This Maharashtra CET Mathematics Practice sample paper is based on Mah CET Maths syllabus.

**Ques:** If cosh y = sec x, then the value of tanh^{2} (y/2) is

(a) tan^{2} (x/2)

(b) cot^{2} x/2

(c) sin^{2} (x/2)

(d) cos^{2} x/2

**Ques.** The triangle formed by the tangent to the curve f(x) = x^{2} + bx – b at the point (1, 1) and the co-ordinate axes, lies in the first quadrant. If its area is 2 then the value of *b* is

(a) –1

(b) 3

(c) –3

(d) 1

**Ques:** The number of values of *c* such that the straight line y = 4x + c touches the curve x^{2}/4 + y^{2} = 1 is

(a) 0

(b) 1

(c) 2

(d) Infinite

**Ques:** The vector b = 3j + 4k is to be written as the sum of a vector b_{1} parallel to a = I + j and a vector b_{2} perpendicular to a. Then b_{1} =

(a) 3/2 (i + j)

(b) 2/3 (I + j)

(c) ½ (I + j)

(d) 1/3 (i + j)

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**Ques:** The number of reflexive relations of a set with four elements is equal to

(a) 2^{16}

(b) 2^{12}

(c) 2^{8}

(d) 2^{4}

**Ques.** The solution of the differential equation sec^{2} x tan ydx + sec^{2} y tan xdy = 0 is

(a) tan x = c tan y

(b) tan x = c tan (x + y)**
**(c) tan x = c cot y

(d) tan x sec y = c

**Ques. **If (1 + x – 2x^{2})^{6} = 1 + a_{1}x + a_{2}x^{2} + a_{3}x^{3} . . . . . then the value of a_{2} + a_{4} + a_{6} + . . . .+ a_{12} will be

(a) 32

(b) 31

(c) 64

(d) 1024

**Ques. **If f’(x) = g(x) and g’(x) = –f(x) for all *x *and f(2) = 4 = f’(2), then f^{2} (24) + g^{2} (24) is

(a) 32

(b) 24

(c) 64

(d) 48

**Ques. **If ^{n}P_{3} + ^{n}C_{n–2} = 14n, then n =

(a) 5

(b) 6

(c) 8

(d) 10

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**Ques:** The triangle formed by the tangent to the curve f(x) = x^{2} + bx – b at the point (1, 1) and the co-ordinate axes, lies in the first quadrant. If its area is 2 then the value of *b* is

(a) –1

(b) 3

(c) –3

(d) 1

**Ques:** 49^{n} + 16n – 1 is divisible by

(a) 3

(b) 19

(c) 64

(d) 29

**Ques:** Two tangents *PQ* and *PR* drawn to the circle x^{2} + y^{2} – 2x – 4y – 20 = 0 from point P (16, 7). If the centre of the circle is *C*, then the area of quadrilateral PQCR will be

(a) 75 *sq*. *units
*(b) 150

*sq*.

*units*

(c) 15

*sq*.

*units*

(d) None of these

**Ques. **The point (4, -3) with respect to the ellipse 4x^{2} + 5y^{2} = 1 is

(a) lies on the curve

(b) is inside the curve

(c) is outside the curve

(d) is focus of the curve

**Ques:** The area of the triangle formed by the line 4x^{2} – 9xy – 9y^{2} = 0 and x = 2 is

(a) 2

(b) 3

(c) 10/3

(d) 20/3

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**Ques:** The differential equation for the family of curves x^{2} + y^{2} – 2ay = 0, where *a* is an arbitrary constant, is

(a) (x^{2} + y^{2})y’ = 2xy

(b) 2(x^{2} + y^{2})y’ = 2xy

(c) (x^{2} – y^{2})y’ = 2xy

(d) 2(x^{2} – y^{2})y’ = xy

**Ques:** If one root of the equation f(x) = 0 is near to x_{o} then the first approximation of this root as calculated by Newton-Raphson method is the abscissa of the point where the following straight line intersects the *x*-axis

(a) Normal to the curve y = f(x) at the point (x_{0}, f(x_{0}))

(b) Tangent to the curve y = f(x) at the point (x_{0}, f(x_{0}))

(c) The straight line through the point (x_{0}, f(x_{0})) having the gradient 1/f’(x_{0})

(d) The ordinate through the point (x_{0}, f(x_{0}))

**Ques. **The coefficient of x^{4} in the expansion of (1 + x + x^{2} + x^{3})^{11} is

(a) 900

(b) 909

(c) 990

(d) 999

**Ques.** Let PQR be a right angled isosceles triangle, right angled at P(2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is

