Maths-3
Maths- 3 XI
Section - A
1) Choose the correct option. (1x15=15)
a) Value of 1/(1+ tan²x) + 1/(1+ cot²x) is
i) 0 ii) 1 iii) 2 iv) -1
b) lim ₓ→₀ (x sin(1/x)) is equal to
a) 1/2 ii) 1 iii) 0 iv) none
c) lim ₓ→₋₂ (1/x+ 1/2)/(x+2) is
i) 1/4 ii) 1/2 iii) 1/-2 iv) -1/4
d) If rth term in the expansion of (2x² - 1/x)¹² is independent of x, then the value of r is
i) 7 ii) 8 iii) 9 iv) 10
e) Determine the centre 2x²+ 2y²= 5x + 7y + 3.
i) (5/2,7/2) ii) (-5/2,-7/2) iii) (5/4,7/4) iv) (-5/4,-7/4)
f) In a function from A to B, Image can have more than one pre image. True or false
g) If α and β are roots of x²+ Kx +12=0 and α - β = 1 then value of k is
i) -6 ii) +7 iii) ±6 iv) ±7
h) The complex number (i¹⁸ + (1/i)²⁵)³ is equal to
i) 3- 3i ii) 2-2i iii) -4+4i iv) none
i) For any two sets A and B, A∩ (AUB) is equal to
i) A ii) B iii) θ iv) A∩B
j) Let n(A)= m and n(B)= n then the number of non empty relations from A to B is
i) mⁿ ii) nᵐ - 1 iii) 2ᵐⁿ -1 iv) 2ᵐⁿ
k) Find derivative : (√x + 1/√x)².
l) In how many ways can 4 ladies and 4 gentlemen be seated at a round table so that all ladies sit together?
m) A book shelf contains 7 different mathematics books and 5 different physics books. How many groups of 3 mathematics and 3 physics books can be selected?
n) Find the equation of Circle whose centre is (0,-4) and which touches the x-axis.
o) The middle term in the expansion of (x²/3 +3)¹⁰, x belongs to R is 252 then find the value of x ?
2) Find the equation of a circle whose centre is the product of intersection of the lines 2x + y = 4 and x - y = 5 an passes through the origin. (2)
OR
Find the parametric equations of the following circles x²+ y²+ 4x - 6x -12=0
3) Which term in the expansion of (x - 2/x)¹¹ contains x⁻³ ? (2)
4) Find the values of cot(-7π/4) + cos(-1710). (2)
5) Find the least positive integeral value of n for which {(1+ i)/(1- i)}ⁿ is a real number. (2)
6) Is the relation f defined by
f(x)= (x², 0≤ x ≤ 3
2x, 3≤x ≤5) a function? Justify your answer. (2)
OR
Find the domain and range of the relation R given by R={(x,y): y= x + 6/x, where x, y ∈ N and x< 6}. 2
7) If x be real, find the maximum and minimum values of (x +2)/(2x² +3x +6). (4)
8) If the coefficient of 2ⁿᵈ, 3ʳᵈ and 4ᵗʰ terms in the expansion of (1+ x)²ⁿ are in AP, then find the value of n. (4)
9) Four different mathematics books, six different physics books and two different chemistry books are to be arranged on a shelf. How many different arrangements are possible if
a) the books in each particular subject must all stand together.
b) the mathematics books must stand together. (4)
OR
In how many ways can the letters of the words DAUGHTER be arranged such that
a) the word starts with D and ends with R
b) no two vowels are together
c) all vowels occur together
d) all vowels never occur together.
10) 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
a) exactly 3 girls
b) atleast 3 girls
c) atmost 3 girls. (4)
OR
In how many ways can final eleven be selected from 15 cricket players if
a) there is no restriction
b) one of them must be included
c) one of them who is in bad form, must always excluded
d) two of them being leg spinners, one and only one leg spinner must be included? (4)
11) For the function f defined by f(x)= x² - 6x - 7, using 1ˢᵗ principle of derivative show that f'(5) - 3 f'(2)= f'(8). (6)
12) Find the equations of the lines which pass through the point (4,5) and make equal angles with the lines 12x - 5y +6=0 and 3x - 4y -7=0.
OR
Find the equation of a line through the point of intersection of the lines 5x - 3y -1 =0 and 2x +3y -23=0 and perpendicular to the line 5x - 3y +1=0.
13) If the coefficient of xʳ⁻¹, xʳ, and xʳ⁺¹ in binomial expansion of (1+ x)ⁿ are in AP, show that n² - n(4r +1)+ 4r² -2=0. (6)
14) Evaluate the limit (if exists): lim ₓ→₀ √(1- cosx)/4x. (6)
OR
Evaluate the limit (if exists): lim ₓ→π/3 √(1- cosx)/{√2(π/3- x)}
Section- B
15) (1x5= 5)
i) Find the distance between the point (-3,4,-6) and its image in XY plane
a) 9 units b) 10 units c) 11 units d) 12 units
ii) Mirror image of the given point (-4,0,1) in the XZ plane
a) (0,0,0) b) (-4,0,-1) c) (4,0,-1) d) (-4,0,1)
iii) Find the coordinate of the point P which is five-sixth of the way from A(-2,0,6) and B(10,-6,-12).
iv) Find the equation of ellipse whose vertex are (±6,0) and foci are (±4,0).
v) Find the eccentricity of the curve 9x²- 16y²= 144.
