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## Karnataka 1st PUC Maths Model Question Paper 3 with Answers

Time: 3.15 Hours

Max Marks: 100

Instructions:

1. The question paper has five parts A, B, C, D and E and answer all parts.

2. Part-A carries 10 marks, Part-B carries 20 marks, Part-C carries 30 marks, Part-D carries 20 marks,

Part-E carries 10 marks.

Section – A

I. Answer ALL the questions. Each question carries one mark. 10 x 1 = 10

Question 1.

Write the set in roster form. A = {x : x : is an integer and -3 < x < 7}

Answer:

A = {-2-1,0,1,2,3,4,5,6}

Question 2.

If A = {7,8}, B = {5,4,2}, find A x B.

Answer:

A x B = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}

Question 3.

Convert \(\frac{7 \pi}{6}\) into degree

Answer:

Question 4.

Find the multiplicative inverse of 2 – 3i.

Answer:

Question 5.

Evaluate \(\frac{n !}{(n-r) !}\), when n = 6 and r = 2

Answer:

Question 6.

If \(a_{n}=\frac{2 n-3}{6}\), then find a_{n}

Answer:

Question 7.

Find the slope of the line 3x – 4y – 2 = 0

Answer:

Question 8.

Evaluate

\(\lim _{x \rightarrow 1}\left(\frac{x^{2}+1}{x+100}\right)\)

Answer:

Question 9.

Write the negation of the statement ’”Every natural number is greater than zero”.

Answer:

Every natural number is not greater than Zero.

Question 10.

A coin is tossed three times. Find the number of elements in ‘Sample Space’.

Answer:

Sample space

S = {HHH, HHT, HTT, HTH, TTH, THH, TTH, THT}

Section – B

II. Answer any TEN Questions (10 x 2 = 20)

Question 11.

Let U = {1,2,3,4,5,6}, A = {2,3}, B = {3,4,5}, find (A∪B)’

Answer:

(A∪B)={2,3.4,5}^{1}={1,6}

Question 12.

If X and Y are such that n(X) = 17, n(Y) = 23. n(X∪Y) = 38, find n(X∩Y).

Answer:

n(X∩Y) = n(X) + n(Y) – n(X∪Y)

= 17 + 23 = 38 = 2

Question 13.

If f(x) = 2x – 5, find the value of f(0) and f(7).

Answer:

f(0) = 2(0) – 5 = -5

f(7) = 2(7) – 5 = 14 – 5 =11

Question 14.

If \(\cos x=\frac{-3}{5}\) lies in 3^{rd }quadrant, find the value of tan x.

Answer:

\(\tan x=+\frac{4}{3}\)

(iii quadrant tan θ is +)

Question 15.

Find the value of sin 75°.

Answer:

sin 75° = sin (45° + 30°) = sin45°cos30° + cos45° sin 30°

Question 16.

Express (1 – i)-(-1 +6i) in a + ib form.

Answer:

(1 – i) – (-1 + 6i) = 1 – i +1 – 6i = 2 – 7i

Question 17.

Solve 5x – 3< 7. Show the graph of the solution on number line,

Answer:

5x – 3 < 7

∴ 5x < 10

x < 2

Question 18.

Find the equation of a line passing through the points (-1, 1) and (2, -4).

Answer:

Question 19.

Find the distance between the parallel lines 3x – 4y + 7 = 0 and 3x – 4y + 5 = 0.

Answer:

Question 20.

Show that the points P(-2,3,5), Q(1,2,3) and 8(7,0,-1) are collinear.

Answer:

Question 21.

Find

\(\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}\)

Answer:

Question 22.

Write the ‘Contrapositive’ and ‘Converse’ of the statement “If X is a prime number then X is odd”.

Answer:

Contrapositive : ~ q → ~ p = If x is not an odd no then X is not a prime.

Converse : q → p If x is are add no then X is a prime no.

Question 23.

Co-efficient of variation of a distribution is 60 and its standard deviation is 21. Find its arithmetic mean.

Answer:

Question 24.

One card is drawn from a well shuffled deck of 52 cards. If each out come is equally likely, calculate the probability that the card will be a diamond.

Answer:

Section – C

III. Answer any TEN of the following questions. Each question carries THREE marks. 10 x 3 = 30

Question 25.

In a survey of 400 students in a school, 100 were listed as taking apple juice, 150 as taking orange juice and 75 were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.

