Students can Download 1st PUC Maths Model Question Paper 4 with Answers, Karnataka 1st PUC Maths Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

## Karnataka 1st PUC Maths Model Question Paper 4 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.

If A = {1, 2), B = {3, 4} Find the number of relations from A to B.

Answer:

Ax B = {(1,3), (1,4); (2,3), (2,4)}

Question 2.

Write the power set of set A – {a, b}

Answer:

{a],{b],{a, b] { } = 2^{2} = 4 power set

Question 3.

Express \(\frac { 5\pi ^{ c } }{ 3 } \) in degree measure.

Answer:

Question 4.

Write(1 – i) – (-1 + i6) in the form of a + ib.

Answer:

(1 – i) – ( – 1 + i6) = 1 – i + 1 – i6 = 2 – 7i

Question 5.

Find ‘n’ if =\(^{n} C_{7}=^{n} C_{6}\)

Answer:

17 = 7 + 6 = 13

Question 6.

Find the tenth term of G P. 5, 25, 125 ……..

Answer:

Question 7.

Write the slope of the tine 3x + 2y + 1 = O.

Answer:

\(m=\frac{-3}{2}=\text { slope }\)

Question 8.

Evaluate \(\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}\)

Answer:

Question 9.

Write the converse of “If a number is divisible by 9 then it is divisible by 3″

Answer:

q → p = If a number is divisible by 3 then it is divisible by 9″

Question 10.

If \(\frac{2}{11}\) is the probability of an event A then what is the probability of the event not A?

Answer:

Section – B

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

Question 11.

If A x B = {(a,1),(a,2),(a,3),(6,1),(A,2),(A,3)} then find A and B.

Answer:

A = {a, b} B = {1,2,3}

Question 12.

If U = {x : a ≤ 10, x ∈ N}

A = {x : x ∈ N and x is prime} and B={x : x ∈ N and x is event}

Write A∩B’ in roster form.

Answer:

u= {1,2,3,4,5,6,7,8,9,10}, A = {2,3,5,7,11,13,…}, B = {2,4,6,8,10….}

∴ A∩B’= {2,3,5,7,11,13,………….. } ∩ {1,4,5,7,9,11,….} ={ } = φ

Question 13.

Kind the domain and range of real function \(f(x)=\sqrt{9-x^{2}}\)

Answer:

Domain of f(a) = [-3 < a < 3}

Range of f(x) is x ∈ R^{+}

Question 14.

A wheel makes 360 revolutions in one minute, through how many radians does it turn in one second.

Answer:

Question 15.

If \(\sin A=\frac{3}{5}\) and A is acute ,then find sin 2A

Answer:

Question 16.

Write the multiplicative inverse of 2 – 3i.

Answer:

Question 17.

Solve 3x – 2 < 2x + 1 and represent the solution graphically on number line.

Answer:

Given

3x – 2 < 2x + 1

3x – 2 < 2 + 1

x < 3

Question 18.

Find the equation of the straight line intersecting y – axis at a distance of 2 units above the origin and making an angle 30° with the positive direction of x – axis.

Answer:

:

Question 19.

Find the angle between the lines \(\sqrt{3} x-y+5=0 \text { and } x-\sqrt{3} y-6=0\)

Answer:

Question 20.

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

Answer:

Question 21.

Evaluate

\(\lim _{x \rightarrow 0} \frac{1-\cos x}{x}\)

Answer:

Question 22.

Find the component statements of the compound statement “All integers are positive or negative”.

Answer:

The component statement are

p : All integral are positive

q : All integral are negative

Question 23.

Write the mean of the given data 6, 7, 10, 12, 13, 4, 6, 12

Answer:

Question 24.

Given P (A) = \(\frac{3}{5}\) and P (B) = \(\frac{3}{5}\) find P (A or B). If A and B are mutually exclusive events.

Answer:

Section – C

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

Question 25.

In a group of 600 students in a school, 150 students were found to be taking tea, 225 taking coffee. Find how many students were taking neither tea nor coffee.

Answer:

n(u) = 600, n(T) = 150, n(C) = 225, n(T ∩ C) = 100. n(C ∪ T) = ?

n(C ∪ T) = n(C) + n(T) – n(T ∩ C) =225 + 150 – 100 = 375 – 100 =225

n(C ∪ T)’ =n(u)-n(C ∪ T) =600 – 225 = 375

Question 26.

If f(x) = x^{2} and g (x) = 2x + 1 be two real valued function then find f + g, f – g, (fg).

Answer:

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

- (f+ g)(x) = f{x) + g{x) =x
^{2}+{2x + 1) = x^{2}+2x + 1 or = (x + 1)^{2 } - f – g(x) =f(x )- g(x) = x
^{2}– 2x – 1 - (fg)(x) =f{g(x)) = f(2x + 1) = (2x +1)
^{2}= 4x^{2}+ 1 + 4x

Question 27.

