6. For proving one-one function M For proving onto function M For finding f M
7. Let (x, y, z ) be foot of perpendicular from (3,, ) in the plane 3x y + 4z = 3x y + 4 z =, and x 3 = y + = z = k M 3 4 x = 3k + 3, y = k, z = 4k + 3(3k + 3) + (k + ) + 4(4k + ) =, hence k = M Foot of perpendicular is ( 3, 3 ½ M Image of (3,, ) is (,, 3) ½ M (,, 3) lies on x + y + z + 4 = ½ M Eq. to plane through intersection of planes is ( + k)x + ( + 3k)y + ( + 4k)z ( + 5k) = M ( + k) ( + 3k) + ( + 4k) =, implies k = 3 The equation to required plane is x z + = Distance from origin to x z + = is 8. Sketch of each curve M Points of intersection Area function and correct limits Calculation of area M 9. Let the no of packets to be transported from A to P is x, and from A to Q is y. P (6) Q (4) R (5) Factory No of packets x y 6 x y A (6) Cost of transportation per packet 5 4 3 Factory B No of packets 6 x 4 y x + y (9) Cost of transportation per packet 4 5 Cost of transportation: Z = 3x + 4y + 45 Constraints: x, y, x + y 6, x 6, y 4, x + y M M ½ M ½ M Corner points Cost x y 3x + 4y + 45 49 4 6 48 4 67 6 63 Cost is minimum when x =, y = Factory A (6) Factory B (9) No of packets No of packets P (6) Q (4) R (5) 5 5 4 ½ M ½ M