Mathematics Test

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Mathematics Test - 1

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Weekly Quiz Competition

Question 1
1 / -0

If log 2 = 0.3010 and log 3 = 0.4771, the value of log₅ 512 is:

A
2.870

B
2.967

C
3.876

D
3.912

Solution

$\begin{array}{l} \log_{5} 512 = \frac{\log 512}{\log 5} \end{array} = \frac{\log 2^{9}}{\log 10 / 2} = \frac{9 \log 2}{\log 10 - \log 2} = \frac{(9 x 0.3010)}{1 - 0.3010} = \frac{2.709}{0.699} = \frac{2709}{0699} = 3.876$
Question 2
1 / -0

If $A = [\begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}]$ and $B = [\begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}]$ then evaluate $A B$ .

A
$[\begin{matrix} 5 & 5 \\ 24 & 25 \end{matrix}]$

B
$[\begin{matrix} 5 & - 5 \\ 24 & 25 \end{matrix}]$

C
$[\begin{matrix} - 5 & 5 \\ - 24 & 25 \end{matrix}]$

D
None of these

Solution

Given-
$A = [\begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}]$
$B = [\begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}]$
$A B = [\begin{matrix} 1 & 1 \\ 4 & 5 \end{matrix}] [\begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}]$
$A B = [\begin{matrix} 1 + 4 & 0 + 5 \\ 4 + 20 & 0 + 25 \end{matrix}]$
$A B = [\begin{matrix} 5 & 5 \\ 24 & 25 \end{matrix}]$
Question 3
1 / -0

What is sin 105° + cos 105° equal to?

A
sin 50°

B
cos 50°

C
$\frac{1}{\sqrt{2}}$

D
0

Solution

$\sin 105^{\circ}=\sin \left(60^{\circ}+45^{\circ}\right)=\sin 60^{\circ} \cos 45^{\circ}+\cos 60^{\circ} \sin 45^{\circ}=\frac{\sqrt{3}+1}{2 \sqrt{2}}$
$\cos 105^{\circ}=\cos \left(60^{\circ}+45^{\circ}\right)=\cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}=\frac{1-\sqrt{3}}{2 \sqrt{2}}$
$\sin 105^{\circ}+\cos 105^{\circ}=\frac{1+\sqrt{3}}{2 \sqrt{2}}+\frac{1-\sqrt{3}}{2 \sqrt{2}}=\frac{1}{\sqrt{2}}$
Question 4
1 / -0

What is the sum of all two-digit numbers which when divided by 3 leave 2 as the remainder?

A
1565

B
1585

C
1635

D
1655

Solution

11,14,17,20, ............98.
29= n-1
$\begin{array}{lrlrlrlrlrl} a_{n} = a + (n - 1) d \\ 98 = 11 + (n - 1) \times 3 \\ 87 = (n - 1) \times 3 \\ 29 = n - 1 \\ s_{n} = \frac{n}{2} [2 a + (n - 1) d] \\ = \frac{30}{2} [2 \times 11 + (29) \times 3] \\ = \frac{30}{2} [22 + 87] \end{array}$
=15[109]
=1635
Question 5
1 / -0

The value of $x + x (x^{x})$ when x = 2 is:

A
10

B
16

C
18

D
36

Solution

Put the value of x = 2 in the above expression we get,
$= 2 + 2 (2^{2})$ $= 2 + 2 (2 \times 2)$ $= 2 + 2 (4)$ $= 2 + 8 = 10$
Question 6
1 / -0

What is the greatest integer among the following by which the number 5⁵ + 7⁵ is divisible?

A
6

B
8

C
11

D
12

Solution

5⁵ + 7⁵ = 3125 + 16807
= 19932
Which is divisible by 12
Question 7
1 / -0

The third term of a geometric progression (GP) is $3$ . What is the product of the first five terms?

