MATH- TEST PAPERS

Wednesday, 16 November 2022

CBSE- MODEL TEST PAPER -1 (XI)

MODEL TEST PAPER -1

Question -1. 1x 10= 10

i) Find the GM between a³b and ab³

A) ab B) ab²C) a²b D) a²b²

ii) Value of (i+ i²+ i³+ i⁴)/(1+ i) is

A) 0 B) 0i C) o+0i D) 1

iii) If (A xB)={(3,2),(3,4),(5,2),(5,4)} find B

A) |2,4| B) [2,4] C) (2,4) D) {2,4}

iv) (A∩B) U(A - B) is

A) A B) B C) A' D) (AB)'

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. 2x110= 20

i) Find the value of sin 105°

ii) Find the nth term of the GP 12, 4, 4/3, 4/9

iii) Find the modulus of (1+ 3i)/(2- i)

iv) The mean deviation of 7,8, 4, 13, 9, 5, 16

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.

Find the number of arrangement of x²y³z⁴

viii) Find the coefficient of x⁷ in the expansion of (x² + 1/x)¹¹

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) Find the eccentricity of an ellipse whose latus rectum is one half of its minor axis.

Find the equation of the hyperbola whose vertices are are (0,±3) and the foci are (0,±5)

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 .

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: y=√{(1- cos 2x)/(1+ cos 2x)).

vii) if the three consecutive vertices of a parallelogram be A(3,4-1), B(7,10,-3) and C(5,-2,7). Find the fourth vertex D.

Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.

viii) √3 sin10 + sin 20 = cos 50.

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.

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) The midpoints of the sides of a triangle are (1,5,-1),(0,4,-2) and (2,3,4) find its vertices

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.

If n be any real integer, find the value of cosec {nπ/2 +(-1)ⁿ π/6}.

iv) In an examination, 56% of the candidates failed in English and 48% failed in science. If 18% failed in both English and science, find the percentage of those who passed in both the subjects.

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|.

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.

Question 5. (20 marks)

i) If the standard deviations of the numbers 2, 3, 2x, 11 is 35, find the possible value of x. (2)

Find the standard deviation of

X: 10 15 18 20 25

F: 3 2 5 8 2

ii) Find the mean deviation of:

Class: 20-30 30-40 40-50 50-60

F: 3 7 12 8 (3)

iii) 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. (4)

iv) Two dice are thrown simultaneously. Find the probability of getting

A) a doublet

B) a multiple of 3 as the sum

C) multiple of 3 and 5

D) multiple of 3 or 5 (4)

v) Find the derivative from the first principle. √(4-x). (3)

vi) Find the equation of a circle concentric with the circle 2x²+ 2y² - 6x + 8y +1= 0. (4)

Saturday, 12 November 2022

MODEL TEST PAPER- 1, 3,4 -XI(ISC) 22/23

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)

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)

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)

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)

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)

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)

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.

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

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.

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

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

__________________________________

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.

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 .

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.

Prove cos²(π - a/2) - cos²(π/8 + a/2)= 1/√2 sin a.

viii) √3 sin10 + sin 20 = cos 50.

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.

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.

Find the fourth term from the end of (x⁴ + 1/x³)¹⁵ .

iii) Prove Cos²A + cos²(A- 120) + cos²(A+ 120) = 3/2.

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

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|.

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)

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)

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)

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.

Friday, 4 November 2022

MODEL TEST PAPER (X) CBSC

MODEL test paper -1

SECTION -A (40 Marks)

1). 1x 10 = 10

i) Value of the quadratic polynomial p(x)= 2x²- 3x + 5 at x= -2

A) 9 B) 10 C) 11 D) 12

ii) Is 3x² - 4x +2 = 2x² - 2x +4 a quadratic equation. T/F

iii) (1- sin²x)sec²x is

A) 0 B) 1 C) sin²x D) tan²x

iv) Cosx sin(90-x)+ sinx cos(90- x) is

A) 0 B) 1 C) - sin x D) - sinx

v) If cos x= 4/5, value of tan x is

A) 1 B) 0 C) 3/2 D) none

vi) Find the diagonal of a cuboid 30cm long, 24cm wide and 18cm high.

A) 40 B) 42.42 C) 50 D) 52.52

vii) The mean of 16 numbers is 8. If 2 is added to every number, what will be the new mean ?

A) 8 B) 16 C) 10 D) 12

viii) Which term of the sequence -1, 3, 7, 11, .... is 95.

A) 30 B) 31 C) 32 D) 33

ix) If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is

A) right angled B) equal C) cyclic D) opposite.

x) A tower is 100√3m hig5. Find the angle of elevation it its top from a point 100m away from its foot.

A) 30 B) 60 C) 45 D) 90

2) 1x 10= 10

i) Write the first three terms in aₙ= n²+1.

ii) How many numbers of two digit are divisibile by 3 ?

iii) The length of a chord of a circle is 4cm. If it's distance from the centre is 1.5cm, determine the radius of the circle.

iv) Find the length of a tangent drawn to a circle with radius 5cm, from a point 13cm from the centre of the circle.

v) An unbiased die is thrown. What is the probability of getting a number between 3 and 6. (1)

vi) One root of x²- 3x - c= 0 is -2, find the value of c.

vii) If the probability of winning a game is 0.3. what is the probability of loosing it ?

viii) Probability of a sure event is __

ix) f(x)= x²+ 3x - 10, g(x)= x -2. Find out whether g(x) is a factor of f(x) or note

x) Find the value of k when f(x)= x³ - 3x² - x + k, if g(x)= x+1 is a factor of f(x)

3) 2x 10= 20

i) Solve x/(x+1) + (x+1/x = 34/15, x≠0, x≠ -1.

ii) Find the sum of the deviations of the variate values 3, 4, 6, 8, 14 from their mean. (2)

iii) Find k if 2x² + Kx + 3= 0 has real roots.

iv) A two digit number is four times the sum and three times the product of its digits. Find the number.

A journey of 240 km would take half an hour less if the speed were increased by 2km per hr. Find the usual speed.

v) Determine the general term of an AP whose 7th term is -1 and 16th term 17.

vi) The 7th term of an AP is 32 and its 13th term is 62. Find AP.

vii) A solid cylinder has total surface area of 462cm². Its curved surface area is one third of its total surface area. Find the volume of the cylinder. (π= 22/7).

viii) In a ∆ABC, AD is the bisector of ang.A, meeting side BC at D . If BD= 2.5cm, AB= 5m and AC= 4.2cm, find DC.

