# Sample Paper – 8 (Mathematics)

12-A Second Floor, Yusuf Sarai New Delhi – 110016
Ph.: 46150335 Mob.: 9810004225
Sample Paper – 8 (Mathematics)
Time : 3 hours
Maximum Marks : 100
General Instructions :
(a) All questions are compulsory.
(b) The question paper consists of 29 questions divided into three sections A, B and C. Section A contains
10 questions of 1 mark each, section B is 12 questions of 4 marks each and section C is of 7 questions of
6 marks each.
(c) There is no overall choice. However internal choice has been provided in section B and C only.
(d) Use of calculator is not permitted. However you may ask for mathematical tables.
Section – A
5 
 3
1. For what value of , the matrix 
 has no inverse?
    1
2. Let A be a non singular matrix of order 3  3 such that |A| = 5. What is |Adj A|?
8  2 7 5
3. Write the matrix x if 3x – 


6 0  0 0
4. Let * be the binary operation defined on R, then of a*b =
(a  b) 2
, which the value of (1*2)*3
3

 1  
5. Write values of : sin sin 1     
 2 3

1
dx
6. Write values of : 
5 x  2 x log x
 /4
7. Write values of :
x

4
sin x dx
 /4
8. What is the value of iˆ.( ˆjxkˆ)  ˆj.(iˆxkˆ)  kˆ.( ˆjxiˆ) ?
9. What is the angle which 3iˆ  6 ˆj  2kˆ makes with z-axis.


15
10. For what value of , a  2iˆ  6 ˆj  5kˆ is parallel to b  3iˆ  ˆj  kˆ ?
2
Section – B
x
y
z
11. Using properties of determinants, prove that x 2 y 2 z 2  ( x  y )( y  z )( z  x)( xy  yz  zx )
yz zx xy
 x3
 | x  3 |  a of

12. For what values of a and b fx   a  b
of
 x3
| x  3 |  2b of

Sample Paper (Maths) – 8 (XII)
1
x3
x  3 is continuous at x = 3
x3
www.eeducationalservices.com
OR
2
d y
dx 2
13. Considered f : R+  [–5, ) given by f(x) = 9x2 + 6x – 5. Show that f is invertible function also find the
inverse of the function f.
14. Using differentials, find the approximate value of 25.3
OR
Find the intervals in which the function defined as f (x) = sin x – cos x, 0 < x < 2 is strictly.
If x = a cos3, y = sin3, then find
x
x
15. Solve the differential equation : y.e y .dx  ( xe y  y )dy
16. Solve the differential equation : cos2x dy = (tan x – y) dx
17. There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items
will include not more than one defective item.
OR
Let x denote larger of the two numbers, when two numbers are chosen from first six positive integers.
Find the probability distribution of x
18. Find the equation of plane passing through the points (2, 3, – 4) and (1, – 1, 3) and parallel to x-axis.
OR

Find the equation of plane passing through (1 – 2, 1) and perpendicular to planes r .(2iˆ  2 ˆj  2kˆ)  0

and r .(iˆ  ˆj  2kˆ)  4
19. Find a vector whose magnitude is 4 and which makes angles


with x=axis, with y-axis and an
4
3
obtuse with z-axis.
20. Solve for x : sin – 1 (1 – x) – 2 sin – 1x =
dy
dx
sin x  cos x

2
21. If xy + yx = a, find
22. Evaluate

9  16 sin 2 x
dx
Section – C
23. Solve the system of linear equations : 4x – 5y – 11z = 12, x – 3y + z = 1, 2x + 3y – 7z = 2
OR
3
1 1
1
2 2
–1
If A =  15 6  5 and B =  1 3
0 find (AB) – 1
5
2 2
0 2 1
24. Using the method of integration, find the area of the region bounded by parabola y2 = x and line x + y =
2.

x tan x
25. Evaluate 
dx
sec x  tan x
0
OR
3
Evaluate  ( x 2  2 x  5)dx, as limit of sum.
1
26. A helicopter is flying along the curve y = x2 + 2. A soldier is placed at the point (3, 2). Find the shortest
distance between the soldier and the helicopter.
27. A girl throw a die, if she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she
gets a 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained
exactly one head, what is eh probability that she threw 1, 2, 3 or 4 with the die?
Sample Paper (Maths) – 8 (XII)
2
www.eeducationalservices.com
28. Find the position vector of foot of perpendicular drawn from point (2iˆ  3 ˆj  5kˆ) to plane

r .(6iˆ  3 ˆj  5kˆ)  74  0.
29. A company manufactures two types of toys A and B. Type A required 5 minutes each for cutting and 10
minutes each for assembling. Type B required 8 minutes each for cutting and 8 minutes each for
assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The
profit is Rs.50 each on type A and Rs.60 each on type B. How many toys of each type should the
company manufacture in a day to maximum the profit?
Sample Paper (Maths) – 8 (XII)
3
www.eeducationalservices.com