A
$216$

B
$226$

C
$243$

D
Cannot be determined

Solution

Given-
The third term of a geometric progression (GP) is $3$ .
$a r^{2} = 3$
Product of first five terms $p = a \times a r \times a r^{2} \times a r^{3} \times a r^{4}$
$\Rightarrow p = a^{5} r^{10}$
$\Rightarrow p = {(3)}^{5}$
$\Rightarrow p = 243$
Question 8
1 / -0

If $\cos α + \cos β + \cos γ = 0$ , where $0 < α \leq π / 2$ and $0 < γ \leq π / 2$ , then what is the value of $\sin α + \sin β + \sin γ$ ?

A
$0$

B
$3$

C
$\frac{5}{\sqrt{2}}$

D
$5 \sqrt{2}$

Solution

Given-
$\cos α + \cos β + \cos γ = 0$
This means $\cos α = \cos β = \cos γ = 0$ which is possible only when $α = β = γ = 90 °$
$\sin α + \sin β + \sin γ$
$= \sin 90 ° + \sin 90 ° + \sin 90 °$
$= 1 + 1 + 1$
$= 3$
Question 9
1 / -0

What is the equation of the plane passing through the points (–2, 6, –6), (–3, 10, –9) and (–5, 0, –6)?

A
2x – y – 2z = 2

B
2x + y + 3z = 3

C
x + y + z = 6

D
x – y – z = 3

Solution

Equation of the plane
$| \begin{array}{ccc} x + 2 & y - 6 & z + 6 \\ x + 3 & y - 10 & z + 9 \\ x + 5 & y - 0 & z + 6 \end{array} | = 0$
$= {(x + 2) [(y - 10) (z + 6) - y (z + 9)] - (y - 6) [(x + 3) (z + 6) - (x + 5) (z + 9)]$
$+ (z + 6) [y (x + 3) - (x + 5) (y - 10)]$
${(x + 2) [- 3 y - 10 z - 60] - (y - 6) [- 3 x - 2 z - 27] + (z + 6) [10 x - 2 y + 50]}$
$= - 18 x + 9 y + 18 z + 18 = 0$
$= 2 x - y - 2 z = 2$
Question 10
1 / -0

If $A + \frac{1}{A} = 5$ then find the value of $A^{4} + \frac{1}{A^{4}}$ .

A
$527$

B
$529$

C
$531$

D
$533$

Solution

Given-
$A + \frac{1}{A} = 5$
Squaring both the sides,
$\Rightarrow A^{2} + \frac{1}{A^{2}} + 2 = 25$
$\Rightarrow A^{2} + \frac{1}{A^{2}} = 23$
Again squaring both the sides,
$\Rightarrow A^{4} + \frac{1}{A^{4}} + 2 = 529$
$\Rightarrow A^{4} + \frac{1}{A^{4}} = 527$

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Re-Attempt Weekly Quiz Competition

Mathematics Test - 1

Result

Mathematics Test - 1

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SHARING IS CARING

Question 1

2.870

2.967

3.876

3.912

Solution

Question 2

552425

5-52425

-55-2425

None of these

Solution

Question 3

sin 50°

cos 50°

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

0

Solution

Question 4

1565

1585

1635

1655

Solution

Question 5

10

16

18

36

Solution

Question 6

6

8

11

12

Solution

Question 7

216

226

243

Cannot be determined

Solution

Question 8

0

3

52

52

Solution

Question 9

2x – y – 2z = 2

2x + y + 3z = 3

x + y + z = 6

x – y – z = 3

Solution

Question 10

527

529

531

533

Solution

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$[\begin{matrix} 5 & 5 \\ 24 & 25 \end{matrix}]$

$[\begin{matrix} 5 & - 5 \\ 24 & 25 \end{matrix}]$

$[\begin{matrix} - 5 & 5 \\ - 24 & 25 \end{matrix}]$

$216$

$226$

$243$

$0$

$3$

$\frac{5}{\sqrt{2}}$

$5 \sqrt{2}$

$527$

$529$

$531$

$533$