Mathematics Mock Test

TIME LEFT -

Question 1
5 / -1

A line makes an angle α, β, γ with the x, y, and z axes. Then sin² α + sin² β + sin² γ is

A
1

B
0

C
2

D
None of the above

Question 2
5 / -1

Order and degree of the differential equation
\(\rm {\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^3}} \right]^{\frac{7}{3}}} = 7\frac{{{d^2}y}}{{d{x^2}}}\) are respectively

A
2, 3

B
3, 2

C
7, 2

D
3, 7

Question 3
5 / -1

Evaluate \(\rm \int_{1}^{\infty} \frac{4}{x^4}dx\)

A
\(\frac 2 3\)

B
\(\frac 4 3\)

C
\(\frac 1 3\)

D
None of the above

Question 4
5 / -1

Suppose P(A) = 0.4, P(B) = P and P(A ∪ B) = 0.7. If A and B are independent events, then the value of P is:

A
0.5

B
0.3

C
0.55

D
0.6

Question 5
5 / -1

Let X be the set of all persons living in a city. Persons x, y in X are said to be related as x < y if y at least 5 years older than x. which one of the following is correct?

A
The relation is an equivalence relation on X.

B
The relation is transitive but neither reflexive nor symmetric.

C
The relation is reflexive but neither transitive nor symmetric.

D
The relation is symmetric but neither transitive nor reflexive.

Question 6
5 / -1

If f: R → R is a function such that, \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} {1,\;if\;x > 0}\\ {0,\;if\;x = 0}\\ { - \;1,\;if\;x < 0} \end{array}} \right.\) then f(x) is a

A
one-one and onto

B
one-one and into

C
many-one and onto

D
many-one and into

Question 7
5 / -1

Let f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| - x ∀ x ∈ R. The (fog) (x) for x < 0 is

A
0

B
4x

C
-4x

D
2x

Question 8
5 / -1

Find the value of \(\rm \displaystyle\int_0^{\pi/2} \dfrac{\sqrt{\sin^8 x}}{\sqrt{\sin^8 x}+ \sqrt{\cos^8 x}}dx\)

A
\(\dfrac{\pi}{4}\)

B
\(\dfrac{\pi}{2}\)

C
0

D
\(\dfrac{3\pi}{4}\)

Question 9
5 / -1

If * is a binary operation on A = {1, 2, 3, 4, 5} given by the table shown below:
*
1
2
3
4
5
1
1
1
1
1
1
2
1
2
1
2
1
3
1
1
3
1
1
4
1
2
1
4
1
5
1
1
1
1
5

Find the value of (3 * 3) * (4 * 4) ?

A
2

B
5

C
1

D
4

Question 10
5 / -1

What is the range of the function \(\rm f(x)=\dfrac{|x|}{x}, \ x \neq 0 \ ?\)

A
Set of all real numbers

B
Set of all integers

C
{-1, 1}

D
{-1, 0, 1}

Question 11
5 / -1

Given f : R → R, f(x) = sin(sinx) and g : R → R, g(x) = e^xthen derivative of gof(x) with respect to x will be -

A
e^sin(sinx)cos(sinx)cosx

B
esin(sinx)cos(sinx)

C
esin(sinx)cosx

D
e^cos(sinx)cos(sinx)cosx

Question 12
5 / -1

Let n >1 be fixed and a, b, c, d be arbitrary integers. If a ≡ b (mod n) and c ≡ d (mod n), then:

A
a + c \(\not\equiv\) b + d (mod n) and ac \(\not\equiv\) bd (mod n)

B
a + c \(\not≡\) b + d (mod n) but ac ≡ bd (mod n)

C
a + c ≡ b + d (mod n) and ac ≡ bd (mod n)

D
a + c ≡ b + d (mod n) but ac \(\not≡\) bd (mod n)

Question 13
5 / -1

The points A(a , b), B(b , a) and C(2a - 3b , 3b - a) will always be -

A
non-collinear

B
collinear

C
vertices of an equilateral triangle

D
vertices of a right angle triangle

Question 14
5 / -1

If A and B are two matrices such that AB = B and BA = A, then A² + B² is equal to

A
2AB

B
2BA

C
A + B

D
AB

Question 15
5 / -1

If the area of the triangle formed by vertices (-k , k), (1 , 0) and (5 , 0) is 8 square units, then k can be -

