1 
The equation has a) No solution b) One solution c) More than one real solution d) Cannot be said
The equation has a) No solution b) One solution c) More than one real solution d) Cannot be said

IIT 1980 
01:57 min

2 
The number of solutions of the equation a) 0 b) 1 c) 2 d) Infinitely many
The number of solutions of the equation a) 0 b) 1 c) 2 d) Infinitely many

IIT 1990 
01:46 min

3 
The number of values of x in the interval (0, 5π) satisfying the equation is a) 0 b) 5 c) 6 d) 10
The number of values of x in the interval (0, 5π) satisfying the equation is a) 0 b) 5 c) 6 d) 10

IIT 1998 
03:17 min

4 
Find the natural number a for which where the function f satisfies the relation f (x + y) = f (x).f(y)for all natural numbers x and y and further f (1) = 2
Find the natural number a for which where the function f satisfies the relation f (x + y) = f (x).f(y)for all natural numbers x and y and further f (1) = 2

IIT 1992 
06:01 min

5 
If α + β = and β + γ = α, then tanα equals a) 2(tanβ + tanγ) b) tanβ + tanγ c) tanβ + 2tanγ d) 2tanβ + tanγ
If α + β = and β + γ = α, then tanα equals a) 2(tanβ + tanγ) b) tanβ + tanγ c) tanβ + 2tanγ d) 2tanβ + tanγ

IIT 2001 
02:03 min

6 
Let n be a positive integer and (1 + x + x^{2})^{n} = a_{0} + a_{1}x + a_{2}x + a_{2}x^{2} + . . . + a_{2n}x^{2n} then prove that
Let n be a positive integer and (1 + x + x^{2})^{n} = a_{0} + a_{1}x + a_{2}x + a_{2}x^{2} + . . . + a_{2n}x^{2n} then prove that

IIT 1994 
06:48 min

7 
The larger of 99^{50} + 100^{50} and 101^{50} is
The larger of 99^{50} + 100^{50} and 101^{50} is

IIT 1982 
04:38 min

8 
Find all solutions of in a) b) c) d)
Find all solutions of in a) b) c) d)

IIT 1984 
03:20 min

9 
Let f (x) = sin x and g (x) = lnx. If the range of the composition functions fog and gof are R_{1} and R_{2} respectively, then a) R_{1} = [ u : −1 ≤ u < 1], R_{2} = [ v : − < v < 0 ] b) R_{1 }= [ u : − < u < 0 ], R_{2} = [ v : −1 ≤ v ≤ 0] c) R_{1} = [ u : −1 < u < 1], R_{2} = [ v : − < v < 0 ] d) R_{1} = [ u : −1 ≤ u ≤ 1], R_{2} = [ v : − < v ≤ 0 ]
Let f (x) = sin x and g (x) = lnx. If the range of the composition functions fog and gof are R_{1} and R_{2} respectively, then a) R_{1} = [ u : −1 ≤ u < 1], R_{2} = [ v : − < v < 0 ] b) R_{1 }= [ u : − < u < 0 ], R_{2} = [ v : −1 ≤ v ≤ 0] c) R_{1} = [ u : −1 < u < 1], R_{2} = [ v : − < v < 0 ] d) R_{1} = [ u : −1 ≤ u ≤ 1], R_{2} = [ v : − < v ≤ 0 ]

IIT 1994 
03:03 min

10 
a) True b) False
a) True b) False

IIT 2002 
02:39 min

11 
Multiple choices y = f ( x ) = then a) x = f (y) b) f (1) = 3 c) y is increasing with x for x < 1 d) f is a rational function of x
Multiple choices y = f ( x ) = then a) x = f (y) b) f (1) = 3 c) y is increasing with x for x < 1 d) f is a rational function of x

IIT 1989 
01:29 min

12 
Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and (0) = 3. Find f (5). a) 1 b) 2 c) 3 d) 6
Let f (x + y) = f (x) f (y) for all x, y. Suppose that f (5) = 2 and (0) = 3. Find f (5). a) 1 b) 2 c) 3 d) 6

