GURU HARKRISHAN PUBLIC SCHOOL, LONI ROAD
PRE-MOCK EXAMINATION-2014-2015
SUBJECT-MATHEMATICS
CLASS XII
Time : 3 Hours Max. Marks : 100
2.
3.
4.
General Instructions
1.
2.
All questions are compulsory.
The question paper consist 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 four marks each and section C comprises of 07 questions of six marks each.
3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
4. 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.
5. Use of calculators is not permitted.
1.
5.
6.
7.
SECTION – A
If the binary operation * defined on Q is defined as a*b=2a+b-ab for all a,b ∈ Q, find the value of 3*4.
Write the value of tan −1 [2 sin(2 cos −1
√3 /2)]
If A is a square matrix of order 3 such that |adj A|=64, find |A|.
For what value of x, the matrix [
5 − 𝑥 𝑥 + 1
2 4
] is singular ?
Write the degree of the differential equation : ( 𝑑𝑦 𝑑𝑥
)
4
+ 3x 𝑑
2 𝑦 𝑑𝑥 2
= 0
Evaluate ∫ 𝜋/2
−𝜋/2 𝑠𝑖𝑛
5 𝑥 dx
SECTION – B
Solve: sin
−1 𝑥 + sin
−1
(1 − 𝑥) = cos
−1 𝑥
8. Let A=N×N and * be a binary operation on A defined by (a,b)*(c,d)=(a+c,b+d). Show that * is commutative and associative. Also, find the identity element for * on A, if any.
OR
Show that the relation R on the set A= {𝑥 ∈ 𝑍: 0 ≤ 𝑥 ≤ 12} ,given by R= {(𝑎, 𝑏): |𝑎 − 𝑏| 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4} is an equivalence relation.
9. Let A= [
0 1
0 0
] , show that (𝑎𝐼 + 𝑏𝐴)
Identity matrix of order 2 and n ∈ N. 𝑛
= 𝑎 𝑛
I + n 𝑎 𝑛−1 bA, where I is the
OR
1

GURU HARKRISHAN PUBLIC SCHOOL, LONI ROAD
Show that A= [
2 −3
3 4
] satisfies the equation 𝑥 2
-6x+17=0. Hence find 𝐴 −1
.
10. Using properties of determinants, prove the following :
|
1 + 𝑎
2
2𝑎𝑏
− 𝑏
2
2𝑏
2𝑎𝑏
1 − 𝑎
2
+ 𝑏
2
−2𝑎
−2𝑏
2𝑎
1 − 𝑎 2 − 𝑏 2
| = (1 + 𝑎
2
+ 𝑏
2
)
3
.
11. If y= sin −1 𝑥
√1−𝑥 2
, 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 (1 − 𝑥
2
) 𝑑 2 𝑦 𝑑𝑥 2
– 3x 𝑑𝑦 𝑑𝑥
– y = 0.
OR
If log( 𝑥
2
+ 𝑦
2
) =2 tan
−1 𝑦 𝑥
, 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑑𝑦 𝑑𝑥
= 𝑥+𝑦 𝑥−𝑦
12. If x=a (cos 𝑡 + log 𝑡𝑎𝑛 𝑡
2
) , y= 𝑎(1 + 𝑆𝑖𝑛𝑡), 𝑓𝑖𝑛𝑑 𝑑 2 𝑦 𝑑𝑥 2
.
13. Find the value of k for which the function
F(x)= {
√1+𝑘𝑥−√1−𝑘𝑥 𝑥
2𝑥+1 𝑥−1
, 𝑖𝑓 − 1 ≤ 𝑥 < 0
, 𝑖𝑓 0 ≤ 𝑥 < 1 is continous at x = 0.
4𝑠𝑖𝑛𝜃
14. Prove that y=
2+𝑐𝑜𝑠𝜃
– 𝜃 is an increasing function of 𝜃 𝑖𝑛 [0, 𝜋
2
] .
15. For the curve y=4 𝑥
3
− 2𝑥
5
, find all the points at which the tangent passes through the origin.
16. ∫ 𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥
√9+16𝑠𝑖𝑛2𝑥 dx
17. ∫
1 cos(𝑥+𝑎)cos (𝑥+𝑏) dx
0
1
18 .
∫ cot
−1 (1 − 𝑥 + 𝑥 2 )𝑑𝑥
OR
∫
0
1 log (1 + 𝑥)
1 + 𝑥 2
19. Show that the differential equation (x-y) 𝑑𝑦 𝑑𝑥
= x + 2y, is homogeneous and solve it.
SECTION-C
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GURU HARKRISHAN PUBLIC SCHOOL, LONI ROAD
20. Find the particular solution of the differential equation :
(x – sin y)dy + (tan y)dx = 0, given that y = 0 when x = 0
21. Prove that tan ( 𝜋
4
+
1
2 𝑐𝑜𝑠
−1 𝑎 𝑏
) + tan ( 𝜋
4
−
1
2 𝑐𝑜𝑠
−1 𝑎 𝑏
) =
2𝑏 𝑎
.
OR
𝑆 how that 2 tan
−1
{𝑡𝑎𝑛 𝛼
2
. 𝑡𝑎𝑛 ( 𝜋
4
− 𝛽
2
)} = tan
−1
{ 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 𝑐𝑜𝑠𝛼+𝑠𝑖𝑛𝛽
} .
22. A dealer in rural area wishes to purchase a number of sewing machines. He has only rupees 560.00 to invest and has space for at most 20 items. Electronic sewing machines cost himrupees 360 and manually operated sewing machine rupees.240 .
He can sell an electronic sewingmachine at a profit of rupees 22 and a manually operated sewing machine at a profit of Rs.18.Assuming that he can sell all the items he can buy, how should he invest his money in orderto maximize his profit. Make it as a Linear Programming Problem and solve it graphically.Justify the values promoted for the selection of the manually operated machines
23. A school wants to award its students for the values of Honesty, Regularity and Hard
Work with a total cash award of Rs 6,000. Three times the award money for Hard
Work added to that given for Honesty amounts to Rs 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity.
Represent the above situation algebraically and find the award money for each value, using matrix method.Apart from these values, namely,Honesty,Regularity and Hard work,suggest one more value which the school must include for awards.
24. A point on the hypotenuse of a right angled triangle is at distances a and b from the sides. Show that the length of hypotenuse is at least (𝑎
2
3
+ 𝑏
2
3
)
3
2 .
OR
Manufacture can sell x items at a price of Rs (5 − 𝑥
100
) each. The cost price of x items is Rs ( 𝑥
5
+ 500) . Find the number of items should sell to earn maximum profit.
25. Evaluate using limit of sums:
1
∫ (3 𝑥 2 + 2𝑥 + 1)𝑑𝑥
0
26. Using integration,find the area of the region enclosed between the two circles 𝑥 2
+ 𝑦 2
=4 and (𝑥 − 2) 2
+ 𝑦 2
= 4.
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