The range of the relation R = {(x, x²) |x is a prime number less than 13} is

(a)

{2,3,5,7}

(b)

{2,3,5,7,11}

(c)

{4,9,25,49,121}

(d)

{1,4,9,25,49,121}

Let f and g be two functions given by
f = {(0,1), (2,0), (3,-4), (4,2), (5,7)}
g = {(0,2), (1,0), (2,4), (-4,2), (7,0)} then the  range of f o g is

(a)

{0,2,3,4,5}

(b)

{–4,1,0,2,7}

(c)

{1,2,3,4,5}

(d)

{0,1,2}

Let f(x) = \(\sqrt { 1+x^{ 2 } } \) then

(a)

f(xy) = f(x).f(y)

(b)

f(xy) ≥ f(x).f(y)

(c)

f(xy) ≤ f(x).f(y)

(d)

None of these

If g = {(1,1), (2,3), (3,5), (4,7)} is a function given by g(x) = αx + β then the values of α and β are

(a)

(-1,2)

(b)

(2,-1)

(c)

(-1,-2)

(d)

(1,2)

f(x) = (x + 1)³ - (x - 1)³ represents a function which is

(a)

linear

(b)

cubic

(c)

reciprocal

(d)

quadratic

If the order pairs (a, -1) and (5, b) blongs to {(x, y) | y = 2x + 3}, then a and b are __________

(a)

-13, 2

(b)

2, 13

(c)

2, -13

(d)

-2,13

If function f : N⟶N, f(x) = 2x then the function is, then the function is ___________

(a)

Not one - one and not onto

(b)

one-one and onto

(c)

Not one -one but not onto

(d)

one - one but not onto

If f(x) = mx + n, when m and n are integers f(-2) = 7, and f(3) = 2 then m and n are equal to ___________

(a)

-1, -5

(b)

1, -9

(c)

-1, 5

(d)

1, 9

If f(x) = ax - 2, g(x) = 2x - 1 and fog = gof, the value of a is ___________

(a)

3

(b)

-3

(c)

\(\frac { 1 }{ 3 } \)

(d)

13

If f(x) = 2 - 3x, then f o f(1 - x) = ?

(a)

5x+9

(b)

9x-5

(c)

5-9x

(d)

5x-9

Euclid’s division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r , where r must satisfy

(a)

1 < r < b

(b)

0 < r < b

(c)

0 \(\le\) r < b

(d)

0 < r \(\le\) b

The sum of the exponents of the prime factors in the prime factorization of 1729 is

(a)

1

(b)

2

(c)

3

(d)

4

7^4k \(\equiv \) ________ (mod 100)

(a)

1

(b)

2

(c)

3

(d)

4

Given F₁ = 1, F₂ = 3 and F_n = F_n-1 + F_n-2 then F₅ is

(a)

3

(b)

5

(c)

8

(d)

11

The next term of the sequence \(\frac { 3 }{ 16 } ,\frac { 1 }{ 8 } ,\frac { 1 }{ 12 } ,\frac { 1 }{ 18 } \), ..... is

(a)

\(\frac { 1 }{ 24 } \)

(b)

\(\frac { 1 }{ 27 } \)

(c)

\(\frac { 2 }{ 3 } \)

(d)

\(\frac { 1 }{ 81 } \)

If 3 is the least prime factor of number and 7 is least prime factor of b, then the least prime factor a + b is ____________

(a)

a + b

(b)

2

(c)

5

(d)

10

The difference between the remainders when 6002 and 601 are divided by 6 is ____________

(a)

2

(b)

1

(c)

0

(d)

3

The first term of an A.P. whose 8^th and 12^th terms are 39 and 59 respectively is ____________

(a)

5

(b)

6

(c)

4

(d)

3

A boy saves Rs. 1 on the first day Rs. 2 on the second day, Rs. 4 on the third day and so on. How much did the boy will save upto 20 days?