IIT 1981 
03:33 min

13 
One or more correct answers In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then a) Altitudes are in arithmetic progression b) Altitudes are in harmonic progression c) Medians are in geometric progression d) Medians are in arithmetic progression
One or more correct answers In a triangle PQR, sin P, sin Q, sin R are in arithmetic progression then a) Altitudes are in arithmetic progression b) Altitudes are in harmonic progression c) Medians are in geometric progression d) Medians are in arithmetic progression

IIT 1998 
03:36 min

14 
The external radii of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression a) True b) False
The external radii of ΔABC are in harmonic progression then prove that a, b, c are in arithmetic progression a) True b) False

IIT 1983 
01:51 min

15 
True / False If f (x) = ( a – x^{n })^{1/n } where a > 0 and n is a positive integer then f ( f ( x ) ) = x. a) True b) False
True / False If f (x) = ( a – x^{n })^{1/n } where a > 0 and n is a positive integer then f ( f ( x ) ) = x. a) True b) False

IIT 1983 
01:23 min

16 
Fill in the blank The domain of the function f (x) = is a) [− 2, − 1] b) [1, 2] c) [− 2, − 1] ⋃ [1, 2] d) None of the above
Fill in the blank The domain of the function f (x) = is a) [− 2, − 1] b) [1, 2] c) [− 2, − 1] ⋃ [1, 2] d) None of the above

IIT 1984 
02:48 min

17 
Both roots of the equation ( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always a) positive b) negative c) real d) none of these
Both roots of the equation ( x – b) ( x – c) + (x – c) ( x – a) + (x – a) (x – b) = 0 are always a) positive b) negative c) real d) none of these

IIT 1980 
02:52 min

18 
Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 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
Two towns A and B are 60 meters apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all the 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

IIT 1982 
01:37 min

19 
If then ab + bc + ca lies in the interval a) b) c) d)
If then ab + bc + ca lies in the interval a) b) c) d)

IIT 1984 
02:29 min

20 
Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are a) a, c b) b, c c) a, b d) a + c, b + c
Let α, β be roots of the equation (x – a) (x – b) = c, c ≠ 0. Then the roots of the equation (x – α) (x – β) + c = 0 are a) a, c b) b, c c) a, b d) a + c, b + c

IIT 1992 
02:15 min

21 
If p, q ε {1, 2, 3, 4}. The number of equations of the form having real roots is a) 15 b) 9 c) 7 d) 8
If p, q ε {1, 2, 3, 4}. The number of equations of the form having real roots is a) 15 b) 9 c) 7 d) 8

IIT 1994 
03:39 min

22 
For all x ε ( 0, 1 ) a) b) ln (1 + x) < x c) sinx > x d) lnx > x
For all x ε ( 0, 1 ) a) b) ln (1 + x) < x c) sinx > x d) lnx > x

IIT 2000 
02:40 min

23 
The number of values of k for which the system of equations (k + 1) x + 8y = 4k kx + ( k + 3 ) y = 3k – 1 has infinitely many solutions is a) 0 b) 1 c) 2 d) Infinity
The number of values of k for which the system of equations (k + 1) x + 8y = 4k kx + ( k + 3 ) y = 3k – 1 has infinitely many solutions is a) 0 b) 1 c) 2 d) Infinity

IIT 2002 
02:56 min

24 
If f (x) = a) f (x) is a strictly increasing function b) f (x) has a local maxima c) f (x) is a strictly decreasing function d) f (x) is bounded
If f (x) = a) f (x) is a strictly increasing function b) f (x) has a local maxima c) f (x) is a strictly decreasing function d) f (x) is bounded

IIT 2004 
02:07 min

25 
Let Δa = Then show that = c, a constant.
Let Δa = Then show that = c, a constant.

IIT 1989 
05:34 min
