## Kerala Plus One Maths Model Question Paper 3

Time Allowed: 2 1/2 hours

Cool off time: 15 Minutes

Maximum Marks: 80

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 the questions and to plan your answers.
- Read instructions carefully .
- Read questions carefully before you answering.
- 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 1 to 7 carry 3 score each. Answer any 6.

Question 1.

a. Write set is the subset of all the given sets?

(a) {1,2,3,….}

(b) {1}

(c) {0}

(d) {}

b. Write down the power set of A = {1,2,3}

Question 2.

In any triangle ABC, prove that

a(SinB- sinC) + b(sinC- sinA) +c(sinA- sinB) = 0

Question 3.

a. Solve x^{2} + 2 = 0

b. Find the multiplicative inverse of 2-3i

a. x^{2} + 2 = 0

Question 4.

a. Solve

b. Find the graphical solution of the above inequality.

Question 5.

a. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of atmost 3 girls,

b. Find n if ^{2n}C_{3}: ^{n}C_{3} = 12 : 1

Question 6.

a. The new coordinates of the points (2, 5) if the origin is shifted to the point (1, 1) by translation of axes.

A) (3, 1) B(1, 3) C (-1, 3) D (3, -1)

b. Find what the equation x^{2 }+ xy – 3y^{2}– y + 2 = 0 becomes when the origin is shifted to the point (1,1)?

Question 7.

Evaluate :

Questions from 8 to 17 carry 4 score each. Answer any 8.

Question 8.

Question 9.

Consider the statement 1.3 + 2.3^{2} + 3.3^{3}

a. Verify whether the statement is true for n =1.

b. Prove the result by using principle of mathematical induction.

Question 10.

a. Express (1 + i)^{3 } + (1 – i)^{3 } in a + ib form.

b. Find the polar form of the complex number – 1 – i.

Question 11.

solve the following linear inequalities graphically:

Question 12.

a. “P_{r }=

b. The letters of the word FATHER be permuted and arranged in a dictionary, find the rank of the word FATHER?

Question 13.

a. The distance of the point P(1,-3) from the line 2y – 3x = 4 is

A) 13

B) \(\frac { 7 }{ 13 } \sqrt { 3 } \)

C) √13

D) None of these

b. Reduce the equation√3x + y + 8 = 0 in to normal form. Find the values of P and ω

imagee

Question 14.

a. A conic with e = 0 is known as

A) a parabola

B) an ellipse

C) a hyperbola

D) a circle

b. Consider the circle x^{2 }+ y^{2 }+ 8x + 10y – 8 = 0

i) Find the centre C and radius ‘r’.

ii) Find the equation of the circle with centre at C and passing through the point (1, 2)

Question 15.

a. What is the perpendicular distance from the point P(6,7,8) from XY plane.

A) 8

B) 7

C) 6

D) 9

b. Find the equation of the set of points P, the sum of whose distances from A(4,0,0) and B(-4,0,0) is equal to 10.

Question 16.

a. Convert 20°40‘ into radian measure.

b. If sin x = \(\frac { 12 }{ 13 } \) and x is an acute angle, find the value of cos 2x.

c. Prove that

\(\frac { sinx-sin\quad 3x }{ { sin }^{ 2 }x-{ cos }^{ 2 }x } \)

= 2 sin x.

Question 17.

a. Write the component statements of the following statement: All prime numbers are either even or odd

b. Verify by the method of contradiction, p = √7 is irrational

Questions from 18 to 24 carry 6 score each. Answer any 5.

Question 18.

a. Write the interval (6,12) in the set-builder form.

b. Draw the Venn diagram of the following sets :

i) A’ ∩ B’

ii)A – B

c. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis? How many like tennis only and not cricket?

Question 19.

a. Write the number of terms in the expansion of (a – b)^{2n}

b. Find the general term in the expansion of (x^{2}– yx)^{12}, x ≠ 0

c. Find the coefficient of x^{6 }y^{3} in the expansion of (x + 2y)^{9}

Question 20.

a. The common ratio of the G.P is \(\frac { -4 }{ 5 } \) and the sum to infinity is \(\frac { -80 }{ 9 } \). Find the first term.

b. Evaluate:

c. Find the sum of first n terms of the series : 0.6+0.66+0.666+…….to n terms.

Question 21.

a. Find the derivative of x^{n} from first principles.