(a)

2¹⁹ + 1

(b)

2¹⁹- 1

(c)

2²⁰- 1

(d)

2²¹- 1

In an A.P if the p^th term is q and the q^th term is p, then its n^th term is ____________

(a)

p+q-n

(b)

p+q+n

(c)

p-q+n

(d)

p-q-n

The solution of the system x + y − 3z = −6, −7y + 7z = 7, 3z = 9 is

(a)

x = 1, y = 2, z = 3

(b)

x = −1, y = 2, z = 3

(c)

x = −1, y = −2, z = 3

(d)

x = 1, y = -2, z = 3

y² + \(\frac {1}{y^{2}}\) is not equal to

(a)

\(\frac {y^{2} + 1}{y^{2}}\)

(b)

\({ \left( y+\frac { 1 }{ y } \right) }^{ 2 }\)

(c)

\({ \left( y-\frac { 1 }{ y } \right) }^{ 2 }+2\)

(d)

\({ \left( y+\frac { 1 }{ y } \right) }^{ 2 }-2\)

Which of the following should be added to make x⁴ + 64 a perfect square

(a)

4x²

(b)

16x²

(c)

8x²

(d)

-8x²

Graph of a linear equation is a ____________

(a)

straight line

(b)

circle

(c)

parabola

(d)

hyperbola

For the given matrix A = \(\left( \begin{matrix} 1 \\ 2 \\ 9 \end{matrix}\begin{matrix} 3 \\ 4 \\ 11 \end{matrix}\begin{matrix} 5 \\ 6 \\ 13 \end{matrix}\begin{matrix} 7 \\ 8 \\ 15 \end{matrix} \right) \) the order of the matrix A^T is

(a)

2 x 3

(b)

3 x 2

(c)

3 x 4

(d)

4 x 3

Which of the following is correct
(i) Every polynomial has finite number of multiples
(ii) LCM of two polynimials of degree 2 may be a constant
(iii) HCF of 2 polynomials may be constant
(iv) Degree of HCF of two polynomials is always less then degree of LCM

(a)

(i) and (ii)

(b)

(iii) and (iv)

(c)

(iii) only

(d)

(iv) only

The HCF of two polynomials p(x) and q(x) is 2x(x + 2) and LCM is 24x(x + 2)² (x - 2) if p(x) = 8x³ + 32x² + 32x, then q(x) ___________

(a)

4x³-16x

(b)

6x³-24x

(c)

12x³+24x

(d)

12x³-24x

Choose the correct answer
(i) Every scalar matrix is an identity matrix
(ii) Every identity matrix is a scalar matrix
(iii) Every diagonal matrix is an identity matrix
(iv) Every null matrix is a scalar matrix

(a)

(i) and (iii) only

(b)

(iii) only

(c)

(iv) only

(d)

(ii) and (iv) only

If \(2A+3B=\left[ \begin{matrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{matrix} \right] \) and \(A+2B=\left[ \begin{matrix} 5 & 0 & 3 \\ 1 & 6 & 2 \end{matrix} \right] \) then B = [hint: B = (A+2B)-(2+3B)]

(a)

\(\left[ \begin{matrix} 8 & -1 & -2 \\ -1 & 10 & -1 \end{matrix} \right] \)

(b)

\(\left[ \begin{matrix} 8 & -1 & 2 \\ -1 & 10 & -1 \end{matrix} \right] \)

(c)

\(\left[ \begin{matrix} 8 & 1 & 2 \\ -1 & 10 & -1 \end{matrix} \right] \)

(d)

\(\left[ \begin{matrix} 8 & 1 & 2 \\ 1 & 10 & 1 \end{matrix} \right] \)

If \(A=\left[ \begin{matrix} y & 0 \\ 3 & 4 \end{matrix} \right] \) and \(I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \) then A² = 16 I for ___________

(a)

y = 4

(b)

y = 5

(c)

y = -4

(d)

y = 16

In a given figure ST || QR, PS = 2 cm and SQ = 3 cm. Then the ratio of the area of \(\triangle\)PQR to the area \(\triangle\)PST is 