A
4

B
-4

C
(i) and (ii) both

D
none of these

Question 16
5 / -1

The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k² has no solution if k equals

A
0

B
1

C
-1

D
-2

Question 17
5 / -1

Consider the given statements -
(i) Simple average of price relative is a weighted index numbering method.
(ii) The index number for base year is always 100.
(iii) If the price index is 25 then the prices have increased by 25%.
(iv) If the price index is 75 then the prices have decreased by 25%

A
(i),(ii) and (iv)

B
(ii) only

C
(ii) and (iii)

D
(ii) and (iv)

Question 18
5 / -1

A manufacturing firm sets the price per unit as p(x) = 400 - αx² where α is a constant and 'x' is the units demanded. The marginal revenue when 10 units were in demand was zero then α will be -

A
16/9

B
4/3

C
25/16

D
5/4

Question 19
5 / -1

The index number of the data given in the below table is to be calculated by simple average of price relative method taking 2010 as the base year.
Item Price
Year 2010 Year 2015
A 100 140
B 80 120
C 160 200
D 220 385
E 112 112
Then the index number will be -

A
126

B
144

C
138

D
136

Question 20
5 / -1

What is \(\rm {\sin ^{ - 1}}(sin \frac{{3\pi }}{5})\) equal to?

A
\(\frac{{3\pi }}{5}\)

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

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

D
None of the above

Question 21
5 / -1

A toy manufacturing firm has its variable cost C(x) = α²x(β√x + αβ) where 'x' is the number of toys produced. The cost of storage and other expenses are Rs. 4500. If the marginal cost is 40 when 144 toys have been produced and αβ = 1, then the possible values of α?

A
2

B
- 20

C
20

D
(i) and (ii) both

Question 22
5 / -1

The function f(x) = tan^-1x - x is monotonically decreasing in the set

A
R

B
(0, ∞)

C
R - {0}

D
None of the these

Question 23
5 / -1

Find the rate of change of volume of the cube when the side of the cube is 10 cm. It is known that the side changes at the rate of 4 cm/s.

A
800 cm³/s

B
1000 cm3/s

C
1200 cm3/s

D
500 cm3/s

Question 24
5 / -1

If y + sin^-1 (1 - x²) = e^x, then \(\rm {dy\over dx}\)

A
\(\rm e^x - {2\over\sqrt{2-x^2}}\)

B
\(\rm e^x + {2\over\sqrt{2-x^2}}\)

C
\(\rm e^x - {1\over\sqrt{2+x^2}}\)

D
\(\rm e^x + {1\over\sqrt{2-x^2}}\)

Question 25
5 / -1

Let f(x) = | x2 - 3x -4 | defined for -1 ≤ x ≤ 4. Then which of the following is correct?

A
The maximum value of f(x) will be 6.25

B
No critical points for this function in -1 ≤ x ≤ 4.

C
The minimum value of f(x) will be 6.25

D
None

Question 26
5 / -1

The function f(x) = sin⁴x + cos⁴ x increases monotonically if

A
\(0 < x < \frac{\pi }{8}\)

B
\(\frac{\pi }{4} < x < \frac{{\pi }}{2}\)

C
\(\frac{{3\pi }}{4} < x < \frac{{5\pi }}{8}\)

D
\(\frac{{5\pi }}{8} < x < \frac{{3\pi }}{4}\)

Question 27
5 / -1

The value of 'c' in Rolle's Theorem for the function f(x) = \(\rm cos \frac{x}{2}\) on [π, 3π]:

A
0

B
2π

C
\(\rm \dfrac{\pi}{2}\)

D
\(\rm \dfrac{3\pi}{2}\)

Question 28
5 / -1

What should be the value of k such that the function \(\rm f(x)=\left\{\begin{matrix} \frac{ksin(π -x)}{π -x} & if & x \neq π \\ 1 & if & x =π \end{matrix}\right.\) is continuous at x = π.

