## Kerala Plus One Maths Previous Year Question Paper 2018

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.

Answer any six from question nunbers 1 to 7. Each carries three answers.

Question 1.

Find the sum to n terms of the sequence

4 + 44 + 444 +…………

Question 2.

Solve : Sin 2x – Sin 4x + Sin 6x = 0

Sin 2x – Sin 4x + Sin 6x = 0

Question 3.

If A and B are events such that P(A)= 1/4;

P(B) =1/2; P(A ∩ B) = 1\6 then find: 2 6

(a) P(A or B)

(b) P(not A and not B)

Question 4.

In a A ABC, prove that

Question 5.

(a) The maximum value of the function f(x) = Sin x is………

(i) 1

(ii) √3/2

(iii) 1/2

(iv) 2

(b) Prove that,

(Sin x + Cos x)2 = 1+Sin 2x.

(c) Find the maximum value of Sin x + Cos x.

Question 6.

\(\underset { x\rightarrow 2 }{ Lim } \) [x] = …….

i. 2

ii. 3

iii. 0

iv. does not exist

(b) Evaluate: \(\underset { x\rightarrow 2 }{ Lim } \) \(\frac { { x }^{ 3 }-\quad { 4x }^{ 2 }+\quad 4x }{ { x }^{ 2 }-4 } \)

Question 7.

Once card is drawn at random from a pack of 52 playing cards . Find the probability that,

(a) the card drawn is black.

(b) the card drawn is a face card.

(c) the card drawn is a black face card

Answer any eight from question numbers 8 to 17. Each carries four scores.

Question 8.

(a) If A={a, b, c}, then write Power Set P(A).

(b) If the numbed of subsets with two elements of a set P is 10, then find the total number of elements in set P.

(c) Find the number of elements in the power set of P.

Question 9.

Consider Venn diagram of the Universal Set

U = {1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

U = {1, 2, 3,4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

(a) Write sets A, B in Roster form.

(b) Verify (A ∪ B)=A ∩ B

(c) Find n(A ∩ B).

Question 10.

Consider the following graphs :

(a) Which graph does not represent a function?

(b) Identify the function f(x)=1/x. from the above graphs.

(c) Draw the graph of the function f(x)=(x-1)^{2} .

Question 11.

The figure shows the graph of a function f(x) which is a semi circle centred at origin.

(a) Write the domain and range of f(x).

(b) Define the function f(x).

Question 12.

a. If 3^{2n+2} – 8n – 9 is divisible by ‘k’ for all n ∈ N is true, then which one of the following is a value of ‘k’ ?

i.8

ii. 6

iii. 3

iv. 12

b. Prove by using the principle of Mathematical Induction

P(n) = 1+3+3^{2}+…………….. +3^{n-1 }= \(\frac { { 3 }^{ n }-1 }{ 2 } \) is true for all n ∈ N.

Question 13.

(a) Solve the inequality

(b) Represent the solution on a number line.

Question 14.

(a) Find the n^{th} term of the sequence 3,5,7……………..

(b) Find the sum to n terms of the series. 3xl^{2} + 5 x2^{2} + 7×3^{2}+ ……..

Question 15.

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

Question 16.

Consider a point A(4, 8, 10) in space.

(a) Find the distance of the point A from XY – plane.

(b) Find the distance of the point A from X – axis.

(c) Find the ratio in which the line segmentjoining the point A and B (6,10, -8) is divided by YZ -plane.

Question 17.

(a) Which one of the following sentences is a STATEMENT?

i. 275 is a perfect square.

ii. Mathematics is a difficult subject.

iii. Answer this question.

iv. Today is a rainy day.

(b) Verify by method of contradiction: ‘√2 is irrational’.

Answer any five from question numbers 18 to 24. Each carries six scores.

Question 18.

Consider the quadratic equation x^{2} + x + 1 = 0

(a) Solve the quadratic equation.

(b) Write the polar form of one of the roots.

(c) If the two roots of the given quadratic are α and β . Show that α^{2} = β .

Question 19.

The graphical solution of a system of linear inequalities is shown in the figure.

(a) Find the equation of the lines L_{1}, L_{2}, L_{3
}(b) Find the inequalities representing the solution region.

Question 20.

