P&S
1. If a, b, c be the pth, qth and rth terms respectively of an A.P. and a G.P. both, then the product of the roots of the
equation is
(a) –1 (b) 1 (c) abc (d) a + b + c
2. The sum of the series is
(a) (b) (c) (d) none of these
3. If x, y are positive real numbers such that then M = xy satisfies the relation
(a) 0 < M < 2 (b) 0 < M 1 (c) 1 M < 2 (d) None of these
4. Given then which of the following are true.
(a) (b)
(c) (d)
5. In a triangle ABC equations of AB and BC are x = k1, y = k2, k1, k2 0 and AD, BE, CF are medians and AB = b,
BC = a. If m1, m2 and m3 are the slopes of AD, BE, CF respectively, (All points in the line AC satisfy
x + y > k1 + k2), then
(a) (b) (c) are in G.P. (d) m2, m1, m3 are in G.P.
6. For n N, let then
(a) P(19) 19 (b) P(18) > 18 (c) P(38) 19 (d) P(38) > 19
7. Let . Then for each n N
(a) (b) (c) (d)
8. If y, z > 0 and y + z = c, then minimum value of is equal to
(a) (b) (c) (d)
9. For the series 21, 22, 23, …., k – 1, k; the A.M. and G.M. of the first and last numbers exist in the given series. If
‘k’ is a three digit number, then ‘k’ can attain
(a) 5 values (b) 6 values (c) 2 values (d) 4 values
10. Let . Then is equal to
(a) (b) (c) (d) None of these
11. Let x, y, z are unequal positive numbers such that . The minimum value of
xy + z + yz + x + zx + y is
(a) 9 (b) 6 (c) 27 (d) 3
12. In a ABC, the incorrect relation is (where symbols have the usual meaning)
(a) s6 27( (b) 2 (sin A + sin B + sin C)
(c) ( s (d) none of these
13. If a1, a2, a3, .., an are positive real numbers whose product is a fixed number c, then the minimum value of a 1 + a2 +
… + an-1 + 2an is
(A) n(2c)1/n (B) (n +1)c1/n
(C) 2nc1/n (D) (n +1)(2c)1/n
14. The value of is equal to
(a) (b) (c) (d)
15. Let . The minimum value of is
(a) 9 (b) 3 (c) (d) 1
16. Value of the expression, terms is (are)
(a) (b) (c) less than (d) less than
17. Length of the tangents from vertices A, B, C to the incircle are in A.P., then
(a) are in A.P. (b) are in A.P.
(c) are in H.P (d)
18. Let , then minimum value of is
(a) 3 (b) 4 (c) 5 (d) 9
19. Let then
(a) (b) (c) (d)
20. If x, y, z be the lengths of perpendicular from the circumcentre on the sides BC, CA and AB of a ABC.
Then which of the following is true
(A) (B)
(C) (D) None of these
21. Which of the following is true for every positive integer n?
(A) (B)
(C) for (D) None of these
22. If pth term of an H.P. is qr and q-th term is pr. Then the rth term is
(A) pqr (B) (C) pq (D) –pqr
23. Let where and are consecutive integers and Then is
(A) always an even integer (B) sometimes an odd integer, sometimes not
(C) always an odd integer (D) sometimes rational, sometimes not
24. The number of ways in which we can select three numbers from the set {10, 11, …, 100} such that they
form a G.P. with common ratio greater than 1, is
(A) 18 (B) 19 (C) 20 (D) none of these
25. If a, b, c and d are distinct integers such that then the real roots of
are
(A) all rational (B) all integers (C) irrational (D) nothing can be said
26. is equal to
(A) (B)
(C) (D)
27. is equal to
(A) (B) (C) (D)
28. If a, b, c are sides of triangle then belong to ___________
(A) (B) (C) (D)
29. Let A be a point inside a regular polygon of 10 sides. Let be the distances of A from the sides
of the polygon. If each side is of length 2, then which of the following is true
(A) (B)
(C) (D)
30. If and let where a and b are relatively prime natural numbers,
then value of is equal to
(A) 49 (B) 48 (C) 50 (D) 39
31. Let a, b, c be the sides of a triangle and then
(A) (B) (C) (D) None of these
Comprehension – 1
Integers from 1 to 1000 one marked in order around a circle beginning from 1, every fifteenth number is marked
(i.e., 1, 16, 31, ........etc). This process is continued until a number is reached which has already been marked.
32. If n is the number of points marked on the circle, then n is
(a) 200 (b) 196 (c) 160 (d) None of these
33. If Sp denotes the number of order pairs (x, y) in and p> 1, then Sn = _________
(a) 20 (c) 50 (c) 35 (d) 45
34. If f is a function defined as f : A A where A = {Sm : m is a prime number} (Sm as defined in above question), then
the function f is
(a) a linear function (b) an identity function (c) identically zero (d) none
35. If a, b, (a > b) are two positive numbers with A and H are their Arithmetic Mean and Harmonic Mean respectively,
then
(a) (b)
(c) ab = Arithmetic Mean of aH and bH (d)
Comprehension – 2
Let D, E, F be points on the sides BC, CA and AB respectively of ABC and let R be circumradius of ABC.
36. If D1, D2 be images of D w.r to AC, AB respectively, then
(a) DE + EF + FD D1D2 (b) DE + EF + FD D1D2
(c) DE + EF + FD > D1D2 (d) DE + EF + FD < D1D2
37. If D, D are midpoints of DD1, DD2 respectively, then DD =
(a) (b) (AD) sin A (c) (d) none of these
38.
(a) (b) (c) (d) none of these
Match the Following
M1. Match the following:
COLUMN – I COLUMN – II
A are in P A.P.
B are in A.P., then p2, q2, r2 are in Q G.P.
p, q, r are in A.P., q, r, s in G.P. r, s, t in H.P, then p, r, t are
C R A.G.P.
in
D 8, 24 and 64 are in S H.P.
Assertion and Reasoning
Directions: The questions given below consists of STATEMENT 1 and STATEMENT 2. Use the following key to
choose the appropriate answer.
(a) Both STATEMENT 1 and STATEMENT 2 are true and STATEMENT 2 is correct explanation of STATEMENT 1
(b) Both STATEMENT 1 and STATEMENT 2 are true but STATEMENT 2 is not correct explanation of
STATEMENT1.
(c) STATEMENT 1 is true, STATEMENT 2 is false.
(d) STATEMENT 1 is false, STATEMENT 2 is true.
STATEMENT
39. For a > 1 and positive integer n,
1:
and
STATEMENT
A.M. of n distinct positive numbers is greater than their G.M.
2:
Numericals
N1: The value of is ____________
N2. If are terms of A.P. such that , then
value of is_____
N3. In a triangle ABC, . If be the harmonic mean of the lengths of the sides BC and CA and the length of
the bisector of is , then the value of is_____(where denotes the greatest integer function).
N4. Let (natural number) then value of is_____.
N5. A, B, C are positive reals with product 1 then the maximum value of
is _____________________________.
N6. Let be non-negative real numbers such that then the minimum value of the
is
N7. is equal to _______________________________.
N8. is equal to _________________________.
N9. and T= then T is equal to __________________________.
N10. is equal to _________________________________.
N11. In any triangle ABC the min. value of
