maths pre found. dpps

SYNCHRO - DIVISIONPRE- FOUNDATION

Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 1 Date : 1.02.2010

Chapter : - Real NumbersTopics1. Euclid’s division Algorithm.2. Fundamental term of Arithmetic.

1. H.C.F. of 23 × 32 × 5, 23 × 33 × 52 and 22 × 3 × 53 × 7 is–(a) 30 (b) 48 (c) 60 (d) 105

2. Five bells begin to toll together and toll respectively at intervals of 6, 7, 8, 9 and 12 seconds. How many time they will toll together in 1 hour excluding one at start?(a) 3 (b) 5 (c) 7 (d) 9

3. If p/q is a rotational number then prime factorization of q, satisfy the property? (a) 2n + 5m (b) 2n / 5m (c) 2n.5m (d) 2n–5m

4. If the L.C.M. of (a, b) = 26460, H.C.F. of (a, b)= 27 and b = 540, find a –(a) 1323 (b) 1325 (c) 1324 (d) None of these

5. Which of the following is equal to

–

(a) (b) (c) (d)

6. If then n2 equal –(a) 92x (b) 33x (c) 272x (d) 9x+1

7. If a = 384, b = 26, compute 2 = (384, 26) as a linear combination of by 384 and 26 –

8. Find the H.C.F. of 300, 540, 890 by applying Euclid’s algorithm–

9. Show that one and only one out of n, n + 4, n + 8, n + 12, n + 16 is divisible by 5, where n is any +ve integer.

10. If a and b are cop rimes, prove that L.C.M. of ma and mb is mab –

11. Prove that if x and y are old positive integers then x2 + y2 is even but not divisible by 4.

12. Prove that n2 – n is divisible by 2 for every positive integer n

13. Prove that one of every three consecutive positive integers is divisible by 3.

14. If be an irrational, prove that is an irrational.

ANSWER KEY1. (c)2. (c)3. (c)

4. (a)5. (d)

6. (d)8. 10

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Chapter :- Real NumberTopics1. Euclid's division Algorithm.2. Proving irrationality of irrational number. L.C.M and H.C.F of numbers.

1. Find the largest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.(a) 17 (b) 18 (c) 16 (d) 15

2. In a school there are two sections, section A and section B of class Xth. There are 32 students in section A and 36 students in section B. Determine the minimum numbers of books required for their class library so that they can be distributed equally among students of section A or section B –(a) 290 (b) 284 (c) 286 (d) 288

3. A circular field has a circumference of 360 km. Three cyclists starts together and can cycle 48, 60 and 72 km a day, round the field, when will they meet again ? (a) 48 Days (b) 54 Days (c) 60 Days (d) 62 Days

4. Three bells in bloodbath temple toll at the interval of 48, 72 and 108 seconds individually. If they have tolled all together at 6:00 AM then how many times these bells will toll together till the 6:00 PM on the same day.(a) 100 times (b) 101 times (c) 99 times (d) 102 times

5. Find the number of numbers lying between 1 and 1000 which are divisible by each of 6, 7 and 15 –(a) 3 (b) 4 (c) 6 (d) 5

6. 105 goats, 104 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boat man has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number of each time. How many animals went in each trip –

7. If a and b are two add positive integers such that a > b then prove that one of the two numbers and is odd and

other is even –

8. Prove that if a positive integer is of the form 6q + 5 then it is of the form 3q + 2 for some integer q, but not conversely

9. Find the H.C.F of 65 and 117 and express it in the form 65m + 117 n.

10. Find the H.C.F of 81 and 237 and express it as linear combination of 81 and 237–

11. Prove that is irrational.

12. Let a, b, c and d be positive rationals. Such that then either a = c and b = d or b and d are square of rationals

13. Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic-wise and height of each stack is the same base. The number of English books is 96, the no. of hindi books is 240 and the number of mathematics is 336. Assuming that the books are of the same thickness determine the number of stacks of English, hindi and mathematics books.

ANSWER KEY1. (a)2. (d)

3. (c)4. (b)

5. (b)13. Hindi – 5, English – 2, Maths – 7

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Chapter : - PolynomialsTopics1. Zeros of polynomials.2. Graphical meaning of zeroes of polynomials3. Relationship between zeroes and co.eff. of polynomials

1. If H.C.F of x2 – x – 6 and x2 + 9x + 14 is x + m then the value of m is –(a) 1 (b) 2 (c) –2 (d) –1

2. The value of P for which (x + 2) is a factor of (x + 1)6 + (2x + P)3 is –

(a) 5 (b) 3 (c) 2 (d) None

3. The set of values of K for which the sum of the roots of P(x) = 4x2 + K2x + K is equal is twice the product of roots is(a) (b) (c) (d)

4. If 2x2 + xy – 3y2 + x + ay – 10 = (2x + 3y + b) (x – y – z), than the value of a and b are – (a) 11 and 5 (b) 1 and –5 (c) –1 and –5 (d) –11 and 5

5. Three number x, y and z are such that x2 + y2 = z2 If z = m2 + n2 , y = 2mn then is –

(a) (b) (c) (d)

6. When y2 + my + 2 is divided by y – 1 the remainder is R1. If the same expression is divided by y + 1. The remainder is R2. If R1 = R2 then m is –(a) 0 (b) 1 (c) –1 (d) 2

7. Write a rational expression whose numerator is a quadratic polynomial with zeroes 2 and –1 and whose denominator is a quadratic polynomial with zeroes ½ and 3.

8. If x + a is the factor of the polynomial x2 + px + q and x2 + mx + n, prove that

9. If x2 – 1 is a factor of ax4 + bx3 + cx2 + dx + e, show that a + c + e = b + d = 0

10. Find the value of p and q such that 1 and –2 are the zeroes of the polynomial x3 + 10x2 + px + q

11. Find the zeroes of polynomial f(x) = abx2 + (b2 – ac) x – bc & verify the relationship between zeroes and co-efficient.

12. If & are the zeroes of the quadratic polynomial f(x) = x2 – px + q , then find the value of –

(i) (ii) (iii) (iv)

(v) (vi)

13. If , are the zeroes of the polynomial f(x) = x2 – 5x + K such that – = 1, find the value of K. –

ANSWER KEY1. (c)2. (d)3. (b)

4. (d)5. (c)6. (a)

7.

10. P = 7 , q = –1813. K = 6

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Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 4 Date : 1.02.2010Chapter : - PolynomialsTopics1. Zeros of polynomials.2. Relation between zeroes and co-efficient of polynomial 3. Division algorithm

1. Let a, b, c be real number. If a + b + c = 7 then has the value equal to –

(a) (b) (c) (d)

2. If are the zeroes of the polynomial f(x) = ax3 + bx2 + cx + d then is equal to –

(a) –b/d (b) c/d (c) –c/d (d) –c/a

3. If one zero of the polynomial f(x) = (k2 + 4)x2 + 13x + 4k is reciprocal of the other, then k =(a) 2 (b) –2 (c) 1 (d) –1

4. If are the zeroes of the polynomial f(x) = ax3 + bx2 + cx + d then

(a) (b) (c) (d)

5. If zeroes of the polynomial f(x) = x3 – 3px2 + qx – r are in A.P, then – (a) 2p3 = pq – r (b) 2p3 = pq + r (c) p3 = pq – r (d) None of these

6. If the polynomial f(x) = ax3 + bx – c is divisible by the polynomial g(x) = x2 + bx + c, then ab =

(a) 1 (b) (c) –1 (d)

