Mathematics Mock Test

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Question 1
5 / -1

Which of the following is true?

A
Two vectors having same magnitude are collinear

B
Two collinear vectors are always equal in magnitude

C
Two collinear vectors having the same magnitude are equal

D
$\vec{a}$ and $- \vec{a}$ are collinear

Question 2
5 / -1

Evaluate
$\int_{1}^{e} \sqrt{x} \ln (x) d x$

A
$\frac{4}{9} \sqrt{e^{3}} + \frac{2}{9}$

B
$\frac{2}{9} \sqrt{e^{3}} - \frac{4}{9}$

C
$\frac{2}{9} \sqrt{e^{3}} + \frac{4}{9}$

D
$\frac{4}{9} \sqrt{e^{3}} - \frac{2}{9}$

Question 3
5 / -1

If $f (x) = [\begin{matrix} x & λ \\ 2 λ & x \end{matrix}]$ then |f(λx) - f(x)| is equal to ?

A
$x (λ^{2} - 1)$

B
$λ^{^{2}} (x^{2} - 1)$

C
$λ (x^{2} - 1)$

D
$x^{2} (λ - 1)^{2}$

Question 4
5 / -1

Determine $| \vec{a} | a n d | \vec{b} |, i f (\vec{a} + \vec{b}) . (\vec{a} - \vec{b}) = 8 a n d | \vec{a} | = 8 | \vec{b} |$ ?

A
$\frac{16 \sqrt{2}}{3 \sqrt{7}}, \frac{2 \sqrt{2}}{3 \sqrt{7}}$

B
$\frac{8 \sqrt{2}}{3 \sqrt{7}}, \frac{2 \sqrt{2}}{3 \sqrt{7}}$

C
$\frac{16 \sqrt{6}}{3 \sqrt{7}}, \frac{2 \sqrt{2}}{3 \sqrt{7}}$

D
$\frac{8 \sqrt{3}}{3 \sqrt{7}}, \frac{2 \sqrt{3}}{3 \sqrt{7}}$

Question 5
5 / -1

if $\cos^{- 1} \frac{3}{5} - \sin^{- 1} \frac{4}{5} = \cos^{- 1} x$ , then x =

A
0

B
1

C
-1

D
2

Question 6
5 / -1

The minors of -4 and 9 and the co-factors of - 4 and 9 in determinant $| \begin{matrix} - 1 & - 2 & 3 \\ - 4 & - 5 & - 6 \\ - 7 & - 8 & 9 \end{matrix} |$ are respectively

A
42, 3; -42, 3

B
6, -3, ; -6, -3

C
42 3; -42, -3

D
42, 3; 42, 3

Question 7
5 / -1

If n (U) = 1000, n (X) = 300, n (Y) = 400 and n (X ∩ Y) = 200 then find n (P (X‘∩ Y’))

A
2²⁰⁰

B
2³⁰⁰

C
2⁴⁰⁰

D
2⁵⁰⁰

Question 8
5 / -1

The real value of k for which the system of equations 2kx - 2y + 3z = 0, x + ky + 2z = 0, 2x + kz = 0, has non trivial solution is

A
2

B
-2

C
3

D
-3

Question 9
5 / -1

The values of x for which the given matrix $[\begin{matrix} - x & x & 2 \\ 2 & x & - x \\ x & - 2 & - x \end{matrix}]$ will be non-singular are

A
-2 ≤ x ≤ 2

B
For all x other than 2 and -2

C
x ≥ 2

D
x ≤ -2

Question 10
5 / -1

If f (1) = 1, f' (1) = 3, then the derivative of f(f(f(x))) + (f(x))² at x = 1 is

A
12

B
33

C
9

D
15

Question 11
5 / -1

If $\int_{- 3}^{2} f (x) d x = \frac{7}{3} a n d \int_{- 3}^{9} f (x) d x = - \frac{5}{6}$ , then what is the value of $\int_{2}^{9} f (x) d x ?$

A
$- \frac{19}{6}$

B
$\frac{19}{6}$

C
$\frac{3}{2}$

D
$- \frac{3}{2}$

Question 12
5 / -1

Determine the vector equation for the line, given the cartesian equation of a line is $\frac{x + 5}{3} = \frac{y - 7}{2} = \frac{z + 3}{2}$ ?

