ASSINGEMENT # 1
PART-I
Objective Type Questions
1. Let x,y and z be real numbers such that x + y + z = 20 and x + 2y + 3z = 16, then the value of
x + 3y + 5z is-
(A) –4 (B) 4 (C) –12 (D) 12
2. The sum of all distinct real solutions of the equation (x – 3) – (4x + 6) + 216 = 18(4x + 6) (3 – x2) is-
2 3 3
(A) –3 (B) 4 (C) 1 (D) –7
3. The number of integral solutions of the equation x+2y = 2xy is (are)
(A) 2 (B) 1 (C) 4 (D) infinitely many
a b c
4. If 5 10 15 6 = , then the value of a + b + c is (where a,b,c are coprime)
b
(A) 10 (B) 11 (C) 12 (D) 13
5. If x = 3
127 10 3 127 10 then x3 + 9x is divisible by
(A) 3 (B) 4 (C) 5 (D) 6
20 n
6. Each set Xr contains 5 elements and each set Yr contains 2 elements and X r S Yr . If each
r 1 r 1
element of S belong to exactly 10 of the Xr's and to exactly 4 of the Yr's, then n is
(A) 10 (B) 20 (C) 100 (D) 50
Subjective Type Questions
7. From 50 students taking examinations in Mathematics, Physics and Chemistry, each of the student has
passed in at least one of the subject, 37 passed Mathematics, 24 Physics and 43 Chemistry. At most 19
passed Mathematics and Physics, at most 29 Mathematics and Chemistry and at most 20 Physics and
Chemistry. What is the largest possible number that could have passed all three examination?
8. Consider the number N = 975X44Y (X and Y are single digit numbers)
(i) If X = 3 and N is divisible by 3
find all possible values of Y.
(ii) If X = 4 and N is divisible by 6, find all possible values of Y.
(iii) If N is divisible by 8, find all all possible values of Y.
9. Find all natural numbers 'n' for which n5 + n4 + 1 is a prime number
10. Factorize following :
(i) 6x3 – 5x2 – 3x + 2
(ii) (1 + x + x2 + x3)2 – x3.
(iii) (x + y – 2xy) (x + y – 2) + (1 – xy)2.
(iv) (x2 + y2 – 2x + 1)2 – (4y – 4xy) (x2 – y2 – 2x + 1).
(v) 16(6x – 1) (2x – 1) (3x + 1) (x – 1) + 25.
(vi) (6x – 1) (4x – 1) (3x – 1) (x – 1) + 9x4.
(vii) (6x – 1) (2x – 1) (3x – 1) (x – 1) + x2.
(viii) (x + 1)4 + (x + 3)4 – 272.
(ix)
x 4 2 7 1 x 7 7 into two quadratic factors
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ASSINGEMENT # 1
11. Factorize : 2xyz + x2y + y2z + z2x + xy2 + yz2 + zx2
12. Simplify x(x 4)(x 8)(x 12) 256 , (x 0).
13. Find the value of 2 2 2 2......
14. The value of the expression 7 48 5 24 3 8 is
15. If a,b,c are positive integer such that a > b > c such that a2 – b2 – c2 + ab = 2011 and
a2 + 3b2 + 3c2 – 3ab – 2ac – 2bc = –1997 find a.
16. Find the value of (a + b) (b + c) (c + a) if a + b + c = 21 and a3 + b3 + c3 = 3213
17. If x2 + 5y2 + 10z2 = 4xy + 6yz + 2z – 1 find x + y + z
18. Positive real numbers a,b and c satisfy the equation a2 + b2 + c2 = 2703 and (a – 1) (b – 1) (c – 1) = abc – 1
compute a + b + c
19. Find 3x + 2y + 6z if x, y, and z satisfy the system of equations
x y z
2 3 5
2x 3y z 16
20. If x, y, z are non zero real numbers such that
yz zx x y yz
, find the value of .
x y z x
(A) –1 or 2 (B) –2 (C) –1 (D) 2 (E) 2 or 1
PART-I
1. D 2. C 3. A 4. A 5. B,C 6. B 7. 14
8. (i) {1,4,7} (ii) {0,6} (iii) {0,8} 9. n=1
10. (i) (x – 1)(3x + 2)(2x – 1) (ii) (1 + x + x2) (1 + x + x2 + x3 + x4) (iii) (x + y – xy – 1)2
(iv) (x2 + 2xy – y2 – 2x – 2y + 1)2 (v) (24x2 – 16x – 3)2 (vi) (9x2 – 7x + 1)2
(vii) (6x2 – 6x + 1)2 (viii) 2(x + 5) (x – 1) (x2 + 4x + 19) (ix) x 2
x 7 x2 x 7 1
1 5
11. (x + y)(y + z)(z + x) 12. x2 + 12x + 16 13. 14. 1 15. 253
2
16. 2016 17. 10 18. 53 19. 84 20. –1 or 2 21. 400
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