SAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII Time allowed : 3 Hours MAX.MARKS 100 Blue Print. Applicatoin.

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1 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

2 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 = 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 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 =.

3 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 =.

4 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.

5 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 = 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, b = = 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 ( ) = 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 M

6 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 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

7 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 ] = [ ] A M [ ] = [ ] A, A = [ ] M For proving one-one function M For proving onto function M For finding f M

8 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 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 Cost is minimum when x =, y = Factory A (6) Factory B (9) No of packets No of packets P (6) Q (4) R (5) ½ M ½ M

y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent

y mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()