Mathematics Multiple Choice Questions for Engineering Entrance Tests IIT, AIEEE, CET and Other Engineering Entrance Test
(1) The area of the region bounded by the curves y = |x – 2|, x = 1, x = 3 and the x-axis is
A. 1
B. 2
C. 3
D. 4
Answer: A. 1
(2) Let A (2, –3) and B(–2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line
A. 2x + 3y = 9
B. 2x – 3y = 7
C. 3x + 2y = 5
D. 3x – 2y = 3
Answer: A. 2x + 3y = 9
(3) The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is –1 is
A. x/2 + y/3 = -1 and x/-2 + y/1 = -1
B. x/2 – y/3 = -1 and x/-2 + y/1 = -1
C. x/2 + y/3 = 1 and x/2 + y/1 = 1
D. x/2 – y/3 = 1 and x/-2 + y/1 = 1
Answer: D. x/2 – y/3 = 1 and x/-2 + y/1 = 1
(4) Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
A. 3/2
B. 5/2
C. 7/2
D. 9/2
Answer: C. 7/2
(5) A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by
A. (3a, 3a, 3a), (a, a, a)
B. (3a, 2a, 3a), (a, a, a)
C. (3a, 2a, 3a), (a, a, 2a)
D. (2a, 3a, 3a), (2a, a, a)
Answer: B. (3a, 2a, 3a), (a, a, a)
(6) Consider the following statements:
(1) Mode can be computed from histogram
(2) Median is not independent of change of scale
(3) Variance is independent of change of origin and scale. Which of these is/are correct?
A. only (1)
B. only (2)
C. only (1) and (2)
D. (1), (2) and (3)
Answer: C. only (1) and (2)
(7) If the straight lines x = 1 + s, y = –3 – Ks, z = 1 + ls and x = , y = 1 + t, z = 2 – t with parameters s and t respectively, are co-planar then K equals
A. –2
B. –1
C. – 1/2
D. 0
Answer: A. –2
(8) The probability that A speaks truth is , while this probability for B is . The probability that they contradict each other when asked to speak on a fact is
A. 3/20
B. 1/5
C. 7/20
D. 4/5
Answer: C. 7/20
(9) The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is
A. 37/256
B. 219/256
C. 128/256
D. 28/256
Answer: D. 28/256
(10) A particle moves towards east from a point A to a point B at the rate of 4 km/h and then towards north from B to C at the rate of 5 km/h. If AB = 12 km and BC = 5 km, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively
A. 17/4 km/h and 13/4 km/h
B. 13/4 km/h and 17/4 km/h
C. 17/9 km/h and 13/9 km/h
D. 13/9 km/h and 17/9 km/h
Answer: A. 17/4 km/h and 13/4 km/h
(1) If A2 – A + I = 0, then the inverse of A is
A. A + I
B. A
C. A – I
D. I – A
Answer: D. I – A
(2) Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12} be a relation on the set A = {3, 6, 9, 12}. The relation is
A. reflexive and transitive only
B. reflexive only
C. an equivalence relation
D. reflexive and symmetric only
Answer: A. reflexive and transitive only
(3) If in a frequently distribution, the mean and median are 21 and 22 respectively, then its mode is approximately
A. 22.0
B. 20.5
C. 25.5
D. 24.0
Answer: D. 24.0
(4) Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid point of PQ is
A. y2 – 4x + 2 = 0
B. y2 + 4x + 2 = 0
C. x2 + 4y + 2 = 0
D. x2 – 4y + 2 = 0
Here 2 is read as square
Answer: A. y2 – 4x + 2 = 0
(5) The system of equations
αx + y + z = α – 1,
x + αy + z = α – 1,
x + y + αz = α – 1
has no solution, if α is
A. -2
B. either – 2 or 1
C. not -2
D. 1
Answer: A. -2
(6) The value of α for which the sum of the squares of the roots of the equation
x2 – (a – 2)x – a – 1 = 0 assume the least value is
A. 1
B. 0
C. 3
D. 2
Answer: A. 1
(7) If roots of the equation x2 – bx + c = 0 be two consectutive integers, then b2 – 4c equals
A. – 2
B. 3
C. 2
D. 1
Answer: D. 1
(8) If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
A. 601
B. 600
C. 603
D. 602
Answer: A. 601
(9) Let f be differentiable for all x. If f(1) = – 2 and f′(x) ≥ 2 for x ∈ [1, 6] , then