16) Calculate the coordinates of the points which trisect the segment AB, given that A(2,1,-3) and B(5,-8,3). (2)
OR
Find the equation of the set of point which are equidistant from the points (1,2,3) and (3,2,-1).
17) A rod of length 15cm rests in between two coordinate axes in such a way that the end point A lies on x-axis and end point B lies on y-axis. A point P(x,y) is taken and the rod in such a way that AP= 6cm. Show that the locus of P is an ellipse. (4)
OR
Find the equation of the ellipse whose eccentricity is 4/5 and whose foci coincide with the focus of the hyperbola 9x²- 16y² + 144=0.
18) An arc is in the form of a parabola with irs axis vertical. The arc is 10m high and 5m wide at the base. How wide it is 2m form the vertex of the parabola? (4)
Section - C
19) 1x5= 5
a) Out of 11 observations arranged in ascending order, the 4th, 5th and 6th observations are 15,18 and 19 respectively. Find the median of all the 11 observations .
i) 19 ii) 18 iii) 15 iv) 14
b) If the mean and mode are 15, 14 find the median
i) 16 ii) 16.5 iii) 17 iv) none
c) Find the median of 10 prime numbers.
d) Find the cov(X, Y) between x and y when ∑xᵢ= 50, ∑yᵢ= 30, ∑xᵢyᵢ= -115 and n= 10.
e) mean≥ median ≥ mode. True or false.
20) Estimate the median for the following data: (2)
Class interval frequency
2.5-3.5 7
3.5-4.5 31
4.5-5.5 33
5.5-6.5 17
6.5-7.5 11
7.5-8.5 1
OR
The coefficient of correlation between two variables X and Y is 0.64. their covariance is 16. The variation of X is 9. Find the standard deviation of Y-Series.
21)
Mid-value frequency
15 12
25 25
35 39
45 54
55 70
65 25
75 10
Find the mode. (4)
OR
Class-interval. frequency
15-25 12
25-35 25
35-45 39
45-55 54
55-65 70
65-75 25
75-85 10
Find the standard deviation.
22) mean age of x and y are 30 and 32 respectively of 10 and 12 members. Find the combined mean. (4)
OR
If n₁= 20, n₂= 30 and σ₁ = 6, σ₂= 7 find the combined standard deviations.
MATHS- 2
F. M= 100
(Attempt all questions from section A and all questions Either from section B or Section C
Section - A (80 Marks)
1) (10 x 2 =20)
a) Let T{x: (x+5)/(x -7) - 5 = (4x -40)/(13- x)}. is T an empty set?
b) Find the domain and the range of the function
f(x)= 1/√(5- x).
c) Show: tan50= tan40+ 2 tan 10.
d) In a triangle ABC, a= 1, b=√3 and c= 7/6. Find the other two angles.
e) If (2+ 3i)/(3- 4i)= a + ib, find the value of a and b.
f) If m,n be the roots of the equation ax²+ bx + c= 0 form an equation whose roots are 1/m. 1/n.
g) Find n if ⁿC₅ = ⁿC₇.
Or cos2A/(1+ sin2A)= tan(π/4 - A).
h) Find dy/dx of (xeˣ + x⁷+ (x +1)√x)/x.
i) Evaluate: lim ₓ→₀ (sinx - 2 sin3x + sin5x)/x.
j) While shuffling a pack of 52 cards, 2 cards are accidentally dropped. Find the probability that the missing cards are of different colours.
2) a) If f(x)= y= (ax - b)/(cx - a), then show that f(y)= x where x≠ a/c.
b) Find the term containing x¹⁰ in expansion of (2x²- 3/x)¹¹. (2+2)
3) a) Show that √{(1+ sinx)/(1- sinx)}+ tan(π/4+ x/2).
b) Solve for x: 4 sin⁴x + cos⁴x = 1. (2+2)
4) Using mathematical induction show that 5ⁿ⁺¹ + 4. 6ⁿ - 9 is divisible by 20 and n ∈ N.
Or
If (z -i)/(z -1) is purely imaginary show that the point z lies on the circle whose centre is the point (1/2) (1+ i) and radius is 1/√2. (4)
5) 7 candidates are to be examined, 2 in mathematics and the remaining in different subjects. In how many ways can they be seated in a row so that the two examinees in mathematics may not sit together.
Or
A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if---
i) They can be of any colour
ii) two must be white and two must be red.
6) Determine whether the expansion of (x²- 2/x)¹⁸ will contain a term containing x¹² ? If it contains also find out the term. (4)
7) Find the equation of the lines through the point (3,2) which make and angle of 45° with the line x - 2y = 3. (4)
8) Find the equation of the circle passing through the point (7,3) having 3 units and whose centre lies on the line y= x -1. (4)
Or
Find the equation of tangents to the circle x²+ y²- 2x - 4y - 20 =0 through the point P(8,1).