(a) Which one of the following has its middle term independent

i.\({ \left( x+\frac { 1 }{ x } \right) }^{ 10 }\)

ii. \({ \left( x+\frac { 1 }{ x } \right) }^{ 9 }\)

iii. \({ \left( { x }^{ 2 }+\frac { 1 }{ x } \right) }^{ 9 }\)

iv. \({ \left( { x }^{ 2 }+\frac { 1 }{ x } \right) }^{ 10 }\)

(b) Write the expansion of \({ \left( { x }^{ 2 }+\frac { 3 }{ x } \right) }^{ 4 }\)

(c) Determine whether the expansion of \({ \left( { x }^{ 2 }+\frac { 2 }{ x } \right) }^{ 18 }\) will contain a term containg x^{10}.

Question 21.

The figure shows an ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } \)= 1 and a line

(a) Find the eccentricity and focus of the ellipse.

(b) Find the equation of the line

(c) Find the equation of the line parallel to line L and passing through any one of the foci.

Question 22.

(a) Find the derivative of y = Sin x from the first principle.

Question 23.

Find n, if

(a) 12 x (n-1) p_{3} =5 x (n+1) p_{3}

(b) If ^{n}p_{r} = 840; ^{n}C_{r} = 35 find r.

(c) English alphabet has 5 vowels and 21 consonants. How many 4 letter words with two different vowels and two different consonants canbe formed without repetition of letters?

Question 24.

Consider the following data:

(a) Find the standard deviation of the distribution.

(b) Find the coefficient of variation of the distribution.

Answers

Answer 1.

S = 4 + 44 + 444 + 4444+ …. to n terms

Answer 2.

sin2x – sin4x + sin6x = 0

sin6x + sin2x – sin4x = 0

Answer 3.

Answer 4.

Answer 5.

(a) 1

(b) (sin x + cosx)^{2
}= sin^{2}x + cos^{2}x + 2sinx cosx

= 1 + sin2x

(c) Maximum value of sin2x = 1

Maximum value of sinx + cosx

= \(\sqrt { 1+1 } =\sqrt { 2 } \)

Answer 6.

Answer 7.

Answer 8.

Answer 9.

(a) A= {3,4,6,10} B = {2,3,4,5,11}

(b) (A∪B)’ = { 1, 7, 8, 9, 12, 13}

A’ = { 1, 2, 5, 7, 8, 9, 11, 12, 13}

B’ = {1, 6, 7, 8, 9, 10, 12, 13 }

A’∩B’ = {1,7, 8, 9, 12, 13}

∴ (A∪B)’ =A’ ∩B’

(c) n(A∩B)’ = n(U)-n(A∩B)=13-2=11

Answer 10.

a. (b) or (c)

b. (a)

Answer 11.

(a) Domain = [-4,4]

Range = [0,4]

(b) x^{2} + y^{2}=16

y^{2}= 16-x^{2
}y= \(\sqrt { 16-{ x }^{ 2 } } \)^{
}i.e.,f (x)= \(\sqrt { 16-{ x }^{ 2 } } \)

Answer 12.

(a) 8

(b) P(n):1 + 3 + 3^{2} + ……..3^{n-1} =\(\frac { { 3 }^{ n }-1 }{ 2 } \)

Let P(1): 1 =\(\frac { { 3 }^{ 1 }-1 }{ 2 } \)=2/2=1

Hence, P(1) is true.

P(k): 1 + 3 + 3^{2} +…… + 3^{k-1}=\(\frac { { 3 }^{ k }-1 }{ 2 } \)

To prove that P(k+1) is true. P(k+1):

1 + 3 +3^{2} + + 3^{i_1} +3^{k+1-1} =\(\frac { { 3 }^{ k+1 }-1 }{ 2 } \)

⇒ P(k)+3^{k} = \(\frac { { 3 }^{ k+1 }-1 }{ 2 } \)

⇒ \(\frac { { 3 }^{ k }-1 }{ 2 } \) + 3^{k}= \(\frac { { 3 }^{ k+1 }-1 }{ 2 } \)

= \(\frac { { 3 }^{ k }-1 }{ 2 } \) + 3^{k}= \(\frac { { { 3 }^{ k } }-1+2\times { 3 }^{ k } }{ 2 } \)

Now, LHS = \(\frac { { { 3\times 3 }^{ k } }-1 }{ 2 } \) = \(\frac { { 3 }^{ k+1 }-1 }{ 2 } \) = RHS

Hence, P(k+1) is true.

