Chapter-1 Equation and In Equation

8/6/2019 Chapter-1 Equation and In Equation

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CHAPTER 1

EQUATION AND INEQUATION

A. FILL IN THE BLANKS:1. The coefficient of in the polynomial is (1982 2 Marks)2. If is a root of the equation , where p and q are real, then (1982-2 Marks)3. If the product of the roots of the equation is 7, then the roots are real for =.

(1984-2 Marks)

4. If the quadratic equation and have a common root, then the numerical vaof is (1986-2 Marks)

5. If and , them = . and = . (1990-2 Marks)6. Let and be positive such that . The number of solutions

integers, satisfying is . (1996 2 Marks)7. The sum of all the real roots of the equation is . (1997-2 Marks)8. The solution of equation is . (1986-2 marks)9. The smallest value of k, for which both the roots of the equation x2 8kx + 16(k2 k + 1 ) = 0 are real, distinct and

have values at least 4, is? (2009)

B. True/False1. For every integer , inequality holds. (1981-2 Marks).2. The equation has an irrational root. (1983-2 Marks).3. If

, then the roots of the equation

are real and distinct.

(1984-1 Marks).

4. If are positive integers, whose sum is an even number, then the number of odd integers among theodd. (1985 -1 Marks)

5. If and , wher then has at least two real roots(1985-1 Marks).

6. If and are positive real numbers and are any positive integers, then (1989-1 Marks).

C. MULTIPLE CHOICE QUESTION WITH ONE CORRECT ANSWER1. The number of real solution of the equation is

(a) 4 (b) 1 (c) 3 (d) 2 (1982-2 Marks).2. Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town

the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built at

(a) town B (b) 45 km from town A (c) town A (d) 45 km from town B

(1982-2 Marks).

3. If p, q, r are any real numbers, then


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(a) (b)

(c) (d) none of these (1982-2 Marks).

4. The largest interval for which is(a) (b) (c) (d) (1984-2 Marks).

5. The equation has (1984-2 Marks).(a) no root (b) one root (c) twp equal roots (d)infinitely many roots

6. (v) if , then lies in the interval

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

(1982-2 Marks).

7. If and are the roots of and are the roots of , then the equation has always

(a) two real roots (b) two positive roots

(c) two negative roots (d) one positive and one negative root (1982-2 Marks).

8. let a, b, c be real numbers, . If is the troot of . is the root of and , then the euation has a root that always satisfies

(a) (b) (c) (d) (1989-2 Marks).

9. The number of solution of the equation is

(a) 0 (b)1 (c)2 (d) Infinitely many (1982-2 Marks).

10. let

be the roots of the equation

. Then the roots of the equation

are

(a) a,c (b)b,c (c) a,b (d) a+c, b+c (1982-2 Marks).

11. The equation has

(a) no solution (b) one solution (c)two solution (d) more than two solution (1997C-2 Marks)

12. If the roots of the equation are real less than 3, then

(a) (b) (c) (d) (1999-2 Marks).

13. If and are the roots of the equation , wher , then

(a) (b) (c) (d) (2000S).

14. If a, b, c are positive real numbers such that , then satifies the relation

(a) (b) (c) (d) (2000S).

15. if , then the equation has

(a) both roots in (a,b) (b)both roots in


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(c) both roots in (d) one root in and the other in (2000S).16. For the equation , if one of the root is square of the other, then is equal to

(a) 1/3 (b)1 (c)3 (d) 2/3 (2000S)

17. If are positive real numbers whose product is a fixed number c, then the minimum value of is(a) (b) (c) (d) (2002S)

18.The set of all numbers x for which , is (2002S)(a) (b) (c) (d)

19. If then

is always greater than or equal to

(a) (b) 1 (c)2 (d) (2003S)

20.For all , then the interval in which a lies is(a) (b) (c) (d) (2004S)

21. If one root is square of the other root of the equation , then the relation between p and q is(a) (b) ) (c) (d) ) (2004S)

22.The number of points of intersection of two curves and is(a) 0 (b)1 (c)2 (d) (1994)

23. If are +ve and are in A.P. , the roots of quadratic equation are all real for(a) (b) (c) all p and r (d) no p and r (1994)

24.Let . The number of equation of the form having real roots is(a) 15 (b) 9 (c) 7 (d) 8 (1994)

25.Let a,b,c, be the sides of a triangle where and R . If the roots of the equation are real then

(a)

(b)

(c)

(d)

(2006-3M,-1)

26.Let, be the roots of the equation x2 px + r = 0 and be the roots of the equation x2 qx + r = 0. Then thevalue of r is

