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 *h*^{2
}(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 a_{i} 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) a_{1} + 2a_{2} + 3a_{3} + … ka_{k}

(b) a_{1} + a_{2} + a_{3} + …, + a_{k}

(c) Zero

(d) None of these

**Ans.** (b)

**Ques.** If the straight line y = mx is outside the circle x^{2} + y^{2} – 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) *x*^{2} + *y*^{2} = 2*a*^{2}

(b) 3*x*^{2} + 3*y*^{2} = 2*a*^{2
}(c) 5*x*^{2} + 5*y*^{2} = 3*a*^{2}

(d) none of these

Ans: (b)

**Ques. **The vertex of the parabola y^{2} = 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 x^{2} + 4xy + 4y^{2} – 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 *R*_{1} are the circum radii of triangles ABC and AOB, then

(a) *R*_{1} > *R*

(b) *R*_{1} = *R
*(c)

*R*

_{1}<

*R*

(d) nothing can be said

Ans. (b)

**Ques. **3x – 4y – 24 = 0 and 3x – 4y -12 = 0 are two parallel lines. If L_{1} makes an intercept of 3 units with these parallel lines then the equation of L_{1} 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 = 5A^{2}. 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

*x*^{2} + *y*^{2} + 8*x* + 6*y* + 16 = 0, is

(a) *x* – *y* + 1 = 0

(b) 3*x* + 4*y* = 25

(c) *x* + *y* – 7 = 0

(d) none of these

Ans. (c)

**Ques. **The number of points with integral coordinates (2*a*, *a* – 1) that fall in the interior of the larger segment of the circle *x*^{2} + *y*^{2} = 25 cut of by the parabola *x*^{2} + 4*y* = 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 e^{sin 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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