D and E are the points on the sides AB and AC respectively of a ∆ABC such that AD= 8cm, DB= 12cm, AE= 6cm and CE= 9cm, prove that BC= 5DE/2.

ix) √{(- sinx)/(1+ sin x)}= secx - tan x.

x) An electric pole is 10m hig8. A steel wire tied to top of the pole is affixed at a point on the ground to keep the pole up right. If the wire makes an angle of 45 with the horizontal the foot of the pole, find the length of the wire. (2)

SECTION - B (40 Marks)

4) Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram. (3)

5) A ladder 15m long reaches a window is 9m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find the width of the street. (3)

6) AB and CD are two parallel chords of a circle whose diameter is AC. Prove that AB= CD. (3) OR

Equal chords of a circle subtends equal angles at the centre.

7) If x sin³a + y cos³a= sina cos a and x sin a = y cos a, prove x² + y² = 1. (3)

8) If A, B, C are the interior angles of a triangle ABC , prove that tan{(A+ C)/2}= cot(B/2). (3)

9) A man standing in the deck of a ship, which is 8m above water level. He observes the angle of elevation of the top of a hill as 60 and the angle of depression of the base of the hill as 30. Calculate the distance of the hill from the ship and the height of the hill. (3)

10) If V is the volume of a cuboid of dimensions a, b, c and S is its surface area, then prove that 1/V = 2/S(1/a + 1/b + 1/c). (3)

11) Class interval. Frequency

0-10 7

10-20 10

20-30 15

30-40 8

40-50 10 find the mean. (4)

Class interval. Frequency

1-10 8

11-20 10

21-30 15

31-40 8

41-50 18 find the median.

12) A vessel in the shape of a cuboid contains some water. If three identical spheres are immersed in the water, the level of water is increased by 2cm. If the area of the base of the cuboid is 160cm² and its height 12cm, determine the radius of any of the spheres. (4)

13)a) If cotx = 1/√3, find the value of (1- cos²x)/(2- sin²x). (2)

b) The radius and slant height of a cone are in the ratio 4:7. If it's curved surface area is 792cm², find its radius. (π= 22/7). (2)

14) a) If P(A)= 1, then A is called

A) certain event

B) impossible event

C) possible event

D) absolute event. (1)

b) A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is

A) red

B) black or white

C) not black. (3)

15)a) 2(a²+ b²)x² +2(a+ b)x+ 1= 0.

b)Find the sum of (-5)+(-8)+(-11)+....+(-230). (1.5+ 1.5)

16)a) AB and CD are two parallel chords of a circle such that AB= 10cm and CD = 24cm. If the chords are on the opposite sides of the centre and the distance between them is 17cm, find the radius of the circle.

b) Two chords AB and CD of a circle intersect each other at P outside the circle. If AB= 5cm, BP=3 cm, and PD= 2cm, find CD.

Thursday, 3 November 2022

MODEL TEST PAPER (WBCHSE) 22/23

Full Marks --80

SET - 1

PART - A (Marks : 10)

1) Choose the correct alternatives:

i) If A={1, 2, 3, 4} and Iₐ be the identity relation on A, then

A) (1,2) belongs to Iₐ

B) (2,2) belongs to Iₐ

C) (2,1) belongs to Iₐ

D) (3,4) belongs to Iₐ

ii) The principle value of tan⁻¹(-√3) is

A) π/3 B) π/4 C) - π/ D) - π/3

iii) If A is an invertible Matrix of order 3 and |A|=5, then the value of |adj A| is equal to

A) 20 B) 21 C) 24 D) 25

iv) If f(x)= log(tan(x/2)), then the value of f'(x) is

A) sin x B) 1/(sin x/2 cos x/2)

C) - cosec x D) Cosec x

v) The value of ∫ |cos x| dx at (π, 0) is

A) 0 B) 1 C) 2 D) none

vi) The area (in square unit) bounded by the curve y =sin x, x-axis and the two ordinates x=π, x= 2π is

A) 1 B) -1 C) -2 D) 2

vii) Angle between the straight lines (x -5)/7 = (y+2)/-5 = z/1 and x/1 = y/2 = z/3 is

A) π/4 B) π/3 C) π/2 D) π

viii) the value of m for which the straight line 3x -2y+z + 3= 0= 4x - 3 + 4z +1 is parallel to the plane 2x - y + mz - 2= 0 is

A) - 2 B) 8 C) -18 D) 11

ix) If the odds against an event are 4 : 5 then the probability the occurrence of the event is

A) 5/9 B) 4/9 C) 4/5 D) 1/9

x) the variance of a binomial distribution with parameters n and p is

A) n/4 B) ≤ n/4 C) > n²/4 D) ≤ n²/4

Part - B (Marks : 70)

2) On the set Q⁺ of all positive rational numbers if the binary operation * is defined by a*b = ab/4 for all a, b belongs to Q⁺, find the identity elements in Q⁺. (2)

3) solve: tan⁻¹(cot x) + cot⁻¹(tan x)= π/4. (2)

4) prove by property

y+ z z y

z z+x x = 4xyz

y x x+ y (2)

If the matrix A= 2 5 and AB=-13 8

1 3 - 8 5 find B.

5) find: lim ₓ→₀ (mᵃˣ - nᵇˣ)/x. (2)

6) if x= t log t, y= (log t)/t, find dy/dx when t= 1. (2)

7) find: ∫(1 - 1/x²) ₑ(x + 1/x) dx. (2)

solve: dy/dx = 1 - x + y - xy

8) Show that the function f(x)= x³/3 - 6x² + 20x - 5 has neither a maximum or minimum value. (2)

9) The radius of a circular plate increases at the rate of 0.002 cm/s. How fast is the area changing when radius is 14 cm? (2)

10) Find the acute angle between the z-axis and the straight line joining the points (3,2,3) and (-3,-1,5). (2)

Prove that the equation of the plane which passes through the point (2,- 3, 5) and which is parallel to the yz- plane is x= 2.

11) Show that the probability that exactly one of the events A and B occurs is P(A)+ P(B) - 2P(AB). (2)

If X is a discrete randomal variable and ' a ' is a constant, show that, E(x - bar x)= 0.

12) Show that the relation R. on the set A={x belongs to z : 0≤ x ≤ 12}, given by R{(a,b): |a - b| is a multiple of 4 and a, b belongs to A} is an equivalence relation on A. (2)

Show that tan⁻¹(1/√3 tan(x/2)) = 1/2 cos⁻¹{(1+ 2 cos x)/(2+ cos x)}.