(a)

25 : 4

(b)

25 : 7

(c)

25 : 11

(d)

25 : 13

In a \(\triangle\)ABC, AD is the bisector \(\angle\)BAC. If AB = 8 cm, BD = 6 cm and DC = 3 cm. The length of the side AC is

(a)

6 cm

(b)

4 cm

(c)

3 cm

(d)

8 cm

The two tangents from an external points P to a circle with centre at O are PA and PB. If \(\angle APB\) = 70^o then the value of \(\angle AOB\) is

(a)

100°

(b)

110°

(c)

120°

(d)

130°

In figure CP and CQ are tangents to a circle with centre at O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 7 cm, then the length of BR is

(a)

6 cm

(b)

5 cm

(c)

8 cm

(d)

4 cm

In figure if PR is tangent to the circle at P and O is the centre of the circle, then \(\angle PQR\) is

(a)

120^o

(b)

100°

(c)

110°

(d)

90°

A line which intersects a circle at two distinct points ic called ____________

(a)

Point of contact

(b)

sccant

(c)

diameter

(d)

tangent

If the angle between two radio of a circle is ^o, the angle between the tangents at the end of the radii is ____________

(a)

50^o

(b)

90^o

(c)

40^o

(d)

70^o

In figure \(\angle OAB={ 60 }^{ o }\) and OA = 6cm then radius of the circle is ____________

(a)

\(\frac { 3 }{ 2 } \sqrt { 3 } cm\)

(b)

2 cm

(c)

\(3\sqrt { 3 } cm\)

(d)

\(2\sqrt { 3 } cm\)

Two concentric circles if radii a and b where a>b are given. The length of the chord of the circle which touches the smaller circle is ____________

(a)

\(\sqrt { { a }^{ 2 }-{ b }^{ 2 } } \)

(b)

\(\sqrt { { a }^{ 2 }-{ b }^{ 2 } } \)

(c)

\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \)

(d)

\(2\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \)

Three circles are drawn with the vertices of a triangle as centres such that each circle touches the other two if the sides of the triangle are 2cm,3cm and 4 cm. find the diameter of the smallest circle.

(a)

1 cm

(b)

3 cm

(c)

5 cm

(d)

4 cm

The slope of the line joining (12, 3), (4, a) is \(\frac 18\). The value of ‘a’ is

(a)

1

(b)

4

(c)

-5

(d)

2

The slope of the line which is perpendicular to a line joining the points (0, 0) and (– 8, 8) is

(a)

–1

(b)

1

(c)

\(\frac13\)

(d)

-8

The equation of a line passing through the origin and perpendicular to the line 7x - 3y + 4 = 0 is

(a)

7x - 3y + 4 = 0

(b)

3x - 7y + 4 = 0

(c)

3x + 7y = 0

(d)

7x - 3y = 0

Consider four straight lines
(i) l₁ : 3y = 4x + 5
(ii) l₂ : 4y = 3x - 1
(iii) l₃ : 4y + 3x =7
(iv) l₄ : 4x + 3y = 2
Which of the following statement is true?

(a)

l₁ and l₂ are perpendicular

(b)

l₁ and l₄ are parallel

(c)

l₂ and l₄ are perpendicular

(d)

l₂ and l₃ are parallel

When proving that a quadrilateral is a parallelogram by using slopes you must find

(a)

The slopes of two sides

(b)

The slopes of two pair of opposite sides

(c)

The lengths of all sides

(d)

Both the lengths and slopes of two sides

If the points (0, 0), (a, 0) and (0, b) are colllinear, then ____________

(a)

a = b

(b)

a + b

(c)

ab = 0

(d)

a ≠ b

Find the slope and the y-intercept of the line \(3y-\sqrt { 3x } +1=0\) is ____________

(a)

\(\frac { 1 }{ \sqrt { 3 } } ,\frac { -1 }{ 3 } \)

(b)

\(-\frac { 1 }{ \sqrt { 3 } } ,\frac { -1 }{ 3 } \)

(c)

\(\sqrt { 3 } ,1\)

(d)

\(-\sqrt { 3 } ,3\)

Find the value of 'a' if the lines 7y = ax + 4 and 2y = 3 - x are parallel

(a)

\(\frac { 7 }{ 2 } \)

(b)

\(-\frac { 2 }{ 7 } \)

(c)

\(\frac { 2 }{ 7 } \)

(d)

\(-\frac { 7 }{ 2 } \)

A line passing through the point (2, 2) and the axes enclose an aream ∝. The intercept on the axes made by the line are given by the roots of ____________

(a)

x²-2-∝x+∝ = 0

(b)

x²+2∝x+∝ = 0

(c)

x²-∝x+2∝ = 0

(d)

none of these

In a right angle triangle, right angled at B, if the side BC is parallel to x axis, then the slope of AB is ___________

(a)

\(\sqrt { 3 } \)

(b)

\(\frac { 1 }{ \sqrt { 3 } } \)

(c)

1

(d)

not defined

tan \(\theta \) cosec²\(\theta \) - tan\(\theta \) is equal to 

(a)

sec\(\theta \)

(b)

\(cot^{ 2 }\theta \)

(c)

sin\( \theta \)

(d)

\(cot\theta \)

A tower is 60 m height. Its shadow is x metres shorter when the sun’s altitude is 45° than when it has been 30°, then x is equal to

(a)

41.92 m

(b)

43.92 m

(c)

43 m

(d)

45.6 m

The angle of depression of the top and bottom of 20 m tall building from the top of a multistoried building are 30° and 60° respectively. The height of the multistoried building and the distance between two buildings (in metres) is

(a)

20, 10\(\sqrt { 3 } \)

(b)

30, 5\(\sqrt { 3 } \)

(c)

20, 10

(d)

30, 10\(\sqrt { 3 } \)

Two persons are standing ‘x’ metres apart from each other and the height of the first person is double that of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the shorter person (in metres) is

(a)

\(\sqrt { 2 } \) x

(b)

\(\frac { x }{ 2\sqrt { 2 } } \)

(c)

\(\frac { x }{ \sqrt { 2 } } \)

(d)

2x

If (sin α + cosec α)² + (cos α + sec α)² = k + tan²α + cot²α, then the value of k is equal to

(a)

9

(b)

7

(c)

5

(d)

3

The value of the expression \(\left[ \frac { { sin }^{ 2 }{ 22 }^{ o }+{ sin }^{ 2 }{ 68 }^{ o } }{ { cos }^{ 2 }{ 22 }^{ 0 }+{ cos }^{ 2 }{ 68 }^{ 0 } } +{ sin }^{ 2 }{ 63 }^{ o+ }{ cos }63^{ 0 }{ sin27 }^{ 0 } \right] \)is ___________

(a)

3

(b)

2

(c)

1

(d)

0

If sec θ + tan θ = n, and sec θ - tan θ = 0, then the value of mn is ___________

(a)

2

(b)

1

(c)

土1

(d)

土2

(cosec²θ - cot²θ) (1 - cos²θ) is equal to ___________

(a)

cosec θ

(b)

cos²θ

(c)

sec²θ

(d)

sin²θ

\(\frac { tan\theta }{ sec\theta } +\frac { tan\theta }{ sec\theta +1 } \) is equal to

(a)

2tanθ

(b)

2secθ

(c)

2cosecθ

(d)

2 tanθsecθ

A ladder of length 14m just reaches the top of a wall. If the ladder makes an angle of 60^o with the horizontal, then the height of the wall is ____________

(a)

\(14\sqrt { 3 } \)

(b)

\(28\sqrt { 3 } \)

(c)

\(7\sqrt { 3 } \)

(d)