Answer:

Question 26.

Find

(i)(f+g) (x)

(ii)(f-g) (x)

(iii) (fg) (x)

Answer:

(i) (f + g)(x) = f(x) +g(x) = x^{2}+2x + 1

(ii) ( f – g) (x) = f(x) – g (x) = x^{2} – 2x – 1

(fg){x) = f{x)-g(x) = x^{2}(2x + 1) = 2x^{3} + x^{2}

Question 27.

Find the general solution of \(\cot x=-\sqrt{3}\)

Answer:

Question 28.

Solve the equation 2x^{2 }+ x + 1 = 0.

Answer:

a = 2, b = 1, c = 1

Question 29.

Represent the complex number \(1+i \sqrt{3} \) in polar form.

Answer:

\(1+i \sqrt{3}=r[\cos \theta+\sin \theta]\)

Question 30.

Find the number of arrangements of the letters of the word “INDEPENDENCE”. In how many of these arrangements.

(i) do the words start with P.

(ii) do the words begin with I and end in P.

Answer:

Indpendence

(i) Wrod begin with P = p ……..

Remaining letter = 11

N repeats = 3

D repeats = 2

F repeats = 4

∴ Total number of arrangment = \(\frac{10 !}{2 ! 3 ! 4 !}[latex]

(ii) (I) …………(P) remaining = 10 letter

N → 3

D → 2

F → 1

[latex]\therefore \text { Total }=\frac{10 !}{3 ! \times 2 ! \times 4 !}\)

Question 31.

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

Answer:

Question 32.

Insert 6 numbers between 3 and 24, such that resulting sequence is an A.P.

Answer:

6 AM’s are A_{l},A_{2},A_{3},A_{4},A_{5},A_{6} between 3 and 24

∴ n = 8, a = 3, a_{n}= 24, d = ?

a_{n}=a + {n-1)d ⇒ 24 = 3+ (8 -1 )d ⇒ 24 – 3 = 7d

21 = 7d ⇒ ∴ d = 3

A_{1} = a + d = 3 + 3 = 6,

A_{2} =9, A_{3} =12, A_{4} =15

A_{5} =18, A_{6}=21.

Question 33.

In a G.P. the 3^{rd} term is 24 and the 6^{th} term is 192. Find the 10^{th} term.

Answer:

Question 34.

Find the centre and radius of the circle x^{2} + y^{2 }– 4x – 8y – 45 = 0.

Answer:

2y = – 4 ⇒ 2f = -8 ⇒ c = -45

∴ g = -2 ⇒ f =-4 ⇒ c = -45

∴ centre = (-g, -f) = (2, 4)

Question 35.

Find \(\lim _{x \rightarrow 0} f(x), \text { where } f(x)=\left\{\begin{array}{cc}

{2 x+3,} & {x \leq 0} \\{3(x+1),} & {x>0}\end{array}\right.[latex]

Answer:

Question 36.

Verify by the method of contradiction \sqrt{2} is irrational.

Answer:

Question 37.

Two dice are thrown and the sum of numbers which come up on the dice is noted. Let us consider the event associated with the experiment A : the sum is even ; B : the sum is multiple of 3. Check whether A and B are mutually exclusive events or not?

Answer:

Sample space =((2,2),(2,3)…………………. (6,6)) 36 ordered pair

A = sum of Even number

(i) A = {(1,1)(1,3),(1,5)(2,2),(2,x),(2,6),(3,1),(3,3),(3,5)}

(4,2),(4,4),(4,6),(5,1)(5,3),(5,5)} (6,2)(6,4)(6,4)} = 18 pair

[latex]\therefore \quad P(A)=\frac { 18 }{ 36 } =\cfrac { 1 }{ 2 } \)

(ii)

Question 38.

Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Ashima will qualify the examination . is 0.10. The probability that both will qualify the examination is 0.02. Find the probability that both Anil and Ashima will not qualify the examination.

Answer:

(Anil qualify the exam) = 0.05 = P(A)

P (Ashima qualify the exam) = 0.10 = P(B)

P (Both will qualify) = 0.02 =P(A∩B)

P (both anil and ashiva will not qualify) =P(A∪B)‘

P(A∪B) = P(A) + P(B)~ P(A∩B) =0.05 + 0.10 – 0.02 = 0. 13

.’. P(A∪B)’ = 1 -(P∪B) =1- 0.13 = 0.87

Section – D

IV. Answer any SIX Questions. 6 x 5 = 30

Question 39.