13) If the matrix

A= 1 x -2

2 2 4

0 0 2 and A² + 2I = 3A. Find x; here I is the unit matrix of order 3. (3)

solve bx matrix method:

2/x - 3/y + 3/z = 10, 1/x + 1/y + 1/z = 10, 3/x - 1/y + 2/z = 13.

14) Prove the determinant. (3)

1 a a²- bc

1 b b²- ca = 0

1 c c² - ab

(a²+ b²)/c c c

a (b²+ c²)/a a = 4abc

b b (c²+ a²)/b

15) If f(x)= tan⁻¹{x/(1+ 20x²)} show that f(x)= {5/(1+ 25x²)} - 4/(1+ 16x²). (3)

Let y= (sin⁻¹x)²+ (cos⁻¹x), show that, (1- x²) d²y/dx² - x dy/dx = 4

16) Evaluate ∫ dx/√(2x³/3 - x² + 1/3). (3)

Evaluate ∫ x²/(x⁴ +1) dx.

17) Solve: (1- x²) dy/dx - xy = x², given y= 2 when x = 0. (4)

Solve: (6x + 9y -7) dx= (2x + 3y -6) dy.

18) AB = 2i - 4j + 5k and BC= i - 2j - 3k in parallelogram ABCD, find a unit vector in direction parallel to the diagonal AC of the parallelogram. (4)

19) prove ∫ log(sin x) dx at (π/2, 0) = π/2 log(1/2). (4)

20) Urn A contains 1 white, 2 black and 3 red balls; Urn B contains two white, one black and one red ball; urn C contains 4 white, 5 black and 3 red balls. One urn is chosen at random and two balls are drawn. These happen to the one white and one red. what is the probability that they come from urn A ? (4)

21) the probability distribution of a random variable X is as follows:

X: 0 1 2 3 4

P(x) 0.20 0.25 0.35 0.14 0.06 Find the probability distribution of the variable Y where y= x² + 5. (5)

22) A company manufacturers two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. He earns a profit of ₹50 each on type A and ₹ 60 each on type B. formulate the above problem as a LPP to maximize the profit. (5)

Solve graphically:

Minimize Z= 3x+ 5y

Subject to the constraints x+3y≥ 3, x+ y ≥ 2, x, y≥ 0

23) A cylindrical tin can, open at the top, of a given capacity, has to be constructed. Show that the amount of the ti required will be least if the height of the can is equal to its radius. (5)

24) Show that lines r= (i+ j+ k)+ t(i - j + k) and r= (3i - k)+ s(4j - 16k) intersect and find the position vector of their point of their point of their point of intersection. (5)

Find the equation of the plane passing through the point (-1, 1,1) and (1,-1,1) and is perpendicular to the plane x+ 2y + 2z = 5.

Wednesday, 2 November 2022

MODEL TEST PAPER ISC(2022/23)

MODEL TEST PAPER -1

Full Marks- 90

SECTION A

1) (any 5) 1x5= 5

Choose the correct alternatives:

i) If A={1, 2, 3, 4} and Iₐ be the identity relation on A, then

A) (1,2) belongs to Iₐ

B) (2,2) belongs to Iₐ

C) (2,1) belongs to Iₐ

D) (3,4) belongs to Iₐ

ii) The principle value of tan⁻¹(-√3) is

A) π/3 B) π/4 C) - π/ D) - π/3

iii) If A is an invertible Matrix of order 3 and |A|=5, then the value of |adj A| is equal to

A) 20 B) 21 C) 24 D) 25

iv) If f(x)= log(tan(x/2)), then the value of f'(x) is

A) sin x B) 1/(sin x/2 cos x/2)

C) - cosec x D) Cosec x

v) The value of ∫ |cos x| dx at (π, 0) is

A) 0 B) 1 C) 2 D) none

vi) If the odds against an event are 4 : 5 then the probability the occurrence of the event is

A) 5/9 B) 4/9 C) 4/5 D) 1/9

i) solve the determinants

x² x 1 (2)

0 2 1 = 28

3 1 4

ii) Solve the equation for x:

cos(tan⁻¹x)= sin(cot⁻¹3/4). (1)

iii) using Matrix rule, solve the system of equations: 5x + 7y = -2, and 4x+ 6y = -3. (2)

iv) Evaluate: lim ₓ→₀{(sin x - x)/x³}. (1)

v) Given that the events A and B are such that P(A)=1/2, P(B)= p, P(A U B)= 3/5. Find p if A and B are

a) mutually exclusive

b) Independent. (2)

vi) Evaluate: ∫ (x sin x)/(1 + sin x) dx at (π, 0). (2)

vii) (1)

For what value of x is the given matrix 2x + 4 4

x + 5 3 a singular Matrix

viii) if y= xʸ, prove that x dy/dx = y²/(1- y log x). (2)

ix) The probability of A, B, C solving a problem 1/2, 1/3, 1/4 respectively. Find the probability that the problem will be solved. (2)

x) If f(x)= 27x³ and g(x)= ³√x, then find gof(x). (1)

3) Using properties of determinant, Prove that 1 + a 1 1

1 1+ b 1

1 1 1+ c = abc(1+ 1/a + 1/b + 1/c) (3)

4) Apply Rolle's theorem to find a point (or points) on the curve y= - 1+ cos x where the tangent is parallel to the x-axis in [0, 2π]. (3)

5) If the following function is differentiable at x= 2, then find the values of a and b.

If the following function is differentiable at x= 2, then find the values of a and b.

f(x)= x², if x ≤ 2

ax+ b, if x > 2 (3)

6) if cos⁻¹(x/a) + cos⁻¹(y/b) = k , prove that x²/a² - (2xy cos k)/ab + y²/b² = sin²k. (3)

7) if y= {x+ √(1+ x²)}ⁿ , then show that (1+ x²) d²y/dx² + x dy/dx = n²y. (3)

8) Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that at least one ball is red ? (3)

9) Evaluate ∫ (3x+1)/√(5 - 2x - x²) dx. (3)

evaluate ∫ √x/{√x + √(a - x)} dx

10) Find the equation of the tangent to the curve y= x² - 2a + 7 which is

a) a parallel to the line 2x - y +9= 0

b) Perpendicular to the line 5y - 15x= 13. (3)

11) Find the interval in which the function f given by f(x)= sin x - cos x, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing. (2)

12) solve: dy/dx + y/x = x². (2)

13) Solve: dy/dx + 1 = eˣ⁺ʸ. (2)

14) Let f: N --> N be a function defined as f(x)= 4x² + 12x + 15, show that f: N --> S is invertible (where S is the range of f). Find the inverse of f and hence find f⁻¹(31) and f⁻¹(87). (3)

15) Using matrix, solve the following equations: 5x+ 3y + z= 1z, 2x+ y+ 3z= 19, x+ 2y+ 4z= 25. (3)

If A= 1 -1 0

2 5 3

0 2 1 find A⁻¹.

