MODEL TEST PAPER 4
1) Value of [i¹⁸ + (1/i)²⁵]³ is
A) 2 B) 1+ i C) 1 - i D) 2(1 - i). (1)
2) Find modulus of (1+ i)²/(3- i). (2)
3) If the roots of the equation ax² + bx + c= 0 be in the ratio 3:4, show that 12b² = 49ac. (2)
4) Simplify ) ¹⁰C₅ + ¹⁰C₄. (1)
5) What is the probability of getting 3 white balls in a draw of 3 balls from a box containing 5 white and 4 black balls? (2)
6) If n be a positive integer, show that n(n +1) is an even positive integer? (1)
7) If f(x)= (x -1)/(2x² - 7x+5)
find f '(1). (4)
8) Four persons are chosen of random from a group consisting of 3men, 2 women and 4 children. Find the probability that exactly two of them will be children. (4)
OR
If m,n are the roots of x²+px+q = 0, then find m⁴+n⁴ in terms of p and q.
9) If 100 times the 100th term of an A. P with non-zero common difference equals to the 50 times 50th term, then find 150th term. (4)
10) Find the total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets atleast one ball. (4)
11) Find the number of ways one or more balls can be selected from 10 white, 9 green and 7 black balls. (3)
12) Find Domain and Range of
2x/(1+2x). (4)
13) Out of 64 students, the number of students taking maths is 45 and the number of students taking both maths and stats is 10. Then find the number of students taking only Statistics. (3)
14) If x=π/7, then find the value of
(tanx. tan2x + tan 2x. tan 4x+
tan 4x. tanx). (2)
OR
Find the coordinates of the point on the y-axis which is equidistant from the point (-5,4) and (3, -2).
15) If the area of the triangle with vertices (p+1,1),(2p+1,1),(2p+2,2p) is 9 square units. Then find the value of p. (4)
16) If cos(x-y)+ cos(y-z)+ cos(x-z)= -3/2 then prove sinx+sin y+sin z= cos x+ cos y+ cos z. (3)
17) Find the co-ordinates of the foci, the eccentricity and the Equations of the directrix of the hyperbola 4x² - 9y² = 36. (4)
OR
Find the focus, the Equation to the directrix and the length of the latus ractum of the parabola y²+22= 4x+ 4y.
18) Find the Quartiles and Quartile Deviation of the daily wages (in Rs.) of 7 persons given bellow:
12, 7, 15, 10, 19, 17, 25. (4)
19) Calculate Karl Pearson's coefficient of correation between X and Y of the following:
X: 5 7 1 3 4
Y: 2 2 4 5 6 (4)
OR
Find Mode of
C. I: 10-20 20-30 30-40 40-50 50-60
F: 5. 12. 18 6 9
20) Express sin 10x in terms of 5x. (4)
21) a) Find the middle term of (2a - b/3)⁹
b) Expand (2/x - x/2)⁵. 2+2
22) If the sum of p terms of an AP is q and the sum of q terms is p. Show that the sum of p+q terms is -(p+q). (4)
OR
The 5th , 8th, and 11th term of a GP are P, Q and S respectively, show that Q²= PS.
23) Find the value of sin 150°. (2)
24) Solve: 2 cos x + cos 3x = 0. (4)
OR
Sec x - cosecx = 4/3.
25)a) if f(x)= x³/2 - x²/2 + x - 16, find f(1/2).
b) if f(x)= 3x²+2 and g(x)= x+1 then find fog. 2+2
26) differentite: (x+1)(2x-3). (2)
MODEL TEST PAPER -3 (XI)
Question 1. 1x10= 10
i) For any set A, (A')' is equal to
A) A' B) A C) null set D) None
ii) If A={1,2,4}, B{2,4,5}, then (A - B) x (B - A) is
A) {(1,2),(1,5),(2,5)} B) {(1,4)}
C) = (1,4) D) none
iii) If D, G And R denotes respectively the number of degrees, grades and radians in an angle then,
A) D/100= G/90= 2R/π
B) D/90= G/100= R/π
C) D/90= G/100= 2R/π
D) D/90= G/100= R/2π
iv) If tan a= x - 1/4x, then sec a - tan a is
A) -2x, 1/2x B) -1/2x, 2x C) 2x D) 2x, 1/2x
v) The value of sin²75 - sin²15 is
A) 1/2 B)√3/2 C) 1 D) 0
vi) The complete set of values of k, for which the quadratic equation x² - Kx + k+2= 0 has equal Roots, consists of
A) 2+√12 B) 2±√12 C) 2- √12 D) - 2- √12
vii) The number of permutations of n different things taking r at a time when 3 particular things are to be included is
A) (n-3) P (r-3) B) (n-3) P r
C) nP (r-3) D) r!. (n-3) P (r-3)
viii) If 7th and 13th term of an AP is 34 and 64 respectively, then its 18th term is
A) 87 B) 88 C) 89 D) 90
ix) One card is drawn from a pack of 52 playing cards. The probability that is the card of a king or spade.