A
$\vec{r} = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})$

B
$\vec{r} = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})$

C
$\vec{r} = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})$

D
$\vec{r} = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})$

Question 13
5 / -1

Determine the vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot (\hat{ı} + \hat{ȷ} + \hat{k}) = 6$ and $\vec{r} . (2 \hat{ı} + 3 \hat{ȷ} + 4 \hat{k}) = - 5$ , and the point (1, 1, 1)?

A
$\vec{r} \cdot (10 \hat{ı} + 13 \hat{ȷ} + 23 \hat{k}) = 69$

B
$\vec{r} \cdot (20 \hat{ı} + 23 \hat{ȷ} + 26 \hat{k}) = 69$

C
$\vec{r} \cdot (30 \hat{ı} + 23 \hat{ȷ} + 13 \hat{k}) = 69$

D
$\vec{r} \cdot (10 \hat{ı} + 23 \hat{ȷ} + 13 \hat{k}) = 69$

Question 14
5 / -1

Determine the direction cosines of the unit vector perpendicular to the plane $\vec{r} \cdot (2 \hat{ı} - 6 \hat{ȷ} - 3 \hat{k}) + 1 = 0$ passing through the origin?

A
- 6/7, 3/7, 2/7

B
- 2/7, 6/7, 3/7

C
- 2/7, 3/7, 6/7

D
- 2/7, 3/7, - 6/7

Question 15
5 / -1

If the plane 2x - y + z = 0 is parallel to the line $\frac{2 x - 1}{2} = \frac{2 - y}{2} = \frac{z + 1}{a}$ , then value of a is

A
4

B
-4

C
2

D
-2

Question 16
5 / -1

If from a point P(a, b, c) perpendiculars PA and PB are drawn to yz and zx planes, then the equation of the plane OAB is

A
bcx + cay + abz = 0

B
bcx + cay - abz = 0

C
bcx - cay + abz = 0

D
-bcx + cay + abz = 0

Question 17
5 / -1

The ratio in which the plane x - 2y + 3z = 17 divides the line joining the points (-2, 4, 7) and (3, -5, 8) is

A
10 : 3

B
3 : 1

C
3 : 10

D
10 : 1

Question 18
5 / -1

If 73 ≡ x mod 13, what is the value of x if 50 < x < 65?

A
60

B
55

C
67

D
54

Question 19
5 / -1

Solve the following Linear Programming Problem Graphically.
Maximize Z = 5x + 8y,
Subject to constraints
2x + 3y ≥ 10
5x + 2y ≥ 14
x, y ≥ 0
For the LPP,

A
feasible region and the solution are bounded

B
feasible region and the solution are unbounded

C
feasible region is unbounded and solution is bounded

D
feasible region is bounded and solution is unbounded

Question 20
5 / -1

A die is thrown 3 times. Events A and B stated below:
4 on the 3rd throw
6 on the 1st and 5 on the 2nd throw
What will be the probability of A knowing that B has already taken place?

A
$\frac{1}{108}$

B
$\frac{1}{36}$

C
$\frac{1}{6}$

D
$\frac{1}{216}$

Question 21
5 / -1

If $g (x) = x \times [\frac{1}{x}]$ , where [.] is the greatest integer function. Then find the value of $\int_{\frac{1}{3}}^{1} g (x) d x$

A
29 / 72

B
17 / 72

C
23/ 72

D
37 / 72

Question 22
5 / -1

The marginal cost is very less than as compared to the marginal revenue for a product. The company selling the product should

A
stop its production immediately

B
sell more products to maximize the profits until the marginal cost becomes greater than marginal revenue

C
sell the product at an increased cost price

D
None of the above

Question 23
5 / -1

If an event E has only one sample point of a sample space, it is called a ______

A
Compound events

B
Sure events

C
Impossible events

D
Simple events

Question 24
5 / -1

A player tosses three coin. If the first toss is head. Find the probability that consecutively two head on three coin.

