Mathematics test Series 3

Q1. The domain of the function f ( x )= sin−1(x−3)√9−x2

is

A) [1,2]B) [2,3]C) [1,3]D) [1,2]

Sol: B)

Q2. If A, B and C are three sets such that A∩B=A∩C And A∪B= A∪C, then

A) B=CB) A∩B=φ

C) A=BD)A=C

Sol: A) B=C

Q3. If zk=cos ( kπ10 )+ isin( kπ10 ), then z1 z2 z3 z4 is equal to

A) -1 B) 1C) -2D)2

Sol: A)

Q4. The value of a for which one root of the quadratic equation

(a2−5a+3 )x2+(3a−1 ) x+2=0 is twice as large as the other is

A) -2/3B) 1/3

C) -1/3D)2/3E) Sol: D) 2/3

Q5. If a, b, c are positive integers such that a> b > c and

|1 1 1a b ca2 b2 c2|=−2

Then 3a +7b +10c equals

A) 10B) 11C) 12D)13

Sol: D)

Assertion – reason Type

Q6.Statement -1: The sum of divisors of

n=210325372112 is

1480

(211−1)(33−1)(54−1)(73−1)(113−1)

Statement -2: The number of divisors of m=P1σ1P2

σ2……… ..Prσr where P1,P2,………Pr

are distinct primes and σ 1 , σ2….σ r are natural numbers is (σ 1+1)(σ 2+1)…….(σ r+1)

A) STATEMENT -1 is true, STATEMENT -2 is true; STATEMENT -2 is a correct explanation for STATEMENT -1

B) STATEMENT -1 is true, STATEMENT -2 is true; STATEMENT -2 is NOT a correct explanation for STATEMENT -1

C) STATEMENT -1 is True, STATEMENT -2 is FalseD)STATEMENT -1 is False, STATEMENT -2 is False

Sol: B)

Q7. Statement :1If n is an odd prime , then greatest integer contained in

(2+√5 )n−2n+1 is divisible by 20 n.

Statement :2 If p is a prime and 1≤ r ≤ p−1, then ( pr ) is divisible by p.

A) STATEMENT -1 is true, STATEMENT -2 is true; STATEMENT -2 is a correct explanation for STATEMENT -1

B) STATEMENT -1 is true, STATEMENT -2 is true; STATEMENT -2 is NOT a correct explanation for STATEMENT -1

C) STATEMENT -1 is True, STATEMENT -2 is FalseD)STATEMENT -1 is False, STATEMENT -2 is False

Sol: A)

Q8. If a1, a2…….., a3 are in A.P. with common difference d≠0 ,then sum of the series sin d [cosec a1cosec a2+cosec a2cosec a3+……….+cosec an−1 cosec an] is

A) Sec a1 – sec an

B) Cosec a1 – cosec an

C) Cot a1- cot an

D)Tan a1 – tan an

Sol: C)

Q9. Let f(x) = { 1 x is rational0 x is irrational , then

A) f is discontinuous for every real xB) f is continuous on RC) f is continuous at the points where x is rational D)f is continuous at the points where x is irrational

Sol: A)

Q10. The number of times the function y= sin -1 (2 x1+ x2) doesn’t exist for

A) all values of x for which |x|<1B) x =-1, 1C) all values of x for which |x|>1D)none of these

Sol: B)

Q11. The curve that passes through the point (2,3) and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact , is given by :

A) 2 y−3 x=0

B) y=6x

C) x2+ y2=13

D)( x2 )2

( y3 )2

=2

Sol: B)

Q12. f ( x )=∫ dxsin 4 x

is a

A) Polynomial of degree 3 in cot xB) Polynomial of degree 4 in cot xC) Polynomial of degree 4 in sin xD)Polynomial of degree 4 in tan x

Sol: A)

Q13. The area of the region bounded by the parabola ( y−2 )2=(x−1), the tangent to the parabola at the point (2,3) and the x-axis is

A) 9

B) 12C) 3D)6

Sol: A)

Q14. The solution for x of the equation

∫√ 2

x dtt √t2−1

= π2

¿¿ is

A) 2B) -√ 2

C) √32

D)2√ 2

Sol: B)

Q15. The curves given by 2 xy y '= y2−x2represents

A) Family of circles with centre on y-axis B) Family of parabola passing through originC) Family of circles with centre on x-axis D)Family of hyperbola

Sol: C)

Q16. If the sum of the slopes of the lines given by x2−2cxy−7 y2=0 is four times their product , then the value of c is

A) 2B) -1C) 1

D)-2

Sol:A)

Q17. Locus of centroid of the triangle whose vertices are (a cost, a sint), (b sin t, -b cos t) and (1,0), where t is a parameter is

A) (3 x−1 )2+(3 y )2=a2+b2

B) (3 x+1 )2+(3 y )2=a2+b2

C) (3 x+1 )2+(3 y )2=a2−b2

D) (3 x−1 )2+(3 y )2=a2−b2

Sol: A)

Q18. If the chord along the line y−x=3 of the circle x2+ y2=k2 subtends an angle of 300 in the major segment of the circle cut off by the chord then k 2=¿

A) 3B) 6C) 9D)36

Sol: B)

Q19. A circle is drawn on a normal chord of the parabola y2=4 x and passes through its vertex . radius of the circle is

A)√ 3

B) 2√3

C) 3√ 3

D)6√ 3

Sol: C)

Q20. The vertices of triangle are A(1,0,0), B(0,2,0), C(0,0,3). If the orthocenter and circumcentre of the triangles are a,b, -111, the a+b is equal to

A) 5B) 10C) 15D)25

Sol: C)

Q21. The two lines x=ay+b , z=cy+d∧x=a' y+b ' z=c' y+d will be perpendicular , if and only if

A) aa’+bb’+cc’=0B) (a+a’)(b+b’)(c+c’)=0C) aa’+cc’+1=0D)aa’+bb’+c’+1=0

Sol:C)

Q22. If A.M., G.M. and H.M. in any series are equal then

A) the distribution is symmetric B) all the values are sameC) the distribution is unimodalD)none of these

Sol: B)

Q23. A and B are two students. Their probabilities of solving a problem correctly are ¼ and 1/5 respectively. If the probability of their making a

common error is 1/40, and they obtain the answer, then the probability of their answer is correct is

A) 1/12B) 1/20C) 10/13D)13/200

Sol: C)

Q24. The acute angle of a rhombus whose side is a mean proportional between its diagonals is

A) 150

B) 200

C) 300

D)800

Sol: C)

Q25. Four persons are selected from a group of 4men, 2 women and 3 children. The probability that exactly two of them are men is

A) 9/11B) 10/23C) 11/24D)10/21

Sol: D)