1. Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number if y isa transcendental number”.
Statement1:
r is equivalent to either q or p .
Statement 2:
r is equivalent to ~(p ~q)
(1) Statement 1 is true , Statement 2 is false
(2) Statement 1 is false, Statement 2 is true
(3) Statement 1 is true, Statement 2 is true:
Statement 2 is a correct explanation for
Statement 1
(4) Statement 1 is true, Statement 2 is true:
Statement 2 is not a correct explanation for
_{ } Statement 1
2. In a shop there are five types of icecreams available. A child buys six icecreams.
Statement 1 :
The number of different ways the child can buy the six icecreams is .
Statement 2 :
The number of different ways the child can buy the six icecreams is equal of different ways of arranging 6 A’s and 4 B’s in a row
(1) Statement 1 is true, Statement 2 is false
(2) Statement 1 is false, Statement 2 is true
(3) Statement 1 is true , Statement 2 is true
Statement 2 is a correct explanation for Statement 1
(4) Statement 1 is true , Statement2 is true :
Statement 2 is not a correct explanation for Statement 1
Ans. x_{1}, x_{2}, x_{3}, x_{4}, x_{5} be the number of icecreams selected from 5 types of icecreams
x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5} = 6
i = 1, 2, 3, 4, 5, 6
solve to get number of ways
=> statement (1) is false but statement (2) is correct.
3. Statement1:
Statement2:
(1) Statement 1 is true, Statement 2 is false
(2) Statement 1 is false, Statement 2 is true
(3) Statement 1 is true, Statement 2 is true
Statement 2 is a correct explanation for Statement 1
(4) Statement 1 is true, Statement2 is true :
Statement 2 is not a correct explanation for Statement 1
Ans.
4. Statement1:
For every natural number
Statement2:
For every natural number
(1) Statement 1 is true, Statement 2 is false
(2) Statement 1 is false, Statement 2 is true
(3) Statement 1 is true, Statement 2 is true
Statement 2 is a correct explanation for Statement 1
(4 Statement 1 is true, Statement2 is true :
Statement 2 is not a correct explanation for Statement 1
Ans.
statement 1 is correct
statement 1 is correct using statement 2
5. Let A be a 2 x 2 matrix with real entries. Let I be the 2 x 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A^{2 }= 1
Statement 1 :
If A 1 and A 1 , then A =  1
Statement 2:
If A 1 and A  1, then tr (A) 0.
(1) Statement 1 is true, Statement 2 is false
(2) Statement 1 is false, Statement 2 is true
(3) Statement 1 is true, Statement 2 is true
Statement 2 is a correct explanation for Statement 1
(4) Statement 1 is true, Statement2 is true :
Statement 2 is not a correct explanation for Statement 1
6. The statement p (q p ) is equivalent to
(1) p (p q)
(2) p (p q)
(3) p
(4) p
7. The value of cost
8. The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is
Ans.
9.
Then which one of the following is true?
Ans.
10. The area of the plane region bounded by the curves x + 2y^{2} = 0 and x + 3y^{2} = 1 is equal to
Ans.
12. AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60^{o}. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45^{o}. Then the height of the pole is
Ans.
13. How many real solutions does the equation x ^{7} + 14x ^{5} + 16x ^{3} + 30x  560 = 0 have ?
(1) 5 (2) 7 (3) 1 (4) 3
Ans.
14.
Then which one of the following is true ?
(1) f is differentiable at x = 1 but not at x = 0
(2) f is neither differentiable at x = 0 nor at x = 1
(3) f is differentiable at x = 0 and x = 1
(4) f is differentiable at x = 0 but not at x = 1
Ans.
= Oscillating between 1 & 1
=> RHD does not exists
Non diff. At x=1
At x=0, f(x) is continuous & diff. As both (x1) and sin(1/x1) are continuous & diff. At x=0
15. The first two terms of a geometric progression add up to 12. The sum if the third and the fourth terms is 48. If the terms of the geometric progression ate alternately positive and negative, then the first term is
(1) 4
(2) –4
(3) –12
(4) 12
Ans. Given b+br = 12 (1)
br^{2} + br^{3} = 48 r<0
b(1+r) = 12 & br^{2} (1+r) = 48
Divide => r^{2} = 4 =>
As r<0, we take r = 2
Replace r in (1) to get
b(12) = 12 => b = 12
16. It is given that the events A and B are such that P(A) =, P(A  B) = and P(B  A) = . Then P(B) is
Ans.
17. A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then is
Ans.
18. Suppose the cubic x^{3} – px + q has three distinet real roots where p> 0 and q > 0. Then which one of the following holds ?
(1) The cubic has maxima at both and
(2) The cubic has minima at and maxima at
(3) The cubic has minima at and maxima at
(4) The cubic has minima at both and
Ans. let f(x) = x^{3}px + q
f ’(x) = 3x^{2}  p
19. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent ?
