1st PUC Maths Model Question Paper 1 with Answers

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Karnataka 1st PUC Maths Model Question Paper 1 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 interval (-3, 0) in set builder form.
Answer:
A -{x: x ∈ z, – 3 < x < 0}

Question 2.
If (x + 1, y – 2) = (3, 1). Find the values of a
Answer:
x +1 = 3      ∴ x = 2
y – 2 – 1       ∴ y = 3

Question 3.
If cos x \(\frac{-3}{5}\) ,x lies in third quadrant, find the value of value of x
Answer:
If cos x \(\frac{-3}{5}\)
In III quadrant only tan a and cot x all + ve

Question 4.
Find the multiplicative inverse of 2 – 3i
Answer:

Question 5.
Find the value of \(\frac{7 !}{5 !}\)
Answer:

Question 6.
Find the sixth term of the sequence \(a_{n}=\frac{n}{n+1}\)
Answer:

Question 7.
Find the slope of the line passing through (3, -2) and (-1, 4).
Answer:

Question 8.
Find the derivative of x² – 2 at x 10.
Answer:
f'(x) = 2x ⇒ ∴ f'{10) = 2(10) = 20

Question 9.
Write the negation of \(\sqrt{2}\) is not a complex number”.
Answer:
p = \(\sqrt{2}\) is a complex number.

Question 10.
Two coins (a one rupee coin and a two rupee coin) are tossed once. Write the sample space
Answer:
Sample space = {HH,HT,TH,TT}

Section – B

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

Question 11.
If U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4} and B = {2,4,6,8} Find (A∪B)’.
Answer:
(A∪B)’ = {1,2,3,4,6,8}¹ = {5,1,9}                                                            .

Question 12.
In a school, there pre 20 teachers who teach mathematics or Physics. Of these, 12 teach Mathematics and 4 teach both Physics and Mathematics. How’ many teach physics?
Answer:

Question 13.
If A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6} find A x (B∪C).
Answer:

Question 14.
Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm
\(\left(\text { use } \pi=\frac{22}{7}\right)\)
Answer:

Question 15.
Find the value of sin 15°.
Answer:

Question 16.
\(\text { If }\left(\frac{1+i}{1-i}\right)^{m}=1\). Then find the least positive integral value of m
Answer:

Question 17.
Solve 5x – 3>3x – 5 and show the graph of the solution on the number line.
Answer:

Question 18.
Find the distance of the points (3, -5) from the line 3x – 4y – 26 = 0
Answer:

Question 19.
Find the equation of the line passes through (1, -1) and (3, 5).
Answer:

Question 20.
Find the ratio in which yz – plane divides the line segment formed b joining the two points (-2, 4, 7) and (3, -5, 8).
Answer:
yz – plane ÷ the line sequent joining the points (-2,4,7) and (3,-5,8) at (x,y,z) ill the ratio k : 1.

Question 21.
Evaluate \( \lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}\)
Answer:

Question 22.
Write the converse and contrapositivc of the statement “If a triangle is equilateral, then it is isosceles”.
Answer:
Converse: q → p = If a triangle is isosceles then it is an equilaterals.
Contrapositive: If a triangle is not an isosceles than it is not an equilateral.

Question 23.
The co-efficient of variation and standard deviation are 60 and 21 respectively. What is the arithmetic mean of the distribution.
Answer:

Question 24.
A card is selected from a pack of 52 cards. Calculate the probability that the card is
(a) An ace
(b) A black card.
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 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee.
Answer:

Question 26.
Solve sin 2x + cos x = 0
Answer:
sin 2x + cos x = 0
2 sinx cosx + cosx = 0
cosx (2 sin x +1) = 0 ⇒ cos x = 0 or 2 sinx + 1 = 0

Question 27.
Prove that sin 3x = 3 sin x – 4 sin³
Answer:
Take sin (A + B) = sin A cos B + cos A sin B Put A = 2x, B – x
sin(2x + x) = sin 2x cos x + cos 2x sinx
sin 3x = (2 sinx cosx.cosx) + (1-2sin² x – sinx
LHS = 2sinx cos²x + sinx – 2sin x
Put (cos²x = 1 – sin²x)
∴ sin 3x = 2 sin x(1 – sin² x) + sin x – 2 sin 3x = 2 sin x – 2 sin³ x + sin x – 2 sin³.
= 3 sinx – 4 sin³ x
= RHS.

Question 28.
Write the polar form of the complex number \(1+i \sqrt{3}\)
Answer:

Question 29.
Solve: 2x² + x +1 = 0.
Answer:
2x²+x + 1 = 0 ⇒ a = 2,b = 1,c = 1

Question 30.
Find \({ ‘n’ }\quad { if }\frac { P_{ 4 } }{ ^{ n-1 }P_{ 4 } } =\frac { 5 }{ 3 } \)
Answer:

Question 31.
Find the co-efficient of x⁵ in (x + 3)⁸.
Answer:

Question 32.
Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.
Answer:
The 3 GM between 1 and 256 are 1,G₁,G₂,G₃, 256
a = 1, a₅= 256, n = 5 ,r = ?