A) 1/26 B) 3/26 C) 4/13 D( 3/13
x) If the equation of a circle is Kx² + (2K -3)y² - 4x + 6y -1= 0, then the coordinates of centre are
A) (4/3,-1) B) (2/3,-1) C) (-2/3,1) D) (2/3,1)
Question 2. 2x10= 20
i) In a group of 800 people, 550 can speak Hindi and 450 can speak English. How many can speak both Hindi and English?
ii) Find the domain of (x-2)/(3- x)
iii) Prove: cos 510 cos 330+ sin 390 cos 120 = -1
iv) Show: √{(1- cos 2x)/(1+ cos 2x)} = tan x
v) Solve: sin x + cosx =√2.
vi) lim ₓ→₀ {√(1+ x + x²) -1}/x.
vii) find dy/dx : x sin x
ix) The odds in favour of an event are 3: 5. Find the probability of occurrence of this event.
x) Find the coordinates of the centre and radius of x² + y² + 6x - 8y -24= 0
xi) If A={1,2,3}, B={4}, C={5}, then verify AX(B UC)= (AXB)U(AXC)
xi) Find the 7th term in the expansion of (3x² - 1/3)¹⁰.
OR
Find the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants
xii) Find the square root of 5 +12i.
xiii) If three points A(h,0), P(a,b) and B(0,k) lie on a line, show that: a/b + b/k = 1.
Question 3. 3x8= 24
i) If sin(A+ B)= 1 and sin(A- B)= 1/2, 0 ≤ A, B≤π/2, then find the value of tan(A+ 2B)
ii) If sinx + sin y= √3(cos y - cos x), prove sin 3x + sin 3y = 0
OR
Show that √[2+√{2+√(2+ 2 cos 8x)}]= 2 cos x
iii) lim ₓ→₀ {(x+y)sec(x+y) - x sec x}/y.
iv) dy/dx of x²/(x sin x + cosx)².
v) Find the probability that in a random arrangement of the letters of the word SOCIAL vowels come together.
Or
How many four different numbers, greater than 5000 can be formed with the digits 1,2,5,9,0 when repetition of digits is not allowed.
vi) How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE ?
vii) Find the equation of a line passing through the point (2,3) and parallel to the line 3x - 4y +5= 0
viii) The sum of first three terms of a GP is 13/12 and their products is -1. Find the GP.
Question 4 (Any one). 2+3= 5
A)
i) lim ₓ→√2 (x⁴ - 4)/(x²+ 3x √2 -8)
ii) Find the equation of a circle concentric with the circle x² + y² - 6x + 12y +15= 0 and double of its area.
B) i) Using the 1st principal find sin 2x
ii) Five cards are drawn from a pack of 52 playing cards. What is the chance that these 5 will contain:
a) just one ace
b) atleast one ace?
C) i) Find two numbers whose arithmetic mean is 34 and the geometric mean is 16.
ii) Solve the equation in R:
|x + 1/3|> 8/3
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MODEL TEST PAPER -1
Question -1. 1x 10= 10
i) The three angles of a right angled triangle are in AP. Then the angle are
A) 30,60,90 B) 35, 65, 80 C) 10, 70, 100 D) none
ii) Value of (i+ i²+ i³+ i⁴)/(1+ i) is
A) 0 B) 0i C) o+0i D) 1
iii) If m and n are the roots of x(x-3)= 4, what is the value of (m²+ n²)
A) 15 B) 16 C) 17 D) 18
iv) 2³ⁿ -1 is divisible by
A) 2 B) 3 C) 7 D) 5
v) In how many ways can the letters of the word BANANA be arranged?
A) 60 B) 72 C) 144 D) 210
vi) Two dice are thrown simultaneously. The probability that the number on both the dice are same
A) 1/6 B) 1/18 C) 1/36 D) none
vii) If cosx= -1/2, what is the general value of x.
A) 30 B) 60 C) 90 D) none
viii) If the distance between the points (-3,3) and (4,y) be 5√2 units, find the value of y.
