1st PUC Maths Model Question Paper 4 with Answers

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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] { } = 2² = 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) = x² and g (x) = 2x + 1 be two real valued function then find f + g, f – g, (fg).
Answer:
Given f (x) = x², g (x) = 2x + 1

• (f+ g)(x) = f{x) + g{x) =x ²+{2x + 1) = x² +2x + 1  or = (x + 1)²
  
• f – g(x) =f(x )- g(x) = x² – 2x – 1
• (fg)(x) =f{g(x)) = f(2x + 1) = (2x +1)² = 4x² + 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 x⁶ y³ 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 A₁ A₂ and A₃
8, A₁ , A₂ ,24
a = 8, n = 5, d = ? a₅ = 24
24 – 8 = 4 ⇒ 24 = 4d
∴ d = 6
∴ A₁ = a + 2d = 8 + 8 = 16
A_{2  =} a + 2d = 8 + 8 = 16
A₃ = a + 3d = 8 + 12 = 20 .

Question 34.
Find the center and radius of the circle x² + y² + 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 + y³ ≤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(x₁,y₁,z₁) and (x₂,y₂,z₂)
Answer:
Proof : Let P (x₁,y₁,z₁) and (x₂,y₂,z₂) 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 = cos²A – sin² A
Answer:
Prove that cos (x+y) = cos x cos y . sin x siny

(ii) Show that cos2x = cos² x-sinx²x
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 = cos² x – sin² 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 F₁ and be the focii, ‘O’ be the mid point of the line segment F₁F₂. ‘O’ be the origin. And a line from O through F₂ be + ve and F₁ be – ve ∴ the co-ordinate of F₁(C₁,0) and f₂(C_2,0)

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