16) Prove that the area of a right angled triangle of given hypotenuse is maximum, when the triangle is isosceles. (3)

Show that of all rectangle inscribed in a given fixed circle, the square has the maximum area.

17) Evaluate: ∫ x² sin⁻¹x dx. (2)

18) ∫ x dx/(x²+ 4x +3). (2)

19) A pair of dice is thrown 4 times. If getting a double is considered a success, find the probability distribution of the number of success and show that its mean is 2/3. (3)

20) One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn from 2nd. Find the probability that the ball drawn is white. (3)

SECTION - B

21) if a= 2i + j + 3k and b= 3i + 5j - 2k represent two adjacent sides of a triangle, then find the area of the triangle. (2)

22) If (8,2,0), B(4,6,-7), C(-3, 1, 2) and D(-9,-2,4) are four given points, then find the angle between AB and CD. (2)

23) Find the vector and Cartesian equation of the plane which bisects the line joining the points (3,-2,1) and (1,4,-3) at right angles. (2)

24) If a= 7i- 2j + 3k, b= i - j + 2k, c= 2i + 8j are three vectors then find a.(b x c) and (a x b). c. (4)

25) find the equation of the plane passing through the line of intersection of the planes x+ 2y+ 3z - 5= 0 and 3x - 2y - z + 1= 0 and cutting off equal intercepts on the OX and OZ axes. (4)

Find the co-ordinate of the points on the line (x -1)/2 = (y+2)/3 = (z - 3)/6 which are at a distance of 3 units from the point (1,-2,3).

26) Find the area of the region included between the parabola y= 3x²/4 and the line 3x- 2y + 12= 0. (4)

27) Find the area bounded by the line y= x, The x-axis and the ordinate8 x= 0 and x= 4. (2)

SECTION- C

28) The fixed cost of a product is ₹ 18000 and the variable cost per unit is ₹550. if the demand function is p(x)= 4000 - 150x, find the break even values. (2)

29) Given x + 4y = 4 and 3x + y= 16/3 are regression lines. Find the line of regression of x on y. (2)

30) The cost function for a commodity is C(x)= ₹(200 +20x - x²/2)

a) find the marginal cost(MC).

b) calculate the marginal cost when x= 4. (2)

31) Two regression lines represented by 2x+ 3y - 10= 0 and 4x + y - 5 = 0. find the line of Regression of y on x. (4)

Fit a straight line to the following data. Treating y as the dependent variable.

X: 1 2 3 4 5

Y: 7 6 5 4 3 Hence, estimate the value of y when x= 3.5

32) the marginal cost function of a firm is MC= 33 log x. Find the total cost function when the cost of producing one unit is ₹11. (4)

If the marginal cost of a commodity is equal to half its average cost, show that fixed cost is zero. If the cost of producing 9 units of the commodity is ₹60, find the cost function.

33) A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours in both machines and each unit of product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at profit of ₹7 and that of B at a profit of ₹4. find graphically the production level per day for maximum profit. 6

Monday, 5 September 2022

REVISION/TEST (C)

A. P

1) If the sum of the three numbers in A. P be 18 then what is the middle term?

A) 6 B) 8 C) 10 D) 25

2) If 10 be added to each term of an AP then, will the new progression be in AP also. ?

A) A. P B) G. P C) H. P D) none

3) If 5 be multiplied with each term of an AP then will the new progression an AP also?

A) A. P B) G. P C) H. P D) none

4) Find the 8th term of an AP., 13, 10, 17, 24......

A) 52 B) 62 C) 54 D) 64

5) Which term of the AP. 7, 11, 15, 19, ....... is 103?

A) 20th B) 22th C) 25th D) 30th

6) If the 7th term of an AP be 20 and the 10th term be 29 then find the first and the common difference.

A) 1,2 B) 2, 3 C) 3,4 D) 4,5

7) Can be any term of the AP 5, 10, 15.....

A) A. P B) G. P C) H. P D) none

8) In an AP 1st term be -5, last term be 25 and number of terms be 10. Find the sum of the terms.

A) 105 B) 102 C) 100 D) 98

9) What do you mean by natural numbers? Find the sum of first 100 natural numbers.

A) 5050 B) 505 C) 55 D) 5550

10) a²+ 3a + 2, 3a²+ 2a+5, 4a²+ 5a + 4 are in AP, then find a.

A) 0 B) 2 C) 4 D) 6

11) Find the arithmetic progression of which nth term is 3n -1.

A) 2, 5, 8......

B) 5, 7, 9 ...

C) 2, 8, 14,....

D) none

12) If the pth term of an AP be q and qth term be p, then find the p+ q th term.

A) 0 B) 1 C) 2 D) 4

13) The sum of first n terms of an AP is n². Find the common difference.

A) 2 B) 4 C) 6 D) 8

14) Find the sum of the first 20 terms of the AP. 20, 17, 14...

A) -170 B) 170 C) 200 D) -200

15) The 5th term and the 11th term of an AP are 41 and 20 respectively. Find the 1st term. What will be the sum of first 11 terms of this AP.

A) 55, 825/2 B) 65, 825 C) 55, 825 D) n

16) The sum of p th term of an AP is 3p²+ 5p; then the series is in

A) AP. B) G P C) HP D) none

17) Insert five A. M between 2 and 20.

18) The n th term of an AP is p. Then the sum of first (2n -1) terms is

A) (2n -1)p B) (2n -1) C) p D) (2n +1)p

19) The p th term of an AP is a, and the q th term is b. Then that the sum of first (p+ q) terms is

A) 1/2 (p+ q)(a+ b + (a- b)/(p - q)).

B) (p+ q)(a+ b + (a- b)/(p - q)).

C) 1/2 (a+ b + (a- b)/(p - q)).

D) 3/2 (p+ q)(a+ b + (a- b)/(p - q)).

20) The sum of how many terms of the series 27 + 24+21+...... will be equal to 132 ?