Question 33.
Find the sum of all numbers between 200 and 400 which are divisible by 7.
Answer:
203, 210, 217, ,…………… ,399
a = 203, d = 7, a_n = 399, n = ?, S_n = ?

Question 34.
Find the co-ordinates of the focus, the equation of directrix and length of latus rectum of the parabola y² = 8x.
Answer:
Equation of parabola y² = 8x compare with the standerd form
y² = 4ax ∴ 4a = 8  ∴ a = 2
∴ focus = (a,0) = (2,0)
Equation of directrix : x = -a
x = -2 or x +2 = 0
Latus rectum = 4a = 8

Question 35.
Find the derivative of sin x from first principle.
Answer:

Question 36.
Verify by the method of contradiction \(“\sqrt { 7 }\) is irrational”.
Answer:
Let us assume that \(\sqrt { 7 }\) in a rational number.

From (1) and (2) 7 divides both a and b a and b have a common factor.
This contradict our-assumption that \(\sqrt { 7 }\) is wrong
∴ \(\sqrt { 7 }\) is an irrational number.

Question 37.
A committee of two persons is selected from two men and two women. What is the probability that the committee will have (a) no man (b) one man (c) two men?
Answer:
Given
2 Men. 2 Women. 4 person Select 2 person

Question 38.
A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow. The discs are similar in shape and size. A disc is drawn at random from the bag. Calculate the probability that it will be (a) Red (b) Yellow (c) not blue.
Answer:
Total ball = 4 Red +3 Blue +2 Yellow = 9 ball
(i) P (red ball) = \(\frac{4}{9}\)
(ii) P(Yellow) = \(\frac{2}{9}\)
(iii) p(blue) = \(\frac { 3 }{ 9 } =\frac { 1 }{ 3 } \)

Section – D

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

Question 39.
Define signum function. Draw the graph of it and w rite down its domain and Range
Answer:
Definition : Let f be a real value function defined f : R → R

then f(x) is called the signum function.
The domain of the signum function is R and the range ={1,0.1}
The graph of the signum function is given below.

Question 40.
Prove that
\(\frac{\sin 5 x-2 \sin 3 x+\sin x}{\cos 5 x-\cos x}=\tan x\)
Answer:

Question 41.
Prove by mathematical induction
\(1^{2}+2^{2}+3^{2}+\ldots \ldots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}\)
Answer:


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

Question 42.
Solve the system of inequality graphically : 2x + y³ < 4, x + y < 3, 2.v – 3y < 6.
Answer:

Question 43.
A Committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of
(i) exactly 3 girls (ii) atleast 3 girls (iii) atmost 3 girls
Answer:

Question 44.
State and prove Binomial Theorem for positive integer ‘n’
Answer:

Question 45.
Derive the formula for the angle between two straight lines with slopes m₁ and m₂. Hence find the slope of the line which makes an angle \(\frac{\pi}{4}\) with the positive direction of x-axis.
Answer:
L₁ and L₂ are two lines wakes an angle θ₁ and θ₂ at A and B of x-axis.
Proof: Let P be the point of interaction
\(A\hat { P } B=\theta \)
In the triangle APB
θ + θ₁ = θ₂
(Sum of teh interior angle = opposite exterior angle).
θ = θ₂ – θ₁
apply tan θ both side

Question 46.
Derive section formula in 3-D for internal division. Also find the co-ordinates of mid points of the line joining the points A(1, -2, 3) and B (3, 4, 8).
Answer:
Proof: Let P(x₁ ,y₁ ,z₁) and Q(x₂ ,y₂ ,z₂) 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 geometrically 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 mean for the following data.

Answer:

Section E

V. Answer any one question .

Question 49.
(a) Prove geometrically that cos(x+y) = cos x cos y – sin x sin y
Answer:
Given cos(x+y) = cos x cos y – sin x sin y

(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) = cosx cosx – sinx sinx
cos 2x = cos² x – sin² x

(b) Find the sum to n terms of the series, \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots . .\)
Answer:

hence the given statement is the for n = 1,2,………. k
For all +ve integers.

Question 50.
(a) Define ellipse as a set of points. Derive its equation in the form
\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)
Answer:
Let F₁ and F₂ be the focli, O’ be the mid point of the line segment F₁ F₂ . O’ be the origin. And a line from O through F₂ be ± e and F₁ be ve .the co-ordinate of F₁(C .0) and F₂(C,0)

(b) Find the derivative of \(\frac{x^{5}-\cos x}{\sin x}\) with respect to x
Answer: