Mathematics Mock Test

TIME LEFT -

Question 1
5 / -1

If \(2f\left( x \right) + f\left( {\frac{1}{x}} \right) = \log x\), for all x > 0, then f(e^x) is

A
x²

B
x

C
2x

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

Question 2
5 / -1

Find the principal value of \(\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\)

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

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

C
\(\frac{5\pi}{2}\)

D
\(\frac{\pi}{5}\)

Question 3
5 / -1

If \(\Delta = \left| {\begin{array}{*{20}{c}} 0\\ {a - b}\\ {a - c} \end{array}\begin{array}{*{20}{c}} {b - a}\\ 0\\ {b - c} \end{array}\begin{array}{*{20}{c}} {c - a}\\ {c - b}\\ 0 \end{array}} \right|\), then \(\triangle\) equals

A
(a + b + c)

B
-(a + b + c)

C
abc

D
0

Question 4
5 / -1

Let \(f\left( x \right) = \frac{{\log \left( {1 - x + {x^2}} \right) + \log \left( {1 + x + {x^2}} \right)}}{{\sec x - \cos x}}\); x ≠ 0 Then the value of f(0) so that f is continuous at x = 0, is

A
0

B
2

C
1

D
\(\frac{1}{2}\)

Question 5
5 / -1

The equation of tangent to the curve x^2/3 + y^2/3= a^2/3 at (a, 0) passes through which of the following points?

A
(0, a)

B
(2, 0)

C
(0, 2)

D
(a, a)

Question 6
5 / -1

Let \(y = f \left ( {\frac{{x + {e^x}}}{{{e^x}}}} \right)\) satisfies f'(1) = 2, Then the value of \(\frac{{dy}}{{dx}}\) at x = 0 equals

A
1

B
2

C
0

D
3

Question 7

5 / -1

Match List I with List II:

List I	List II
A. \(\int\limits_0^{\frac{π }{2}} {\frac{{\sin 2x}}{{{{\sin }^4}x + {{\cos }^4}x}}dx} \)	I. \(\frac{π }{4}\)
B. \(\int\limits_0^{\frac{π }{2}} {\frac{{\sin x}}{{\sin x + \cos x}}dx} \)	II. π
C. \(\int\limits_0^{2π } {\frac{1}{{1 + {e^{\sin x}}}}dx} \)	III. \(\frac{π }{{12}}\)
D. \(\int\limits_0^{\frac{π }{2}} {\frac{1}{{9{{\sin }^2}x + 4{{\cos }^2}x}}dx} \)	IV. \(\frac{π }{2}\)

Choose the correct answer from the options given below:

A - IV, B - II, C - I, D - III

A - II, B - I, C - III, D - IV

A - IV, B - I, C - II, D - III

A - II, B - I, C - IV, D - III

Question 8
5 / -1

If x = a(cos θ + θ sin θ), y = a(sin θ - θ cos θ), then at \(\theta = \frac{\pi }{4}\), we have

A
\(\frac{{dy}}{{dx}} = 1\)

B
\(\frac{{{d^2}y}}{{d{x^2}}} = \frac{{8\sqrt 2 }}{{a\pi }}\)

C
\(\frac{{dy}}{{dx}} = - 1\)

D
\(\frac{{{d^2}y}}{{d{x^2}}} = \frac{{ - 8\sqrt 2 }}{{a\pi }}\)

Question 9
5 / -1

If f(x + y) = f(x) . f(y) for all x, y ∈ R such that f(5) = 2 and f(0) = 3, then the value of f'(5) is

A
6

B
2

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

D
3

Question 10
5 / -1

If A = {1, 2, 3} and R is a relation defined on A such that R = {(1, 1), (1, 2), (2, 3), (2, 2), (3, 3), (3, 1)}. Then the relation R is ?

A
Reflexive and symmetric but not transitive

B
Reflexive and transitive but not symmetric

C
Reflexive but neither symmetric nor transitive

D
Reflexive, symmetric and transitive

Question 11
5 / -1

In order to establish a marriage hall, Ashok bought the tables and chairs. The cost of a table and a chair is 2500 and 500 Rs respectively. At a party, there is limited space and also a budget problem. The maximum number of chairs and tables that can be used for that party is 60 and the Ashok can spend a maximum of money 60000 Rs for these chairs and tables. If the objective of Ashok is to maximize the profit and the profits are 25 Rs per chair and 75 Rs per table then the maximum profit that the Ashok will earn?