7. If & are the zeroes of the quadratic polynomial f(x) = kx2 + 4x + 4 such that , find the value of k. –

8. If & are the zeroes of the polynomial f(x) = 2x2 + 5x + k satisfying the relation , then find the value

of k –

9. If & are the zeroes of the quadratic polynomial f(x) = 2x2 – 5x + 7, find a polynomial whose zeroes are and

10. Find a quadratic polynomial whose zeroes are reciprocal of the zeroes of the polynomials f(x) = ax2 + bx + c

11. If & are the zeroes of the quadratic polynomial p(x) = x2 – px + q prove that

12. Find the condition that the zeroes of the polynomial f(x) = x3–3px2+ qx – r may be in arithmetic progression –

13. By applying division algorithm prove that the polynomial g(x) = x2 + 3x + 1 is a factor of the polynomial f(x) = 3x4 + 5x3 – 7x2 + 2x + 2

14. Find the value of a & b so that x4 + x3 + 8x2 + ax + b is divisible by x2 + 1.

15. What must be subtracted from the polynomial f(x) = x4 + 2x3 – 13x2 – 12x + 21. So that the resulting polynomial is exactly divisible by x2 – 4x + 3

ANSWER KEY1. (b)2. (c)3. (a)4. (d)5. (a)

6. (a) 7. k = –1, 2/38. k = 2

9.

10. k (cx2 + bx +a)12. 2p3 = pq – r14. a = 1, b = 715. 2x – 3

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Pre-Foundation Faculty : Subject : Mathematics D.P.P. No. 5 Date : 1.02.2010

Chapter : - Linear equation in two variable Topics1. Solution of linear equation in two variable 2. Conditions for solvability (or consistency)3. Algebraic Methods for finding solutions 4. Word problems based on linear equation two variable

1. The value of k for which the system of equation kx – y = 2, 6x – 2y = 3 has a unique solution is –(a) = 3 (b) 3 (c) 0 (d) = 0

2. If am bl, then the system of equation ax + by = c, lx + my = n (a) Has a unique solution (b) Has no solution

(c) Infinitely many solution (d) May or may not have a solution

3. If 2x – 3y = 7 and (a + b) x – (a + b – 3)y = 4a + b represent coincident lines then a and b satisfy the equation. (a) a + 5b = 0 (b) 5a + b = 0 (c) a – 5b = 0 (d) 5a – b = 0

4. A person is standing on a staircase. He walks down 4 steps, up 3 steps, down 6 steps, up 2 steps, up 9 steps and down 2 steps. Where is he standing in relation to the step on which he started –(a) 2 steps above (b) 1 steps above (c) The same place (d) 1 step below

5. There are n questions in question papers A student answers 15 of the first 20 correctly of the remaining question. He answers one third correctly. All question carry equal marks. If the student get 50% marks. what is the value of n ?(a) 20 (b) 50 (c) 60 (d) None of these

6. The graphs of 2x + 3y – 6 = 0, 4x – 3y = 6, x = 2, y = 2/3 intersect in. (a) Four points (b) One point(c) In no point (d) In infinite number of point

7. Find graphically the vertices of a triangle. Whose sides are y = x, y = 2x, x + y = 6.

8. A person invested some amount at the rate of 12% simple interest and the remaining at 10%. He received yearly interest of Rs.130. But if he had interchanged the amounts invested. He would have receive Rs. 4 more interest. How much money did he invest at different rates –

9. If a student had walked 1 Km/h faster, he would have taken 15 minutes less to walk 3 Km. find the rate at which he was walking.

10. Out a group of swans, 7/2 times the square root of the number are playing on the shore of a tank the two remaining one's are playing with amorous fight, in the water what is the total no. of swans –

11. A Peacock is sitting on the top of a pillar which is 9 m high. From a point 27 m away from the botton of the pillar, a snake is coming to its hole at the base of the pillar. Secing the snake the peacock pounces on it. if their speeds are equal at what distance from the hole is the snake caught –

12. in ABCA = x°, B = (3x –2)°, C = y° and C – B = 9°. Determine the three angles A = 25° , B = 73°, C =82° –

ANSWER KEY1. (b)2. (a)3. (c)4. (a)

5. (a)6. (b)8. 500 at 12% and 700 at 10%

10. 1611. 12 Metre12. A = 25°, B = 73°, C = 82°

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Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 6 Date : 1.02.2010Chapter : - Linear equation in two variable Topics

1. Algebraic methods for finding the solution of linear equation2. Word problems based on linear Equation in two variable

1. Twice the measure of the supplement of an angle is added to three trines the measure of the of complement of the same angle. The sum is the measure of an interior angle of a regular nine-sided polygon. The measure of the supplement of the angle is – (a) 82° (b) 72° (c) 98° (d) 108°

2. It the system of equations 2x + 3y = 7; 2ax + (a+b)y = 28 has infinitely many solutions, then –(a) a = 2b (b) b = 2a (c) a + 2b = 0 (d) 2a + b = 0

3. If the ordered pairs (0, 4) and (0, 7) lie on a line than the equation of that line is –(a) x = 0 (b) y = 0 (c) y = 4 (d) y = 7

4. For what value of k, the solution of the system of linear equations 2x + ky = 1; 3x – 5y = 7 is impossible?

(a) (b) (c) k = 3 (d) None of these

5. The difference of the ages of two brothers is 5 years whereas the sum of their present age is 25 years. Find their present ages.(a) 10, 5 (b) 15, 10 (c) 20, 15 (d) 12, 7

6. Avinash travels a distance of 820 km partly by car and partly by train. If he travels a distance of 700 km by car and rest distance by train then he had taken 12 hours. But if he travels 350 km by car and rest distance by train then he takes 50 minutes more find the separate speed of car and train.

7. A train starts from P towards Q with a uniform speed. After covering only a distance of 120 km. Some defect crop up in the rail engine and therefore its speed is reduced to 3/4 of its original speed. Consequently the train reaches its destination late by 30 minutes. Had this defect crop up in at a distance of 90 km from P, the train would have reached Q late by 10 minutes more. Determiner the original speed of the train and distance between P and Q.

8. Some quantity of 80% purity and some quantity of 95% purity are mixed to prepare 24 liters solution of 90% purity find quantity of each solution.

9. Some monkeys are sitting on two trees. If two monkeys go form the first tree to the second one, then the number of monkeys become equal on both trees. But if 4 monkeys go from the second tree to the first one, then the number of monkeys on the first tree is twice number monkeys on the second one. Find the no. of monkeys on two trees.

10. On selling a shirt at 6% loss and a trouser at 10% gain, a readymade cloth seller gains Rs. 5. If he sells the shirt at 10% gain and trouser at 5% gain, he gains Rs. 22. Find the actual prices of the shirt and trouser.

11. The average score of boys in an examination of a school is 71 and that of girls is 73. The average score of the school in that examination is 71.8. Find the ratio of the number of boys to the number of girls appeared in the examination.

12. Railway honors senior citizen, a person of 65 year age or more, by giving 25% concession in the railway fare. A man travels from patna to Delhi with his wife in second class sleeper and he spends Rs. 471 in purchased tickets. It he had traveled alone then he would have spent only Rs. 204 in purchasing the ticket. If the age of man is 68 years and the age of his wife is 62 years. Then find the full fare of second class of an adult from patna to delhi and the amount of reservation fee. Railway does not permit any concession in reservation fee.