A) 8 B) 11 C) 8 or 11 D) 8 and 12

21) If the sum of n th term of an AP is n² + 3n, then eight term is equal to the sum of ___

A) 10th B) 11th C) 13th D) none

22) Find the sum upto n terms of the following series. 1+ 5 + 12+ 22+ 35+ .......

A) {n²(n+1)/2}. B) n(n+1)/2

C) n²(n+1) D) {n²(n-1)/2}

23) If the sum of the p, q and r terms of an AP be x, y, z respectively then value of

x/p (q- r)+ y/q (r- p)+ z/r (p- q) is

A) 0 B) 1 C) 2 D) 15

24) Find the following sum:

1²+ 3² + 5²+ ...... upto n terms.

A) n/3 (4n² - 1). B) n (4n² - 1).

C) n²/3 (4n² - 1). D) n/3 (4n² + 1).

25) The middle term of an AP having 11 terms is 12. Find the sum of the 11 terms of that progression.

A) 12 B) 21 C) 33 D) none

26) Find the sum of n terms of 2.4 + 6.8 + 10.12+ .....

A) 4n/3 (n+1)(4n -1)

B) 4n (n+1)(4n -1)

C) 4n/3 (n+1)(4n +1) D) none

27) The p th, q th and r th terms of an AP are P, Q , R respectively. Then the value of p(Q - R) + q(R - P) + r(P - Q)

A) 0 B) 1 C) 2D) none.

28) The sum of the three numbers in AP is 21 and their product is 231. Find the numbers.

A) 3, 7,11 B) 11, 7, 3 C) both A and B C) either A or B

29) There are n arithmetic means between 4 and 31. If the second mean: last mean= 5: 14 then find the value of n.

A) 8 B) 10 C) 12 D) 32

30) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then the value of sum of the first P+ Q terms is

A) zero. B) 1 C) 2 D) 4

31) The sum of n, 2n and 3n of an AP are a, b, c respectively. then the value 3(b - c).

A) a B) b C) c D) none

32) Find the sum of 1² - 2²+ 3²- 4² + 5² - 6²+ ......

A) n/2 (n+ 1). B) n (n+ 1)

C) n/2 (n- 1). D) none

33) If the sum of p terms of an AP is to the sum of q terms as p² : q², then the value of (p th term)/(q th term)

A) (2p -1)/(2q -1). B) 2p/3q

C) n/2 (n+ 1). D) none

34) The interior angles of a polygon are in AP. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.

A) 9 B) 6 C) 9 or 6 D) 9 and 6

35) The sides of a right angled triangle are in AP. If the smallest side is 5cm then find the largest side.

A) 2 B) 5 C) 3 D) 25/3

38) A man divided ₹24000 among his four sons in such a way that the amounts of four sons are in AP. The product of the amounts of 1st and 3rd sons is to the product of amounts of 2nd and 4th sons as 7: 15. Calculate the amounts received by each son.

39) Two posts are offered to a man. In the one, the starting salary is ₹1200 and increases annually by ₹80; in the other, the salary starts at 850 and increases annually by ₹120. The man decides to accept the post that will give him better total income in the first 16 years. Which post will be accept ?

A) 1st offer B) 2nd offer C) both F) either A or B

40) A polygon has 25 sides, the lengths of which starting from the smallest side are in AP. If the perimeter of the polygon is 2100 cm, and the length of the largest side is 20 times that of the smallest, find the length of the smallest side and the common difference of the AP. 8, 19/3

41) The first term of an AP is a and the sum of first p terms is zero. Show that the sum of next q terms is -{aq(p+q)}/(p - 1).

42) If the sum of p-terms of an AP be q and the sum of q terms be p then show that the sum of its (p+ q) terms is -(p+ q).

Logarithm

1) What is the value of x, if y= log₇x and 5(y/log7⁵)

A) 7 B) 28 C) 49 D) 14 E) 21

2) If logₐx = Logₘˣ what is the value of logₐm if x≠ 1 ?

A) x B) x² C) 1 D) 1- x E) 2 - x

3) if Log₁₀x = y and logₑ10= 1/m, what is x (in terms of y, e and m) ?

A) y/m B) logₘy C) eʸ⁾ᵐ D) m logₑy E) m/y

4) Which of the following is true?

A) 1/4> log₁₀ 2 > 1/8

B) 1/2 > log₂₀ 3 > 1/3

C) 1/9 < log₁₀ 3 < 1/3

D) 1/4> log₁₀ 4 > 1/4

E) 1/5 < log₁₀ 2 < 1/4

5) solve for x: log₂ log₂ logₓ6561= 2 and x> 0

A) 3 B) 2 C) √7 D) 9 E) √3

6) If x= 1+ logₐmc, y= 1+ logₘca, k = 1 + log꜀am what is the value of xy + yk + kx

A) logₓᵧₖamc B) xyk

C) (logx + log y+ log k)/log amc

D) 1 E) 2xyk

7) Solve for x: if logₓ[log₅{√(x +5) +√x)}] = 0

A) 1 B) 9 C) 12 D) 3 E)4

8) If aˣ = nʸ = cᵐ and y² = xm what is the value of ( logₙa)/(log꜀n)

A) 1 B) logᵧx /logₘy C) xm/y D) xym E) xy/m

9) If a is a perfect square what is the value of a given aˣ = 1/8 and x= 3/2[(log₂a) - 3]?

A) 1 B) 64 C) 4 D) 16 E) 8

10) It is given that f(x) =(e²ˣ + e⁻²ˣ)/2, what is the value of f(logₑ3) ?

A) e⁶ B) e⁶ᶻ C) 1 D) 41/9 E) 9e⁶

11) If a, m, c are distinct values, what is the value of abc if (logₘa) (log꜀a)+ (logₐm)(log꜀m)+ (logₐc) (logₘc) - 3 = 0 ?

A) 2 B) 0 C) 1 - log a - log b - log c D) 3 E) 1

12) If log₁₂ 27= a, what is the value of log₆16 ?

A) 4(3- a)/(3+a)

B) 3(9- 4a)/(3- 2a)

C) a⁴ - 9 D) (3 + a)/(3 - a)

E) (3- a)/(3+a)

13) Express log[³√a⁴/√(b³c) ÷ (a²b³/c³)¹⁾⁶] in terms of log a, log b, and log c.

A) log a - log b

B) 1/2 log a +1/3 log b - log c

C) 1/3 log a + 1/2 log c

D) log a - 2 log b

E) none

14) If z log y + log(log x) = log[log x + log y+ log z], which of the following is true, given that x, y and z are distinct.