A
Rs 2200

B
Rs 2250

C
Rs1500

D
Rs 1800

Question 12
5 / -1

The system of linear equations
x + y + z = 4
x + 2y + 3z = 7
x + 4y + λz = μ
has a unique solution if-

A
\(\lambda=7,\mu=13\)

B
\(\lambda\ne7\)

C
\(\mu\ne13\)

D
\(\lambda=7,\mu\ne13\)

Question 13
5 / -1

If the difference between the greatest and the least value of the function \(f(x)=\sin^2x+\cos x\) on [0,\(\pi\)] is \(\frac{a-4}{4}\), then \(a\) equals to -

A
5/4

B
1

C
-1

D
13

Question 14
5 / -1

Consider \(f(x)=2x^3-9x^2+12x+6\).Then

A
The absolute maximum of the function is 15 in the interval [1,3].

B
The absolute minimum of the function is 11 in the interval [1,3].

C
The absolute maximum of the function is 11 in the interval [1,3].

D
The absolute minimum of the function is 15 in the interval [1,3].

Question 15
5 / -1

Let m ∈ Z and consider the relation R_m defined by a R_mb if and only if a ≡ b mod m. Then R_m is -

A
reflexive relation

B
symmetric relation

C
transitive relation

D
equivalence relation

Question 16
5 / -1

For the function \(f(x)=\sin x-\sqrt{3}\cos x-x\) on the interval [0,π] the stationary points will be -

A
2π/3

B
2 π

C
x = 2π/3, 2 π both

D
None

Question 17
5 / -1

Which of the following is one of the feasible solutions for LPP with constraints?
X + 2y ≥ 2, 5x + 4y ≤ 20, 2x –y ≤ 4, x ≥ 0, y ≥ 0
A. (1,0), B (0,2), C (3,2), B (1,5)

A
(1,0)

B
(0,2),

C
(3,2)

D
((1,5)

Question 18
5 / -1

Consider a system of linear equations.
4x - 2y = 3
6x + α y = 5
For what values of constant α the system of linear equations has a unique solution -

A
α = -3

B
α ≠ - 3

C
α ≠ 3

D
α = 3

Question 19
5 / -1

Directions For Questions

Directions: For the next two (02) items that follow:
Consider the plane passing through the points
A (2, 2, 1), B(3, 4, 2) and C (7, 0, 6)

...view full instructions

Which one of the following points lies on the plane?

A
(1, 0, 0)

B
(1, 0, 1)

C
(0, 0, 1)

D
None of the above

Question 20
5 / -1

Directions For Questions

Directions: For the next two (02) items that follow:
Consider the plane passing through the points
A (2, 2, 1), B(3, 4, 2) and C (7, 0, 6)

...view full instructions

What are the direction ratios of the normal to the plane?

A
< 1, 0, 1>

B
<0, 1, 0>

C
<1, 0, -1>

D
None of the above

Question 21
5 / -1

Maximize z = 4x + 6y
subject to
3x + 2y ≤ 12, x + y ≥ 2, x, y ≥ 0

A
24 at (6, 0)

B
36 at (0, 6)

C
24 at (0, 4)

D
16 at (4, 0)

Question 22
5 / -1

The number of straight lines that are equally inclined to the three dimensional co-ordinate axes, is

A
2

B
4

C
6

D
8

Question 23

5 / -1

Match List I with List II :

List I	List II
A. \(f\left( x \right) = \frac{1}{{\sqrt {x - 2} }} + \frac{1}{{\sqrt {3 - x} }}\)	I. Domain f(x) = [2, 3]
B. \(f\left( x \right) = \sqrt {\frac{{x - 2}}{{3 - x}}} \)	II. Domain f(x) = (2, 3]
C. \(f\left( x \right) = \sqrt {\frac{{3 - x}}{{x - 2}}} \)	III. Domain f(x) = [2, 3)
D. \(f\left( x \right) = \sqrt {\left( {x - 2} \right)\left( {3 - x} \right)} \)	IV. Domain f(x) = (2, 3)