ANSWER KEY1 . (a)2. (b)3. (c) 4 (a)

5. (b)6. Speed of car = 70 km/h speed of train = 60 km/h7. Original speed = 60 km/h, distance = 210 km8. 80% purity = 8 litre, 95% Purity = 16 litre

9. 20 monkeys on the first tree 16 are on second tree10. 150, 14011. 3 : 212. Fare = Rs. 252, Reservation fee = Rs. 15

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Chapter – Quadratic EquationsTopic :- Nature of roots

Method of salving quadratic equation by factorisation Completing square method Quadratic Formula

1. If the equations x2 – ax + 1 = 0 has two distinct roots then –(a) | a | = 2 (b) | a | < 2 (c) | a | > 2 (d) None

2. If the equation ax2 + 2x + a = 0 has two distinct roots. If(a) a = + 1 (b) a = 0 (c) a = 0, 1 (d) a = -1, 0

3. The value of is –

(a) 4 (b) 3 (c) -2 (d) 3.5

4. If 2 is a root of the equation x2 + bx + 12 = 0 and the equation x2 + bx + q = 0 has equal roots, then q =(a) 8 (b) –8 (c) 16 (d) –16

5. If the equation (a2 + b2)x2 –2 (ac + bd) x + c2 + d2 = 0 has equal roots then –(a) ab = cd (b) ad = bc (c) (d)

6. If x = 1 is a common root of the equation ax2 + ax + 3 = 0 and x2 + x + b = 0, then ab =(a) 3 (b) 3.5 (b) 6 (d) –3

7. Solve

8. Solve for x each of the following (i) 9x2 – 6a2x + (a4 – b4) = 0(ii) 36x2 – 12ax + a2 – b2 = 0(iii) abx2 + (b2 – ac) x – bc = 0

(iv)

(v) 12abx2 – (9a2 –8b2) x – 6ab = 0

9. Solve the quadratic equation by factorization method

(i)

(ii) a2b2x2 + b2x – a2x – 1 = 0

(iii)

10. Solve the following by using quadratic formula.(i) p2x2 + (p2 – q2) x – q2 = 0(ii) 9x2 – 9 (a + b) x + (2a2 + 5ab + 2b2) = 0

11. Prove that the equation x2 (a2 + b2) + 2x (ac + bd) + (c2 + d2) = 0 has real root if

12. If the roots of the equation (c2 – ab) x2 – 2 (a2 – bc) x + (b2 – ac) = 0 are equal. Prove that either a = 0 or a3 + b3 + c3 = 3abc.

13. If the equation (1 + m2) x2 + 2mcx + (c2 – a2) = 0 has equal roots, Prove that c2 = a2 (1 + m2)

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ANSWER KEY1. (a)2. (a)3. (b)4. (c)5. (d)6. (a)7. –1, 1

8.(i)

(ii) (iii)

(iv) x = –a, –b (v)

9. (i) (ii)

10. (i) (ii)

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Chapter :- Quadratic equationTopic :- Roots of quadratic equation

Algebratic, Methods for finding the solution of quadratic equation Word problems based on quadratic equations Equations reducible to quadratic equations.

1. If the sum of first n even natural numbers is 420, find the value of n(a) 20 (b) 25 (c) 30 (d) 35

2. If the roots of quadratic equation x2 + px + q = 0 differ by unify then –(a) p2 = 4q + 1 (b) p2 = 4q – 1 (c) q2 = 4p + 1 (d) q2 = 4P – 1

3. The value of P for which (x + 2) is a factor of (x + 1)6 + (2x + P)3 is –(a) 5 (b) 3 (c) 2 (d) None

4. If the roots of x2 = bx + c = 0 are sin and cos then b2 =(a) C (b) 1 + 2c (c) 1 + c (d) 1 – c

5. One hundred Monkeys have 100 apples be divide. Each adult gets three apples while three children share one. Number of adult monkeys are –(a) 20 (b) 25 (c) 30 (d) 33

6. The roots of the equation x2 + Ax + B = 0 are 5 & 4. The roots of x2 + Cx + D = 0 are 2 and 9. Which of the following is the of x2 + Ax + D = 0.(a) 3 and 9 (b) –6 and –3 (c) 6 and 9 (d) 3, 3

7. A person on tour has Rs. 360 for his daily expanses. If he exceed his tour programme by 4 days, he must cut down his daily expanses Rs. 3 per day. Find the number of days of his tour programme.

8. If the list price of a book is reduced by Rs. 5. A person can buy 5 more books for Rs. 300. Find the original list price of the book.

9. A line segment AB of 2m length is divided at C into two parts such that AC2 = AB × CB. Find the length of CB.

10. Abhishek 6 days, than the time taken by anubhar to finish a piece of work. If both of them together finish the work in 4 days. Find the time taken by anubhar alone to finish the work.

11. The angry Arjun carried sons arrows for fighting with bheeshm with half he cut down the arrows throuen by bharhm on him and with six arrows he killed the charioter of bheeshm with one arrow each he knocked down respectively the rath, flag and bow of bheeshm. finally with one more than four times the square root of arrows he killed bhecshm unconscious of an arrow-bed. Find the total no of arrows arjun had.

12. A swinging pool is filled with three pipes with uniform flow. The first two pipes operating simultaneous, fill the pool in same time during which the pool is filled by third pipe alone. The second pipe fills the pool five hours faster than first pipe and four hours slower than the third pipe. Find the time required by each five to fill the pool separately.

ANSWER KEY1. (a)2. (a)3. (d)4. (b)

5. (b)6. (b)7. 20 days8. Rs. 20

9.

10. 12 days11. 10012. 15Hrs., 10Hrs & 6 Hrs.

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Chapter – Quadratic EquationsTopics :- Nature of roots of quadratic equation

Methods of solving quadratic equation Word problems

1. The set of values of K for which the sum of the roots of P(x) = 4x2 + k2x + k is equal is twice the product of roots is(a) {–2} (b){–2, 0} (c) {0} (d) {2}

2. If the minimum value of a(x – 2)2 + b (x – 2) + c is 8. Then the minimum value of ax2 + bx + c will be (a) 16 (b) 12 (c) a + b + c (d) 8

3. The roots of x3 + 6x2 + 8x + 5 = 0 are a, b, c and those of x3 + px2 + qx + r = 0 are a – 6, b – 6, c – 6 then value of P is (a) 12 (b) 18 (c) 24 (d) –24

4. A man rows a boat upstream a certain distance and then returns back to the same place. If time taken by him in going upstream is twice the time taken in rowing down stream then the speed of boat in still water and the speed of stream are in the ratio(a) 1 : 3 (b) 3 : 1 (c) 1 : 2 (d) 2 : 1

5. If the roots of a quadratic equation x2 – 5x + 6 = 0 are reduced by 1, then the quadratic equation is –(a) x2 – 3x + 6 = 0 (b) x2 – 3x + 2 = 0 (c) x2 – 7x + 12 = 0 (d) x2 – 7x + 6 = 0

6. If the ratio of the roots of the quadratic equation lx2 + nx + n = 0 be p : q then prove that

7. If and are the roots of the equation x2 – 2x + 3 find the value of P and whenP = 3 – 32 + 5 – 2 and = 3 – 2 + + 5

8. The product of two numbers is 12 and the sum of those numbers and their squares is 32. Find the numbers.

9. If the roots of the equation (b – c) x2 + (c – a) x + (a – b) = 0 be equal, then prove that 2b = a + c

10. Prove that the roots of equation(x – a) (x –b) + (x – b) (x – c) + (x – a) = 0 are always real and they will be equal if a = b = c

11. In a farewell party of class X of a school, each student shakes hand with his/her fellow students. If the total no. of hand shakes is 2256. Find how many student participated in fare well party

12. Out of a number of saras birds, one fourth of the number are moving about in lots, coupled with as will as 7

times the square roat of the number move a hill, 56 birds remain in vakula tree. What is the total no. of birds ?