A. f(6) ≥ 8
B. f(6) < size=”2″>
Answer: A. f(6) ≥ 8
(10) If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in
A. G.P.
B. A.P.
C. Arithmetic − Geometric Progression
D. H.P.
Answer: B. A.P.
(1) The area enclosed between the curve y = loge (x + e) and the coordinate axes is
A. 1
B. 2
C. 3
D. 4
Answer: A. 1
(2) If x dy/dx = y (log y − log x + 1), then the solution of the equation is
A. y log(x/y) = cx
B. x log(y/x) = cy
C. log(y/x) = cx
D. log(x/y) = cy
Answer: C. log(y/x) = cx
(3) The line parallel to the x−axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx − 2ay − 3a = 0, where (a, b) ≠ (0, 0) is
A. below the x−axis at a distance of 3/2 from it
B. below the x−axis at a distance of 2/3 from it
C. above the x−axis at a distance of 3/2 from it
D. above the x−axis at a distance of 2/3 from it
Answer: A. below the x−axis at a distance of 3/2 from it
(4) ∫ cosx
A. tanx
B. secx
C. sinx
D. -sinx
Answer: C. sinx
(5) The angle between the lines 2x = 3y = − z and 6x = − y = − 4z is
A. 0 Degree
B. 90 Degree
C. 45 Degree
D. 30 Degree
Answer: B. 90 Degree
(6) If the plane 2ax − 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres
x2 + y2 + z2 + 6x − 8y − 2z = 13 and
x2 + y2 + z2 − 10x + 4y − 2z = 8, then a equals
Here 2 is read as Square
A. − 1
B. 1
C. − 2
D. 2
Answer: C. − 2
(7) If non-zero numbers a, b, c are in H.P., then the straight line x/a + y/b + z/c always passes through a fixed point. That point is
A. (-1, 2)
B. (-1, -2)
C. (1, -2)
D. (1, -1/2)
Answer: C. (1, -2)
(8) If a vertex of a triangle is (1, 1) and the mid-points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is
A. (-1, 7/3)
B. (-1/3, 7/3)
C. (1, 7/3)
D. (1/3, 7/3)
Answer: C. (1, 7/3)
(9) If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct points P and Q then the line 5x + by – a = 0 passes through P and Q for
A. exactly one value of a
B. no value of a
C. infinitely many values of a
D. exactly two values of a
Here 2 is read as Square
Answer: B. no value of a
(10) A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is
A. an ellipse
B. a circle
C. a hyperbola
D. a parabola
Answer: D. a parabola
(1) Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is
A. 5/9
B. 1/9
C. 8/9
D. 4/9
Answer: B. 1/9
(2) A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard will catch the insect after
A. 19 s
B. 1 s
C. 21 s
D. 25 s
Answer: C. 21 s
(3) If both the roots of the quadratic equation x2 – 2kx + k2 + k – 5 = 0 are less than 5, then k lies in the interval
Here 2 read as Square
A. (5, 6]
B. (6, ∞)
C. (-∞, 4)
D. [4, 5]
Answer: C. (-∞, 4)
(4) A plane passes through (1, − 2, 1) and is perpendicular to two planes 2x − 2y + z = 0 and x − y + 2z = 4. The distance of the plane from the point (1, 2, 2) is
A. 0
B. 2
C. Square Root of 3
D. 2 Square Root of 2
Answer: D. 2 Square Root of 2
(5) A tangent drawn to the curve y = f(x) at P(x, y) cuts the x-axis and y-axis at A and B respectively such that BP : AP = 3 : 1, given that f(1) = 1, then
A. equation of curve is x dy/dx – 3y = 0
B. normal at (1, 1) is x + 3y = 4
C. curve passes through (2, 1/8)
D. equation of curve is x dy/dx + 3y = 0