Choose the correct answer from the options given below:

A - I, B - III, C - II, D - IV

A - IV, B - III, C - II, D - I

A - I, B - II, C - III, D - IV

A - IV, B - II, C - III, D - I

Question 24
5 / -1

The solution of differential equation cos(x + y) dy = dx is given by

A
\(y = x\sec \left( {\frac{y}{x}} \right) + c\)

B
\(y + {\cos ^{ - 1}}\left( {\frac{y}{x}} \right) = c\)

C
\(y = \tan \left( {\frac{{x + y}}{2}} \right) + c\)

D
\(y = \cot \left( {\frac{{x + y}}{2}} \right) + c\)

Question 25
5 / -1

Let \(\int {\frac{{\sqrt x }}{{\sqrt {1 - {x^3}} }}dx = \frac{2}{3}} \). gof(x) + c. Then

A
f(x) = √x

B
\(f\left( x \right) = {x^{\frac{3}{2}}}\)

C
g(x) = sin x

D
\(g\left( x \right) = \sqrt[3]{{\sin x}}\)

Question 26
5 / -1

Directions For Questions

Let L be a line that is perpendicular to the plane x - 3y + 2z = k and passes through the point (1, 2, -2) and let Pbe the plane passing through the intersection of planes P₁ : x + y + z - 6 = 0 and P₂ : 2x + 3y + 4z + 5 = 0

...view full instructions

The point on the line L that is nearest to the y-axis is -

A
(8/5, 1/5, -4/5)

B
(-8/5, 1/5, -4/5)

C
(8/5, -1/5, -4/5)

D
1(8/5, 1/5, 4/5)

Question 27
5 / -1

Directions For Questions

Let L be a line that is perpendicular to the plane x - 3y + 2z = k and passes through the point (1, 2, -2) and let Pbe the plane passing through the intersection of planes P₁ : x + y + z - 6 = 0 and P₂ : 2x + 3y + 4z + 5 = 0

...view full instructions

If the plane P contains origin then the equation of the plane P is -

A
17x - 23y - 29z = 0

B
17x + 23y + 29z = 0

C
17x - 23y + 29z = 0

D
17x + 23y - 29z = 0

Question 28
5 / -1

Directions For Questions

Let L be a line that is perpendicular to the plane x - 3y + 2z = k and passes through the point (1, 2, -2) and let Pbe the plane passing through the intersection of planes P₁ : x + y + z - 6 = 0 and P₂ : 2x + 3y + 4z + 5 = 0

...view full instructions

The reflection of the origin in the plane P₂ is -

A
\((-\frac{20}{29},-\frac{30}{29},-\frac{40}{29})\)

B
3\((\frac{20}{29},\frac{30}{29},\frac{40}{29})\)

C
\((-\frac{20}{29},\frac{30}{29},-\frac{40}{29})\)

D
\((\frac{20}{29},-\frac{30}{29},\frac{40}{29})\)

Question 29
5 / -1

Directions For Questions

Let L be a line that is perpendicular to the plane x - 3y + 2z = k and passes through the point (1, 2, -2) and let Pbe the plane passing through the intersection of planes P₁ : x + y + z - 6 = 0 and P₂ : 2x + 3y + 4z + 5 = 0

...view full instructions

The unit vector perpendicular to L and the normal to the plane P₁ is -

A
±  \(\frac{{( 5 \widehat i + \widehat j -4 \widehat k)}}{{\sqrt {42} }}\)

B
n ±  \(\frac{{( 5 \widehat i - \widehat j +4 \widehat k)}}{{\sqrt {42} }}\)

C
±  \(\frac{{( 5 \widehat i - \widehat j -4 \widehat k)}}{{\sqrt {42} }}\)

D
x ±  \(\frac{{( 5 \widehat i + \widehat j +4 \widehat k)}}{{\sqrt {42} }}\)

Question 30
5 / -1

If the probability of A to fail in an examination is 0.2 and that for B is 0.3, then, the probability that either A or B fails is:

A
0.5

B
0.38

C
0.8

D
0.25

Question 31
5 / -1

A line joining the points (1, 2, 0) and (4, 13, 5) is perpendicular to a plane. Then the coefficients of x, y and z in the equation of he plane are respectively