13. A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over when he increased the size of the square by one student he found that he was short of 25 students find no. of students.

ANSWER KEY1. (b)2. (d)3. (c)4. (b)

5. (b)6. (b)7. P = 1, Q= 28. 3, 4 or –6, –2

11. 4812. 57613. 600

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Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 10 Date : 1.02.2010Chapter :- Arithmetic progressionTopic :- nth term of an A.P.

Sum of n terms of an A.P.

1. If the average of 148, 146, 144, 142, 140 ...... in A.P. is 125. Then the total numbers in the series will be(a) 18 (d) 24 (c) 30 (d) 48

2. A person saves each year Rs. 200 more than he saves in the preceding year, and he saves Rs. 400 in the first year hour many year would it take for his savings not including interest to amount 200, 000 ? (Approx)(a) 40 year (b) 43 year (c) 36 year (d) None

3. The product of the first and fifth terms of an arithmetic progression is less than the square of the third term by 100 twice the second term exceeds the fourth term by 3 then the first term of the progression is –(a) 10 (b) 12 (c) 9 (d) 8

4. The difference between the 6th and 3rd terms of a sequence in A.P. is 12 the common diff. of the sequence is –(a) 2 (b) 3 (c) 4 (d) 12

5. The sum of first 7 terms of an A.P. is 91 whereas the sum of its first 6 terms is 66, then its 7th term is –(a) 25 (b) 29 (c) 21 (d) 27

6. In a class the ages of the students from and A.P. whose common difference is 3 months. If the ages of the youngest and oldest students be respectively 8 years and 14 years, find the number of student in that class.

7. Find fine numbers in A.P. whose sum is 25 and the sum of whose square is 135.

8. If a, b, c are in A.P., show that

a2(b + c) + b2(c + a) + c2 (a + b) = (a + b + c)3

{Hint use 2b = a + c}

9. A gardner waters 15 trees which are in a line at an interval of 3 metres. There is a well at a distance of 7 metres from the first tree in the line of the trees. If the gardner bringes water from the well for each tree seprately, what distance he covers in order to water all the trees beginning with the first assuming that he starts from the well.

10. A man was appointed in the scole of Rs. 5000-200-8000 Find the total amount he gets after his 5 years of services.

11. If the roots of the equation (b – c) x2 + (c – a)x + (a – b) = 0 are equal, then show that a, b, c are in A.P.

12. Two cars start together in the same direction from the same places. The first goes with uniform speed of 10 km/h. The second goes at a speed of 8 km/h in the first hour and increases the speed by ½ km in each succeding hour. After how many hours will the second car overtake the first car if both car go non-stop ?

13. Along a road lie an odd number of stones place at intervals of 10 metres. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried the job with one of the end stones by carrying them in succession. In carrying all the stones he covered a distance of 3 km. Find the number of stones.

14. A man is employed to count Rs. 10710. He counts at the rate of Rs. 180 per minute for half an hour. After this he counts at the rate of Rs. 3 less every minute than the preceding minute. Find the time take by him to count the entire amount.

ANSWER KEY1. (b)2. (b)3. (d)4. (c)

5. (a)6. 257. 3, 4, 5, 6, 7 or 7, 6, 5, 4, 39. 791 meter

10. 32400012. 9 Hrs13. 25 14. 89 minute

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Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 11 Date : 05.02.2010Chapter: - Triangles.Topics

Criteria for similarity of triangles.Area of similar triangle.

1. A vertical stick 20 m long casts a shadow 10 m long on the ground. At the same time, a tower cats a shadow 50 m long on the ground. The height of tower is –(a) 100 m (b) 120 m (c) 25 m (d) 200 m

2. In ABC, point D is on the side AB and point E is on side AC, such that BCED is a trapezium. If DE : BC = 3:5, then area (ADE) : Area ( BCED) –(a) 3 : 4 (b) 9 : 16 (c) 3 : 5 (d) 9 : 25

3. If ABC and DEF are similar such that 2AB = DE and BC = 8 cm, then EF = (a) 16 cm (b) 12 cm (c) 8 cm (d) 4 cm

4. In ABC and DEF , , AB : ED = AC : EF and then (a) 35° (b) 65° (c) 75° (d) 85°

5. ABC is such that AB = 3 cm , BC = 2 cm and CA = 2.5 cm. If DEF -ABC and EF = 4 cm then perimeter of DEF is –(a) 7.5 cm (b) 15 cm (c) 22.5 cm (d) 30 cm

6. ABC and BDE are two equilateral triangles such that D is the mid point of BC. The ratio of the areas of triangles ABC and BDE is –(a) 2 : 1 (b) 1 : 2 (c) 4 : 1 (d) 1 : 4

7. In figure ABCD is a rectangle with sides x and y and an equilateral triangle are placed on side AB. If perimeter of figure is

and area of figure is , then find x + y = ?

8. ABCD is a parallelogram, P is a point on side BC and DP, when produced meets AB produced at L. Prove that –

(i) (ii)

9. In ABC, if AD is the bisector of , prove that

10. Two triangles BAC and BDC, right angled at A and D respectively are drawn on the same side BC and on the same side of BC. If AC and DB intersect at P. Prove that AP × PC = DP × PB.

11. In ABC, AB = AC and , prove that BD2 – CD2 = 2AD.CD

12. In trapezium ABCD, AB||DC and DC = 2AB EF is drawn parallel to AB cuts AD in F and BC in E such that

diagonal DB intersect EF at G. Prove that 7EF = 10 AB

13. Prove that Perimeters of similar triangles are in the ratio of their corresponding sides –

ANSWER KEY1.(a) 2.(d) 3.(d) 4.(c) 5.(b) 6.(c)

7.

12

SYNCHRO - DIVISION

PRE- FOUNDATION Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 12 Date : 05.02.2010

Chapter: - TrianglesTopics1. Criteria for similarity of Triangles.2. Area of two similar Triangles.3. Pythagoras Theorem

1. In ABC, ; If Then which angle is a right angle ?

(a) ABC (b) BAC (c) CAD (d) BAD

2. ABC is a right triangle right angled at A and . Then

(a) (b) (c) (d)

3. In ABC , perpendicular AD from A on BC meets BC at D. If BD = 8 cm, DC = 2 cm and AD = 4 cm, then –(a) ABC is isosceles (b) ABC is equilateral (c) AC = 2AB (d) ABC is right angled at A

4. In ABC , A = 90°, AB = 5 cm and AC = 12 cm if , then AD =

(a) (b) (c) (d)

5. In figure DE || BC. If AD = X, DB = X – 2, and EC = X – 1. Find the value of X

(a) 4 (b) 6 (c) 2 (d) None

6. In figure x is in terms of a, b, c is –

(a) (b) (c) (d)

7. Let x be any point on the side BC of a triangle ABC. If XM , XN are drawn parallel to BA and CA meeting CA, BA in M and N respectively, MN meets BC produced in T, prove that TX2 = TB × TC

8. Let ABC be a triangle and D and E be two points on the side AB such that AD = BE. If DP || BC and EQ || AC. Then prove that PQ || AB –

9. ABC and DBC lie on the same side of the base BC from a point P on BC, PQ || AB and PR || BD are drawn. They meet AC in Q and DC in R respectively. Prove that OR || AD –

10. Two poles of height a meters and b meters are P meters apart. Prove that the height of point of intersection of the lines joining

the top of each pole to the foot of the opposite pole is given by meters –

11. In ABC, C is obtuse produced and produced prove that AB2 = AC.AE + BC.BD

ANSWER KEY1.(b) 2.(b) 3.(d) 4.(b) 5.(a) 6.(b) 7. 8. 9.