Question 22.

a. Find the point of intersection of the lines 2x + y = 5 and x + 3y + 8 = 0.

b. Find the equation of a line passing through the point of intersection of the above lines and parallel to the line 3x + 4y =7.

c. Find the distance between these two paralle lines.

Question 23.

a. Find the Mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from mean is ………

b. Calculate mean, variance and standard deviation of the following distribution:

Question 24.

a. In aleap year the probability of having 53 Sundays is ……..

b. Events E and F are such that P (not E or not F) = 0.25, state whether E and F are mutually exclusive.

c. How many points are there in a sample space, if a card is selected from a pack of 52 cards.

Answer

Answer 1.

a. d) { }

b. P(A)={{1,2,3}, {1,2}, {2,3}, {1,3}, {1}, {2}, {3}, ø}

Answer 2.

LHS = a (sinB-sinC) + 6(sinC-sinA) +c (sinA-sinB)

= 2R sin A(sinB-sinC) + 2R sin B (sinC-sinA)+2R sinC (sinA – sinB)

= 2R [sin A sin B – sinAsinC + sinB sinC – sinBsinA + sinCsinA – sin C sinB]

= 2R x 0 = 0 = RHS

Answer 3.

Answer 4.

Answer 5.

Answer 6.

a. B) (1, 3) since [2 – 1, 5 – 2]

b. Let the coordinates of a point P changes from (x,y) to (X,Y) when origin is shifted to (1,1)

∴ x = X + 1, y = Y + 1 Substituting in the given equation, we get

(X+1)^{2} + (X + 1) (7 + 1) -3 (7 + 1)^{2 }-(7 + 1) + 2 = 0

⇒ X^{2}+2X + 1 + XY + X+ Y+ 1-3 (P + 27 + 1) – (7+ 1) + 2 – 0

⇒ X^{2} – 3Y^{2} + XY + 3X- 6Y = 0

∴ Equation in new system is

X^{2 }– 3Y^{2} + XY + 3X -6Y = 0

Answer 7.

Answer 8.

Answer 9.

Answer 10.

Answer 11.

Answer 12.

a. D

b. The alphabetical order of the word ‘ FATHER are A, E, F, H, R, T No. of words beginning with

Answer 13.

Answer 14.

a. D

b. Comparing x^{2} + y^{1} – 8x +10 y – 12 = 0 with

Answer 15.

a. A. 8 [ || perpendicular distance of a point from XY plane = z coordinate

b. Let P(x, y, z) be the point such that PA + PB = 10

Answer 16.

Answer 17.

a. p: All prime numbers are even. q: All prime numbers are odd

b. Let us assume that √7 is a rational number

∴ √7 = , where a and b are co-prime, i.e. a and b have no common factors, which implies that 7b^{2 }– a^{2} ⇒ 7 divides a.

∴ there exists an integer ‘k’ such that a = 7k

a^{2 }= 49k^{2} ⇒ 7b^{2 }= 49k^{2}⇒b^{2}= 7k^{2} 7 divides b.

i.e., 7 divides both a and b, which is contradiction to our assumption that a and b have no common factor.

∴ our supposition √7 is wrong, is an irrational number.

Answer 18.

a. {x : x ∈ R, 6 < x ≤ 12}

Answer 19.

a. Number of terms in expansion can be given by 2n + 1

b. General term can be given by

Answer 20.

Answer 21.

Answer 22.

a. 2x + y =5 ……………. (1)

x + 3y = -8………………….. (2)

(1) x 3 – (2) ⇒

6x + 3y = 15 x + 3y = -8

(-) (-) (+)

————-

5x = 7 7

∴ x = 7/5

b. Slope of the required line, m = \(-\frac { 3 }{ 4 } \)

|| slope of the parallel line Equation of the line is y – y = m (x – x_{1})

Answer 23.

a. b

c. The parallel lines 15x + 20y – 57 = 0 and

3x + 4y – 7 = 0 ⇒15x + 20y – 35=0 x ing by 5

Parallel distance

Answer 24.

a. (b) 7/3

b. (not E or not F)

*= *P(E ‘∪ F’) = P(E ∩ F)’ = 1 = 1P(E ∩ F)

⇒ 0.25 = 1 -P(E n F) ⇒ p(E ∩ F) = 1 -0.25 = 0.75 ≠ 0

⇒ E and F are not mutually exclusive

c. 52