SECTION A 1. Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is 2iˆ

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1 Session: Subject: Mathematics Class XII Duration: 3 hr. M.M: 100 General Instructions: (i) All questions are compulsor. (ii) This question paper contains 9 questions. (iii) Question 1 in Section A are ver short-answer tpe questions carring 1 mark each. (iv) Question 5 1 in Section B are short-answer tpe questions carring marks each. (v) Question 13 3 in Section C are long-answer-i tpe questions carring marks each. (vi) Question 9 in Section D are long-answer-ii tpe questions carring marks each. (vii) Use of calculators is not permitted. SECTION A 1. Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is iˆ 3ˆj kˆ.. If A is a square matri such that A = 5. Write the value of AA t. 3. If f() = + 7 and g() = 7, R, then find fog (7).. Find the integrating factor of the differential equation: 1. SECTION B e d d

2 Let A, B 3 and C 7. Find a matri D such that CD AB = If, then find that d d 1 log 7. Sand is pouring from a pipe at the rate of 1 cm /sec. The falling sand form a cone on the ground in such a wa that the height of the curve is alwas one-sith of the radius of the base. How fast in the height of the sand cone increasing when the height is cm? 8. Find the approimate change in the volume V of a cube of side metres caused b increasing the side b %. 9. Evaluate: 9 log d. 10. Form the differential equation of the famil of ellipses having foci on - ais and centre at the origin. 11. If the sum of two unit vectors is a unit vector, show that magnitude of their difference is A die is thrown twice and the sum of the numbers appearing is observed to be 7. What is the conditional probabilit that the number has appeared atleast once? SECTION C 13. Evaluate: d

3 1 Evaluate: cot 1 1 d 0 1. Find the shortest distance between the following lines: r ( 1 t)ˆ i ( t ) ˆj (3 t) kˆ and r ( s 1)ˆ i (s 1) ˆj (s 1) kˆ Find the equation of the plane passing through the line of intersection of the planes + z = 3 and z + 9 = 0 and is parallel to the line z Evaluate: d Prove that sin tan Solve the equation for : cos (tan -1 ) = sin (cot -1 3/). 17. For what value of k is the following function continuous at? f ( ) 3 sin cos k,, 18. The monthl income of Aran and Babban are in the ratio 3: and their monthl ependiture are in the ratio 5:7. If each saves Rs per

4 month, find their monthl incomes using matri method. Which value is reflected in this problem? 19. Differentiate tan w. r. t.cos Find the general solution of the differential equation: (1 + tan) (d d) + d = 0. Solve the following differential equation: 1 e d e 1 d If a b c d and a c b d, show that a d is parallel to b c, where a d and b c.. In a certain college % of bos and 1% of girls are taller than 1.75 metres. Furthermore, 0% of that students in the college are girls. A student is selected at random from the college and is found to be taller than 1.75 metres. Find the probabilit that the selected student is a girl. 3. Prove that if E and F are independent events, then so are the events E and F. SECTION D. Find the equation of the plane which contains the line of intersection of the planes + + 3z = 0 and + z + 5 = 0 and whose intercept is twice its z intercept. Hence write the vector equation of a plane passing through the point (, 3, -1) and parallel to the plane obtained above.

5 5. Show that the semi-vertical angle of the cone of the maimum volume and of given slant height is sin 1. 3 If the function f() = 3 9m + 1m + 1, where m > 0 attains its maimum and minimum at p and q respectivel such that p = q, then find the value of m.. On the set {0, 1,, 3,, 5, } a binar operation * is defined as: a b, if a b 7 a * b a b 7, if a b 7 Write the operation table of the operation * and prove that zero is the identit for this operation and each element a 0 of the set is invertible with 7 a being the inverse of a. 7. Using propert of determinants, prove that z z z z z z 3 z z If 1 0 A 0 1 and A 3 A + 7A + ki3 = 0 find k A compan manufactures two tpes of cardigans: tpe A and tpe B. It costs Rs 30 to make a tpe A cardigan and Rs 10 to make tpe B cardigan. The compan can make at most 300 cardigans and spend at most Rs 7000 a da. The number of cardigans of tpe B cannot eceed

6 the number of cardigans of tpe A b more than 00. The compan makes a profit of Rs 100 for each cardigan of tpe A and Rs 50 for ever cardigan of tpe B. Formulate this problem as a linear programming problem to maimize the profit of a compan. Solve it graphicall and find maimum profit. 9. Using integration, find the area of the curves 1, which is eterior to the parabola = 0.