13



Chapter: - Co-ordinate GeometryTopics1. Distance between two points.2. Section formula.3. Co-ordinates of mid points of a line segment.

1. Find the points on the x-axis which is equidistant from (2, –5) and (–2, 9)(a) (–7, 0) (b) (–6, 0) (c) (7, 0) (d) None

2. Find values of y for which the distance between the points P(2, –3) and Q(10, y) is 10 units –(a) 3 or –6 (b) 3 or –9 (c) 3 or –5 (d) 3 or 2

3. If the distance of P(x, y) from A(5, 1) and B(–1, 5) are equal then

(a) (b) 3x = 2y (c) (d) All of these

4. Find the value of P for which the points (–1, 3) , (2, P) and (5, –1) are collinear –(a) 2 (b) 3 (c) 1 (d) 4

5. If the co-ordinates of the middle point of the line segment joining the points (2, –1) and (1, –3) be then the condition satisfies –(a) 6 + – 8 = 0 (b) 6 + – 5 = 0 (c) + 6 = 8 (d) None

6. If (–1,2) , (2, –1) and (3, 1) are any three vertices of a parallelogram, then –(a) a = 2, b = 0 (b) a = –2, b = 0 (c) a = –2, b = 6 (d) a = 6, b = 2

7. Find the reflection (image) of point (7, –5) in x axis

8. If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, prove that 3x = 2y

9. Find the point of intersection of y – axis and the perpendicular bisector of the line segment joining the points (3, 6) and (–3, 4)

10. Find the ratio in which the point P(m, 6) divides the line segment joining the points A(–4, 3) and B(2, 8). Also find the value of m

11. If A(5, –1), B(–3, –2) and (–1, 8) are vertices of triangle ABC, find the length of median through A and the co-ordinates of the centoid

12. If (x, y) be on the line joining the two points (1, –3) and (–4, 2) prove that x + y + 2 = 0

ANSWER KEY

1.(a) 2.(b) 3.(d) 4.(c) 5.(a) 6.(c) 7.(7,5) 8. (0, 5) 9. 3 : 2, m = –2/5 10. 3 : 2, m = –2/5

11.

14



Chapter: - Co-ordinate GeometryTopics1. Co-ordinate of centered of Triangle.2. Area of Triangle.3. Cullinarity of points.

1. If the centroid of a triangle is (1, 4) and two of its vertices are (4, –3) and (–9, 7) than the area of the Triangle is –

(a) 183 sq. unit (b) sq. unit (c) 366 sq. unit (d) sq. unit

2. If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a3 + b3 + c3 is equal to–(a) abc (b) 0 (c) a + b + c (d) 3 abc

3. If point (1, 2), (–5, 6) and (a, –2) are collinear than a is equal to –(a) –3 (b) 7 (c) 2 (d) –2

4. Write the co-ordinates of the reflection of point (3, 5) in x-axis –(a) (3, –5) (b) (–3, 5) (c) (–3, –5) (d) None

5. If A(5, 3), B(11, –5) and P(12, y) are the vertices of a right triangle right angled at P, then y =(a) –2 , 4 (b) –2 , –4 (c) 2 , –4 (d) 2 , 4

6. If points (a, 0), (0, b) and (1, 1) are collinear then is –

(a) 1 (b) 2 (c) 0 (d) –1

7. If the centroid of the triangle formed by points P(a, b), Q(b, c) and R(c, a) is at the origin. Find the value of a + b + c

8. If the centroid of the triangle formed by points P(a, b), Q(b, c) and R(c, a) is at the origin, what is the value of

9. The co-ordinates of A,B,C are (6, 3), (–3, 5) and (4, –2) respectively and P is any point (x, y). Show that

10. If P and Q are two points whose co-ordinates are (at2, 2at) and respectively and S is the point (a, 0). Show that

is independent of t –

11. If two vertices of a triangle ABC are A(3, 2), B(–2, 1) and its centroid G has the co-ordinates , find the co-ordinates

of the third vertex –

12. If the points (a, b), (a' , b') and [(a–a'), (b–b')] are collinear, show that ab' = a'b

ANSWER KEY

1.(b) 2.(d) 3.(b) 4.(a) 5.(c) 6.(a) 7. 0 8. 3 11. (4, –4)

15


Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 15 Date : 05.02.2010Chapter: - Trigonometry Topics1. Trigonometric ratios.2. Trigonometric ratio of complementary angles.3. Trigonometric identities.

1. If sin = P, cos = q then which is a true statement –(a) P2 + q2 = 1 (b) P2 – q2 = 1 (c) P2 – q2 = 1 (d) None

2. is equal to –

(a) 1 (b) 0 (c) –1 (d) 2

3. If cosec A = 2 find the value of

(a) 1 (b) 3 (c) 2 (d) 4

4. Find the value of x if tan3x = sin45°. cos45° + sin30°

(a) (b) (c) (d)

5. If x cos A = 1 and tan A = y then find the value of x2 – y2 (a) 0 (b) 1 (c) 2 (d) –1

6. If and b cot A = 1 then find the value of a2 – b2

(a) 0 (b) 2 (c) 1 (d) 3

7. Prove that

8. Prove that (1– sin + cos)2 = 2(1 + cos– sin

9. Prove that

10. If x = a cos3 and y = b sin3 then prove that

11. If , show that

12. If tan + sin = a and tan – sin = b show that

13. Without using tables, evaluate (i) sin(50 + ) – cos(40 – ) + tan1° tan10° tan20° tan70° tan80° tan89°

(ii) + 2tan15° tan37° tan53° tan60° tan75°

14. Prove that cot. tan (90°–) – sec(90°–) cosec+ sin225° + sin265° – (tan5° tan45° tan85°)

ANSWER KEY1. (a) 2. (b) 3. (c) 4. (a) 5. (b) 6. (c)

16


Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 16 Date : 1.02.2010Chapter : - Application of TrigonometryTopics1. Problems releated to heights and distance

1. An observer on the top of a 120 m tall tower finds that the angle of depression of a hut on the level ground is 60° find the distance of the hut from the foot of the tower –(a) 69.28 m (b) 67.28 m (c) 65.08 m (d) None

2. ABCD is a rectangle 60 m × 30m in which P and Q are points produced in BC and DC respectively. If

Find AP + AQ

(a) 90° (b) 120° (c) 160° (d) 180 m

3. Two persons are a metres apart and the height of one is double that of the other. If from the middle point of the line joining their feet, an observer finds the angular elevation of their tops to be complementary then the height of the shorter post is -

(a) (b) (c) (d)

4. If the angle of elevation of a tower from two points distance a and b (a > b) from its foot and in the same straight line from it are 30° and 60° then the height of the tower is –

(a) (b) (c) (d)

5. A tower subtends an angle of 30° at a point on the same level as its foot. At a second point h metres above the first, the depression of the foot of the tower is 60°. The height of the tower is –

(a) (b) (c) (d)

6. A bridge across a river makes an angle of 45° with the river bank. If the length of the bridge across a river is 150 m. What is the width of the river

7. A boy standing on the ground and is flying a kite with 100 m of string at an elevation of 30°. Another boy is standing on the roof of a 10 m high building and is flying his kite at an elevation of 45°. both the boys are on opposite sides of the kites find the length of the string that the second boy must have so that two kites meet –

8. The height of a house subtends a right angle at the opposite window. The angle of elevation of the window from the base of the house is 60°. If the width of the road is 6m. Find the height of the house.