A
π

B
1

C
-1

D
0

Question 29
5 / -1

Let f be a differentiable function defined for all x ∈ R such that f(x³) = x⁵ for all x ∈ R, x ≠ 0. Then the value of \(\dfrac{df}{dx} (8)\) is:

A
5/3

B
None of these

C
20

D
20/3

Question 30
5 / -1

Suppose the function f(x) = xⁿ, n ≠ 0 is differentiable for all x. Then n can be any element of the interval

A
[1, ∞)

B
(0, ∞)

C
\(\left( {\frac{1}{2},{\rm{\;}}\infty } \right)\)

D
None of the above

Question 31
5 / -1

What is the slope of the tangent to the curve x = t² + 3t - 8, y = 2t² - 2t - 5 at t = 2?

A
7/6

B
6/7

C
1

D
5/6

Question 32
5 / -1

If \(f'\left( x \right) = \frac{{{x^2}}}{2} - kx + 1\), such that f(0) = 0 and f(3) = 15. Find the value of k.

A
5 / 3

B
3 / 5

C
– 5 / 3

D
– 3 / 5

Question 33
5 / -1

For the function f(x) = \(\rm x+\frac1x\), x ∈ [1, 3], the value of c for mean value theorem is:

A
1

B
√3

C
2

D
None of these.

Question 34
5 / -1

What is the integral of f(x) = 1 + x² + x⁴ with respect to x²?

A
\(\rm x + \frac{x^3}{3}+\frac{x^5}{5}+C\)

B
\(\rm \frac{x^3}{3}+\frac{x^5}{5}+C\)

C
\(\rm x^2 + \frac{x^4}{4}+\frac{x^6}{6}+C\)

D
\(\rm x^2 + \frac{x^4}{2}+\frac{x^6}{3}+C\)

Question 35
5 / -1

The area cut off the parabola 4y = 3x² by the straight line 2y = 3x + 12 is

A
15 sq. units

B
18 sq. units

C
24 sq. units

D
27 sq. units

Question 36
5 / -1

If the angle between \(\rm \vec{a} \ and \ \vec{b} \ is \ \dfrac{2\pi}{3}\) and the projection of \(\vec{a}\) in the direction of \(\vec{b}\) is -2, then \(|\vec{a}|=\)

A
4

B
3

C
2

D
1

Question 37
5 / -1

Let L₁ and L₂ be two parallel lines with the equations \(\rm \vec{r}=\vec{a}_1 +\lambda \vec{b}\) and \(\rm r=\vec{a}_2 + \mu\vec{b}\) respectively. The shortest distance between them is:

A
\(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)

B
\(\rm d=\left|\dfrac{\vec{b}\cdot (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)

C
\(\rm d=\left|\dfrac{\vec{a}_1\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)

D
\(\rm d=\left|\dfrac{\vec{a}_2\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)

Question 38
5 / -1

What is the perpendicular distance from the point (2, 3, 4) to the line \(\rm \frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0} \ ?\)

A
6 units

B
5 units

C
3 units

D
2 units

Question 39
5 / -1

Find the angle between the line \(\vec r = \left( {\hat i + 2\hat j - \;\hat k} \right) + \lambda \;\left( {\hat i - \;\hat j + \;\hat k} \right)\) and the plane \(\vec r \cdot \left( {2\hat i - \;\hat j + \;\hat k} \right) = 6\) ?

A
\({\sin ^{ - 1}}\left( {\frac{{2\sqrt 2 }}{3}} \right)\)

B
\({\sin ^{ - 1}}\left( {\frac{{\sqrt 2 }}{3}} \right)\)

C
\({\sin ^{ - 1}}\left( {\frac{{2}}{3}} \right)\)

D
None of these

Question 40
5 / -1

Consider the given problem:
5x + y ≤ 100 ... (1)
x + y ≤ 60 ... (2)
x ≥ 0 ... (3)
y ≥ 0 ... (4)
If we solve the above linear equations by the graphical method of Linear Programming, then the following point ____ will not form the the boundary of the feasible region.

A
(60, 0)

B
(20, 0)

C
(0, 60)

D
(10, 50)

Question 41
5 / -1

If four dice are thrown together, then what is the probability that the sum of the numbers appearing on them is 25?

A
0

B
1/2

C
1

D
1/1296

Question 42
5 / -1

A die is thrown twice and the sum of the dots on the two faces appeared, is observed to be 7. The conditional probability that number 2 has appeared at least once is

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

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

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

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

Question 43
5 / -1

Which one of them is not a property of Linear programming problem (LPP)?

