Plus One Maths Previous Year Question Paper 2018

Kerala Plus One Maths Previous Year Question Paper 2018

Time Allowed: 2½ 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 familiarize yourself with the questions and plan your answers.
• Read instructions carefully.
• Read questions carefully before you answer.
• 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 nonprogrammable calculators, are not allowed in the Examination Hall.

Answer any six from the question numbers 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

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 ΔABC, prove that \(\tan \left(\frac{B-C}{2}\right)=\frac{b-c}{b+c} \cot \frac{A}{2}\)

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 the card is drawn randomly 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 number 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 a Venn diagram of the Universal Set 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 the origin.

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

Question 12.
(a) If 3²ⁿ⁺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+3²+…………….. +3^n-1 = \(\frac { { 3 }^{ n }-1 }{ 2 } \) is true for all n ∈ N.

Question 13.
(a) Solve the inequality \(\frac{2 x-1}{3} \geq \frac{3 x-2}{4}-\frac{2-x}{5}\).
(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. 3 × 1² + 5 × 2² + 7 × 3²+ ……..

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 the XY-plane.
(b) Find the distance of the point A from the X-axis.
(c) Find the ratio in which the line segm­entjoining the point A and B (6, 10, -8) is divided by the 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 x² + 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 α² = β.

Question 19.
The graphical solution of a system of linear inequalities is shown in the figure.

(a) Find the equation of the lines L₁, L₂, L₃
(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 x¹⁰.

Question 21.
The figure shows an ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } \)= 1 and a line L.

(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.
(b) Find \(\frac{d y}{d x}\) if y = \(\frac{x^5-\cos x}{\sin x}\)

Question 23.
Find n, if
(a) 12 × (n-1) p₃ =5 × (n+1) p₃
(b) If ⁿp_r = 840; ⁿC_r = 35 find r.
(c) The English alphabet has 5 vowels and 21 consonants. How many 4 letter words with two different vowels and two different consonants can be 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.

Answer 2.

Answer 3.

Answer 4.

Answer 5.
(a) 1
(b) (sin x + cosx)² = sin²x + cos²x + 2 sinx cosx = 1 + sin2x
(c) Maximum value of sin2x = 1
Maximum value of sinx + cosx = \(\sqrt { 1+1 } =\sqrt { 2 } \)

Answer 6.
(a) (iv) Does not exist.

Answer 7.

Answer 8.
(a) P(A) = {φ, {a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}
(b) nC2 = 10
⇒ \(\frac{n(n-1)}{2}\) = 10
⇒ n2 – n – 20 = 0
⇒ (n – 5)(n + 4) = 0
⇒ n = 5, -4
Total number of elements in set P is 5
(c) Number of elements in power set of P is 25 = 32

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) There are two answers (ii) and (iii).
(vertical line interest at more than two points)
(b) (i)
(c) The graph of f(x) = (x – 1)2 is obtained by shifting the graph of f(x) = x2 to right 1 units.

Answer 11.
(a) Domain = [-4,4]
Range = [0,4]
(b) x² + y²=16
y²= 16-x²
y= \(\sqrt { 16-{ x }^{ 2 } } \)
i.e.,f (x)= \(\sqrt { 16-{ x }^{ 2 } } \)

Answer 12.
(a) 8
(b) P(n):1 + 3 + 3² + ……..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 + 3² +…… + 3^k-1=\(\frac { { 3 }^{ k }-1 }{ 2 } \)
To prove that P(k+1) is true. P(k+1):
1 + 3 +3² + + 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.
(a) \(\frac { 2x-1 }{ 3 } \ge \frac { 5(3x-3)-4(2-x) }{ 20 } \)
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 1², 2², 3²…….. )
= [3+(n-1)²].n² = (2n+1)n² = 2n³ + n²

Answer 15.
Let the equation of the circle be
(x – h)² + (y – k)² = r²
Since the circle passes through (4, 1) and (6,5), we have
(4-h)² + (1 -k)² = r²……….. (1)
and (6 -h)² + (5 – k)² = r²………… (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 = r²
r²=10
Hence, the equation of the required circle is
(x-3)²+1 (y-4)²= 10
x²-6x + 9 + y²-8y + 16-10 = 0
x² + y² – 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 divides the YZ plane, its x coordinate is zero.
So =\(\frac { { mx }_{ 1 }+{ nx }_{ 1 } }{ m+n } \)
⇒ mx₂ + nx₁=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.
p² = 2q²
i.e., 2 divides p² is 2 divides p
∴ p = 2k, p² = 4k²
2q² = 4k², q² = 2k²
2 divides q² is 2 divides q.
p and q have a common factor 2.

Answer 18.

Answer 19.
(a) L1 line passes through (2, 2) and (4, 0)
Slope = \(\frac{2-4}{2-0}\) = -1
Equation is (y – 0) = -1(x – 4) ⇒ x + y = 4
L2 is parallel to the x-axis and passes through (0, 1).
Hence the equation is y = 1
L3 passes through (0, 0) and (2, 2).
Hence the equation is y = x
(b) The inequalities that form the shaded region are x + y ≤ 4; y ≥ 1; y ≤ x

Answer 20.

Answer 21.

Answer 22.

Answer 23.
(a) 12 x (n-1)P₃ = 5 x (n+1)P₃
12 x (n-1 )(n – 2)(n – 3) = 5 x (n+1 )n(n-1)
12n² – 6Qn + 72 – 5n² – 5n = 0
7n²-65n+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) ⁿP = r! x ⁿC_r   840=r! x 35
r! = 24 ⇒ r = 4
Two different vowels can be selected in ⁵C₂ ways. Two different consonants can be selected in ^2IC₂ Ways.
Total selection 4 letters = ⁵C₂ x ²¹C₂
Total words = 4! x ⁵C₂ x²¹C₂
= 10 x 210 x 24 = 50400

Answer 24.

Plus One Maths Previous Year Question Papers and Answers