1st PUC Maths Model Question Paper 2 with Answers

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Karnataka 1st PUC Maths Model Question Paper 2 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.
Define power set.
Answer:
All possible subsets of a given set is called power set.

Question 2.
If G = {7, 8} and H = {5,4, 2}. Find G x H.
Answer:
G x H = {(7.5).(7.4),(7,2)(8.5),(8,4).(8,2)}

Question 3.
Convert 520° into radian measure.
Answer:

Question 4.
Express \((-5 i)\left(\frac{1}{8} i\right)\) in the form of a +ib
Answer:

Question 5.
If \(^{n} C_{8}=^{n} C_{2} \text { Find }^{n} C_{2}\)
Answer:

Question 6.
Write the 5^th terms of the sequences whose n^th term is a_n -2ⁿ
Answer:
a₅ =2⁵ = 32

Question 7.
Find the slope of the lines making inclination of 60° with the positive direction of x-axis.
Answer:
Slope = m = tan 60° = \(\sqrt{3}\)

Question 8.
Evaluate
\(\lim _{x \rightarrow 0}\left[\frac{\cos x}{\pi-x}\right]\)
Answer:

Question 9.
Write the contrapositive; If a triangle is equilateral then it is isosceles.
Answer:
Contrapositive – q → – p = “If a triangle is not isosceles then the triangle is equilateral.”

Question 10.
If two coin are tossed once. Find a sample space.
Answer:
Two coin tossed
∴ S = {HH,HT,TH.TT}

Section B

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

Question 11.
Given A = {2, 3}, B = {x : x is solution of x² + 5x + 6 = 0} Find A∪B.
Answer:
A = {2,3}, B = {-2.-3}, A∪B ={ } = φ

Question 12.
If X and Y are two sets such that X∪Y has 50 elements, X has 28 elements and Y has 32 elements, how many elements docs X∩Y have?
Answer:
h(X∩Y) = n(X) + n(Y) – n(X∪Y) = 28 + 32 – 50 = 10

Question 13.
If (x + 1, v – 2) = (3, 1) Find the values of x and y.
Answer:
x +1 = 3 ∴ x = 2
y – 2 = 1 ∴ y = 3

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

Question 15.
Find the values of sin 75°.
Answer:
sin 45^o = sin(45^o + 30°) =sin 45⁰ cos 30⁰ + cos 45⁰ sin 30°

Question 16.
If \(x+i y=\frac{a+i b}{a-i b}\) ,prove that x² + y² = 1
Answer:

Question 17.
Solve 5x + 1 > – 24, 5x – 1 < 24 and represent the solution graphically on number line
Answer:

Question 18.
Show that two lines with slope m₁ and m₂ are parallel if and only if m₁ = m₂.
Answer:

two lines l₁ = and l₂= makes angle
α and β w. r. t  x – axis
∵  α = β , tan α = tan β
∴ m₁ = m₂ ⇒ l₁ || l₂
∴ If m_x – m₂ ⇒ two lines are parallel.

Question 19.
Find the equation of the line passing through the points (-1, 1) and (2, -4).
Answer:

Question 20.
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, -1).
Answer:
Let P(x, y,z) be a point equidistant. From
A(1,2,3) and B(3,2,-l)
PA = PB
From distance plunkle.

Question 21.
Evaluate
\(\lim _{x \rightarrow 0}\left[\frac{1-\cos x}{x}\right]\)
Answer:

Question 22.
Construct the truth table of p ∧ q.
Answer:

Question 23.
Write the mean of the given data: 6, 7, 10, 12, 13, 4, 8, 12.
Answer:

Question 24.
Given \(P(A)=\frac{3}{2} \text { and } P(B)=\frac{1}{5} \) and 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 = 3

Question 25.
In survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked product C and A. 14 people liked product B and C and 8 liked all the three products. Find how many liked product C only?
Answer:

Question 26.
Let A = {1, 2, 3), B = (3, 4) and C = {4, 5, 6) find (A x B)∩(A x C).
Answer:
(A x B)(A x C) = {(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}
{(1,4),(1,5),(1,6)}
(2,4),(2,5)(2,6),(3,4),(3,5),(3,6)
={(1,4),(2,4),(3,4)}.

