Engineering Entrance Sample Papers

IPUCET Mathematics Test Sample Paper

IPUCET Mathematics practice questions

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)

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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)

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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)

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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)

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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)

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