A
5, 15, 5

B
3, 11, 5

C
3, -11, 5

D
-5, -15, 5

Question 32
5 / -1

The area of the region bounded by Y-axis, y = cos x and y = sin x; \(\rm 0 \le x \le \dfrac{\pi}{4}\) is

A
√2 - 1 Sq. units

B
√2 Sq. units

C
√2 + 1 Sq. units

D
2 - √2 Sq. units

Question 33
5 / -1

Below is a graph showing the various inequality constraints, its corner points A, B, and C, and subsequently, the feasible region which is colored in red.
At what point, the objective function Z = x + y will achieve its maximum value?

A
A

B
B

C
All points on the line joining A and C

D
A and C

Question 34
5 / -1

The present value of a perpetuity is Rs 50,000 when the payment of Rs 1,000 is done at the end of each period. Find the present value of the perpetuity for the same amount of payment, interest rate, and time period, except this time, the payments are done at the beginning of each period

A
50,000

B
51,000

C
Information not sufficient

D
49,000

Question 35
5 / -1

The point that lies in the region bounded by the lines 7x + y ≥ 40 and 2x + 3y ≤ 25 is,

A
(0, 0)

B
(8, 5)

C
(3, 7)

D
(6, 4)

Question 36
5 / -1

If the co-ordinates of A and B be (1, 2, 3) and (7, 8, 7), then the projections of the line segment AB on the co-ordinate axes are

A
6, 6, 4

B
4, 6, 4

C
3, 3, 2

D
2, 3, 2

Question 37
5 / -1

A coin is tossed 3 times to record the outcomes. If X is a random variable representing number of tails in the experiment, then the mathematical expectation of X will be equal to,

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

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

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

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

Question 38
5 / -1

Nisha needs a total of Rs 4,00,000 for her daughter to provide her with a good education which includes all the college expenses. At the same time, the college announced that 50% of the funding from the state government will be provided for girls’ college expenses. In a timeline of 4 years, estimate the amount Nisha needs to save for her daughter’s education if the interest rate is 8%. Assume \(\frac{1.08^{4}-1}{0.08}\) = 4.5.

A
50,000 Rs

B
55,000 Rs

C
45,000 Rs

D
44,444 Rs

Question 39
5 / -1

An objective function Z = ax + by is maximum at points (9, 3) and (5, 7). If a, b ≥ 0 and ab = 16, then the maximum value of the function is given by,

A
48

B
78

C
108

D
148

Question 40
5 / -1

A battery manufacturing company produces ‘x’ units a year and the variable cost is V(x) = 3x² + 3500, where x is the number of batteries produced. The fixed cost paid on the warehouse facility is Rs 2000. So, the marginal cost of producing 50 battery units will be given by:

A
400

B
200

C
300

D
700

Question 41
5 / -1

A linear programming problem is as follows:
Maximize Z = 30x - 18y
subject to the constraints,
3x + 4y ≤ 60
5x - 3y ≥ 20
x, y ≥ 0
In the feasible region, the maximum value of Z occurs at

A
Two points

B
One point

C
Zero points

D
Infinite number of points

Question 42
5 / -1

Find the area between the curves y = 16x² and y - 9 = 0

A
5

B
6

C
7

D
9

Question 43
5 / -1

A solar panel receives annual average sunlight of 5 units/m². There was a sudden surge in the electricity production from the panels during the month of February. A study was undertaken to show that the increase in electricity production is caused due to the fact that the average sunlight received in February was greater than the annual average sunlight received. What will be the null hypothesis for this problem?

A
Average sunlight received in February > Annual average sunlight received

B
Average sunlight received in February = Annual average sunlight received

C
Average sunlight received in February < Annual average sunlight received

D
None of these

Question 44
5 / -1

The matrix A = \(\begin{bmatrix} 6 & -3 \\ 4 & -2 \end{bmatrix}\) and matrix B = \(\begin{bmatrix} 2 & y\\ x & 7 \end{bmatrix}\), if AB = 0 find the value of x and y

A
3, 3

B
4, 3.5

C
3, 1.5

D
4, 2

Question 45
5 / -1

A company produces and sells ‘x’ units of a product in a given time period. The general formula for marginal cost is given by 5x² + 20x + 7, while the marginal profit is given by 200 -4x. The condition where the company needs to produce to maximize the profit is given by:

A
Marginal Cost < Marginal Revenue

B
Marginal Cost = Marginal Revenue

C
Marginal Cost > Marginal Revenue

D
None of the above

Question 46
5 / -1

Directions For Questions

Study the pie chart given below and answer the question that follow. The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures
(A) Agriculture
(B) Industry
(C) Education
(D) Employment
(E) Miscellaneous
The total allocation is 36000 (in crore of rupees).