A
Proportionality

B
Additivity

C
Negativity

D
Certainty

Question 44
5 / -1

The chances of a defective screw in three boxes A, B, C are \(\frac{1}{5},{\rm{\;}}\frac{1}{6}\) and \(\frac{1}{7}\) respectively. A box is selected at random to be defective. Find the probability that it came from box A.

A
\(\frac{{40}}{{107}}\)

B
\(\frac{{41}}{{107}}\)

C
\(\frac{{42}}{{107}}\)

D
\(\frac{{43}}{{107}}\)

Question 45
5 / -1

A medicine is known to be 75% effective to cure a patient. If the medicine is given to 5 patients, what is the probability that at least one patient is cured by this medicine?

A
\(\frac{1}{{1024}}\)

B
\(\frac{{243}}{{1024}}\)

C
\(\frac{{1023}}{{1024}}\)

D
\(\frac{{781}}{{1024}}\)

Question 46
5 / -1

Given that x ~ B ( n = 10, p) if E(x) = 8 find the value of P is

A
0.6

B
0.7

C
0.8

D
0.4

Question 47
5 / -1

Let Z denote the number of hours you study on a monday. Also it is known that
P(Z = z) = \(\rm \left\{\begin{matrix} 0.4 & if z = 0\\ kz & if z = 1 \: or \: 2\\ 0 & otherwise \end{matrix}\right. \)
where k is constant.
What is the probability that you study atleast two hours?

A
0.5

B
0.1

C
0.4

D
0.3

Question 48
5 / -1

If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6, then P(A ∪ B) is equal to:

A
0.24

B
0.3

C
0.48

D
0.96

Question 49
5 / -1

If A = \(\left( {\begin{array}{*{20}{c}} 2&2\\ 9&4 \end{array}} \right)\)and I = \(\left( {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right)\), then 10A^-1 is equal to

A
4I - A

B
A - 6I

C
6I - A

D
A - 4I

Question 50
5 / -1

The differential equation representing the family of curves y = a sin (λx + α) is:

A
\(\rm \frac {d^2y}{dx^2} + \lambda^2y = 0\)

B
\(\rm \frac {d^2y}{dx^2} - \lambda^2y = 0\)

C
\(\rm \frac {d^2y}{dx^2} + \lambda y = 0\)

D
None of the above

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Question Analysis

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Unanswered - 50

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Item	Price
Item	Year 2010	Year 2015
A	100	140
B	80	120
C	160	200
D	220	385
E	112	112

Mathematics Moc...

Question 1

1

0

2

None of the above

Question 2

2, 3

3, 2

7, 2

3, 7

Question 3

\(\frac 2 3\)

\(\frac 4 3\)

\(​​\frac 1 3\)

None of the above

Question 4

0.5

0.3

0.55

0.6

Question 5

The relation is an equivalence relation on X.

The relation is transitive but neither reflexive nor symmetric.

The relation is reflexive but neither transitive nor symmetric.

The relation is symmetric but neither transitive nor reflexive.

Question 6

one-one and onto

one-one and into

many-one and onto

many-one and into

Question 7

0

4x

-4x

2x

Question 8

\(\dfrac{\pi}{4}\)

\(\dfrac{\pi}{2}\)

0

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

Question 9

2

5

1

4

Question 10

Set of all real numbers

Set of all integers

{-1, 1}

{-1, 0, 1}

Question 11

esin(sinx)cos(sinx)cosx

esin(sinx)cos(sinx)

esin(sinx)cosx

ecos(sinx)cos(sinx)cosx

Question 12

a + c \(\not\equiv\) b + d (mod n) and ac \(\not\equiv\) bd (mod n)

a + c \(\not≡\) b + d (mod n) but ac ≡ bd (mod n)

a + c ≡ b + d (mod n) and ac ≡ bd (mod n)

a + c ≡ b + d (mod n) but ac \(\not≡\) bd (mod n)

Question 13

non-collinear

collinear

vertices of an equilateral triangle

vertices of a right angle triangle

Question 14

2AB

2BA

A + B

AB

Question 15

4

-4

(i) and (ii) both

none of these

Question 16

0

1

-1

\(\frac 1 3\)

e^sin(sinx)cos(sinx)cosx

e^cos(sinx)cos(sinx)cosx

800 cm³/s