A
1/3

B
1/2

C
1/8

D
1/4

Question 25
5 / -1

A 10 km race is organized at 800 m circular racecourse. P and Q are the contestants of this race. If the ratio of the speeds of P and Q is 5 : 4, the difference between their speed is 1 m/sec, how many times will the winner overtake the loser?

A
4 times

B
1 time

C
2 times

D
3 times

Question 26
5 / -1

In (- 4, 4) the function $f (x) = \int_{- 10}^{x} {(t^{4} - 4) e}^{- 4 t} d t$ has

A
No extrema

B
One extremum

C
Two extrema

D
Four extrema

Question 27
5 / -1

A family has two children if the first child is a boy then find prob that other also a boy?

A
1/2

B
1/3

C
1/4

D
None

Question 28
5 / -1

Let P(h, k) be a point on the curve y = x²+ 7x +2, nearest to the line y = 3x - 3. Then the equation of the normal to the curve at P is

A
x + 3y - 62 = 0

B
x + 3y + 26 = 0

C
x - 3y - 11 = 0

D
x - 3y + 22 = 0

Question 29
5 / -1

Which of the following method is used to analyze the simple trend of the time series data?

A
Moving average method

B
Method of least square

C
Only 1

D
Both1 and 2

Question 30
5 / -1

If sin^-1 a + sin ^-1 b + sin^-1 c = π, then the value of a $a \sqrt{(1 - a^{2})} + b \sqrt{(1 - b^{2})} + c \sqrt{(1 - c^{2})}$ will be

A
2abc

B
abc

C
$\frac{1}{2} a b c$

D
$\frac{1}{3} a b c$

Question 31
5 / -1

A water filling Pipe P is 5 times fast as second Pipe Q. If P fills a cistern in 30 minutes. How long will it take to fill the tank, when both the pipes are kept in operation simultaneous?

A
27 min

B
26 min

C
28 min

D
25 min

Question 32
5 / -1

The number of vectors of unit length prependicular to vectors a = (1, 1.0) and b = (0, 1.1) is

A
Three

B
One

C
Two

D
Infinite

Question 33
5 / -1

If $A = [\begin{matrix} 1 & - 2 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}]$ , then A. adj (A) is equal to

A
$[\begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix}]$

B
$[\begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}]$

C
$[\begin{matrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{matrix}]$

D
$[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

Question 34
5 / -1

Thea area bounded by curve y = |x| - 1 and y = -|x| + 1 is:

A
4

B
$\sqrt{2}$

C
2

D
$2 \sqrt{2}$

Question 35
5 / -1

The real values of x satisfying the inequalities:
$\log_{0.5} (x^{2} - 25) - \log_{0.5} (x - 5) < \log_{0.5} 3$

A
x < 2

B
x > 5

C
x > 1

D
x < 5

Question 36
5 / -1

In what ratio must tea at Rs. 50 per kg be mixed with tea at Rs. 75 per kg so the mixture must be worth Rs. 60 per kg?

A
3 : 1

B
3 : 2

C
4 : 3

D
5 : 3

Question 37
5 / -1

If X & Y two independent Binomial Random variables such that X ∼ B (2, p) Y ∼ B (4, p) if p (x ≥ 1) = 5/9
i. p (y = 4)
ii. V(2x - y) respectively,

A
$\frac{1}{27}, \frac{24}{9}$

B
$\frac{1}{81}, \frac{24}{9}$

C
$\frac{1}{81}, \frac{25}{9}$

D
None

Question 38
5 / -1

Rajesh comes across a person who tells him about a bond having Rs 3,500 as its par value, and redeemable at 7% at the end of 8 years. What will be the present value of the bond if the annual yield rate is 8%? Take (1.08)-8 = 0.54

A
3,500

B
3310.23

C
3625.97

D
3298.75

Question 39
5 / -1

If $y = 1 + \frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}} + . . .$ to ∞ with |x| > 1 then $\frac{d y}{d x} =$

A
$\frac{x^{2}}{y^{2}}$

B
x²y²

C
$\frac{y^{2}}{x^{2}}$

D
$\frac{- y^{2}}{x^{2}}$

Question 40
5 / -1

Shyam wants to have a minimum of 150 units of carbohydrates and 120 units of proteins. Product A provides 16 units of carbohydrates and 11 units of proteins, while product B provides 10 units of carbohydrates and 20 units of proteins. The cost of product A is Rs 50 per unit while the cost of product B is Rs 60 per unit. Formulate this given situation in a linear programming problem. Consider x and y as units of products A and B consumed respectively.