(a) 3x^{2} – 3y^{2} + 8xy + 20x + 10y + 25 = 0

(b) 3x^{2} – 3y^{2} + 8xy – 20x – 10y + 25 = 0

(c) 3x^{2} – 3y^{2} + 8xy + 10x + 15y + 20 = 0

(d) 3x^{2} – 3y^{2} – 8xy – 10x – 15y – 20 = 0

**Ques.** Let function f (x) = x^{2} + x + sin x – cos x + log (1 + |x|) be defined over the interval [0, 1]. The odd extensions of f(x) to interval [–1, 1] is

(a) x^{2} + x + sin x + cos x – log (1 + |x|)

(b) –x^{2} + x + sin x + cos x – log (1 + |x|)

(c) –x^{2} + x + sin x – cos x + log (1 + |x|)

(d) None of these

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**Ques. **The product of lengths of perpendiculars from any point of the hyperbola x^{2} – y^{2} = 8 to its asymptotes is

(a) 2

(b) 3

(c) 4

(d) 8

**Ques. **The roots of the equation 3^{2x} – 10.3^{x} + 9 = 0 are

(a) 1, 2

(b) 0, 2

(c) 0, 1

(d) 1, 3

**Ques. **The curve y – e^{xy} + x = 0 has a vertical tangent at the point

(a) (1, 1)

(b) at no point

(c) (0, 1)

(d) (1, 0)

**Ques. **If are in A.P. 10^{ax + 10} , 10^{bx + 10} , 10^{cx + 10} are in

(a) A.P.

(b) G.P. when *x* > 0

(c) G.P. for all *x*

(d) G.P. when *x* < 0

**Ques. **If log 2, log (2^{n} – 1) and log (2^{n} + 3) are in A.P., then *n *=

(a) 5/2

(b) log_{2} 5

(c) log_{3} 5

(d) 3/2

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**Ques. **The normal chord of a parabola y^{2} = 4ax at (x_{1},x_{1}) subtends a right angle at the

(a) Focus

(b) Vertex

(c) (–*a*, 0)

(d) None of these

**Ques.** The binary equivalent of octal number (5473.64)_{8} is

(a) (110100111011.110101)_{2}

(b) (101100111011.110100)_{2}

(c) (100100111011.110100)_{2}

(d) (101100111101.110100)_{2
}

**Ques. **The area of the triangle formed by the line x + y = 3 and the angle bisectors of the pair of straight lines x^{2} – y^{2} + 2y = 1 is

(a) 2

(b) 4

(c) 6

(d) 8

**Ques. **If iz^{3} + z^{2} – z + i = 0 then the value of |z| is

(a) 1

(b) 2

(c) < 1

(d) > 1

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**Ques.** Ranks of 10 students of a class in two subjects are (1, 10), (2, 9), (3, 8), (4, 7), (5, 6), (6, 5), (7, 4), (8, 3), (9, 2), (10, 1), then rank correlation coefficient is

(a) 0

(b) – 1

(c) 1

(d) 0.5

**Ques.** A particle is dropped under gravity from rest from a height h(g = 9.8 m/sec^{2}) and then it travels a distance 9h/25 in the last second. The height *h* is

(a) 100 *metre*

(b) 122.5 *metre
*(c) 145

*metre*

(d) 167.5

*metre*

**Ques. **The number of distinct terms in the expansion of (x + 2y – 3z + 5w – 7u)^{n} is

(a) n + 1

(b) ^{n+4}C_{4
}(c) ^{n+4}C_{n+2}^{
}(d) none of these

**Ques.** cot x – tan x =

(a) cot 2x

(b) 2 cot^{2} x

(c) 2 cot 2x

(d) cot^{2} 2x

**Ques.** The tangents are drawn from the point (4, 5) to the circle x^{2} + y^{2} – 4x – 2y – 11 = 0. The area of quadrilateral formed by these tangents and radii, is

(a) 15 *sq*. *units*

(b) 75 *sq. units
*(c) 8

*sq*.

*units*

(d) 4

*sq*.

*units*

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**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

**Ques.** If the roots of the equations x^{2} – bx + c = 0 and x^{2} – cx + b = 0 differ by the same quantity, then b + c is equal to

(a) 4

(b) 1

(c) 0

(d) –4

**Ques.** The coefficient of x^{5} in the expansion of (1 + x^{2})^{5} (1 + x)^{4} is

(a) 30

(b) 60

(c) 40

(d) None of these

**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

**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

**Ques.** A man of mass 80 *kg*. is travelling in a lift. The reaction between the floor of the lift and the man when the lift is ascending upwards at 4 *m*/*sec*^{2} is

(a) 1464.8 *N*

(b) 1784.8 *N*

(c) 1959.8 *N*

(d) 1104.8 *N
*

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**Ques.** The value of sin^{2} 5^{o} + sin^{2} 10^{o} + sin^{2} 15^{o} + … + sin^{2} 85^{o} + sin^{2} 90^{o} is equal to