A) ₓzʸ = 1 B) ₓyᶻ = cyz

C) xʸ = z D) y² = z E) ᵧxᶻ = 1

15) If log₆ 161= a, log 23= b, what is the value of log₇6 in terms of a and b ?

A) a)b B) a+ b C) 1/(a - b) D) b/a E) 1/(a+ b)

16) If log₄3= 1/a, what is log√₃ 36 + log√₃ 36 + ......20 times ?

A) 9a/4 B) 80+ 40a C) √3 a/36 D) (9a+ 4)/√3 E) 4/9

17) 1/(log 2/log x + log 4/log x + log 8/log x) = log₂y, then what is the relation between x and y.

A) x= y⁶ B) x= y³ C) x= y² D) x= y E) x= y⁴

18) Solve for x: if log₂(1- (1/2)ˣ)= x - 2.

A) 0 B) 3 C) 2 D) 4 E) 1

19) If logₓ2 logₓ/₁₆2 = logₓ/₆₄2 , then the value of x is

A) 2 B) 4 C) 8

D) choice B or choice C

E) choice A or choice C

20) If log₂[5(2ˣ)+ 1)], log₄(2¹⁻ ˣ +1), and 1 are in AP, then what is the value of x ?

A) 1 B) log₂√5 C) log₂(2/5) D) log√₂ 5 E) log₂5

21) Value of 1/logₐx + 1/log√ₐx¹⁾⁴ + 1/log³√ₐx¹⁾⁹ + ....+ 1/log²⁰√ₐx¹⁾⁴⁰⁰ is

A) logₓa²¹⁰ B) x²⁰ + a C) logx D) logₐx⁴⁴ E) logₓa

₂

22) value of x: ₄₉log₇5+ log₂2² = 31 + log ₁₀ˣ.

A) 0 B) 1 C) 10 D) 10⁵ E) 10¹⁰

23) If log₁₀1234= 3.0913,

log₁₀769874= 5.88642 find the value of ⁸√0.000000001234

A) 0.0769874 B) 0.030913

C) 0.00769874 D) 0.0030913 E) 0.00678947

24) The logarithm of 7623 is 3.8821. find the number whose logarithm is -0.1179.

A) 0.08821 B) 0.7623 C) 0.07623 D) 0.8821 E) 0.6732

25) How many digits are there in (441)⁵⁰ if log 3= 0.4771 and log 7= 0.845

A) 50 B) 21 C) 442 D) 133 E) 213

26) The number of zeros immediately after the decimal of N is 5. If 0< N < 1,

Monday, 22 August 2022

Revision Maths -X(CBSC) 20/23

18/9/22

PROBABILITY

1) A fair die is thrown once. Find the probability that the number on it is:

A) an odd number. 1/2

B) greater than 4. 1/3

C) at the most 4. 2/3

D) between 7 and 10 0

2) Two fair coins are tossed. Find the probability that

A) head turns up exactly once. 1/2

B) Tail turns up at least once. 3/4

C) Tail does not turn up at all. 1/4

3) A coin is tossed thrice. Find the probability that

A) tail turns up at least twice. 1/2

B) Head does not turn up at all. 1/8

C) tail comes on the second toss. 1/2.

D) Head turns up at least once. 7/8

4) Two fair dice are tossed. Find the probability that

A) The sum of the score is 9. 1/9

B) the product of the score is 12.1/9

C) the score on the second die is greater than the score on the first die. 5/12

C) the Sum of the scores is a prime number. 5/12

D) the sum of the score is a multiple of 4. 1/4

E) the sum of of the scores is a perfect square. 7/36

5) One card is drawn from a pack of pack of well shuffled pack of cards. Find the probability that the card drawn is :

A) A Queen. 1/13

B) not a queen. 12/13

C) a face card. 3/13

D) red card . 1/2

E) bears a number lesson than 4. 3/13

F) king or black black card. 7/13

6) In a match between A and B the probability of winning of A is 0.43. What is the probability of winning of B ? 0.57

7) A bag contains 3 red, 4 white, 5 blue marbles. All the marbles are identical in shape and and size. One marble is drawn at random from the bag. find the probability that the marble drawn is.

A) Red. 1/4

B) red or white. 7/12

8) An unbiased coin is is tossed. Find the probability that

A) A head turns up. 1/2

B) A tail turns up. 1/2

C) both head and tail turn turn tail turn turn up 0

D) neither head nor tail turns up. 0

9) A Uniform die is thrown. find the the probability of the event A, B and E where

A: score is an even number. 1/2

B: score is a number number less than 5 but less than 2. 1/2

E: score is a number that is a multiple of 3 or 5. 1/2.

10) A perfect cubic die is thrown. find the probability that

A) A prime number comes up. 1/2.

B) A perfect square comes up. 1/3

11) Two coins are tossed simultaneously. Find the probability of getting exactly one head. 1/2

12) Three coins are tossed simultaneously. Find the probability of getting at least one head. 7/8.

13) Three unbiased coins are tossed. find the probability of of getting at least two heads up. 1/2

14) A coin is tossed 3 times. Write the sample space. find the probabilities of

E: getting two or more more heads. 1/2

F: the second is a not a head. 1/2

15) Two unbiased dice are thrown in the air. find the probability that the sum of the scores is a multiple of 3 multiple of 3 a multiple of 3 multiple of the scores is a multiple of 3 multiple of 3 a multiple of 3 multiple of is a multiple of 3 multiple of 3. 1/3

16) two dice are are thrown. What is the probability that the sum of the points obtained is greater than 4 ? 5/6

17) Two unbiased dice are thrown. Find the probability that the sum of the numbers on their faces is at most 5 5 5. 5/18

18) two fair dice dice are thrown, find the probability of getting the same score on the first die as on the the second. 1/6

19) two unbiased dice are thrown in the air. Find the probability that the sum of the scores is greater than 9 or an even number. 5/9

20) if two fair dice are thrown, find the probability that the sum of the points the points points on their uppermost face is a perfect square or a multiple of 3 3. 5/12

21) a box contains 5 red, 11 white and 7 black balls. one ball is drawn at random, find the probability that ball drawn is a white white ball. 11/23

22) a bag contains 6 contains 6 red, 5 blue, 3 white and 4 Black Black balls. A ball is drawn at random. find the probability that the ball is red or black . 5/9

23) in a bag there are six six black, 4 white and 3 yellow balls. A ball is taken at random. find the probability of of getting a yellow or a white ball. 7/13

24) A box contains 7 red, 5 white and 8 green balls identical in all respects except colour. one ball is drawn at random. find the probability that that it is not white. 3/4

25) a card is drawn from a pack of well shuffled 52 playing cards. find the probability that the card drawn is.

