Kerala Plus Two Maths Model Question Paper 3
Time : 2 1/2 Hours
Cool off time : 15 Minutes
Maximum : 80 Score
General Instructions to Candidates :
- There is a ‘Cool off time’ of 15 minutes in addition to the writing time.
- Use the ‘Cool off time’ to get familiar with questions and to plan your answers.
- Read questions carefully before you answering.
- Read the instructions careully.
- When you select a question, all the sub-questions must be answered from the same question itself.
- Calculations, figures and graphs should be shown in the answer sheet itself.
- Malayalam version of the questions is also provided.
- Give equations wherever necessary.
- Electronic devices except non programmable calculators are not allowed in the Examination Hall.
QUESTIONS
Question 1 to 7 carry 3 scores each. Answer any six questions only
Question 1.
a. Let f : R → R be a function defined by f (x) = x3 + 5. Then f1 (x) is
i. (x+5)1/3
ii. (x-5)1/3
iii. (5-x)1/3
iv. 5-x
b. Let * be a binary operation defined on Q
a*b = a-b + ab. Check whether
i. It is commutative?
ii. Is * associative ?
Question 2.
Question 3.
Question 4.
Question 5.
Prove that the function f given by f (x)= log sin x is strictly increasing on ( 0,\(\frac { \pi }{ 2 } \))
Question 6.
Question 7.
Question 8 to 17 carry 4 scores each. Answer any eight questions only
Question 8.
a. Show that the relation R in set of real numbers defined as R = {(a,b): a < b2} is neither reflexive nor symmetric not transitive.
b. Show that the operation * on Q, defined by a*b = a+b-ab is commutative, and ex-its and identity elements find it.
Question 9.
a. The principal value of the expression cos-1 cos (680) is …………..
Question 10.
Question 11.
Question 12.
Question 13.
Question 14.
Question 15.
a. Find the distance between the planes x-y + z-5 = 0 and 2x-2y + 2z = 0.
b. Write the vector equation corresponding to Cartesian equation of a line
Question 16.
Find the shortest distance between the lines
Question 17.
Question 18 to 25 carry 6 scores each. Answer any 5 questions only
Question 18.
Question 19.
a. Use differential to approximate (0.999)1/10
b. A window is in the form of rectangle sur-mounted by a semicircular opening. The total perimeter of the windows is 1 Om. Find the dimensions of the window to admit maximum light through the whole opening.
Question 20.
Find the area lying above x axis and included between the circle x2 + y2 = 8x and inside of the parabola y2 = 4x. Also draw a neat diagram.
Question 21.
Evaluate:
Question 22.
Minimize and maximize Z = 5x + 10 y subject to x + 2y < 120, x + y > 60, x – 2y > 0, x, y > 0
a. Draw the feasible region
b. Find the comer points
c. Find the maximum and minimum profit.
Question 23.
a. Find the distance of the point (-1, -5, -10) from the point of intersection of the line
Question 24.
a. Two numbers are selected at random (with-out replacement) from the Pt six positive integers.
Let X denote the larger of the two numbers obtained. Find E(X) and Var(X)
b. A card from a pack of 52 cards is lost from the remaining X cards of the pack, two cards are drawn and are found to be both spades. Find the probability of the lost card being a spade.
ANSWERS
Answer 1.
a. ii
b. a * b = a – b + ab
b*a = b- a + ba = b – a + ab
∴ * is not commutative.
(a,b) * c = d * c = d- c + dc
= a- b + ab – c + ac – bc + abc
= a- b + ab – c + ac – bc + abc
= a – b – c + ab – bc + ca + abc
a * (b * c) = a * d = a- d + ad
= a – (b – c + be) + a (b – c + be)
= a- b + c – bc + ab – ac + abc
= a – b + c ab – be + ca + abc
∴ * is not associative.
Answer 2.
Answer 3.
We are giving that
Answer 4.
Answer 5.
Answer 6.
Answer 7.
Answer 8.
a. R = {(a,b): a < b2}
Relation R is defined in the set of real numbers.
i. Reflexive
Consider a ∈ R
If a ∈ R ⇒ a < a2 which is false (a, a) ∈ R
R is not reflexive.
ii. Symmetric
Let a,b ∈ R and
(a,b) ∈ R ⇒ a < b2 and b < a2,
which is false ⇒ (a,b) ∈ R, but (b,a) ∈ R
∴ R is not symmetric.
iii. Transitive
Let a, b, c ∈ R
Answer 9.
Answer 10.
Answer 11.
Answer 12.
Answer 13.
Answer 14.
Answer 15.
Answer 16.
Answer 17.
Answer 18.
Answer 19.
Answer 20.
The given equation of the circle x2 + y2 = 8x can be expressed as (x – 4)2 + y2 = 16. Thus, the centre of the circle is (4,0) and radius is 4. Its intersection with the parabola y2 = 4x gives
x2 + 4x = 8x
or
x2 – 4x = 0
or
x(x-4) = 0
or
x = 0, x = 4
Thus, the point of intersection of these two curves are 0 (0,0) and P (4,4) above the x-axis.
From the above the required area of the region OPQCO included between these two curves above x-axis is
= (area of the region OCPO)
+ (area of the region PCQP)
Answer 21.
Answer 22.
a. The feasible region determine! by the constraints,
x + 2y < 120, x + y > 60, x – 2y > 0,
x > 0 and y > 0
is as follows.
b. The comer points of the feasible are region are A(60,0), C(60,30) and D (40,20).
The values of Z at these comer points are as follows.
Corner point | Z=5x + 10y | |
A (60,0) | 300 | → Minimum |
B (120,0) | 600 | → Maximum |
C (60,30) | 600 | → Maximum |
D (40,20) | 600 |
a. The minimum value of Z is 300 at (60,0) and the maximum value of Z is 600 at all the points on the line segment joining (120,0) and (60,30)
Answer 23.
Answer 24.
a. The two positive integers can be sele-cted from the fist six positive integers without replacement in 6 x 5 = 30 ways.
X represents the larger of the two numbers obtained. Therefore, X can take the value of 2,3,4,5 or 6.
For X=2, the possible observations are (1,1)and(2,1)
∴ P (x = 2) = \(\frac { 2 }{ 30 } \) = \(\frac { 1 }{ 15 } \)
For X = 3 the possible observations are (1,3), (2,3), (3,1) and (3,2).
∴ p (x = 3) = \(\frac { 4 }{ 30 } \) = \(\frac { 2 }{ 15 } \)
For x = 4 the possible observations are
(1,4), (2,4), (3,4), (4,3), (4,2) and (4,1).
∴ p (x = 4) = \(\frac { 6 }{ 30 } \) = \(\frac { 1 }{ 5 } \)
For X = 5, the possible observations are (1.5) , (2,5), (3,5), (4,5), (5,4), (5,3), (5,2) and (5,1).
∴ p (x = 5) = \(\frac { 8 }{ 30 } \) = \(\frac { 4 }{ 15 } \)
For X = 6, the possible observations are (1.6), (2,6), (3,6), (4,6), (5,6), (6,4), (6,3), (6,2) and (6,1)
∴ p (x = 6) = \(\frac { 10 }{ 30 } \) = \(\frac { 1 }{ 3 } \)
Therefore, the required probability distribution is as follows.
b. Let E and E, be the respective events of choosing a spade card and a card which is not spade. Out of 52 cards, 13 cards are spade and 39 cards are not spade.