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MODEL TEST PAPER No.1: 2014-15 SUBJECT-MATHEMATICS

CLASS-XII

TIME ALLOWED: 3 HOURS MAX. M.: 100 GENERAL INSTRUCTIONS

(i) All questions are compulsory. (ii) The question paper consists of 26 questions divided into three sections

A, B and C. Section A comprises of 6 questions of one mark each, Section B comprises of 13 questions of 4 marks each and Section C comprises of 7 questions of six marks each.

(iii) There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You

have to attempt only one of the alternatives in all such questions. (iv) Use of calculators is not permitted.

SECTION-A

Question numbers 1 to 6 carry one mark each.

1. A square matrix A, of order 3 has |A|=5, find |A.adj A|.

2. Differentiate 𝑒log 𝑠𝑖𝑛 𝑥 with respect to x.

3. Find the value of x+y from the following equation:

2 𝑥 57 𝑦 − 3

+ 3 −41 2

= 7 6

15 14

4. If 0 𝑎 −45 0 3𝑐 −3 𝑏

is a skey symmetric matrix, find the values of a, b and c.

5. Find the value of 𝜆 such that the line 𝑥−2

9=

𝑦−1

𝜆=

𝑧+3

−6is perpendicular to the plane 3𝑥 − 𝑦 − 2𝑧 = 7.

6. Find 𝑥3 sin (tan −1 𝑥4)

1+𝑥8 𝑑𝑥

SECTION-B Question numbers 7 to 19 carry 4 marks each.

7. If 𝑎 , 𝑏 , 𝑐 are the three vectors such that 𝑎 × 𝑏 = 𝑐 and 𝑏 × 𝑐 = 𝑎 . Prove

that 𝑎 , 𝑏 , 𝑐 are mutually perpendicular vectors and | 𝑏 | = 1, | 𝑐 | = | 𝑎 |.

8. Prove that tan 𝜋

4+

1

2cos−1 𝑎

𝑏 + 𝑡𝑎𝑛

𝜋

4−

1

2cos−1 𝑎

𝑏 =

2𝑏

𝑎

OR

Solve for x:

2

sin−1(1 − 𝑥) − 2 sin−1 𝑥 =𝜋

2

9. If a, b and c are real numbers and

∆= 𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏𝑐 + 𝑎 𝑎 + 𝑏 𝑏 + 𝑐𝑎 + 𝑏 𝑏 + 𝑐 𝑐 + 𝑎

= 0

Show that either a+b+c= 0 or a=b=c.

10. Show that f(x)= |x–3|, ∀ 𝑥 ∈ 𝑅, is continuous but not differentiable at x=3.

11. If 𝑥𝑦 + 𝑦𝑥 = 𝑎𝑏 , find 𝑑𝑦

𝑑𝑥.

OR

If 𝑦 = 𝑎(𝑠𝑖𝑛 𝜃 − 𝜃 cos 𝜃) 𝑎𝑛𝑑 𝑦 = 𝑎 (cos 𝜃 + 𝜃 𝑠𝑖𝑛 𝜃), find 𝑑2𝑦

𝑑𝑥2.

12. For the curve 𝑦 = 4𝑥3 − 2𝑥5, find all points at which the tangent passes

through origin. OR

Find the intervals in which the function f given by f(x)=sin x – cos x, 0 ≤ 𝑥 ≤ 2𝜋is strictly increasing or strictly decreasing.

13. Find log(log 𝑥) + 1

(log 𝑥)2 𝑑𝑥

OR

Evaluate |𝑥3 − 𝑥|𝑑𝑥2

−1

14. Solve the following differential equation:

1 + 𝑥2 + 𝑦2 + 𝑥2𝑦2 + 𝑥𝑦 𝑑𝑦

𝑑𝑥= 0

15. Find the general solution of the differential equation

𝑥2 + 1 𝑑𝑦

𝑑𝑥+ 2𝑥𝑦 = 𝑥2 + 4

16. Show that the vector 𝑎 , 𝑏 , 𝑐 are coplanar if and only if 𝑎 + 𝑏 , 𝑏 + 𝑐 and

𝑐 + 𝑎 are coplanar.

17. Find the distance between the lines 𝑙1𝑎𝑛𝑑 𝑙2 given by

𝑟 = 𝑖 + 2𝑗 − 4𝑘 + 𝜆 (2𝑖 + 3𝑗 + 6𝑘 )and

𝑟 = (3𝑖 + 3𝑗 − 5𝑘 + 𝜇 2𝑖 + 3𝑗 + 6𝑘 .

18. Find the equation of the plane that contains the point (1, –1, 2) and is

perpendicular to each of the planes 2x+3y–2z=5 and x+2y–3z=8.

19. Three balls are drawn one by one without replacement from a bag

containing 5 white and 4 green balls. Find the probability distribution of number of green balls drawn.

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SECTION-C

Question numbers 20 to 26 carry 6 marks each.

20. A helicopter of enemy is flying along the curve y=x2+2. A soldier is

positioned at the point (3, 2). Find the nearest distance between the soldier and the helicopter.

21. Using elementary transformations, find the inverse of the matrix:

0 1 21 2 33 1 1

OR

If A= 1 −1 12 1 −31 1 1

, find A-1 and hence solve the system of linear equations:

x+2y+z=4, –x+y+z=0, x–3y+z=2.

22. Draw a rough sketch of the region enclosed between the circlesx2+y2=4 and

(x–2)2+y2=4.Using integration, find the area of the enclosed region.

23. Consider the binary operation ∗: 𝑅 × 𝑅 → 𝑅 and 𝑜: 𝑅 × 𝑅 → 𝑅 defined as

a∗b=|a–b|and a 𝑜b=a for all a, b ∈ 𝑅. Show that * is commutative but not associative, o is associative but not commutative. Further show that * distributes over the operation o. Does o distribute over * ? Justify your answer.

24. Find 𝑑𝑥

sec 𝑥+sin 𝑥

OR

Find tan 𝑥 𝑑𝑥

25. Assume that the chances of a patient having a heart attack is 40%.

Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance 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. Interpret the result and state which of the above stated methods is more beneficial for the patient.

26. If a young man rides his motorcycle at 25km/hr, he has to spend `2/km on petrol, if he rides it at a faster speed of 40km/hr, the petrol cost increases to `5/km. He has `100 to spend on petrol and wishes to find the maximum

distance he can travel within one hour. Express this as L.P.P. and solve it graphically.