A
Z = 10x + 20y, 16x + 150y ≥ 10, 11x + 120y ≥ 20

B
Z = 50x + 60y, 16x + 10y ≥ 150, 11x + 20y ≥ 120

C
Z = 16x + 11y, 150x + 10y ≥ 16, 120x + 20y ≥ 11

D
Z = 150x + 120y, 16x + 10y ≥ 150, 11x + 20y ≥ 120

Question 41
5 / -1

The solution of $\frac{d y}{d x} = \frac{x + y}{x - y}$ is

A
$\log x = t a n^{- 1} \frac{y}{x} - \frac{1}{2} l o g (1 + (\frac{y}{x})^{2}) + c$

B
$\log x = t a n \frac{y}{x} + \frac{1}{2} l o g (1 + (\frac{y}{x})^{2}) + c$

C
$\log x = t a n^{- 1} \frac{y}{x} + \frac{1}{2} l o g (1 - (\frac{y}{x})^{2}) + c$

D
$\log x = t a n \frac{x}{y} + \frac{1}{2} l o g (1 + (\frac{x}{y})^{2}) + c$

Question 42
5 / -1

The minimum value of $e^{(2 x^{2} - 2 x + 1) s i n^{2} x}$ is

A
e

B
1\e

C
1

D
0

Question 43
5 / -1

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f'(2) = 5, f'(x) ≥ 1 and f''(x) ≥ 4, for all x ∈ (1, 6) then

A
f(5) ≤ 10

B
f'(5) + f''(5) ≤ 20

C
f(5) + f'(5) ≥ 28

D
f(5) + f' (5) ≤ 26

Question 44
5 / -1

The angle between the curves y = sin x and y = cos x is

A
tan^-1(2√2)

B
tan^-1(3√ 2)

C
tan^-1(3√ 3)

D
tan^-1(5√ 2)

Question 45
5 / -1

If $\cos x = \frac{1}{\sqrt{1 + t^{2}}}$ and $\sin y = \frac{t}{\sqrt{1 + t^{2}}}$ , then $\frac{d y}{d x} =$

A
- 1

B
$\frac{1 - t}{1 + t^{2}}$

C
$\frac{1}{1 + t^{2}}$

D
1

Question 46
5 / -1

Let $f (x) = \sqrt{x - 1} + \sqrt{x + 24 - 10 \sqrt{x - 1;}} 1 < x < 26$ be real valued function, them f'(x) for 1 < x < 26 is,

A
0

B
-2

C
1

D
None of these

Question 47
5 / -1

If X = $[\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ , then the value of Xⁿ is

A
$[\begin{matrix} 3 n & - 4 n \\ n & - n \end{matrix}]$

B
$[\begin{matrix} 2 + n & 5 - n \\ n & - n \end{matrix}]$

C
$[\begin{matrix} 3^{n} & {(- 4)}^{n} \\ 1^{n} & {(- 1)}^{n} \end{matrix}]$

D
None of these

Question 48
5 / -1

The function $y = \frac{1}{1 + x^{2}}$ is decreasing in the interval

A
(-∞, -1]

B
(-∞, 0)

C
[1, ∞)

D
(0, ∞)

Question 49
5 / -1

If $D_{p} = | \begin{matrix} p & 15 & 8 \\ p^{2} & 35 & 9 \\ p^{3} & 25 & 10 \end{matrix} |, t h e n$ D₁ + D₂ + D₃ + D₄ + D₅ =

A
0

B
25

C
625

D
None of these

Question 50
5 / -1

The differential form of the equation (y - a)² = 2bx is:

A
2x $\frac{d^{2} y}{d x^{2}}$ + $\frac{d y}{d x}$ = 0

B
2 $\frac{d^{2} y}{d x^{2}}$ + x $\frac{d y}{d x}$ = 0

C
$\frac{d^{2} y}{d x^{2}}$ + 2x $\frac{d y}{d x}$ = 0

D
x $\frac{d^{2} y}{d x^{2}}$ + $\frac{d y}{d x}$ = 0

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Mathematics Moc...