9) Differentiate the function (x³+ 2x) by first principal of differentiation.
10)a) If the equation 3x²+ px +1=0 and 2x²+ ax +1=0 have a common root, show that 2p²+ 3q²- 5pq +1= 0.
b) Show that for all real values of x the expression (x²- 2x +4)/(x²+ 2x +4) has the greatest value 3 and least value 1/3. (3+3)
11)a) 1/(x + y), 1/2y, 1/z are in AP, then show that y is the geometric mean between x and z.
Or
Find the sum of the series to the n terms 1 +3 +7+ 15 +31+... to n terms. 6
12) The diameter of circle (in mm) drawn in a design are given below:
Diameter (in mm) no of circles
33-36 15
37-40 17
41-44 21
45-48 22
49-52 25
Calculate the standard deviation and mean diameter of the circles . 6
Section - B
13) (3x2=6)
a) Find the equation of the parabola with focus (6,0) and directrix x= -6.
b) Construct the table: ~ (p v ~ q)
c) Using contrapositive method proved that is n² is an even integer the x is also an even integer the x is also an even integer .
14) a) Find the equation of the ellipse if its foci are (±2,0) and the length of the latus rectum is 10/3.
Or
Find the equation of the hyponovaloids whose foci are (0,±13) and length of the conjugate axis is 24. (4)
15) Find the coordinates of the point P which is five-sixth of the way from A(-2,0,6) to B(10,-6,-12).
Or
The centroid of ∆ ABC is at the point (1,1,1). if the co-ordinates of A and B are (3,-5,7) and (-1,7,-6) respectively , find the co-ordinate of the point C. 6
16) A double olimate of the parabola y²= 4ax is the length 8a. Prove that the lines from the vertex to its ends are at right angles.
Section - C. (20 Marks)
16)a) Let mean monthly salary paid to all employees of a company is Rs 8300. The mean monthly salary paid to male and female employees was Rs 8000 and Rs 9000 respectively. Determine the percentage of males and females employed by the company. (2)
b) Find the values of a and b from the following data: (4)
Marks. No of students
00-05 7
05-10 a
10-15 25
15-20 30
20-25 b
Total. 100 given that the third decile is 1.1 (4)
Or
Calculate the mode of the following data:
C-I frequency
05-15 6
15-25 11
25-35 21
35-45 23
45-55 14
55-65 5
17) a) If in a sample of n observation given that ∑ d²= 55 and rank correlation r= 2/3. Then find the value of n. (2)
b) The mathematical aptitude score of 10 computer programmers with their job performance is given below: (4)
Person math(sc) job rating
A 7 8
B 5 16
C 1 8
D 4 9
E 3 5
F 0 4
G 2 3
H 6 8
I 8 17
J 9 12
Calculate spearman's rank correlation.
Or
Find Karl Perason's coefficient of correlation between X and Y for the following data:
X: 5 4 3 2 1
Y: 4 2 10 8 6
18) Find the consumer price index for 2007 on the basis of 2005 from the following data using weighted average of price relative method. (4)
Items: Food Rent cloth fuel
Price ('05): 200 100 150 50
Price ('07): 280 200 120 100
Weight: 30 20 20 10
19) Obtain the three year moving average for the following series of observations.
Year: Sales(Rs '0000)
1995 3.6
1996 4.3
1997 4.3
1998 3.4
1999 4.4
2000 5.4
2001 3.4
2002 2.4
Represent these graphically. (4)
TEST PAPER - 2
1) If ⁿC₂ = 44::2, find the value of the n.
2) Find the term free x in the expansion of (4x²/3- 3/2x)⁹.
3)
4) A card is drawn from a pack of 52 playing cards. Find the probability that the card is a face card.
5)
6) Find mean of the following: 1,2,3.....upto 10th term.
7) Find the coefficient of the x⁻¹¹ in the expansion of (x²+ 1/x³)¹².
8) In how many ways can the letter of the word ARRANGE be arranged such that the two 'R's never occur together?
9) Find the S. D of following:
Age(years): 70 60 50 40 30
No of men: 51 140 153 132 64
10) Find the coefficient of correlation from the following data:
X: 10 9 8 7 6 5 4 3 2 1
Y: 10 14 18 22 26 30 34 38 42 46
28/10/25
TEST PAPER -11) In how many ways the letter of the word STRANGE be arranged so that the vowels never together?
2) If the binomial co-efficient of the 3rd term from the end of the expansion (√x + √y)ⁿ equal 45, calculate its 6th term.
3) If the values of the variates x are multiplied by 3, show rthe final AM will also be multiplied by the same number.
4) Calculate the S. D of 7,9,16,24,26(kg).
5) The coefficient of (1+ x)¹⁰ the co-efficient of (2r +1)th term is equal to the the value co-efficient of (4r +5)th term. Find r.
6) There are 10 points in a plane, no three of which are collinear except 4. Find the number of triangles formed by joining them.
7) Find the AM and standard deviation of first n natural numbers.
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