A) 3 B) 4 C) 5 D) 6 units
ix) Find the equation of straight line whose x-intercept and y-intercept are 3 and - 4 respectively.
A) x/3 + y/4 = 1 B) x/3 - y/4 = 1
C) - x/3 + y/4 = 1 D) x/3 + y/4 = -1
x) The middle term of (x/y - y/x)¹⁰ is
A) 5th B) 6th C) 7th D) 4th
Question 2. 2x10= 20
i) Find the value of sin 105°
ii) nth terms of the series 16, 8, 4, 2.
iii) Find the modulus of (1+ 3i)/(2- i)
iv) A. M and G. M of two numbers is 6.5, 6 respectively. Find the numbers.
v) Find the value of cos(-1170).
vi) Find the radius and centre of the circle x²+ y² = 36 is
vii) Find the number of arrangement of word MONDAY.
OR
Find the number of arrangement of x²y³z⁴
viii) Prove by induction 1+2+3+ ....n = n(n+1)/2
ix) Find the value of sin 75 + cos 75
x) Find the area of the triangle whose vertex are (3,2,(4,-2),(-4,7) respectively.
Question 3. 3x10= 30
i) How many term of the series e+(-6)+12 +(-24)+.... must be added from the first so that the sum may be -1023?
ii) Find the square root of 5 - 12i
iii) a) lim ₓ→₁ (x²-1)/(x +1).
b) lim ₓ→₂ (x³ +1)/(x²+1+ 3x)
iv) How many numbers of four digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7 .
OR
Prove cos 3π/32 = 1/2 √[2+√{2+ √(2+ √2)}].
v) A die is thrown and at the same time a card is drawn from a pack of 52 playing cards. Find the probability of getting 5 and Ace of hearts.
vi) find dy/dx: (x²+3x+ 1)/(x²+ 2x -1)
vii) Expand: (x - 2/x²)⁵.
OR
Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.
viii) √3 sin10 + sin 20 = cos 50.
OR
Prove cos 9 cos 27 cos 63 cos 81 = 1/16
ix) Prove that the straight lines is concurrent x+ y+5= 0, x- y+1= 0, 3x - y+ 7= 0.
x) If 2 cos a= x+ 1/x, then show that 2 cos 3a = x³ + 1/x³.
OR
A 4 digited number is written by the digits 1,2,3 and 4 and where no digits is repeated in any number. Find the probability that the number is
A) odd
B) mutiple of 4.
Question 4. 4x5= 20
i) If one of the roots of the equation x² - px+ q= 0 be double the other, then prove 2p² = 9q.
OR
Find the fourth term from the end of (x⁴ + 1/x³)¹⁵ .
ii) In how many can a committee of 6 persons be formed taken atleast 3 gentlemen and 2 ladies from 10 gentlemen and 7 ladies, where two particular ladies refuse to serve in the same committee together.
iii) Prove Cos²A + cos²(A- 120) + cos²(A+ 120) = 3/2.
OR
If n be any real integer, find the value of cosec {nπ/2 +(-1)ⁿ π/6}.
iv) The value of (2x²- 2x+4)/(x²- 4x+3) doesn't lie between -7 and 1
OR
Which term in the expansion of (2x² - 1/x)¹² is independent of x ? Find the value of that term.
v) The coordinates of the points A, B and C are (6,3),(-3,5) and (4,-2) respectively and that of the point P(x,y). Show that the ratio of the area of PBC and ABC is |(x+ y -2)/7|.
Or
Find the equation of the straight line which passes through the point of intersection of the straight lines y- 2x +2= 0 and y - 3x +5= 0 and is at a distance of 7/√2 units from the origin.
SECTION- B
SECTION - C
1) Find the 3 yrs moving average of 2, 3, 4, 3, 4, 5, 6, 3, 5, 6, 4,3,2. (2)
2) Mean age of 20 boys and 20 girls are 30 and 40. Find the combined mean age. .(3)
3) Find correlation coefficient between x and y of following:
X: 2 3 5 6 4
Y: 1 3 5 3 4 (5)
OR
Find the mode of:
Class: 20-30 30-40 40-50 50-60
F:. 5 12 25 7
4) Find the 52 percentile of:
Class: 1-10 11-20 21-30 31-40
F: 12. 35. 15. 3 (5)
OR
Find median of the above question
5) Find the standard deviation of
X: 20-30 30-40 40-50 50-60
F:. 12 25 6 10 (5)
Or
Find the combined standard deviations of 10 numbers when mean of x and y are 20, 30 respectively. And standard deviations of x and y are 3 and 4 respectively.