Mathematics Entrance Test

Mathematics Entrance Test

Mathematics Aptitude Questions with Answers for the preparation for Engineering Entrance Test like AIEEE, IIT-JEE, CET etc.

Test 1

(1) 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 P (A ∪ B) is

(a) 3/5
(b) 0
(c) 1
(d) 5/2

Answer (c) 1

(2) A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi−major axis is

(a) 4/3
(b) 8/3
(c) 7/3
(d) 5/3

Answer (b) 8/3

(3) A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at

(a) (0, 2)
(b) (0, 1)
(c) (1, 0)
(d) (2, 0)

Answer (c) (1,0)

(4) The point diametrically opposite to the point P (1, 0) on the circle x2 + y2 + 2x + 4y − 3 = 0 is

(a) (− 3, − 4)
(b) (-3, 4)
(c) (3, 4)
(d) (-4, -1)

Answer (a) (-3, -4)

(5) The conjugate of a complex number is 1/i-1. Then the complex number is

(a) -1/i-1
(b) 1/i+1
(c) 1/i-1
(d) -1/i+1

Answer (d) -1/i+1

(6) Let R be the real line. Consider the following subsets of the plane R × R. S = {(x, y) : y = x + 1 and 0 < t =" {(x,">
(a) neither S nor T is an equivalence relation on R
(b) both S and T are equivalence relations on R
(c) S is an equivalence relation on R but T is not
(d) T is an equivalence relation on R but S is not

Answer (d) T is an equivalence relation on R but S is not

(7) The perpendicular bisector of the line segment joining P (1, 4) and Q (k, 3) has y−intercept − 4. Then a possible value of k is

(a) 1
(b) -4
(c) 3
(d) 2

Answer (b) -4

(8) 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?

(a) a = 0, b = 7
(b) a = 5, b = 2
(c) a = 3, b = 4
(d) a = 2, b = 4

Answer (c) a = 3, b = 4

(9) The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz−plane at the point (0, 17/2, -13/2) Then

(a) a = 2, b = 8
(b) a = 4, b = 6
(c) a = 6, b = 4
(d) a = 8, b = 2

Answer (c) a = 6, b = 4

(10) Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that A2 = I.
Statement −1: If A ≠ I and A ≠ − I, then det A = − 1.
Statement −2: If A ≠ I and A ≠ − I, then tr (A) ≠ 0.

(a) Statement −1 is false, Statement −2 is true
(b) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
(c) Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
(d) Statement − 1 is true, Statement − 2 is false.

Answer (d) Statement − 1 is true, Statement − 2 is false.

(11) The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

(a) -2
(b) -4
(c) -12
(d) 8

Answer (c) -12

(12) How many real solutions does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have?

(a) 1
(b) 4
(c) 7
(d) 5

Answer (a) 1

(13) The statement p → (q → p) is equivalent to

(a) p → (p → q)
(b) p → (p ∨ q)
(c) p → (p ∧ q)
(d) p → (p ↔ q)

Answer (b) p → (p ∨ q)

(14) The value of cot(cosec-1 5/3 + tan-1 2/3) is

(a) 2/17
(b) 6/17
(c) 7/17
(d) 3/17

Answer (b) 6/17

(15) The area of the plane region bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is equal to (y2 = y square)

(a) 3/5
(b) 4/3
(c) 7/3
(d) 1

Answer (b) 4/3

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