cbse.entrancei.com CLASS 12 MATHEMATICS SAMPLE PAPER – 1 SECTION – A (Questions number 1 to 10 carry 1 marks each) 1 1. Find the principal value of tan 2. Solve tan 1 (1 x) tan 1 (1 x) 2cos 2sin 2 1 1 2 . ( 1 x 1). d2y 3. Find the order and degree of the differential equation. 5 log e 1 dx 2 0 4. Without actual expanding write the value of 54 98 5. If A' 2 3 1 2 and B 6. Differentiate w.r.t. ‘x’, 1 0 1 2 x. 54 98 0 67 . 67 0 ' , then find A 2B . cos(2log2 x ) . 7. Find the distance between the parallel planes r .(2iˆ ˆj 3kˆ) 4 and 6x 3 y 9z 13 0 . 1 0 1 8. A k 1 2 , then write 3 A and find the value of k, if A is singular matrix. 0 1 4 ˆj kˆ having magnitude 8 units. 10. Find the sine of the angle between the vectors a 2iˆ ˆj 3kˆ and b iˆ 3 ˆj 2kˆ . 9. Find a vector in the direction of given vector iˆ SECTION-B (Questions number 11 to 22 carry 4 marks each.) 11. Let f : R R and f ( x) 1 x 0 0 x 0 and g(x)= x , then does fog and gof coinside in [-1,0)? 1 x 0 OR P(X) be defined as A*B = (A-B) , let : P(X) P(X) (B-A), A, B P(X) Given X show that is the identity element for the operation * and all the elements A of P(X) are invertible with A-1=A. 12. By using the properties of determinants prove that : 1 a 1 1 1 1 1 b 1 abc a 1 1 1 c 1 13. Show that tan x 1 1 x 1 x 1 x 1 x 4 1 b 1 cos 1 x. where 2 www.cbse.entrancei.com 1 1 . c 1 2 x 1. cbse.entrancei.com OR 1 Show that , sin 12 13 cos 4 5 1 tan 1 63 16 dy cos 2 (a y ) . 14. If cos y x cos(a y ), prove that dx cos a 15. Verify the applicability of the Rolle’s Theorem for the function f ( x) sin 4 x cos 4 x on [0, /2]. 16. For what value of a and b, the function defined as: 3ax b ; if x 1 11 ; if x 1 f ( x) is continuous at x 1. 5ax 2b ; if x 1 17. Evaluate 18. Evaluate 19. Evaluate sin 8 x cos8 x dx. 1 2sin 2 x.cos 2 x 5x 3 x 2 dx. What should be the ideal role of youth in national integration? 4 x 10 2 log(tan x cot x) dx. 0 20. ˆ b = 3iˆ - 2 ˆj + 7kˆ and c = 2iˆ - ˆj + 4k, ˆ find a vector d which is perpendicular Let a = ˆi + 4 ˆj + 2k, . to both a and b, and c · d =15. "Directionless youth is the burden on nation" comment on it. OR A girl walks 5 km towards west, and then she walks 3 km in a direction 60° east of north and stops. Determine the girl’s displacement from her initial point of departure. Respect the girl implies respect the nation comment on it. 21. Find the length of the perpendicular drawn from the point (2, -3,1) to the line x 1 2 3 y 3 2z 4 . 2 22. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a number greater than 4’ given that ‘there is at least one tail’. OR Let E and F be two independent events. The probability that exactly one of them occurs is 11 2 and the probability of none of them occurring is . If P(T) denotes the probability of 25 25 the occurrence of the event T, then find the value of P(E) and P(F). 2 www.cbse.entrancei.com cbse.entrancei.com SECTION-C (Question number 23 to 29 carry 6 marks each) 23. Using elementary transformations, find the inverse of 1 3 2 3 0 5 . 2 0 5 24. The total cost function and demand function of an item are given x3 7 x 2 111x 50 and p 100 x respectively. Write the total revenue by C ( x) 3 function. Find the number of items when the profit will be maximum. Find the maximum profit also. OR Find the equation of tangents to the curve y = cos (x + y), 2 x 2 that are parallel to the line x + 2y = 0. 25. Find the area lying above x-axis and included between the circle x 2 y 2 8 x and inside the parabola y 2 4 x . 26. Solve the differential equation e 2 y y x dy y dx 1 ; (y 0) and y(1) 2 . OR Solve the differential equation ( x.dy y.dx) y.sin y x ( y.dx x.dy ).x.cos y . x 27. Assume that the chance of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga? Why Meditation and Yoga is necessary and sufficient thing for peace in mind and for good health. 28. Find the vector equation of the line passing through (1, 2, 3) and parallel to the ˆ ˆ ˆ plane r .(i j 2k ) 5 and 3x y z 6. 29. An aero plane can carry a maximum of 200 passengers. A profit of 1000 is made on each executive class ticket and a profit of 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit? How one should respect the hard earn money of parents/guardians in a best economical way. 3 www.cbse.entrancei.com

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