Ans. MPP SSSS
Arrange MPP in
Between 7 letter there are 8 possibilities for 4
select 4 position out of 8 in ways
20. The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has yintercept –4. Then a possible value of k is
(1) 4
(2) 1
(3) 2
(4) –2
Ans.
21. A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at
(1) (2, 0)
(2) (0, 2)
(3) (1, 0)
(4) (0, 1)
Ans.
distance of vertex from directrix = distance from focus
=> V=(1, 0)
22. The point diametrically opposite to the point P(1, 0) on the circle x^{2} + y^{2} + 2x + 4y – 3 = 0 is
(1) (3, 4)
(2) (3, 4)
(3) (3, 4)
(4) (3, 4)
23. A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is . Then the length of the semimajor axis is
Ans.
24. The solution of the differential equation
satisfying the condition y(1) = 1 is
(1) y = x ln x + x
(2) y = ln x + x
(3) y = x ln x x + x^{2}
(4) y = x e^{(x1)}
Ans.
25. Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cz, and z = bx +ay. Then a ^{2} + b ^{2} + c ^{2} + 2abc is equal to
(1) 1
(2) 2
(3) –1
(4) 0
Ans. x  cy  bz = 0
cx  y + az = 0
bx + ay  z = 0
It is given that system of equations is consistent.
i.e. posses a solution.
It is given that not all x, y, z are zero.
=> Non trivial solution exist
=> D = 0
Evalute 1(1  a^{2}) + c(c  ab)  b(ac + b) = 0
=>1  a^{2 } c^{2 } abc  abc  b^{2} = 0
=> a^{2 }+ b^{2 }+ c^{2 }+ 2abc = 1
26. Let A be a square matrix all of whose entries are integers. Then which one of the following is true ?
(1) If det A = , then A^{1} need not exist
(2) If det A = , then A^{1} exists but all its entries are not necessarily integers
(3) If det A , then A^{1} exists and all its entries are nonintegers.
(4) If det A = , then A^{1} exists and all its entries are integers
Ans. For example, let
det(A) = 1
A^{1} exists & all entries are integers.
27. The quadratic equations x^{2} – 6x + a = 0 and x^{2} – cx + 6 = 0have one root in common. The other roots of the first and second equations are integers in the ration 4: 3. Then the common root is
(1) 2 (2) 1
(3) 4 (4) 3
Ans.
28. The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b ?
(1) a = 3, b = 4
(2) a = 0, b = 7
(3) a = 5, b = 2
(4) a = 1, b = 6
Ans.
simplify (1) to get:
23 + a + b=30 => a + b = 7 (3)
simplify (2) to get
(a  6)^{2 }+ (b  6)^{2 }+ 4 + 1 + 16=34
=> (a  6)^{2 }+ (7 a  6)^{2} = 13
(a^{2}12a + 36) + (a^{2 }+1  2a) = 13
2a^{2}14a + 24 = 0
a^{2 } 7a +12 = 0 => (a4) (a3) = 0 => b = 3, or 4
29. The vector lies in the plane of the vectors and and bisects the angle between Then which one of the following gives possible values of
Ans.
30. The nonzero vectors are related by Then the angle between is
Ans.
31. The line passing through the points (5, 1, a) and (3, b, 1) crosses the yzplane at the point Then
(1) a = 8, b = 2
(2) a = 2, b = 8
(3) a = 4, b = 6
(4) a = 6, b = 4
Ans.
Equation of line is
let coordinates of intersection of l with yz plane be [2k+5, (1b)k+1, (a1)k+a]
it lies on yz plane,
2k+5 = 0 => k = 5/2
Also (1b)k+1=17/2 => 1b = 15/2(2/5) = 3 => b =4
Also (a1)k+a =13/2
A(1+k)k = 13/2 => a(15/2) = 13/25/2 = 9
=> 3/2 a = 9 => a = 6
32. If the straight lines
intersect at a point, then the integer k is equal to
(1) 2
(2) –5
(3) 5
(4) 2
Ans.
33. The conjugate of a complex number is Then that complex number is
Ans.
34. Let R be the real line. Consider the following subsets of the plane R R:
S = {(x, y) : y = x + 1 and 0 < x < 2}
T = {(x, y) : x – y is an integer}.
Which one of the following is true ?
(1) T is an equivalence relation on R but S is not
(2) Neither S nor T is an equivalence relations on R
(3) Both S and T are equivalence relation on R
(4) S is an equivalence relation on R but T is not
35. Let f : N Y be a function defined as
f(x) =4x + 3 where
Y = {y N : y = 4x + 3 for some x N}. Show that f is invertible and its inverse is
Ans. To find inverse of f, replace y by x and x by y.
i.e.
x = 4f ^{1}(x)+3 => f ^{–1}(x) = g(x) = x3/4
Interims of y
g(y) = y3/4