Question 27.
Two trees, A and B are on the same side of a river. From a point C in the river the distance of the trees A and B is 250 m and 300m respectively. If the angle C is 45°. Find the distance of the trees.
Answer:

Question 28.
Express \(-1+i \sqrt{3}\) in the polar form
Answer:

Question 29.
Solve the equation
\(x^{2}+x+\frac{1}{\sqrt{2}}=0\)
Answer:

Question 30.
In how many ways can 5 girls and 3 boys be selected in a row so that no two boys are together?
Answer:
Given 5 girls, 3 boys seated in a low no two boys together

Question 31.
Find the 13^th term in the expansion of \(\left( 9x-\frac { 1 }{ 3\sqrt { x } } \right) ^{ 18 }:x\neq 0\)
Answer:

Question 32.
Find the 20 ^th and n^th term of the \({ G.P. }\frac { 5 }{ 2 } ,\frac { 5 }{ 4 } ,\frac { 5 }{ 8 } \)
Answer:

Question 33.
Find the 20^th term of the series 2 x 4+4 x 6+6 x 8+………… n^th terms.
Answer:
2.4.6………….. 2n,  a_n = n^th term , 4.6.6…………….. (2n + 2)
∴ a_n-2n {2n + 2), a_n= 4n² + 4n
∴ a₂₀ = 4(20) +4(20) = (4 x 400)+ 80 = 1680
a_n =4+(n-1)2 = 4+(n – 1)2 = 4 + 2n – 2
= 2n + 2­

Question 34.
Find the coordinates of foci, the vertices length of major axes of the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)
Answer:

Question 35.
Find the derivative of (tan x) w.r.t. x from first principle method.
Answer:

Question 36.
Write the statement in three different ways conveying the same meaning. “If a triangle is equiangular then it is an obtuse angled triangle’’.
Answer:

• Converse: If a triangle is an obtuse angle triangle than it is an equilateral.
• Inverse: If a triangle is not a equiangles then it is not on obtuse angle triangle.
• Contrapositive : If a triangle is not an obtuse angle triangle then it is not an equilateral triangle.

Question 37.
A coin is tossed twice, .what is the probability that atleast one tail occurs?
Answer:
S = {HH,HT,TH,Tl} = 4 cases
atleast and tail ={TH,HT,TT} = 3
⇒ ∴ P (atleast one T) = 3/4

Question 38.
There are four men and six women on the city council. If one council member is . selected for a committee at random, how likely is it that it is a women?
Answer:
Given 4 men 6 women
Select 1 woman
∴ number of selection = ⁴C₀ x ⁶C₃ = 6 ways.

Section – D

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

Question 39.
Draw the graph of the function F(x) = x² and write its domain and Range.
Answer:
y = f(x) = x²

Question 40.
Prove that cos² 2x – cos² 6x = sin 4a . sin 8x
Answer:

Question 41.
Prove that \(1^{2}+2^{2}+3^{2}+\ldots \ldots n^{2}=\frac{n(n+1)(2 n+1}{6} \forall n \in N\)
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 inequalities graphically, 2x+y >6, 3x + 4y < 12.
Answer:

Question 43.
In an examination, a question paper consists of 12 question divided into two parts, Part I and Part II containing 5 and 7 questions, respectively. A student is required ‘ to attempt 8 questions in all selecting atleast 3 from each part. In how many ways can a student select the questions?
Answer:
Given 12 question
Total number of selection
(i) ⁵C₃x ⁷C₅ =10 x 21 = 210
(ii) ⁵C₄ x ⁷C₄ =5 x 35 = 175
(iii) ⁵C₅x ⁷C₃ =1 x 35 = 35

Question 44.
Prove Binomial theorem for any positive integer
Answer:

Question 45.
Prove that the equation of a line ‘P’ as the length of the perpendicular drawn from origin and ‘w’ as the angle made by this perpendicular with positive direction of x- axis is x cos (ω) + y sin (ω) = P.
(equation in the Normal form).
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 46.
Derive the section formula in 3-D for internal division.
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 that \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\) Where ‘x’ measures in radian.
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 1 – 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 cos(x + y) = cos cos y – sin x. sin y and hence prove that
cos(x – y) = cos.x cos y + sin x. sin y using unit circle concept.
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 of the n terms to the series 5²+6² + 7² +…………………………….. 20²
Answer:
5² +6² +7² +8² + 2D² find the sum
Let S_n =25 + 36 + 49 + 64 +………. + 400 …(1)

Question 50.
(a) Derive the equation of an ellipse in the form of \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)
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 derivation \(\frac{x+\cos x}{\tan x} \) with respect to x
Answer: