Galgotias University entrance exam Mathematics practice question paper contains 30 multiple choice questions.

**Ques. **For the function f(x) = 1 + 3^{x} log 3, find the antiderivative F which assumes the value 7 for x = 2. At what value of x does the curve y = F(x) cut the abscissa?

(a) x = 3

(b) x = 1

(c) x = 0

(d) None of these

Ans. (b)

**Ques. **If *T*_{1} is the period of the function *y* = *e*^{3(x }^{– [x]) } and *T*_{2} is the period of the function

*y* = *e*^{3x }^{– [3x]} ([.] denotes the greatest integer function), then

(a) *T*_{1} = *T*_{2}

(b) *T*_{1} = T_{2}/3

(c) *T*_{1} = 3*T*_{2}

(d) none of these

Ans. (c)

**Ques. **If [y] = [sin x] ([.] denotes the greatest integer function) and y = cos x are two given equations, then the number of ordered pair (x, y) is

where [ ] denotes the greatest integer function.

(a) 2

(b) 4

(c) Infinitely many

(d) none of these

Ans. (c)

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**Ques. **f(x) = x^{3} +ax^{2} + bx + 5sin^{2}x is an increasing function in the set of real numbers if a and b satisfy the condition

(a) a^{2} – 3b – 15 > 0

(b) a^{2} – 3b + 15 > 0

(c) a^{2} – 3b + 15 < 0

(d) a > 0 b > 0

Ans. (c)

**Ques. **Sum of coefficients in the expansion of (x + 2y + z)^{10} is

(a) 2^{10}

(b) 3^{10}

(c) 1

(d) none of these

Ans. (d)

**Ques. **16 R^{2} r r_{1} r_{2} r_{3} =

(a) *abc*

(b) a^{3} b^{3} c^{3}

(c) a^{2} b^{2} c^{2}

(d) a^{2} b^{3} c^{4
}Ans. (c)

**Ques. **The number of straight lines that can be formed by joining 20 points no three of which are in the same straight line except 4 of them which are in the same line

(a) 183

(b) 186

(c) 197

(d) 185

Ans. (d)

**Ques. **The expansion of (8 – 3x)^{3/2} in terms of power of x is valid only if

(a) x < 3/8

(b) |x| < 8/3

(c) x > 8/3

(d) none of these

Ans. (b)

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**Ques.** The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball, is

(a) 10 !

(b) 10! / 5!

(c) 10! / (5!)^{2}

(d) None of these

**Ans.** (c)

**Ques. **If a_{1}, a_{2}, a_{3}, ….. a_{n} are n distinct odd natural numbers not divisible by any prime greater than or equal to 7, then the value of 1/a_{1} + 1/a_{2} + … + 1/a_{n} is always less than

(a) 1

(b) 15/8

(c) ½

(d) ¾

Ans. (b)

**Ques. **A father with 8 children takes them 3 at a time to the Zoological gardens, as often as he can without taking the same 3 children together more than once. The number of times he will go to the garden is

(a) 336

(b) 112

(c) 56

(d) None of these

Ans. (c)

**Ques. **If the angles A, B, C are the solutions of the equation tan^{3 }x –3k tan^{2} x –3 tan x + k = 0 then the triangle ABC is

(a) isosceles

(b) equilateral

(c) acute angled

(d) such a triangle does not exist

Ans. (d)

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**Ques. **The circle x^{2} + y^{2 }–2x –6y + 2 = 0 intersects the parabola y^{2} = 8x orthoganally at the point P. The equation of the tangents to the parabola at P can be

(a) x –y –4 = 0

(b) 2x + y –2 = 0

(c) x + y –4 = 0

(d) 2x –y +1 = 0

Ans. (d)

**Ques. **The equation of a parabola which passes through the intersection of a straight line

x + y = 0 and the circle x^{2} + y^{2} + 4y = 0 is

(a) y^{2} = 4x

(b) y^{2} = x

(c) y^{2} = 2x

(d) none of these

Ans. (c)

**Ques. **A coin is tossed n times. The probability of getting head at least once is greater than 0.8, then the least value of n is

(a) 2

(b) 3

(c) 5

(d) 4

Ans. (b)

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**Ques. **The vertices of a triangle *ABC* are (1, 1), (4, – 2) and (5, 5) respectively. The equation of perpendicular dropped from *C* to the internal bisector of angle *A* is

(a) *y* – 5 = 0

(b) *x* – 5 = 0

(c) 2*x* + 3*y* – 7 = 0

(d) none of these

Ans. (b)

**Ques. **An aircraft gun takes a maximum of fourshots at an enemy’s plane moving away from it. The probability of hitting the plane at first, second, third and fourth shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

(a) 1

(b) 0.550

(c) 0.6976

(d) none of these

Ans. (c)

**Ques. **If a > b > c and the system of equations ax + by + cz = 0, bx + cy + az = 0 and cx + ay + bz = 0 has a non trivial solution, then both the roots of the quadratic equation at^{2} + bt + c = 0 are

(a) at least one positive root

(b) opposite in sign

(c) positive

(d) imaginary

Ans. (a)

**Ques. **Number of ways in which 3 boys and 3 girls (all are of different heights) can be arranged in a line so that boys as well as girls among themselves are in decreasing order of height (from left to right), is

(a) 1

(b) 6!

(c) 20

(d) none of these

Ans. (c)

**Ques. **If 2z_{1} – 3z_{2} + z_{3} = 0 then z_{1}, z_{2}, z_{3} are represented by

(a) three vertices of a triangle

(b) three collinear points

(c) three vertices of a rhombus

(d) none of these

Ans. (b)

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**Ques. **If the distances of two points P and Q from the focus of a parabola y^{2 }= 4x are 4 and 9 respectively, then the distance of the point of intersection of tangents at P and Q of the parabola from the focus is

(a) 8

(b) 6

(c) 5

(d) 13

Ans. (b)

**Ques. **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

Ans. (a)

**Ques. **From a point A(1, 1) on the circle *x*^{2} + *y*^{2} – 4*x* – 4*y* + 6 = 0 two equal chords AB & AC of length 2 units are drawn. The equation of chord BC is

(a) 4*x* + 3*y* = 12

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

(c) 3*x* + 4*y* = 4

(d) *x* + *y* = 6

Ans. (b)

**Ques. **The minimum marks required for clearing a certain screening paper is 210 out of 300. The screening paper consists of ‘3’ sections each of Physics, Chemistry and Maths. Each section has 100 as maximum marks. Assuming there is no negative marking and marks obtained in each section are integers, the number of ways in which a student can qualify the examination is (Assuming no cut-off limit for individual subject)

(a) ^{210}C_{3} – ^{90}C_{3}

(b) ^{93}C_{3
}(c) ^{213}C_{3}

(d) (210)^{3
}Ans. (b)

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**Ques. **A pregnancy test is done on 100 pregnant women and 100 non-pregnant women. Test suggests out of 100 pregnant woman, 2 are pregnant and 8 out of 100 non-pregnant woman are pregnant. A woman is selected randomly and test is done which says woman is pregnant. What is the probability woman is non-pregnant.

(a) ½

(b) 92/100

(c) 8/100

(d) none of these

Ans. (c)

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