(a) 7

(b) 8

(c) 9

(d) 9 ½

**Ques.** The co-ordinates of the point from where the tangents are drawn to the circles x^{2} + y^{2} = 1, x^{2} + y^{2} + 8x + 15 = 0 and x^{2} + y^{2} + 10y + 24 = 0 are of same length, are

(a) (2, 5/2)

(b) (–2, –5/2)

(c) (–2, 5/2)

(d) (2, –5/2)

**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

**Ques.** Two trains *A* and *B*, 100 *kms* apart, are travelling to each other with starting speed of 50 *km/hr *for both. The train *A* is accelerating at 18 *km/hr*^{2} and *B* is decelerating at 18 *m*/*h*^{2}. The distance where the engines cross each other from the initial position of *A* is

(a) 50 *kms*

(b) 68 *kms
*(c) 32

*kms*

(d) 59

*kms*

**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

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**Ques.** The sums of n terms of three A.P.’s whose first term is 1 and common differences are 1, 2, 3 are s_{1}, S_{2}, S_{3} respectively. The true relation is

(a) S_{1} + S_{3} = S_{2}

(b) S_{1} + S_{3} = 2S_{2
}(c) S_{1} + S_{2} = 2S_{3}

(d) S_{1} + S_{2} = S_{3}

**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

**Ques.** The equation of the common tangent to the curves y^{2} = 8x and xy = –1 is

(a) 3y = 9x + 2

(b) y = 2x + 1

(c) 2y = x + 8

(d) y = x + 2

**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

**Ques.** A man in a balloon, rising vertically with an acceleration of 4.9 *m/sec*^{2} releases a ball 2 *seconds* after the balloon is let go from the ground. The greatest height above the ground reached by the ball is

(a) 14.7 *m*

(b) 19.6 *m*

(c) 9.8* m*

(d) 24.5* m
*

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**Ques.** The figure formed by the lines x^{2} + 4xy + y^{2} = 0 and x – y = 4, is

(a) A right angled triangle

(b) An isosceles triangle

(c) An equilateral triangle

(d) None of these

**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

**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

**Ques.** The root of the equation x^{3} + x – 3 = 0 lies in interval (1, 2) after second iteration by false position method, it will be in

(a) (1.178, 2.00)

(b) (1.25, 1.75)

(c) (1.125, 1.375)

(d) (1.875, 2.00)

**Ques.** The number of solution of the following equations x_{2} – x_{3} = 1, –x_{1} + 2x_{3} = –2, x_{1} – 2x_{2} = 3 is

(a) Zero

(b) One

(c) Two

(d) Infinite

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**Ques.** If ^{56}P_{r+6} : ^{54}P_{r+3} = 30800 : 1, then r =

(a) 31

(b) 41

(c) 51

(d) None of these

**Ques.** The resultant of two forces *P* and *Q* is *R*. If *Q* is doubled, *R* is doubled and if *Q* is reversed, *R* is again doubled. If the ratio P^{2} : Q^{2} : R^{2} = 2 : 3 : x, then *x* is equal to

(a) 5

(b) 4

(c) 3*
*(d) 2

**Ques.** If x = log_{3} 5, y = log_{17} 25, which one of the following is correct

(a) x < y

(b) x = y

(c) x > y

(d) None of these

**Ques.** The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex is x – y + 1 = 0 is

(a) x^{2} + y^{2} – 2xy – 4x + 4y – 4 = 0

(b) x^{2} + y^{2} – 2xy + 4x – 4y – 4 = 0

(c) x^{2} + y^{2} + 2xy – 4x + 4y – 4 = 0

(d) x^{2} + y^{2} + 2xy – 4x + 4y + 4 = 0

**Ques.** A root of the equation x^{3} – 3x – 5 = 0 lies between 2 and 2.5. Its approximate value, by applying bisection method 3 times is

(a) 2.0625

(b) 2.3125

(c) 2.3725

(d) 2.4225

**Ques.** Given that the equation z^{2} + (p + iq)z + r + I s = 0, where p, q, r, s are real and non-zero has a real root, then

(a) pqr = r^{2} + p^{2} s

(b) prs = q^{2} + r^{2} p

(c) qrs = p^{2} + s^{2} p

(d) pqs = s^{2} + q^{2} r

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**Ques.** If the product of roots of the equation x^{2} – 3kx + 2e^{2 log k} – 1 = 0 is 7, then its roots will real when

(a) k = 1

(b) k = 2

(c) k = 3

(d) None of these

**Ques.** If cos (u + iv) = x + iy, then x^{2} + y^{2} + 1 is equal to

(a) cos^{2} u + sinh^{2} v

(b) sin^{2} u + cosh^{2} v

(c) cos^{2} u + cosh^{2} v

(d) sin^{2} u + sinh^{2} v

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