Hence, P(n) is true for all n∈N

Answer 13.

\(\frac { 2x-1 }{ 3 } \ge \frac { 5(3x-3)-4(2-x) }{ 20 } \)

40x-20 ≥ (15x-10-8+4x)^{3}

40x-20 ≥ -54

40x-57x ≥ -54 + 20

-17x ≥-34

x ≤ 2 ⇒ x∈(-∞, 2)

Answer 14.

(a) t_{n}= 3+(n-1)^{2} = 2n+1

(b) Let T_{n} denotes the nth term of the given series.

T_{n} = (n^{lh} term of 3, 5, 7……….. )(n^{th} term of 1^{2}, 2^{2}, 3^{2}…….. )

= [3+(n-1)^{2}].n^{2} = (2n+1)n^{2} = 2n^{3} + n^{2
}

Answer 15.

Let the equation of the circle be

(x – h)^{2} + (y – k)^{2} = r^{2
}Since the circle passes through (4, 1) and (6,5), we have

(4-h)^{2} + (1 -k)^{2} = r^{2}……….. (1)

and (6 -h)^{2} + (5 – k)^{2} = r^{2}………… (2)

Also since the centre lies on the line 4x + y = 16, we have

4h + k = 16…………. (3)

Solving the equations (1), (2) and (3), we get

h = 3 and k = 4

(4 – 3)^{2} + (1 – 4)^{2} = r^{2
}r^{2}=10

Hence, the equation of the required circle is

(x-3)^{2}+1 (y-4)^{2}= 10

x^{2}-6x + 9 + y^{2}-8y + 16-10 = 0

x^{2} + y^{2 }– 6x – 8y + 15 = 0

Answer 16.

(a) 10 [Z coordinate of the point]

(b) Let P(4,0,0) be a line on the X-axis.

distance = \(\sqrt { { (4-4) }^{ 2 }+{ (8-0) }^{ 2 }+{ (10-0) }^{ 2 } } \)

= \(\sqrt { 64+100 } =\sqrt { 164 } =2\sqrt { 41 } \)

(c) Since the line segment divided the YZ plane, its x coordinate is zero.

So =\(\frac { { mx }_{ 1 }+{ nx }_{ 1 } }{ m+n } \)

⇒ mx_{2} + nx_{1}=0*
*\(\frac { m }{ n } =\frac { { -x }_{ 1 } }{ { x }_{ 2 } } =\frac { -2 }{ 3 } \)

⇒m : n=-2 : 3

Answer 17.

(a) 275 is a perfect square,

(b) Let -√2 is rational.

\(\sqrt { 2 } =\frac { p }{ q } \) , p and q have no common factor.

p^{2} = 2q^{2
}i.e., 2 divides p^{2} is 2 divides p

∴ p = 2k, p^{2} = 4k^{2
}2q^{2} = 4k^{2}, q^{2} = 2k^{2
}2 divides q^{2} is 2 divides q.

p and q have a commonfactor 2.

Answer 18.

Answer 19.

b. The inequalities are: x + y ≤ 4, y ≥ 1, y ≤ x

Answer 20.

Answer 21.

In (1),3x + 5y-12 = 0

Answer 22.

a. Let f[x) = sinx

f(x + h) = sin(x + h)

f (x + h) – f(x)= sin(x + h) – sinx

Answer 23.

(a) 12 x (n-1)P_{3} = 5 x (n+1)P_{3
}12 x (n-1 )(n – 2)(n – 3) = 5 x (n+1 )n(n-1)

12n^{2} – 6Qn + 72 – 5n^{2} – 5n = 0

7n^{2}-65n+72 = 0

\(\frac { 65+\sqrt { 4225-2016 } }{ 14 } =\frac { 67647 }{ 14 } \)

\(\frac { 112 }{ 14 } ,\frac { 18 }{ 14 } =8,\frac { 9 }{ 7 } \)

But n cannot be a fraction. n = 8

(b) ^{n}P = r! x ^{n}C_{r} 840=r! x 35

r! = 24 ⇒ r = 4

Two different vowels can be selected in ^{5}C_{2} ways .Two different consonents can be selected in ^{2I}C_{2 }Ways.

Total selection 4 letters = ^{5}C_{2} x ^{21}C_{2
}Total words = 4! x ^{5}C_{2} x^{21}C_{2
}= 10 x 210 x 24 = 50400

Answer 24.

a.