Answer Question No. 1 (compulsory) from Part I and ten questions from Part II, choosing four questions from Section A, two questions from Section B, two questions from Section C and two questions from Section D.
All working including rough work should be done on the same sheet as the rest of the answer.
The intended marks for questions or part of questions are given in brackets [ ].
Mathematical tables and squared paper are provided.
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PART I
Answer all questions
Question 1. (i) Using Bionomial Theorem, find the value of :
(√2 + 1)6 + (√2 − 1)6
Answer:Answer:198
how to solve it
(Contributed by : saad99)
(ii) Use the principle of mathematical induction to prove that :
4 + 8 + 12 + … + 4n = 2n (n + 1)
Answer:Answer:
(iii) In how many ways can three prizes be distributed among four boys when no boy gets more than one prize?
Answer:Answer:24
(iv) Use Cramer's rule to solve the following system of equations :
2x − y = 17, 3x + 5y = 6
Answer:Answer:x = 7, y = −3
(v) If A =
⌈ ⌊
3 −2 4 −2
⌉ ⌋
is a 2 × 2 matrix, find the scalar k such that A2 = k A − 2 I.
Answer:Answer:1
(vi) If the minor axis of an ellipse is equal to the distance between its focii, prove that its eccentricity is 1/√2.
Answer:Answer:
(vii) Differentiate √(sin x3)w.r.t. x.
Answer:Answer:
dy
dx
=
3 x2 cos x3
2 √(sin x3)
(viii) Evaluate :
∫
tan (sin−1x)
√(1 − x2)
dx
Answer:Answer:log | sec (sin−1x) | + C = −½ log | 1 − x2 | + C
(ix) Calculate the mean deviation from the mean of the following data :
x:
2
4
6
8
10
f:
1
4
6
4
1
Answer:Answer:1.5
6
(Contributed by : aryas)
(x) Three dice are thrown together. Find the probability of getting a sum of at least 5.
Answer:Answer:53/54
PART II
SECTION A
Answer any four questions
Question 2. (a) Find the tenth term in the expansion of (2x2 + 1/x)12.
Answer:Answer:1760 x−3
(b) Out of 5 men and 2 women, a committee of 3 is to be formed. In how many ways can it be done so as to include at least 1 woman?
Answer:Answer:25
I DON'T KNOW THW ANSWER BUT THE ANSWER 25 IS NOT RIGHT OR GIVE EXPLANATION HOW IT COMES
(Contributed by : aryas)
Question 3. (a) Using properties of determinants, prove that :
| | | | |
1 b + cb2 + c2
1 c + ac2 + a2
1 a + b a2 + b2
| | | | |
= (b − c) (c − a) (a − b).
Answer:Answer:
(b) Find the coordinates of vertex, focus and the length of latus rectum of the parabola y2 = 6(x + y).
Answer:Answer:Vertex is (−3/2, 3); Focus is (0, 3); Length of latus rectum = 6 units
latus rectum-3 focus-(-9/8,3)
(Contributed by : tanyadutta)
Question 4.
If A =
⌈ | | | ⌊
0 1 2
1 2 3
3 1 1
⌉ | | | ⌋
(i) Find A−1.
Answer:Answer:
A−1 = −½
⌈ | | | ⌊
−1 1 −1
8 −6 2
−5 3 −1
⌉ | | | ⌋
(ii) Hence, solve the following system of equations : y + 2z + 8 = 0, x + 2y + 3z + 14 = 0, 3x + y + z +8 = 0
Answer:Answer:x = −1, y = −2, z = −3
Question 5. (a) If y = [x + √(x2 + a2)]n, prove that
dy
dx
=
ny
√(x2 + a2)
Answer:Answer:
(b) If y = etan−1x, show that
(1 + x2)
d2y
dx2
+ (2x − 1)
dy
dx
= 0.
Answer:Answer:
Question 6. Evaluate the following : (a)
∫
cos x
√(sin2 x + 2 sin x + 3)
dx
Answer:Answer:log | sin x + 1 + √(sin2 x + 2 sin x + 3) | + C
(b)
∫
x + 3
√(2 − x2)
dx
Answer:Answer:−√(2 − x2) + 3 sin−1x/√2 + C
Question 7. (a) Show that the height of a closed cylinder of given surface area and maximum volume is equal to the diameter of its base.
Answer:Answer:COMING SOON!
(b) Differentiate (sin x)cos−1x w.r.t. x.
Answer:Answer:(sin x)cos−1x [cot x cos−1x − log(sin x)/√(1 − x2)]
SECTION B
Answer any two questions
Question 8. Evaluate the following : (a)
1 ∫ 0
sin−1xdx
Answer:Answer:π/2 − 1
-1
(Contributed by : tanyadutta)
(b)
π/2 ∫ 0
sin3/2x
sin3/2x + cos3/2x
dx
Answer:Answer:π/4
Question 9. Find the area enclosed by the curve y = 3x− x2 and the x-axis. If this area is rotated through four right angles about the x-axis, find the volume of the solid generated.
Answer:Answer:Area = 4.5 square units; Volume = 8.1π cubic units
Question 10. Solve the following differential equations: (a)
x
dy
dx
−3y = x2
Answer:Answer:y = Cx3 − x2
(b)
dy
dx
= 1 − x + y − xy
Answer:Answer:log | 1 + y | = x − x2/2 + C
SECTION C
Answer any two questions
Question 11. (a) Calculate the standard deviation of the following data :
Class :
1 - 3
3 - 5
5 - 7
7 - 9
9 - 11
11 - 13
13 - 15
Frequency :
3
9
25
35
17
10
1
Answer:Answer:2.47
(b) Using 2000 as base year, the index numbers for the price of a commodity in 2001 and 2002 are 118 and 125 respectively. Calculate the index numbers for 2000 and 2002, if 2001 is taken as base year.
Answer:Answer:Index number for 2000 = 84.75; Index number for 2002 = 105.93
Question 12. Calculate Karl Pearson's coefficient of correlation between the marks in English and Hindi obtained by 9 students :
Marks in English (x) :
25
17
21
23
12
18
22
15
19
Marks in Hindi (y) :
22
20
24
25
17
11
18
25
16
[Take the assumed mean for x as 12 and that for y as 17].
Answer:Answer:0.284
Question 13. (a) A problem is given to three students whose chances of solving it are 1/2, 1/3 and 1/4 respectively. What is the probability that the problem will not be solved by any of them?
Answer:Answer:1/4
(b) If 10 unbiased coins are tossed, what is the probability of getting not more than 3 heads?
Answer:Answer:11/64
SECTION D
Answer any two questions
Question 14. (a) Calculate the banker's gain on Rs. 2500 due in 6 months at 5% per annum.
Answer:Answer:Rs. 1.52
(b) A bill for Rs. 1000 drawn on May 7 for 6 months was discounted on August 29 of the same year, for a cash payment of Rs. 988. Find the rate of interest charged by the bank.
Answer:Answer:6% per annum
Question 15. (a) A person borrowed a sum of money and returned it in three equal quarterly instalments of Rs. 4630.50 each. If the rate of interest charged was 20% per annum, find: (i) the sum of money borrowed
Answer:Answer:Rs. 12610
(ii) the total interest charged.
Answer:Answer:Rs. 1281.50
(b) A retailer asks his supplier to make out a single bill in lieu of the following bills: Rs. 2528 payable on January 6, 2007, Rs. 3860 on February 13, 2007, Rs. 5000 on February 26, 2007 and Rs. 1650 on March 9, 2007. What would be the due date of the single bill?
Answer:Answer:February 14, 2007
Question 16. (a) The fixed cost of a new product is Rs. 30000 and the variable cost per unit is Rs. 800. If the demand function is given by p(x) = 4500 − 100x, find the break-even points.
Answer:Answer:12, 25
(b) The cost function of a firm is given by C(x) = 300x − 10x2 + x3/3. Calculate the output x at which the marginal cost is minimum.