Define modulus function. Write the graph, domain and range of the function.

Answer:

Definition of modulus function:

The function f: R →R defined by 1(x) = x. For each X∈R is called modulus function. For each non-negative value of n. f(x) = x. But for negative value

Question 40.

Prove that

\(\frac{\cos 4 x+\cos 3 x+\cos 2 x}{\sin 4 x+\sin 3 x+\sin 2 x}=\cot 3 x\)

Answer:

Question 41.

For all n> 1, Prove that

\(1^{2}+2^{2}+3^{2}+\ldots \ldots \ldots n^{2}=\frac{n(n+1)(2 n+1)}{6}\)

Answer:

Hence the given series is true for n = 1, 2…………. k,k + 1…………. for all positive integer of n

Question 42.

Solve the following system of inequation graphically

x +2y ≤ 8

2x + y ≤ 8

x ≥ 0, y ≥ 0

Answer:

Question 43.

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has

(i) no girl

(ii) atleast one boy and one girl. atleast 3 girls

Answer:

Given 4 girls, 7 boys Select 5 members

Question 44.

For any positive integer n . prove that

\((a+b)^{n}=^{n} c_{0} a^{n}+^{n} c_{1} a^{n-1} \cdot b^{1}+^{n} c_{2} \cdot a^{n-2} \cdot b^{2}+\ldots \ldots \ldots+n+n+n\)

Answer:

Question 45.

Derive equation of a line in intercept form.

Answer:

Intercept form

A line l intercept x – axis at A(a ,0) and y – axis at B(0,b) using

Two points form of the line eqn

Question 46.

Let the two points be P(x_{1},y_{1,}z_{l}) and Q(x_{2},y_{2},z_{2}). Let R(x,y,z) divide PQ internally in the ratio m : n. Find the co-ordinates of R

Answer:

Proof: Let P(x_{1} ,y_{1 },z_{1}) and Q(x_{2} ,y_{2 },z_{2}) be the given points.

Let R(x,y,z) divide PQ intenal in the ratio m : n

Draw PL, QM, RN perpendicular to xy-plane.

∴ PL || RN || QM

PL,RN,QM lie in one plane

So that the points L, N, M lie in a straight line which is the intersection of the plane and XY plane.

Through the point R draw a line AB || to the line LM. The line AB intersect the line LP externally at A and the line MQ at B.

Triangle APR and BQR are similar.

Question 47.

Prove that \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\), where x is measured in radians.

Answer:

\(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\)

Proof: Consider a circle with centre ‘O’ and radius ‘r’. Mark two point A and l-3 on the

circumference of the circle so that \(\angle A O B=\theta \) radian.

At ‘A’ draw a tangent to the circle produce

OB to cut the tangent at C. Joint AB.

Draw BM ⊥ OA,

Here OA = OB = r

From the figure

Area of triangle OAB <area of the sector AOB < area of triangle OAC

Question 48.

Find the mean deviation about the mean for the following data.

Answer:

Section E

V. Answer any one question .

Question 49.

(a) Prove that cos(x + y) = cos x . cos y-sin x . sin y. Hence find cos(x – y)

Answer:

Given cos(x+y) = cos x cos y – sin x sin y

(ii) Show that cos2x = cos^{2} x-sinx^{2}x

Take cos (x + y) = cos x cos y – sin x sin y Put y = x

cos(x + x) = cosx cosx – sinx sinx

cos 2x = cos^{2} x – sin^{2} x

(b) Find the sum to n terms of the series, whose n^{th} term is n(n + 3).

Answer:

Given a_{n}=n(n + 3) ⇒ a_{n} = n^{2} +3n .

apply Σ on both side

Σa_{n} = Σn^{2} + 3n

Question 50.

(a) Define an ellipse. Derive equation of an ellipse in the form\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)

Answer:

(a) Let F_{1 }and F_{2} be the focli, O’ be the mid point of the line segment F_{1 }F_{2} . O’ be the origin. And a line from O through F_{2 }be ± e and F_{1 }be ve .the co-ordinate of F_{1}(C .0) and F_{2}(C,0)

(b) Find the derivative of \(\frac{x-\cos x}{\tan x}\) with respect to x

Answer:

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