Question 1

Two vectors having same magnitude are collinear

Two collinear vectors are always equal in magnitude

Two collinear vectors having the same magnitude are equal

a→ and −a→ are collinear

Question 2

49e3+29

29e3−49

29e3+49

49e3−29

Question 3

x(λ2−1)

λ2(x2−1)​

λ(x2−1)

x2(λ−1)2

Question 4

16237,2237

8237,2237

16637,2237

8337,2337

Question 5

0

1

-1

2

Question 6

42, 3; -42, 3

6, -3, ; -6, -3

42 3; -42, -3

42, 3; 42, 3

Question 7

2200

2300

2400

2500

Question 8

2

-2

3

-3

Question 9

-2 ≤ x ≤ 2

For all x other than 2 and -2

x ≥ 2

x ≤ -2

Question 10

12

33

9

15

Question 11

−196

196

32

−32

Question 12

r→=(−3i^+5j^−6k^)+λ(2i^+4j^+2k^)

r→=(−5i^+7j^−3k^)+λ(2i^+4j^+2k^)

r→=(−5i^+7j^−3k^)+λ(3i^+2j^+2k^)

r→=(−3i^+5j^−6k^)+λ(3i^+2j^+2k^)

Question 13

r→⋅(10ı^+13ȷ^+23k^)=69

r→⋅(20ı^+23ȷ^+26k^)=69

r→⋅(30ı^+23ȷ^+13k^)=69

r→⋅(10ı^+23ȷ^+13k^)=69

Question 14

- 6/7, 3/7, 2/7

- 2/7, 6/7, 3/7

- 2/7, 3/7, 6/7

- 2/7, 3/7, - 6/7

Question 15

4

-4

2

-2

Question 16

bcx + cay + abz = 0

bcx + cay - abz = 0

bcx - cay + abz = 0

$\vec{a}$ and $- \vec{a}$ are collinear

$\frac{4}{9} \sqrt{e^{3}} + \frac{2}{9}$

$\frac{2}{9} \sqrt{e^{3}} - \frac{4}{9}$

$\frac{2}{9} \sqrt{e^{3}} + \frac{4}{9}$

$\frac{4}{9} \sqrt{e^{3}} - \frac{2}{9}$

$x (λ^{2} - 1)$

$λ^{^{2}} (x^{2} - 1)$

$λ (x^{2} - 1)$

$x^{2} (λ - 1)^{2}$

$\frac{16 \sqrt{2}}{3 \sqrt{7}}, \frac{2 \sqrt{2}}{3 \sqrt{7}}$

$\frac{8 \sqrt{2}}{3 \sqrt{7}}, \frac{2 \sqrt{2}}{3 \sqrt{7}}$

$\frac{16 \sqrt{6}}{3 \sqrt{7}}, \frac{2 \sqrt{2}}{3 \sqrt{7}}$

$\frac{8 \sqrt{3}}{3 \sqrt{7}}, \frac{2 \sqrt{3}}{3 \sqrt{7}}$

2²⁰⁰

2³⁰⁰

2⁴⁰⁰

2⁵⁰⁰

$- \frac{19}{6}$

$\frac{19}{6}$

$\frac{3}{2}$

$- \frac{3}{2}$

$\vec{r} = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})$

$\vec{r} = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})$

$\vec{r} = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})$

$\vec{r} = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})$

$\vec{r} \cdot (10 \hat{ı} + 13 \hat{ȷ} + 23 \hat{k}) = 69$

$\vec{r} \cdot (20 \hat{ı} + 23 \hat{ȷ} + 26 \hat{k}) = 69$

$\vec{r} \cdot (30 \hat{ı} + 23 \hat{ȷ} + 13 \hat{k}) = 69$

$\vec{r} \cdot (10 \hat{ı} + 23 \hat{ȷ} + 13 \hat{k}) = 69$

$\frac{1}{108}$

$\frac{1}{36}$

$\frac{1}{6}$

$\frac{1}{216}$

$\frac{1}{2} a b c$