9. A man on the deck of a ship is 16 m above water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the base is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

10. At the foot of a mountain, the elevation of its summit is . After ascending 1 km towards the mountain up an inclination of , it is found that the angle of elevation is Show that is height of the mountain is

11. Prove that

ANSWER KEY1. (a)2. (d)3. (d)

4. (b)5. (c)6. meter

7. 8. 13.856 meter9. 27.712, 43.712

17



Chapter :- CirclesTopics1. Properties of tangents of circle

1. In the following figure, O is the centre of the circle and then –A

CBm°

n°

(a) m + n = 90° (b) m + n = 180° (c) m + n = 120° (d) m + n = 150°

2. Two congruent circles of radius r intersect such that each passes through the centre of the other, then the length of the common chord is given by –(a) r (b) 2r (c) (d)

3. If tangents PA and PB from a point p to a circle with centre O are inclined to each other at angle of 80°, then is equal to –

(a) 50° (b) 60° (c) 70° (d) 80°

4. A circle is inscribed in a having sides 8 cm, 10 cm, 12 cm as shown in fig. find CF.

(a) 5 cm (b) 6 cm (c) 7 cm (d) None

5. In the fig. PA and PB are tangents to the circle drawn from an external point P. CD is a third tangent touching the circle at Q. If PB = 10 cm, and CQ = 2cm. What is the length of PC –

(a) 4 cm (b) 6 cm (c) 8 cm (d) 10 cm

6. The minimum distance of a chord from the centre of the circle is 10 cm. If the radius of the circle is 26 cm then the length of the chord is –(a) 24 cm (b) 20 cm (c) 52 cm (d) 48 cm

7. In the given figure, AR and RT are tangents O is the centre, B is the mid point of QS and AOBT is a straight line. Then the true statement is given by –

(a) OA = OR (b) OQTS is cyclic quad. (c) SQ = AR (d) All of the above

8. EF is parallel to side BC of the and meets AB at E and AC at F. Prove that the circum-circle of touch each other at A.

18

9. PQ is a line segment and R is its mid point. Semicircle are drawn in the same side of PQ, by taking PR, RQ and PQ as

diameter. A circle C(o, r) is drawn which touches the three semicircles. Prove that

ANSWER KEY1. (a)2. (d)

3. (a)4. (c)

5. (c)6. (d)

19



Chapter : - CirclesTopics4. Properties of tangents to a circle.5. Circle through collinear points.

1. Though three collinear points, the number of circles that can be drawn passing through all the points is/are –(a) One (b) Two (c) Three (d) None of these

2. A rectangle PQRS is inscribed in a quadrant of a circle, with P at the centre and R on the circumference. If PS is 12 cm and

PQ is 5 cm, then the diameter of the circle is –(a) 26 cm (b) 29 cm (c) 28 cm (d) None

3. A square ABCD is inscribed in a circle of radius r. Another circle is inscribed in ABCD and a square EFGH is inscribed in this circle then the side EF is equal to –(a) r (b) (c) r/2 (d)

4. r1 and r2 are the radii of two circles with centers P1 and P2, with circles touching each other internally. If r2 > r1, P1 P2 is equal to–

(a) r1 + r2 (b) r2 – r1 (c) (d)

5. In given figure, QS is the diameter and APT the tangent at P, then is equal to –

(a) 60° (b) 30° (c) 40° (d) 50°

6. In the figure which one is true –

(a) PS = QR (b) (c) T1T2 || RS (d) T1T2 || PQ

7. In the figure PQ is common tangents to both the circles. T1R and T2S are tangents than the true statements will be

(a) T1R || T2S (b) PT1 : PR = PT2 : PS (c) Both (a) & (b) (d) Neither (a) nor (b)

8. Each of the congruent circles shown is externally tangent to other circles and/or to the side(s) of the rectangle as shown. If each circle has circumference 16 then the length of the diagonal is –

9. The incircle of touches the sides BC, CA and AB at D, E and F respectively show that

20

AF + BD + CE = AE + BF + CD = perimetre.

10. In figure c(O, r) and c(O', r/2) touch internally at a point A and AB is a chord of the circle c(O, r) intersecting c(O', r/2) at c. Prove that AC = CB

O

CO1

11. Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

12. From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP-diameter of the circle show that is equilateral.

13. Find the length of the transverse common tangent to two circles of radii 8 cm and 3 cm the centers of the circles be at a distance of 13 cm–

ANSWER KEY1. (d)2. (a)3. (a)

4. (b) 5. (a)6. (b)

7. (c)8. 8013. cm

21


Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 19 Date : 1.02.2010Chapter : - Area releated to circlesTopics1. Area of sector of circle.2. Area of segment of circle.1. Each of congruent circles shown is externally tangent to other circles and/or to the side(s) of the rectangle as shown.

If each circle has circle has circumference 16 Then the length of the diagonal is –

(a) 80 (b) 40 (c) 20 (d) 15

2. If a rectangle having width 6 cm and length 8cm inscribed in a circle then radius of circle.(a) 2 cm (b) 5 cm (c) 7 cm (d) 10 cm

3. The circumference of a circle is 100 cm then the side of a square inscribed in this circle is

(a) (b) (c) (d)

4. The radius of a circle is 20 cm. Three more concentric circles are drawn inside it in such a way that it is divided into four parts of equal area, then radius of one of the three concentric circles is(a) (b) (c) (d)

5. If a square and a circle have the same perimeter then – (a) The area of circle is greater than that of a square(b) The area of the square is greater than that of circle

(c) The area of square is times that of the circle

(d) Their areas are equal

6. The area of largest triangle that can be inscribed in a semi circle whose radius r cm is –

(a) 2r cm2 (b) r2 cm2 (c) 2r2 cm2 (d)

7. A wheel has diameter 84 cm find how many complete revolution must it take to cover 792 metres –

8. A lawn is in the shape of semicircle of diameter 35 m. The lawn is surrounded by a flower bed of width 3.5 m all around find the area of the flower bed in m2.

9. A copper wire when bent in the form of a square enclosed an area 121 cm 2. If the same wire is bent into the form of a circle. Find the area of circle

10. In the given fig. the square ABCD is divided into five equal parts all of the same area. The central part is circular and the line AE, GC and BF, HD lie along the diagonal AC and BD of the square. If AB = 11 cm calculate(i) The radius and the circumference of the central part.(ii) The perimeter of the part ABFE

A

CD 11cm

11cm

22

11. In fig. diameter of the biggest semicircle is 108 cm and the diameter of the smallest circle is 36 cm. Calculate area of shaded portion.

B C

D

12. ABCD is a rectangle in which AB : BC = 2 : 1. M is the mid point of AB. MD and MC are one fourth of the circles with

centers at A and B respectively. find the ratio of the area of the rectangle ABCD and the shaded portion.

13. A circular disc of radius 10 cm is divided into sectors with angles 120° and 150°. then find the ratio of the areas of two sectors.

14. In fig. two circles cut at A and B. P and Q are the centers of the circles. If and . Find the area of shaded region. If AP = 4 cm.

ANSWER KEY1. (a)2. ()3. (b)4. (c) 5. (a)

6. (b)7. 300 8. 358.75 m2

9. 154 cm2

10. (i) 2.78 cm, 17.47 cm2 (ii) 25.37 cm

11. 1272.9 cm2

12. 4 : 4–13. 4 : 514. 14.38 cm2

23



Chapter : - Area related to circlesTopics

1. A cord in the form of a square enclosed the area 's' cm2. If the same cord in bent into the form of a circle, then the area of the circle is –

(a) (b) (c) (d)

2. A circle in inscribed in an equilateral triangle and a square in inscribed in the circle then the ratio of the area of the triangle to the area of the square is – (a) (b)

(c) (d)

3. If the area of a circle is halved when its radius is decreased by n, then the radius is equal to – (a) (b) (c) (d)

4. In the adjoining figure, the radius of the inner circle if other circles are of radii m units is –

(a) (b) (c) (d)

5. The total length of the curve in the adjoining figure is –

100 cm

50

(a) 314 cm (b) 315 cm (c) 316 cm (d) 317 cm

6. The area of the shaded region in the figure is –

(a) /3 sq. unit (b) /2 sq. unit(c) /4 sq. unit (d) 2 sq. unit

7. ABC is equilateral triangle and six equal circles are inscribed in it. The side of ABC is a cm. Find the radius of each circle.