(A) (p - q)(2q - p) (B)

(q - p)(2p - q) (C)

(q - 2p)(2q - p) (D)

(2p - q)(2q - p) (2007)

27.Let p and q be real numbers such that p 0, p3 q and p3 -q. If are nonzero complex numbers satisfying then a quadratic equation having

as its roots is (2010)

(a) (b)


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(c) (d)

D. MULTIPLE CHOICE QUESTION WITH ONE OR MORE THAN ONE CORRECT ANSWER :1. For real

, the function

will assume all real values provided

(a) (b) (c) (d) (1984)2. If S is the set of all real such that is positive, then S contains (1986)(a) (b)

(c)

(d)

(e) none of these

3. If a, b and c are distinct positive numbers, then the expression is (1986)(a) positive (b) negative (c)non-positive (d) non-negative (e) none of these

4. If a, b, c, d and p are distinct real numbers such that tha, b, c, d (1987)

(a) are in A.P. (b) are in G.P. (c) are in H.P. (d)satisfy ab = cd (e) satisfy none of these

5. The equation has(a) at least one real solution (b) exactly three solutions

(c)exactly one irrational solution (d) complex roots (1989)

6. The product of positive numbers is unity then their sum is (1991)(a) a positive integer (b) divisible by n (c) equal to (d) never less than n

7. Number of divisor of the form of the integer 240 is(a) 4 (b) 8 (c)10 (d)3 (1998)

8. Root(s) of the expression 2sin2 + sin22 = 2 (IITJEE 2009)(A)

(B)

(C)

(D)

(E)

E. SUBJECTIVE PROBLEMS :1. Show that the equation has no teal solution. (1992)2. squares of equal size are arranged to from a rectangle of dimension m by n, where m and n are natural numbers.

square will be called neighbours if they have exactly one common side. A natural number is written in each square s

that the number written in any square is the arithmetic mean of the numbers written in its neighboring squats. Shows

that this is possible only if all the numbers used are equal. (1982)

3. If one root of the quadratic equation is equal to the n-th power of the other, then show that

(1983)

4. Find all real values of which satisfy and (1983)


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5. Solve for (1985)6. For , determine all real roots of the equation (1986)7. Find the set of all for which (1987)8. Lat be the roots of and respectively. If the system of equations

abd has a nontrival solution, then prove that

.

9. Solve (1988)10.Let a, b, c be real. If has two real roots and , where and then show that

. (1995)

11.Find the set of all solutions of the equation 12.Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. if a, b, c and d denot

the lengths of the sides of the quadrilateral, prove that

13. If are the roots of and are the roots of forsome constant , then prove that

(2000)

14.Let a, b, c be real numbers with and let be the roots of the equation . Express the roots in terms of .

15. If where then find the values of a for which equation has unequal real rofor all values of b. (2003)16. If a, b, c are positive real numbers. Then prove that (2004)17.Let and be the roots of the equation and those of are c, d then the

value of a + b + c + d, when , is (2006)

F. Reason Assertion Typw

1. Let a, b, c, p, q be real numbers. Suppose, are the roots of the equation x2 + 2px + q = 0 and,

are the roots of the equation ax2 + 2bx + c = 0, where2{1, 0, 1}. (2008)

STATEMENT1 : (p2 q) (b2 ac) 0

and

STATEMENT2 : b pa or c qa

(A) STATEMENT1 is True, STATEMENT2 is True; STATEMENT2 is a correct explanation for STATEMENT1

(B) STATEMENT1 is True, STATEMENT2 is True; STATEMENT2 is NOT a correct explanation for

STATEMENT1.

(C) STATEMENT1 is True, STATEMENT2 is False

(D) STATEMENT1 is False, STATEMENT2 is True


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ANSWERS

A. 1. -5050

B. 1. T 2. F 3. T 4. F 5. T 6. FC. 1. (a) 2. (c) 3. (b) 4. (b) 5. (a) 6. (c) 7. (a) 8. (d) 9. (a) 10. (c) 11. (a) 12. (a) 13. (b) 14. (a)

15. (d) 16. (c) 17. (a) 18. (b) 19. (a) 20. (b) 21. (a) 22. (a) 23. (b) 24. (c) 25. (a) 26. (d) 27. (b)

D. 1. (c,d) 2. (a,d) 3. (b) 4. (b) 5.(a,b,c) 6. (d) 7. (a) 8.(b,d)E. 4. 5. 6.

7. 9. 11.

14. 15. 16. 1210

F. 1 (b)

Chapter-1 Equation and In Equation

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