Answer: C. curve passes through (2, 1/8)
(6) Suppose a population A has 100 observations 101, 102, … , 200, and another population B has 100 observations 151, 152, … , 250. If VA and VB represent the variances of the two populations, respectively, then VA/VB is
A. 1
B. 9/4
C. 4/9
D. 2/3
Answer: A. 1
(7) The number of values of x in the interval [0, 3π] satisfying the equation 2sin2x + 5sinx − 3 = 0 is
A. 4
B. 6
C. 1
D. 2
Answer: A. 4
(8) Let W denote the words in the English dictionary. Define the relation R by : R = {(x, y) ∈ W × W | the words x and y have at least one letter in common}. Then R is
A. not reflexive, symmetric and transitive
B. reflexive, symmetric and not transitive
C. reflexive, symmetric and transitive
D. reflexive, not symmetric and transitive
Answer: B. reflexive, symmetric and not transitive
(9) A particle has two velocities of equal magnitude inclined to each other at an angle θ. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then θ is
A. 90°
B. 120°
C. 45°
D. 60°
Answer: B. 120°
(10) A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4s prior to passing through P. If g = 10 m/s2, then the height above the point P from where the body began to fall is
A. 720 m
B. 900 m
C. 320 m
D. 680 m
Answer: A. 720 m
(1) A straight line through the point A(3, 4) is such that its intercept between the axes is bisected at A. Its equation is
A. x + y = 7
B. 3x − 4y + 7 = 0
C. 4x + 3y = 24
D. 3x + 4y = 25
Answer: C. 4x + 3y = 24
(2) In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is
A. 3/5
B. 1/2
C. 4/5
D. 7
Answer: A. 3/5
(4) The function f(x) = x/2 + 2/x has a local minimum at
A. x = 2
B. x = −2
C. x = 0
D. x = 1
Answer: A. x = 2
(5) The set of points where f(x) = x / 1+|x| is differentiable is
A. (−∞, 0) ∪ (0, ∞)
B. (−∞, −1) ∪ (−1, ∞)
C. (−∞, ∞)
D. (0, ∞)
Answer: C. (−∞, ∞)
(6) At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is
A. 5040
B. 6210
C. 385
D. 1110
Answer: C. 385
(7) Differentiation of logx.sinx
A. sinx.1/x
B. cosx.sinx + logx
C. sinx.1/x + logx.cosx
D. cosx.(-1/x) + 1/logx
Answer: C. sinx.1/x + logx.cosx
(8) y = sinx then evaluate dy/dx = ? then what is Integration of ?
A. sinx
B. cosx
C. -sinx
D. -cosx
Answer: A. sinx
(9) What is the value of factorial Zero (0!)
A. 10
B. 0
C. 1
D. -1
Answer: C. 1
(10) y = sinx + cosx – 5a what is dy/dx
A. cosx – sinx
B. cosx + sinx -5
C. sinx – secx
D. sinx + cosx + 5
Answer: A. cosx – sinx
(1) If the system of linear equations
x + 2ay + az = 0
x + 3by + bz = 0
x + 4cy + cz = 0
has a non-zero solution, then a, b, c
A. are in A. P.
B. are in G. P.
C. are in H.P.
D. satisfy a + 2b + 3c = 0
Answer: C. are in H.P.
(2) If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then a/c, b/a, and c/b are in
A. arithmetic progression
B. geometric progression
C. harmonic progression
D. arithmetic-geometric-progression
Answer: C. harmonic progression
(3) A student is to Answer: 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
A. 140
B. 196
C. 280
D. 346
Answer: B. 196
(4) The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by
A. 6! * 5!
B. 50
C. 5! * 4!
D. 7! * 5!
Answer: A. 6! * 5!
