9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

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1 Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the domain R ( 1 1) Find if = ,. 4. If A is a squae mati of ode such that adj A = T 64, find A. 5. If A is a squae mati A = I,then what is the invese of A? 6. If f ( ) = 0 sin, find dy d. 7. What is the degee of the following diffeential equation? d y 3 3 dy y = d y + d 3 d d 8. If a and b epesent the two adjacent sides of a paallelogam, the wite the aea of the paalogam in tems of a and b. 9. Find the angle between the vectos a and b. if a = 3, b = 4 and a b = Find the diection of cosines of a line passing though oigin and lying in the fist octant, making equal angles with thee coodinate aes. Section B

2 Available Online Tutoing fo students of classes 4 to 1 in Physics, 11. Show that the elation R in the set A= { : ε z, 0 < <1} given by { } = a, b : a b is divisible by 4 is an equivalence elation. Find the set of all R ( ) elements elated to Solve fo ( ) = ( ) π tan sin tan sec, 0 < < y + y tan sin + cos = y 1 y 13. If none of a, band c is zeo, using popeties of deteminants, pove that: bc b + bc c + bc a + ac ac c + ac = ( bc + ca + ab) 3 dy 1 y 1 1 y = a y, pove that = d If + ( ) m 15. If y = + 1 d y dy then show that ( + ) + = 1 m y 0. d d 16. Find the points of discontinuity of the function f ( ) = on [ 1, ], whee [ ] denotes the geatest intege function.

3 Available Online Tutoing fo students of classes 4 to 1 in Physics, Diffeentiate sin 1 with espect to cos Evaluate 1 d ( log ) cos ( a)( b) d 18. The dot poduct of a vecto with the vectos ι-3k, ι-k and ι + j + 4k ae 0, 5 ad 8 espectively. Find the vecto. π d 19. Using popeties of definite integate evaluate. 0 4 cos 0. Find the equation of a plane passing though the point (1,, 1) and pependicula to the line joining the points (1, 4, ) and (, 3, 5). Also find the pependicula distance of the plane fom the oigin. 1. Find the distance of the pependicula dawn fom the point P(, 4, -1) to the line z 6 = = A biased die is twice, is likely to show an even numbe than an odd numbe. The die is olled 3 times. If occuence of a even no. is consideed a success, the wite the pobability distibution of the numbe of successes. Also find the mean vaiance of the distibution.

4 Available Online Tutoing fo students of classes 4 to 1 in Physics, Section C 3. Using maties, solve the following system of equations: = 4; = 0; = y Z y Z y Z 0, y 0, z 0 4. Show that the nomal at any point Q to the cuve = a cosθ + a θ sinθ and y = a sinθ a cosθ is at the constant distance fom the oigin. 5. Find the aea of the egion {(, y) : 0 y 0, 0 y + ; 0 3} 6. Find the paticula solution of the diffeential equation. y y dy yd y yd dy cos ( ) sin = ( ) given that y = π when = Find the equation of the plane passing though the point (1, 1, 1) and contains the line = 3ι + j + 5k + λ 3ι j + 5k ( ) ( ) Also show that the plane contains the line = ( ι + j + 5k) + λ( ι j 5k). 8. A company sells two diffeent poducts A and B. The two poducts ae poduced in a common poduction pocess which has a total capacity of 500 man hous. It takes 5h to poduce a unit of A and 3 h to poduce a unit of B. The demand in the maket shows that the maimum no. of units of A that can be sold is A and that of B is 15. Pofit on each of A is Rs. 0 and on B is Rs.

5 Available Online Tutoing fo students of classes 4 to 1 in Physics, 15. How many units of A and B should be poduced to maimize the pofit. Fom an LPP and solve it gaphically. 9. Two bags and B contain 4 white and 3 black balls and white and black balls espectively. Fom Bag A, balls ae dawn at andom and the tansfeed to Bag B. A ball is then dawn fom Bag B and is found to be black ball. What is the pobability that the tansfeed balls ae 1 white and 1 black? 30. In an eamination, 10 questions of tue-false type ae asked. A student tosses fai coin to detemine his answe to each question. If the coins falls head, he answes tue and if the coin falls tail, he answes false. Show that the pobability that he answes at least 7 questions coectly is