Find the general solution of sin x + six 3x + sin 5.v = 0.

Answer:

Find the G.S. of sin x + six 3x + sin 5x = 0

(sin 5x + e sin x) + sin 3x = 0

Question 28.

Express \(1+i \sqrt{3}\) in polar form

Answer:

Question 29.

Solve the equation :

\(x^{2}+\frac{x}{\sqrt{2}}+1=0\)

Answer:

Question 30.

Find ‘r’ \(5 \times^{4} P_{r}=6 \cdot^{5} P_{r-1}\)

Answer:

Question 31.

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

Question 32.

The sum of first three terms of a G . P. is \(\frac{39}{10} \)and their product is 1. find the common ratio and the terms.

Answer:

Question 33.

Insert three Arithematic mean between 8 and 24.

Answer:

Let the 3 Am are A_{1} A_{2} and A_{3
}8, A_{1} , A_{2} ,24

a = 8, n = 5, d = ? a_{5} = 24

24 – 8 = 4 ⇒ 24 = 4d

∴ d = 6

∴ A_{1} = a + 2d = 8 + 8 = 16

A_{2 = }a + 2d = 8 + 8 = 16

A_{3} = a + 3d = 8 + 12 = 20 .

Question 34.

Find the center and radius of the circle x^{2} + y^{2} + 8x + 10y – 8 = 0

Answer:

2g = 8, 2f = 10, c = -8

∴ g = 4, f = 5,c = -8

Question 35.

Compute the derivative of sin x using first principle method.

Answer:

Question 36.

Verify by the method of contradiction that \(“\sqrt { 2 } \) is an irrational”

Answer:

Question 37.

If E and F are two events such that \(P(E)=\frac { 1 }{ 4 } ,P(F)=\frac { 1 }{ 2 } { and }P(E{ \quad and\quad }F)=\frac { 1 }{ 8 } \)

Answer:

Question 38.

4 cards are drawn from a well-shuffled deck of 52 cards what is the probability of obtaining 3 diamonds and one spade.

Answer:

Section – D

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

Question 39.

Define modulus function. Draw the graph of modulus function. Write down its domain and range.

Answer:

Definition of modulus function: The function f : R→ R defined by f(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

\(\frac{\sin 9 x+\sin 7 x+\sin 3 x+\sin 5 x}{\cos 9 x+\cos 7 x+\cos 3 x+\cos 5 x}=\tan 6 x\)

Answer:

Question 41.

Using mathematical induction prove that

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

Answer:

Question 42.

Solve the system of inequality graphically : 2x + y^{3} ≤4, x + y ≤ 3, 2x – 3y ≤ 6.

Answer:

Question 43.

A group consists of 7 boys and 5 girls. Find the number of ways in which a team of 5 members can be selected so as to have at least one boy and one girl.

Answer:

7 boys, 5 girl, Select 5 members – atleast 1 boy and 1 girl Way of selection

Question 44.

State and prove Binomial theorem for positive integral index of ‘n’

Answer:

Question 45.

If P is the length of perpendicular from the origin to the line whose intercepts on the axes are ‘a’ and ‘b’ then prove that \(\frac{1}{p^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}\)

Answer:

Question 46.

Derive section formula in three dimensions for internal division. Also find the coordinates of the midpoint of the line joining the points P(x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2})

Answer:

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

Let R{x,y,z) divide PQ internally 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 \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\) (φ is being in radius)

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 geometrically that cos (A + B) = cosA CosB – sinA sinB, hence find cos 2A = cos^{2}A – sin^{2} A

Answer:

Prove that cos (x+y) = cos x cos y . sin x siny

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

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

cos(x + x) = cos x cos x-sin x sin x

cos 2x = cos^{2} x – sin^{2} x

(b) Find the sum to n terms of the series, 5 + 11 + 19 + 29 + 41 +……….

Answer:

The given series is not in GP.

assume S_{n} = 5 + 11 + 15 + 29 + 41 +……….

S_{n} = 5 + 11 + 19 + 29 +………. + a_{n }_ _{2} + a_{n t} + a_{n
}0 = 5 + {6 + 8 + 10 + 12 +…… + (n – 1) term} – a_{n}

0 = 5 + [6 + 8 +10 +12 +…….. + (n -1)] – a_{n
}∴ a_{n} = 5+ (n – 1)(n + 4) ⇒ a_{n }= 3n -1

Apply Σ on both sides

Question 50.

(a) Define ellipse and derive its equation in the form \(\frac { x^{ 2 } }{ a^{ 2 } } +\frac { y^{ 2 } }{ b^{ 2 } } =1\)

Answer:

Let F_{1} and be the focii, ‘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 + ve and F_{1} be – ve ∴ the co-ordinate of F_{1}(C_{1},0) and f_{2}(C_{2,}0)

(b) Find the derivative of \(f(x)=\frac{x^{5}-\cos x}{\sin x}= w.r.t\)

Answer: