Model Set C

F.M. 75
Attempt all questions
Group A [ 1×11=11¿
Write the correct option in your answer sheet.
1. For any two real numbers x and y, which one of the following is incorrect?
a)|x + y|≤| x|+| y| b)|x− y|≥|x|+| y|
c) |x− y|≤|x|+| y| ||
x
d) y =
|x|
| y|
1 1 1
2. The sum of the series 1 - 2 + 4 − 8 + …is
3 2 1 1
a) 2 b) 3 c) 2 d) 3
−1 −1 π
3. If cos x +cos y= 2 then x2 + y2 = ….

a. 1 b. 2 c. 3 d. 4

4. Which one of the following expression is a trigonometric equation?

2 tanx tanθ+ tan 2θ
a. 2sinx. cosx = b. 1−tanθ . tan2 θ =tan 3θ
1+ tan 2 x

c. sin2 x+ cos2 x=1 d. 2cos2x – 1 = 0

5. If the vectors 3 + -  and λ - 4  +4 are collinear vectors then the value
of λ is
a. 11 b. 12 c. -12 d. -11
6. The value of k so that the equation
x2 + kxy + 2y2 +3x +5y +2 = 0 to represent a line pair is
−7 5 9 −9
a) 2 or 3 b) 1 or 3 c) 3 or 2 d) 2 or 2
7. If P(A) = 0.4 , P(B) = 0.35 and P(A∪ B) = 0.55 then P(A∩ B ¿is
a) 0.40 b) 0.10 c) 0.30 d) 0.20
( 2 x−1 )6 ( 3 x−1 )4
8. The value of 10 is
(2 x +10)
81 91 71 81
a) 13 b) 17 c) 15 d) 16
9. The interval in which the function f(x) = 2x3 -15x2 +36x +1, decreasing is,
a) (1, 2) b) (2, 3) c) (-∞ , 2) d) (3, ∞ )
−1
dx
10. ∫ =¿ ¿
0 x+2

a.5log2 + C b.2log3 c.-log2 d.4log2+ C

π
2 1
11. The value of ∫ √ sinx dx, n = 4 using Simpson's 3 rule up to 4 places of
0

decimal is
a) 1.1782 b) 2.2375 c) 1.3421 d) 3.2146
Group B [ 5× 8=30 ¿
12.(a) Draw the graph of the function f(x) = x2 + 4x +3 indicating all its
characteristics.
logx logy logz x y z
(b) If y−z = z −x = x− y , prove that x y z =1[3+2]
13.(a)Define implication of two statement. Construct the truth table for
~[ p∨(~ q )] [2]

| || |
a+b b+ c c +a a b c
(b)Prove that b+c c +a a+ b =2 b c a [3]
c+ a a+b b+ c c a b
−1 −1 −1
14.(a) If sin x +sin y +sin z=π ,prove that
x √ 1−x 2 + y √ 1− y 2 + z √ 1−z 2 =2 xyz [3]
(b) Show that the lines joining the origin to the point of intersection of
x2+hxy - y2+gx+fy=0 and fx-gy = λ are at right angle for all values of λ≠0
15. A lot contains 10 items of which 3 are defectives. Three items are chosen
from the lot at random one after another without replacement. Find the
probability that (i) all three are defective (ii) all three are non-defective [2]
(b) Determine the karl Pearson’s coefficient of skewness from the following
frequency distribution [3]
Daily sales 0-10 10-20 20-30 30-40 40-50
No.of shops 2 9 10 7 2

ax+b
16. a. Find from first principle the derivative of √ x [3]
 lim xcos θ−θ cos x
b. Evaluate the limit x →θ x−θ [2]

17. Determine where the graph of the function f(x) = x4 - 8x3+18x2 - 24 is

concave upward and concave downward .

18. Find the root of the equation f(x) = 2x2 + x – 4 = 0, x0= 1 correct to 5 decimal
places with error less than 10-5, by Newton – Raphson method.
dx
∫ 2 2
19. (a) Find the indefinite integral of x √ x +1

(b) Find the condition for the two quadratic equations to have one root
common . [3+2]

Group C [8×3=24 ¿

20(a) Define tautology and contradiction with suitable examples. [2]

(b) Define absolute value of a complex number. For any two complex numbers z
and w prove that |z +w|≤|z|+|w| [1+3]
(c) If A and B are two subsets of universal set U then prove that
A−( B ∪C )=( A−B)∩(A−C ) [2]
21(a) Express r⃗ =(−2 , 16 , 2) as the linear combination of
⃗
a⃗ =( 0 ,3 , 4 ), b=(0 ,0 ,−2) and c⃗ =(1 ,−5 , 0) [3]
(b) Find the angle between the pair of lines
x −¿ 2xy cotθ− y =0 [2]
2 2

(c) Find the direction cosines of l, m, n of two lines which satisfy the equations
l + m + n = 0 and 2lm – m n +2nl = 0 [3]
22(a) Define definite integral of a function.
π
2
Evaluate: ∫ x cosx dx [3]
0

dy [2]
(b) Find dx if x = acos2θ , y=b sin 2 θ
2 2
x y
2
+ 2 =1
(c) Find the area of the ellipse a b [3]