A) a diamond 1/4

B) A red card. 1/2

C) A king 1/13

D) an ace or a Queen. 2/13

E) a face card card 3/13

F) A card bearing a number between and including 2 and 6. 5/13

26) Six tokens bearing numbers 1 to 6 are placed in one box and seven tokens bearing 1 to 7 are placed in another box. If one token is drawn from each box, what is the probability that the sum of of the number is

A) six 5/42

B) 11 1/14

27) If P(A)= 0.95, find P(not A). 0.05

28) a bag contains a certain number of a blue blue balls. A ball is drawn, find the probability that the ball drawn is.

A) black. 0

B) blue. 1

29) the probability that two boys do not have the same birthday is 0.394. What is the probability that the two boys have the same have the same same birthday ? 0.606

30) which of the following cannot be the probability of an event

A) 5/7

B) 0.28

C) √2

D) - 2.4 √2 and - 2.4

31) From a deck of 52 cards, all the face cards are removed and then remaining cards are shuffled. now one card is drawn from the remaining deck. find the probability that the card is.

A) a black card. 1/2

B) 8 of red colour. 1/20

C) A king of black colour. 0

32) a box contains thousand bulbs out of Prit Prit 25 defective, it is not possible to just look at the bulb and tell whether or not it is defective. one bulb is taken out at random from the box. calculate the probability that the bulb taken out is:

A) a good one. 39/40

B) a defective one. 1/40

33) a bag contains 50 identical cards which are numbered from 1 to 50. If one card is drawn at random from the bag, find the probability that it bears

A) a perfect square number. 7/50

B) a number divisible by 4 divisible by 4. 6/25

C) A number divisible by 5. 1/5

D) A number divisible by 4 or 5. 2/5

E) a number divisible by 4 and 5. 1/25.

34) A bag contains 6 red, 8 white and x blue balls which are identical in shape and size. the probability that a ball drawn at random is blue or white is 5/7. find x. 7.

35) Two fair dice are thrown. Find the probability of getting

A) the same score on the first die as on the second. 1/6

B) the score of second dice is greater than the score of first. 5/12

36) from a group of 5 men and 3 women A committee of 4 persons persons is to be selected selected be selected selected randomly. find the probability that

A) there is a majority of man. 1/2

B) there is majority of women. 1/14

37) A bag contains 6 white and 9 black balls, if 3 balls are drawn at random, find the probability that all of them are black. 12/65

38)A coin is tossed once. Find the probability of getting head

39) A dice is thrown once. What is the probability of getting a prime number.

40)a) A die is thrown once. What is the probability of getting a number other than 4 ?

b) A ‘3’

c) A ‘4’

c) An odd number

d) A Number greater than 4

41) Three unbiased coins are tossed simultaneously. Find the probability of getting

a)All heads

b) All tails

c) No head.

d) No tail

e) Atleast one head

f) Atleast one tail

g) All not heads

h) Atmost one tail

I) Two or more tails

j) More than two tails

k) Less than one head

l) Heads and tails

m) Heads are the two extremes

n) Heads will come in the 1st row

o) Heads will exceed the number of tails in a particular throw.

p) Exactly 2 heads

q) Atmost two heads.

r) Atleast two heads

5) Two coins are tossed simultaneously. What is the probability

a)All heads

b) All tails

c) No heads

d) No tails

e) All not heads

f) at least one head

g) Exactly one head

h) At Most one head.

i) Atleast two heads

j) Heads will come in first row.

k) Heads and tails will occur alternately.

42) One card is drawn at random from well shuffled pack of cards. What is the probability of

drawing a

a)A King.

b) A queen

c) An eight

d) A black card

e) The six of the clubs.

f) A spade.

g) A King of red suit

h) A queen of black suit

i) A pack of hearts.

j) A red of face’card.

k) A King or a jack

l) A non-Ace

m) A red card

n) Neither king nor a queen

o) Neither a red card nor a queen.

p) It is either a King or a knave

q) It is neither a King nor a Knave

r) It is neither a heart nor a diamond

s) It is neither an Ace, nor a King, nor a Queen, nor a Knave.

t) A spade or an Ace not of a spade.

43) Two dice are rolled simultaneously. What is the probability of getting

a)8 as the sum of two numbers that turn up

b) A doublet

c) Sum is 7

d) Sum is 11

e) It is either 7 or 11.

f) It is neither 7 nor 11.

g) Sum is odd number more

than 3.

h) Sum is a multiple of 4.

i) Sum is a multiple of 3 and 4

j) Sum is multiple of 3 or 4.

k) Sum is atleast 8

l) Sum is Atmost 7

m) The product of the faces is 12.

n) Sum of the faces is more

than 12

44) There are 35 students in a class of whom 20 are boys and 15 are girls. From these students one is chosen at random. What is the probability that the chosen student is a

a)A Boy

b) A girl.

45) Seventeen cards numbered 1,2,3...16,17 are put in a box and mix thoroughly. One person draw a card from the box. Find probability that the number on the card is

a)A prime

b) Divisible by 3

c) Divisible by 2 and 3 both

d) Not divisible by 2

e) Divisible by 2 but not by 3

f) Divisible by 3 but not by 2

g) Divisible by either 3 or 2

h) Neither Divisible by 2 nor 3

i) multiple of 4

10) A bag contains 6 Red,8 white balls, 5 green and 3 black balls . One ball is drawn at random from the bag. Find the probability that the ball is:

a) White

b) Red or White

c) Not green

d) Neither white nor black

e) A pink ball

46) From a pack of cards jacks, queens,kings and aces of red colours are removed. From the remaining, a card is drawn random. Find the probability:

a)A black queen

b) A red card

c) A picture card

47) The probability that will rain today is 0.84. what is the probability that will not rain today ?

48) What is the probability that an ordinary year has 53 Sundays

49) Find the probability of getting 53 Friday in a leap year

50) In a lottery there are 10 prizes and 25 blanks . What is the probability of getting a prize ?

51) It is known that a box of 200 electric bulbs contains 16 defective bulbs. One bulb is taken out random from the box. What is the probability that the bulb drawn is

a) Defective

b) not defective.

52) A bag contains 3 green and 8 white balls. If one ball is drawn at random. Find the probability that:

a)It is green

b) It is white

53) There are 17 numbered 1 to 17 in a bag, If a person selects one ball is drawn randomly. Find the probability that the number printed on the ball will be an even number greater than 9.