(5) Let f (x) be a polynomial function of second degree. If f (1) = f (- 1) and a, b, c are in A. P., then f’ A., f’ B. and f’ C. are in
A. A.P.
B. G.P.
C. H. P.
D. arithmetic-geometric progression
Answer: A. A.P.
(6) If x1, x2, x3 and y1, y2, y3 are both in G.P. with the same common ratio, then the points (x1, y1) (x2, y2) and (x3, y3)
A. lie on a straight line
B. lie on an ellipse
C. lie on a circle
D. are vertices of a triangle
Answer: A. lie on a straight line
(7) The real number x when added to its inverse gives the minimum value of the sum at x equal to
A. 2
B. 1
C. – 1
D. – 2
Answer: B. 1
(8) The area of the region bounded by the curves y = |x – 1| and y = 3 – |x| is
A. 2 sq units
B. 3 sq units
C. 4 sq units
D. 6 sq units
Answer: A. 2 sq units
(9) The degree and order of the differential equation of the family of all parabolas whose axis is x-axis, are respectively
A. 2, 1
B. 1, 2
C. 3, 2
D. 2, 3
Answer: B. 1, 2
(10) Locus of centroid of the triangle whose vertices are (a cos t, a sin t), (b sin t, – b cos t) and (1, 0), where t is a parameter, is
A. (3x – 1)2 + (3y)2 = a2 – b2
B. (3x – 1)2 + (3y)2 = a2 + b2
C. (3x + 1)2 + (3y)2 = a2 + b2
D. (3x + 1)2 + (3y)2 = a2 – b2
Here 2 is read as Square
Answer: B. (3x – 1)2 + (3y)2 = a2 + b2
(1) Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is
A. a function
B. reflexive
C. not symmetric
D. transitive
Answer: C. not symmetric
(2) Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
A. x2 + 18x + 16 = 0
B. x2 – 18x – 16 = 0
C. x2 + 18x – 16 = 0
D. x2 – 18x + 16 = 0
Here x2 read as x Square
Answer: D. x2 – 18x + 16 = 0
(3) If (1 – p) is a root of quadratic equation x2 + px + (1-p) = 0 , then its roots are
A. 0, 1
B. -1, 2
C. 0, -1
D. -1, 1
Answer: C. 0, -1
(4) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?
A. 120
B. 480
C. 360
D. 240
Answer: C. 360
(5) If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of ‘q’ is
A. 49/4
B. 4
C. 3
D. 12
Answer: A. 49/4
(6) A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is and when he retires 40 meter away from the tree the angle of elevation becomes . The breadth of the river is
A. 20 m
B. 30 m
C. 40 m
D. 60 m
Answer: A. 20 m
(7) The graph of the function y = f(x) is symmetrical about the line x = 2, then
A. f(x + 2)= f(x – 2)
B. f(2 + x) = f(2 – x)
C. f(x) = f(-x)
D. f(x) = – f(-x)
Answer: B. f(2 + x) = f(2 – x)
(8) A point on the parabola y2 = 18x at which the ordinate increases at twice the rate of the abscissa is
A. (2, 4)
B. (2, -4)
C. (-9/8, 9/2)
D. (9/8, 9/2)
Answer: D. (9/8, 9/2)
(9) The normal to the curve x = a(1 + cosq), y = asinq at ‘q’ always passes through the fixed point
A. (a, 0)
B. (0, a)
C. (0, 0)
D. (a, a)
Answer: A. (a, 0)
(10) If 2a + 3b + 6c =0, then at least one root of the equation ax2 + bx + c lies in the interval
A. (0, 1)
B. (1, 2)
C. (2, 3)
D. (1, 3)
Answer: A. (0, 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 + 14×5 + 16×3 + 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