\(35\sqrt { 3 } \)

The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is

(a)

60\(\pi\) cm²

(b)

68\(\pi\) cm²

(c)

120\(\pi\) cm²

(d)

136\(\pi\) cm²

The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be

(a)

12 cm

(b)

10 cm

(c)

13 cm

(d)

5 cm

If the radius of the base of a cone is tripled and the height is doubled then the volume is

(a)

made 6 times

(b)

made 18 times

(c)

made 12 times

(d)

unchanged

A spherical ball of radius r₁ units is melted to make 8 new identical balls each of radius r₂ units. Then r₁:r₂ is

(a)

2:1

(b)

1:2

(c)

4:1

(d)

1:4

The volume (in cm³) of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and height 5 cm is

(a)

\(\frac{4}{3}\pi\)

(b)

\(\frac{10}{3}\pi\)

(c)

\(5\pi\)

(d)

\(\frac{20}{3}\pi\)

How many balls, each of radius 1 cm, can be made from a solid sphere of lead of radius cm?

(a)

64

(b)

216

(c)

512

(d)

16

The ratio of the volumes of two spheres is 8 : 27. If r and R are the radii of sphere respectively, Then (R - r) : r is ___________

(a)

1:2

(b)

1:3

(c)

2:3

(d)

4:9

A solid frustum is of height 8 cm. If the radii of its lower and upper ends are 3 cm and 9 cm respectively, then its slant height is ___________

(a)

15 cm

(b)

12 cm

(c)

10 cm

(d)

17 cm

A solid is hemispherical at the bottom and conical above. If the curved surface areas of the two parts are equal, then the ratio of its radius and the height of its conical part is ___________

(a)

1:3

(b)

\(1:\sqrt { 3 } \)

(c)

1:1

(d)

\(\sqrt { 3 } :1\)

A floating boat having a length 3m and breadth 2m is floating on a lake. The boat sinks by 1 cm when a man gets into it. The mass of the man is (density of water is 10000 kg/m³)

(a)

50 kg

(b)

60 kg

(c)

70 kg

(d)

80 kg

Which of the following is not a measure of dispersion?

(a)

Range

(b)

Standard deviation

(c)

Arithmetic mean

(d)

Variance

The sum of all deviations of the data from its mean is

(a)

Always positive

(b)

always negative

(c)

zero

(d)

non-zero integer

The standard deviation of a data is 3. If each value is multiplied by 5 then the new variance is

(a)

3

(b)

15

(c)

5

(d)

225

If the standard deviation of x, y, z is p then the standard deviation of 3x + 5, 3y + 5, 3z + 5 is

(a)

3p + 5

(b)

3p

(c)

p + 5

(d)

9p + 15

A page is selected at random from a book. The probability that the digit at units place of the page number chosen is less than 7 is

(a)

\(\frac{3}{10}\)

(b)

\(\frac{7}{10}\)

(c)

\(\frac{3}{9}\)

(d)

\(\frac{7}{9}\)

which of the following is true?

(a)

0 ≤ p(∈) ≤ 1

(b)

p(∈) > 1

(c)

p(∈) < 0

(d)

\(-\frac { 1 }{ 2 } \ge P(\in )\le \frac { 1 }{ 2 } \)

IF the probability of the non-happening of a event is q, then the probability of happening of that event is 

(a)

1-q

(b)

q

(c)

q/2

(d)

∝q

The variance of 5 values is 16. If each value is doubled them the standard deviation of new values is_______ 

(a)

4

(b)

8

(c)

32

(d)

16

The range of first 10 prime number is ___________

(a)

9

(b)

20

(c)

27

(d)

5

When three coins are tossed, the probability of getting the same face on all the three coins is ___________

(a)

\(\frac { 1 }{ 8 } \)

(b)

\(\frac { 1 }{ 4 } \)

(c)

\(\frac { 3 }{ 8 } \)

(d)

\(\frac { 1 }{ 3 } \)