...view full instructions

Which head is allocated maximum funds?

A
Agriculture

B
Industry

C
Employment

D
Miscellaneous

Question 47
5 / -1

Directions For Questions

Study the pie chart given below and answer the question that follow. The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures
(A) Agriculture
(B) Industry
(C) Education
(D) Employment
(E) Miscellaneous
The total allocation is 36000 (in crore of rupees).

...view full instructions

How much money (in crore) is allocated to Education?

A
3000

B
6000

C
9000

D
10800

Question 48
5 / -1

Directions For Questions

Study the pie chart given below and answer the question that follow. The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures
(A) Agriculture
(B) Industry
(C) Education
(D) Employment
(E) Miscellaneous
The total allocation is 36000 (in crore of rupees).

...view full instructions

How much money (in crore) is allocated to both Agriculture and Employment?

A
20000

B
21000

C
24000

D
27000

Question 49
5 / -1

Directions For Questions

Study the pie chart given below and answer the question that follow. The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures
(A) Agriculture
(B) Industry
(C) Education
(D) Employment
(E) Miscellaneous
The total allocation is 36000 (in crore of rupees).

...view full instructions

How much excess money (in crore) is allocated to Miscellaneous over Education ?

A
3600

B
42000

C
4500

D
4800

Question 50
5 / -1

The three vertices of a parallelogram ABCD are A (1, 2, 3), B (-1, -2, -1) and C (2, 3, 2). The coordinates of the fourth vertex D are

A
(-2, -3, 0)

B
(4, 7, 6)

C
(2, 3, 4)

D
(-2, -3, 4)

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Mathematics Moc...

Question 1

x2

x

2x

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

Question 2

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

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

\(\frac{5\pi}{2}\)

\(\frac{\pi}{5}\)

Question 3

(a + b + c)

-(a + b + c)

abc

0

Question 4

0

2

1

\(\frac{1}{2}\)

Question 5

(0, a)

(2, 0)

(0, 2)

(a, a)

Question 6

1

2

0

3

Question 7

A - IV, B - II, C - I, D - III

A - II, B - I, C - III, D - IV

A - IV, B - I, C - II, D - III

A - II, B - I, C - IV, D - III

Question 8

\(\frac{{dy}}{{dx}} = 1\)

\(\frac{{{d^2}y}}{{d{x^2}}} = \frac{{8\sqrt 2 }}{{a\pi }}\)

\(\frac{{dy}}{{dx}} = - 1\)

\(\frac{{{d^2}y}}{{d{x^2}}} = \frac{{ - 8\sqrt 2 }}{{a\pi }}\)

Question 9

6

2

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

3

Question 10

Reflexive and symmetric but not transitive

Reflexive and transitive but not symmetric

Reflexive but neither symmetric nor transitive

Reflexive, symmetric and transitive

Question 11

Rs 2200

Rs 2250

Rs1500

Rs 1800

Question 12

\(\lambda=7,\mu=13\)

\(\lambda\ne7\)

\(\mu\ne13\)

\(\lambda=7,\mu\ne13\)

Question 13

5/4

1

-1

13

Question 14

The absolute maximum of the function is 15 in the interval [1,3].

The absolute minimum of the function is 11 in the interval [1,3].

The absolute maximum of the function is 11 in the interval [1,3].

The absolute minimum of the function is 15 in the interval [1,3].

Question 15

reflexive relation

symmetric relation

transitive relation

equivalence relation

Question 16

2π/3

2 π

x = 2π/3, 2 π both

x²

α ≠ - 3

α ≠ 3

\((\frac{20}{29},-\frac{30}{29},\frac{40}{29})\)