8. Three horses are grazing within a semi circular field with centre O. Horses are tied up at P, R and S such that PO and RO are the radii of two semicircles drawn with centre at P and R respectively. S is the centre of circle touching the two semicircle

24

with diameters AO and OB. The horses tied at P and R can graze within the respective semicircles and the horse tied at S can graze within the circle centred at S. Find the percentage of the area of the semicircle with diameter AB that can not be grazed by the horse in nearest –

9. In Fig. ABC is a right angled triangle, AB = 28 cm, and BC = 21 cm. with AC is a diameter a semicircle is drawn and with BC as radius a quarter circle is drawn. Find the area of the shaded region correct to two decimal places.

B C

A

28 cm

10. A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF it is 5 cm. Find (i) Radii of two circles (ii) The area of shaded region.

11. Three circles of radius 2 cm touch another externally. These circle are circumscribe by a circle of radius R cm. Find the value of R and the area of the shaded region in terms of and

A

B

12. There are two circles intersecting each other another smaller circle with centre O, is lying between the common region of two larger circles. Centers of the circle (i.e. A, O and B) are lying on a straight line. AB = 16 cm and the radii of the larger circles are 10 cm each. What is the area of the smaller circle –

BO

13. A square ABCD is inscribed in a circle of radius 'r'. Another circle is inscribed in ABCD and a square EFGH is inscribed in this circle. Then find the length of side EF.

ANSWER KEY

1. (d)2. (d)3. (a)4. (a) 5. (a)

6. (a)

7.

9. 1872.5 cm2

10. (i) 25cm, 20.5 cm (ii) 643.5 cm2

11.

12. 4cm2

13. r

25


Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 21 Date : 19.02.2010Chapter No. : - Surface Area and Volumes.Topics 1. Surface Area and Volume of Cylinder, Cone and sphere2. Surface Area volume of combination of solids

1. The areas of three adjacent faces of a cuboid are x, y and z then the volume of the cuboid is –

(a) xy (b) xyz (c) (d)

2. A rectangular paper of dimensions 6 cm and 3 cm is rolled to form a cylinder with height equal to width of the paper, then its base radius is –

(a) (b) (c) (d)

3. A conical container of base radius ‘r’ and height ‘h’ is full of water which is poured into a cylindrical container of radius mr, then it will occupy a height equal to –

(a) 3m2h (b) (c) (d)

4. If the volume in m3 and the surface area in m2 of a sphere are numerically equal, then the radius of the sphere in m is –(a) 4 (b) 2 (c) 3.5 (d) 3

5. If S1 and S2 be the whole surface of a sphere and the curved surface of the circumscribed cylinder then S1 is equal to –

(a) S2 (b) 2S2 (c) (d)

6. A right circular cone and a cylinder have a circle of unit radius as base and their heights are equal to the radius it self and a hemisphere has the same radius then their volumes are proportional respectively to –(a) 1 : 2 : 3 (b) 3 : 2 : 1 (c) 2 : 1 : 3 (d) 1 : 3 : 2

7. A cone, hemisphere and a cylinder stands on equal bases and have the same height, the height being equal to the radius of the circular base then their whole surface areas are in the ratio(a) (b)

(c) (d)

8. A cube of 9 m edge is immersed completely in a rectangular vessel containing water. If the dimensions of the base are 15 cm by 12 cm find the rise in water level in the vessel

9. Water flowes in a tank 150 × 100 m at the base through a pipe whose cross section is 2 dm by 1.5 dm at the speed of 15 km/hr. in what time, will the water be 3 m deep

10. One filling pipe A is six times faster than second pipe B. If B can fill a cistern in 28 minutes then find the time when the cistern will be full if both pipe are opened together

11. A circular pipe is designed in such a way that water flowing through it at a velocity of 7m/min. is collected at its open end at the rate of 11 cubic meter per min find the radius of pipe

12. Two rectangular sheets of paper each 30 cm × 18 cm are made into two right circular cylinders, one by rolling the paper along its length and the other along its breadth. Find the ratio of the volume of the two cylinder formed.

13. From a circular sheet of paper of radius 10 cm a sector of area 40% is removed. If the remaining part is used to make a conical surface then the ratio of the radius & the height of the cone will be.

14. The areas of three faces of a rectangular block are in the ratio 2 : 3 : 4 and its volume is 9000 sq. metre find the length of the shortest side

ANSWER KEY

26

1. (c)2. (c)3. (b)4. (d) 5. (a)

6. (d)8. 6 m 9. 5 : 3 10. 4 min

11.

12. 5 : 313. 3 : 414. 15 m

27


Pre-Foundation Faculty Subject : Mathematics D.P.P. No. 22 Date : 19.02.2010Chapter – Surface area and volumes Topic :- 1. Surface Area and V of cylinder, cone and sphere.2. Surface Area and volume of frustum of cone.3. Conversion of solid from one shape to another.

1. A wire is in the shape of an equilateral triangle, encloses an area ‘S’ cm2. If the same wire is bent to form a circle then the area of the circle will be –

(a) (b) (c) (d)

2. The ratio of the height of a circular cylinder to the diameter of its base is 1 : 2, then the ratio of the areas of its curved surface to the sum of the areas of its two ends is –(a) 1 : 1 (b) 1 : 2 (c) 2 : 1 (d) 1 : 3

3. The volume of greatest sphere cut off from a cylindrical wood of base radius 1 cm and height 5 cm is –

(a) (b) (c) (d)

4. The length of a rectangle is 1 cm more than its breadth. A square of side ‘P’ has bean cut out of it. If P is one third the breadth of the rectangle the remaining area is (a) (8P2 + 3P) cm2 (b) (3P2 + 8P) cm2

(c) (8P2 + 8) cm2 (d) (3P2 + 8) cm2

5. From a right circular cylinder of radius 6 cm and height 11 cm, one hemisphere of the same radius is removed, then the volume of the remaining part is –(a) 792 cm3 (b) 892 cm3 (c) 992 cm3 (d) None of these

6. The slant height of a frustum of a cone is 4 cm and the perimeters of its circular ends are 18 cm and 6 cm find the curved surface area of the frustum.(a) 42 cm2 (b) 48 cm2 (b) 50 cm2 (d) None

7. From a right cylindrical solid of radius 7 cm and height 12 cm, a right cone half the height of the cylinder is removed. The base of the cone is one of the plane faces of the cylinder, their centers being the same. If the radius of the base of the cone

is half of the radius of the base of the cylinder find the total surface area solid left.

8. The diameter of the circular base and the height of a right cylinder are equal. The cylinder is filled with water to the brim four equal solid spherical ball are put into it. The diameter of each sphere is equal to the radius of the base of the cylinder. What percentage water will flow out.

9. A cylindrical can whose base is horizontal and the internal radius 3.5 cm contains sufficient water so that whom a solid sphere is placed in the can, water just covers the sphere. Given that the sphere just fits in the can. Calculate the depth of water in the can before the sphere was put into the can.

