Engineering Entrance Sample Papers

IPUCET Mathematics Test Sample Paper Online MCQs for Guru Gobind Singh Indraprastha University Common Entrance Test (IPUCET) Mathematics with answers.

Ques. The equation 3(x – 1)2 + 2h (x – 1) (y – 2) + 3(y – 2)2 = 0 represents a pair of straight lines passing through the point (1, 2). The two lines are real and distinct if h2
(a) is greater than 3
(b) is greater than 9
(c) equals 7
(d) is grater than 7
Ans. (b)

Ques. If a, b, c are in A.P., then the straight line ax + by + c = 0 will always pass through the point
(a) (–1, –2)
(b) (1, –2)
(c) (–1, 2)
(d) (1, 2)
Ans. (b)

Ques. The value of k, for which (cos x + sin x)2 + k sin x cos x – 1 = 0, is an identity, is
(a) – 1
(b) – 2
(c) 1
(d) 0
Ans. (b)

Ques. The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is
(a) cos–1 (2/3)
(b) cos-1 (3/4)
(c) cos-1 (4/5)
(d) cos-1 (5/6)
Ans: (c)

Ques. Two circles, each of radius 5, have a common tangent at (1, 1) whose equation is 4x + 3y – 7 = 0, then the centres are
(a) (–5, 4), (3, –2)
(b) (–3, 4), (5, –2)
(c) (5, 4), (–3, –2)
(d) (4, 2), (–2, 0)
Ans: (c)

Ques. In a certain test ai students gave wrong answers to at least i questions where i = 1, 2, 3, …, k. No student gave more than k wrong answers. The total numbers of wrong answers given is
(a) a1 + 2a2 + 3a3 + … kak
(b) a1 + a2 + a3 + …, + ak
(c) Zero
(d) None of these
Ans. (b)

Ques. If the straight line y = mx is outside the circle x2 + y2 – 20y + 90 = 0, then
(a) m > 3
(b) m < 3
(c) |m| > 3
(d) |m| < 3
Ans: (d)

Ques. If the letters of the word ‘DATE’ be permuted and the words so formed be arranged as in a dictionary.  Then the rank of ‘DATE’ is
(a) 12
(b) 13
(c) 14
(d) 8
Ans. (d)

Ques. A circle circumscribing an equilateral triangle with centroid at (0, 0)  of side a is drawn and a square is drawn whose four sides touch the circle. The equation of the circle circumscribing the square is
(a) x2 + y2 = 2a2
(b) 3x2 + 3y2 = 2a2
(c) 5x2 + 5y2 = 3a2
(d) none of these
Ans: (b)

Ques. The vertex of the parabola y2 = 4(a’ – a)(x – a) is
(a) (a‘, a)
(b) (a, a’)
(c) (a, 0)
(d) (a’, 0)
Ans: (c)

Ques. Function f : R – – R, f(x) = [x] is
(a) One-one onto
(b) One-one into
(c) Many-one onto
(d) Many-one into
Ans. (d)

Ques. The equation x2 + 4xy + 4y2 – 3x – 6y – 4 = 0 represents
(a) circle
(b) pair of lines
(c) parabola
(d) none of these
Ans: (b)

Ques. Let ABC be a triangle and O be its orthocentre. If R and R1 are the circum radii of triangles ABC and AOB, then
(a) R1 > R
(b) R1 = R
(c) R1 < R
(d) nothing can be said
Ans. (b)

Ques. 3x – 4y – 24 = 0 and 3x – 4y -12 = 0 are two parallel lines. If L1 makes an intercept of 3 units with these parallel lines then the equation of  L1 may be given as
(a) x = 1
(b) y = 1
(c) x = 2y + 3
(d) none of these
Ans. (a)

Ques. Let the determinant of a 3 ´ 3 matrix A be 6, then B is a matrix defined by B = 5A2. Then determinant of B is
(a) 180
(b) 100
(c) 80
(d) none of these
Ans. (a)

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Ques. The number of ways that 8 beads of different colours be string as a necklace is
(a) 2520
(b) 2880
(c) 5040
(d) 4320
Ans. (a)

Ques. Equation of the straight line, passing through the point (3, 4) and farthest from the circle
x2 + y2 + 8x + 6y + 16 = 0, is
(a) xy  + 1 = 0
(b) 3x + 4y = 25
(c) x + y – 7 = 0
(d) none of these
Ans. (c)

Ques. The number of points with integral coordinates (2a, a – 1) that fall in the interior of the larger segment of the circle x2 + y2 = 25 cut of by the parabola x2 + 4y = 0, is
(a) one
(b) two
(c) three
(d) none of these
Ans. (c)

Ques. A hyperbola passing through origin has 3x – 4y – 1 = 0 and 4x – 3y – 6 = 0 as its  asymptotes.  Then the equations of its transverse and conjugate axis are
(a) x – y – 5 = 0 and x + y + 1 = 0
(b) x – y = 0 and x + y + 5 = 0
(c) x + y – 5 = 0 and x – y – 1 = 0
(d) x + y – 1 and x – y – 5 = 0
Ans. (c)

Ques. The number of real roots of the equation esin x – e–sin x – 4 = 0 are
(a) 1
(b) 2
(c) infinite
(d) none of these
Ans. (d)

Ques. Numbers of zeros at the end of 300 ! is equal to
(a) 75
(b) 89
(c) 74
(d) 98
Ans. (c)

Ques. If A is a non–singular matrix of order 3, then |adj(adj A)| equals
(a) | A|4
(b) | A |6
(c) | A |3
(d) none of these
Ans. (a)

Ques. The vertices of a triangle are (0, 3), (–3, 0) and (3, 0). The co-ordinates of the orthocentre are
(a) (1, 2)
(b) (1/2, 2)
(c) (2, 3)
(d) (0, 3)
Ans. (a)

Ques. A die is formed in such a way that the probability of occurrence of an even face is twice of the probability of occurrence of an odd face. The probability of occurrence of a prime number is
(a) 1/3
(b) 4/9
(c) 5/9
(d) ½
Ans. (b)

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Ques. The number of solutions for the equations |z – 1| = |z – 2| = |z – i| is :
(a) one solution
(b) 3 solutions
(c) 2 solutions
(d) no solution
Ans. (a)