MOCK TEST PAPER — 2

SECTION — A

Q.1. Consider the binary operation * on Q by :a * b = a + 12b + ab for ,a b Q∈

Find 23b

∗ Ans. 203

Q.2. Find λ such that a

and b

perpendicular, where 2a i j k= λ + + and 5 9 2b i j k= − +

Ans. 165

λ =

Q.3. Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and z– axis respectively.Ans. 6x + 4y + 32 = 12

Q.4. Evaluate : 1

2

1tan1

xe x dxx

−⎛ ⎞+⎜ ⎟⎝ ⎠+∫ Ans. extan–1x+c

Q.5. Using properties of determinants, evaluate : 43 1 635 7 4 017 3 2

=

Q.6. Evaluate : / 2

5 2

/ 2

cosx xdxπ

−π∫ Ans. (0)

Q.7 What is the principal value of

1 12 2cos cos sin sin3 3

− −π π⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Ans. π

Q.8. If ; ;a i j b j k c k i= + = + = + find a unit vector in the direction of a b c+ +

.

Ans. 1 ( )2

i j k+ +

Q.9. Evaluate x, if 2 3 34 5 2 5

xx

= Ans. x = 2

Q.10. If is a square matrix satisfying A2 + A – I = 0, then find A–1. Ans. A–1 = A + I

2 CBSE MATHEMATICS – XII

SECTION — B

Q.11. Let ‘*’ be the binary operation on N given by : a * b = l.c.m. of a and b.Find : (i) 5 * 7, 20 * 16

(ii) Is * cummtative ?(iii) Is * associative ?(iv) Find the identity of * in N.(v) Which elements of N are invertible for the operation ?

Ans. (i) 35, 80 (ii) Yes (iii) Yes (iv) 1 (v) 1

Q.12. Show that : 1 1 13 8 84sin sin cos5 17 85

− − −− =

OR

Show that : 2 2

1 1 2

2 2

1 1 1tan cos4 21 1

x x xx x

− −+ + − π= +

+ − −

Q.13. Using properties of determinants, prove that :

4y z z y

z z x x xyzy x x y

++ =

+

Q.14. Find the value of a and b if the function : 3 1

( ) 11 13 2 1

ax b ifxf x ifx

ax b ifx

+ >⎧⎪ =⎨⎪ − <⎩

Ans. a = 3, b = 2

Q.15. If x = a cos log tan2tt⎡ ⎤+⎢ ⎥⎣ ⎦

and y = a sin t, then find dydx Ans. tan t

OR

If x = 1sina t− , y = 1cosa t

− , show that :

dy ydx x

−=

Q.16. Integrate : 2

2

1( 1)

x xe dxx

⎛ ⎞+⎜ ⎟+⎝ ⎠∫ Ans. 1

1xx e c

x−

++

OR

Evaluate : 20

sin1 cos

x x dxx

π

+∫ Ans. 2

4π

MOCK TEST PAPER – 2 3

Q.17. Obtain the differential equation of the family of the ellipses having foci on y-axis andcurve at the origin.

Ans. 22

2 0d y dy dyxy x ydx dxdx

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

Q.18. Solve the differential equation :

(x + y) dy + (x–y ) dx = 0, given that y = 1 when x = 1 Ans. 1log log 24 2

x π− + +

Q.19. If 0a b c+ + =

, show that a b b c c a× = × = ×

Q.20. Find the co-ordinates of the foot of the perpendicular drawn from the point A (1, 8. 4) to

the line joining B (0, –1, 3) and C (2, –3, –1) Ans. 5 2 19, ,3 3 3−⎛ ⎞

⎜ ⎟⎝ ⎠

Q.21. Prove that 4sin(2 cos )

y θ= − θ

+ θ is an increasing function in 0,

2π⎡ ⎤

⎢ ⎥⎣ ⎦OR

Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1Q.22. Five dices are thrown simultaneously. If the occurrence of an even number is considered a

‘success’, find the probability of at the most 3 successes. Ans. 136

SECTION — C

Q.23. Express the matrix A as the sum of symmetric and a skew-symmetric matrix, where :

A =

3 1 02 0 31 1 2

−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦

ORSolve the following system of linear homogeneous equations :3x + 2y + 7z = 07x –3y – 2z = 05x + 9y + 23z = 0x = k, y = 2k, z = –k; k being any real number.

Q.24. Show that the height of the cylinder, open at the top of given surface area and greatestvolume is equal to the radius of its base.

Q.25. Calculate the area of the region enclosed between the circles :

x2 + y2 = 1 and (x–1)2 + y2 = 1 Ans. 2 33 2

⎛ ⎞π−⎜ ⎟⎜ ⎟

⎝ ⎠ sq.unit.

4 CBSE MATHEMATICS – XII

Q.26. Evaluate : tan xdx∫ Ans. 1 tan 2 tan 11 tan 1 1tan log2 2 2 tan 2 2 tan 2 tan 1

x xx Cx x x

− − +−⎛ ⎞+ +⎜ ⎟

⎝ ⎠ − +Q.27. Find the distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to

the line

2 3 6x y z= = Ans. 1 unit

Q.28. Suppose that reliability of a HIV test is specified as follows:Of people having HIV, 90% of the test detected but 10% go undetected of people free ofHIV, 99% of test are judged HIV +ve but 1% are diagnosed as showing HIV+ve. From alarge population of which only 0.1 to have HIG, one person is selected at random, giventhe HIV test, and the pathologist reports him/her as HIV+ve. What is the probability thatthe person actually has HIV ? Ans. 0.083 app.

Q.29. The demand in the market shows that the minimum number of units of A that can sold is70 and that of B is 120. Profit on each unit of A is Rs. 20 on B is Rs. 15. How many unitsof A and B should be produced to maximise the profit. Form an L.P.P and solve it graphically.

Ans. Maximum profit is Rs. 237525 units of A; 125 units of B