54) One card is drawn from a pack of cards. Find the probability that

a)Either a spade or a diamond

b) Either a spade or a king.

c) Neither king nor diamond.

d) Either red or queen.

55) One counter is drawn at random from a bag contain 70 counters marked with the first 70 numerical. Find the chance that:

a) multiple of 8 or 9

b) either multiple of 3 or 4

56) What is the probability that in a group of 3 people

a) 3 people having same Birthday

b) 2 people have same birthday

c) All have different birthdays

d) atleast one have same birthday

e) Atleast 2 have same birthday

f) Atmost 2 have same birthday.

57) the face cards are removed from a full pack. Out of the remaining 40 cards, 4 are drawn at random.find probability that they belong to different suits?

58) In a throw of two dice, find the probability of getting one prime and one composite number? ⅓

59) Find the probability that a leap year chosen at random will have 53 Sundays. 2/7

60) An integer is chosen at random from the first 100 integers. What is the probability that this number will not be divisible by 5 or 8? 7/10

61) In throwing a fair dice, what is the probability of getting the number' 3' ⅙

62) Find the probability of throwing atleast one ace in a sample throw with two dice. 11/36

63) From a pack of 52 Cards, two are drawn at random. Find the probability that one is a knave and the other a queen. 8/663

64) If a card is picked up at random from a pack of 52 Cards. Find the probability that it is

A) a spade. 1/4

B) a king or queen. 2/23

C) a spade or king or a queen. 19/52

65) Three coins are tossed. What is the probability of getting

A) 2 Tails and 1 Head.. 3/8

B) 1 tail and 2 heads. 3/8

C) neither 3 heads nor tails. 3/4

D) three heads. 1/8

E) atleast one tail. ⅞

66) Two fair dice are thrown. Find the probability of getting

A) a number divisible by 2 or 4. 1/2

B) a number divisible by 2 & 4. 1/4

C) a prime no. less than 8. 13/36

67) What is the probability of throwing a number greater than 2 with a fair dice. ⅔

68) Two fair coins are tossed. Find the probability of getting

A) 2 heads. 1/4

B) 1 head and 1 tail. 1/2

C) 2 tails. ⅔

69) In rolling two dices, find the probability that

A) atleast one '6'. 11/16

B) the sum is 5. 22/36

70) Two fair dice are thrown. What is the probability of

A) throwing a double. 1/6

B) the sum is greater than 10. 1/12

C) The sum is less than 10. ⅚

71) From a pack of 52 Cards, three cards are drawn at random. Find the probability of drawing a king, a queen and Jack. 16/5525

72) A bag contains 20 balls marked 1 to 20. One ball is drawn at random. Find the probability that it is marked with a number multiple of 5 or 7.

73) Two fair dice are thrown. Given that the sum of the dice is less than or equal to 4, find the probability that only one dice shows two. ⅓

74) Out of all the 2digit Integers between 1 to 200, a 2digit number has to be selected at random. What is the probability that the selected number is not divisible by 7 ? 77/90

75) Two dice are thrown. If the total on the faces of the two dices are 6, find the probability that there are two odd numbers on the faces. ⅗

ARITHMETIC PROGRESSION

1) If the sum of the three numbers in A. P be 18 then what is the middle term?

2) If 10 be added to each term of an AP then, will the new progression be in AP also. ? Y

3) If 5 be multiplied with each term of an AP then will the new progression an AP also?

4) Find the 8th term of an AP., 13, 10, 17, 24......

5) Which term of the AP. 7, 11, 15, 19, ....... is 103?

6) If the 7th term of an AP be 20 and the 10th term be 29 then find the first and the common difference.

7) In an AP 1st term be -5, last term be 25 and number of terms be 10. Find the sum of the terms.

8) What do you mean by natural numbers? Find the sum of first 100 natural numbers.

9) a²+ 3a + 2, 3a²+ 2a+5, 4a²+ 5a + 4 are in AP, then find a.

10) Find the arithmetic progression of which nth term is 3n -1.

11) If the pth term of an AP be q and qth term be p, then find the p+ q th term.

12) The sum of first n terms of an AP is n². Find the common difference.

13) Find the sum of the first 20 terms of the AP. 20, 17, 14...

14) The 5th term and the 11th term of an AP are 41 and 20 respectively. Find the 1st term. What will be the sum of first 11 terms of this AP.

15) The sum of p th term of an AP is 3p²+ 5p; show that it is an AP.

16) Insert five A. M between 2 and 20.

17) The n th term of an AP is p. Show that the sum of first (2n -1) terms is (2n -1)p.

18) The sum of how many terms of the series 27 + 24+21+...... will be equal to 132 ?

19) If the sum of n th term of an AP is n² + 3n, then eight term is equal to the sum of first eleven terms.

20) Find the sum upto n terms of the following series. 1+ 5 + 12+ 22+ 35+ ....... {n²(n+1)/2}

21) Find the following sum:

1²+ 3² + 5²+ ...... upto n terms.

22) The middle term of an AP having 11 terms is 12. Find the sum of the 11 terms of that progression.

23) The sum of the three numbers in AP is 21 and their product is 231. Find the numbers.

24) There are n arithmetic means between 4 and 31. If the second mean: last mean= 5: 14 then find the value of n.

25) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then show that the sum of the first P+ Q terms is zero.

26) The first term of an AP is a, the second term is b and the last term is c. Show that the sum is {(a+ c)(b+ c - 2a)}/{2(b - a)}.

27) The interior angles of a polygon are in AP. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.

28) The sides of a right angled triangle are in AP. If the smallest side is 5cm then find the largest side.

29) A man divided ₹24000 among his four sons in such a way that the amounts of four sons are in AP. The product of the amounts of 1st and 3rd sons is to the product of amounts of 2nd and 4th sons as 7: 15. Calculate the amounts received by each son. 3000, 5000, 7000, 9000

30) Two posts are offered to a man. In the one, the starting salary is ₹1200 and increases annually by ₹80; in the other, the salary starts at 850 and increases annually by ₹120. The man decides to accept the post that will give him better total income in the first 16 years. Which post will be accept ? 1st offer

31) A polygon has 25 sides, the lengths of which starting from the smallest side are in AP. If the perimeter of the polygon is 2100 cm, and the length of the largest side is 20 times that of the smallest, find the length of the smallest side and the common difference of the AP.

**32) If the sum of p-terms of an AP be q and the sum of q terms be p then show that the sum of its (p+ q) terms is -(p+ q).

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