10. A sector of a circle of radius 15 cm has the angle 120°. It is rolled up so that two bounding radii are joined together to form a cone find the volume of the cone

11. A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface area of the

remainder is of the C.S.A. of the whole cone, find the ratio of the line segments into which the cone’s altitude is divided

by the plane.

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12. If h, C, V are respectively the height curved surface area and volume of cone. Prove that 13. The height of a right circular cone is trisected by two planes drawn parallel to the base. Show that the volume of the three

portions starting from the top are in the ratio 1 : 7 : 19.

ANSWER KEY1. (d)2. (a)3. (b)4. (a) 5. (a)

6. (b)7. 873.9 cm2 8. 62.5 %

9. cm

10. 370.9 cm3 11. 1 : 2

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Chapter :- StatisticsTopic :- 1. Mean of the freq. distribution 2. Direct method3. Assumed mean method 4. Step-deviation method

1. The statistical data are of two types. These two types are –(a) Technical data and presentation data(b) Primary data and secondary data(c) Primary data and personal data(d) None of these

2. If the mean of x and is M, then the mean of x2 and is –

(a) M2 (b) M2/4 (c) 2M2 – 1 (d) 2M2 + 1

3. If the mean of x and is M, then the mean of x3 and is –

(a) (b) M(4M2 – 3) (c) M3 (d) M3 + 3

4. The mean of x1, x2 ..... x50 is M. If every xi, i = 1, 2...... 50 is replaced by xi/50 then the mean is –

(a) M (b) (c) (d)

5. The mean of cubes of first n natural numbers is –

(a) (b) (c) (d) n2 + n + 1

6. The mean of first n natural numbers is –

(a) (b) (c) (d)

7. The arithmetic mean of the set of variable a, a + d, a + 2d, a + 3d, ...... a + 2nd is –(a) a + nd (b) a – nd (c) (a + n)d (d) ad + n

8. A frequency distribution of the life times of 400 T.V. picture tubes tasted in a tube company is given below find the average life of tape

Life time (in hrs.) Frequency Life time (in hrs.) Frequency300 – 399 14 800 – 899 62400 – 499 46 900 – 999 48500 – 599 58 1000 – 1099 22600 – 699 76 1100 – 1199 6700 – 799 68

9. The mean of the following frequency table is 50. But the frequencies f1 and f2 are missing. Find the missing frequencies.

Class 0 – 20 20 – 40 40 – 60 80 – 100 80 – 100 TotalFrequency 17 f1 32 f2 19 120

10. Find the mean marks of the students from the following cumulative frequency distribution.Marks Below Below Below Below Below Below Below Below Below Below

10 20 30 40 50 60 70 80 90 100No. of 5 9 17 29 45 60 70 78 83 85

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StudentsANSWER KEY

1. (b)2. (c)3. (b)4. (d) 5. (b)

6. (b)7. a + nd 8. 715 hour. 9. f1= 28, f2= 24

10. 48.41

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Chapter: - StatisticsTopic :-

1. Median and mode of the frequency distribution.2. Cumulative frequency curve (ogive curve)

1. Which of the following can not be determined graphically?(a) Mean (b) Median (c) Mode (d) None of these

2. The median of a given frequency distribution is found graphically with the help of –(a) histogram (b) frequency curve (c) frequency polygon (d) ogive

3. One of the methods of determining mode is –(a) Mode = 2 Median – 3 Mean (b) Mode = 2 Median + 3 Mean(c) Mode = 3 Median – 2 Mean (d) Mode = 3 Median + 2 Mean

4. Mode is –(a) Least frequency value (b) Middle most value (c) Most frequent value (d) None of these

5. The mean of n observations is . If the first item is increased by 1, second by 2 and so on, then the new mean is –

(a) (b) (c) (d) None of these

6. Find the value of x, if the mode of the following data is 25 –15, 20, 25, 18, 14, 15, 25, 15, 18, 16, 20,25, x, 18(a) 25 (b) 20 (c) 18 (d) 16

7. The median of the following data is 525 find the missing frequency, if it is given that there are 100 observations in the data.

Class Interval Frequency Class Interval Frequency0 – 100

100 – 200200 – 300300 – 400400 – 500

25f11217

500 – 600600 – 700700 – 800800 – 900900 – 1000

20f2974

8. Compute the median from the following data:

Mid value : 115 125 135 145 155 165 175 185 195Frequency : 6 25 48 72 116 60 38 22 3

9. Calculate the value of mode for the following frequency distribution :Class : 1-4 5-8 9-12 13-16 17-20 21-24 25-28 29-32 33-36 37-40Frequency : 2 5 8 9 12 14 14 15 11 13

10. The frequency distribution of scores obtained by 230 candidates in a medical entrance test is as follows:Scores : 400-450 450-500 500-550 550-600 600-650 650-700 700-750 750-800Number of candidates :

20 35 40 32 24 27 18 24

Draw cumulative frequency curve by less than and more than method on the same axes. Also draw the two types of cumulative frequency polygons.

ANSWER KEY1. (a)2. (d)3. (c)4. (c) 5. (c)

6. (a)7. f1 = 9, f2 = 158. 135.89. 24.5

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Chapter: - Probability Topic :- Problems related to coins, dice, cards

1. Two dice are thrown simultaneously find the probability getting.

(i) The sum as a prime number.

(ii) A doublet of even number.

(iii) A multiple of 2 on one dice and a multiple of 3 on the other.

2. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of the red ball. Find the

number of blue balls in the bag.

3. A bag contains 12 balls out of which x are white –

(i) It one ball is drawn at random what is the probability that it will be a white ball?

(ii) It 6 more white balls are put in the bag, the probability of drawing a white ball will be double than that in (i) find x.

4. What is the probability that an ordinary year has 53 Sunday –

5. What is the probability that a leap year has 53 Sunday and 53 Monday –

6. A and B throwing a pair of dice. If A throues 9, find B's chance of throwing a higher number –

7. A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is –

(i) A black king (ii) Either a black card or a king (iii) A jack, queen or a king (iv) Neither a heart nor a king

8. A dice is thrown twice. What is the probability that (i) 5 will not come up either time (ii) 5 will come up at least once.

9. Two dice are thrown. Determine the probability of getting a multiple of two on one die and a multiple of three on the other

10. A bag contain 20 balls of different colours. The probability of drawing a green ball is 2/5. How many green balls are there in

a bag.

11. Two customers shyam and ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely

to visit the shop on any day and so on another day. What is the probability that the both will visit the shop on (i) The same

day (ii) Consecutive days (iii) Different days.

12. A die is numbered in such a way that its faces shows the numbers 1,2,2,3,3,6. It is thrown two times and total score in two

throws are noted. What is the probability that the total score is –

(i) Even (ii) 6 (iii) At least 6.

13. A bag contains 80 envelops of which 30 are airmail and the rest are ordinary. Out of the 80 envelops in the bag 48 are

stamped and the rest are unstamped. There are 20 unstamped ordinary envelopes in the bag. If one envelop is chosen at

random from the bag. Then the probabilities that this is an unstamped airmail envelope is –

14. An old man while dialing a 7 digit telephone number. After having dialed the first five digits suddenly forget the last two

digits. But he remains that the last two digit are different. What is the probability of correct telephone number?

ANSWER KEY1. (i) 1/2 (ii) 1/12 (ii) 11/36 2. 10 3. (i) x/12 (ii) 34. 1/7 5. 1/7 6. 1/67. (i) 1/26 (ii) 7/13 (iii) 3/13 (iv) 9/13 8. (i) 25/36 (ii) 11/36 9. 11/36 10. 8. 13. 3/20 14. 1/90

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