Mathematics Test

Result

Mathematics Test - 22

/

Score
-

Rank

Time Taken: -

Question 1/10
1 / -0

Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2).

Correct

Wrong

Skipped
A
$(2, \frac{1}{3})$

Correct

Wrong

Skipped
B
$(3, \frac{11}{2})$

Correct

Wrong

Skipped
C
$(\frac{9}{7}, \frac{12}{7})$

Correct

Wrong

Skipped
D
$(\frac{3}{2}, \frac{11}{2})$

Solutions
Question 2/10
1 / -0

Find the unit digit in $7^{65} \times 6^{41} \times 3^{57}$ =?

Correct

Wrong

Skipped
A
6

Correct

Wrong

Skipped
B
7

Correct

Wrong

Skipped
C
5

Correct

Wrong

Skipped
D
8

Solutions

Unit digit in $(7)^{65}$
$=$ Unit digit in $7^{4 \times 16 + 1}$
$=$ Unit digit in $7^{1} = 7$ Unit digit in $6^{41} = 6 [∵ unit digit in (6)^{n} = 6]$
Unit digit in $3^{57}$
$=$ Unit digit in $3^{4 \times 14 + 1}$
$=$ Unit digit is $3^{1} = 3$ Thus, unit digit in $7^{65} \times 6^{41} \times 3^{57}$
$=$ Unit digit in $7 \times 6 \times 3$
$=$ Unit digit in $126 = 6$

Unit digit in $(7)^{65}$
$=$ Unit digit in $7^{4 \times 16 + 1}$
$=$ Unit digit in $7^{1} = 7$ Unit digit in $6^{41} = 6 [∵ unit digit in (6)^{n} = 6]$
Unit digit in $3^{57}$
$=$ Unit digit in $3^{4 \times 14 + 1}$
$=$ Unit digit is $3^{1} = 3$ Thus, unit digit in $7^{65} \times 6^{41} \times 3^{57}$
$=$ Unit digit in $7 \times 6 \times 3$
$=$ Unit digit in $126 = 6$
Question 3/10
1 / -0

$What is \int \frac{\sin \sqrt{x}}{\sqrt{x}} d x is equal to?$

Correct

Wrong

Skipped
A
$\frac{\cos \sqrt{x}}{2} + c$

Correct

Wrong

Skipped
B
$2 \cos \sqrt{x} + c$

Correct

Wrong

Skipped
C
$\frac{- \cos \sqrt{x}}{2} + c$

Correct

Wrong

Skipped
D
$- 2 \cos \sqrt{x} + c$

Solutions

$\begin{array}{l} Let I = \int \frac{\sin \sqrt{x}}{\sqrt{x}} d x \\ Put t = \sqrt{x} \Rightarrow d t = \frac{1}{2 \sqrt{x}} d x \\ I = \int 2 \sin t d t = - 2 \cos t + c \\ = - 2 \cos \sqrt{x} + c \end{array}$
Question 4/10
1 / -0

For a frequency distribution, mean, median and mode are connected by the relation?

Correct

Wrong

Skipped
A
Mode = 3 Mean – 2 Median

Correct

Wrong

Skipped
B
Mode = 2 Median – 3 Mean

Correct

Wrong

Skipped
C
Mode = 3 Median –2 Mean

Correct

Wrong

Skipped
D
Mode = 3 median + 2 Mean

Solutions

Mode = 3 Median –2 Mean

Hence correct answer is option C
Question 5/10
1 / -0

What is $\int_{0}^{1} x e^{x} d x$ equal to?

Correct

Wrong

Skipped
A
1

Correct

Wrong

Skipped
B
-1

Correct

Wrong

Skipped
C
0

Correct

Wrong

Skipped
D
E

Solutions

Let $I = \int_{0}^{1} \begin{aligned} x & e^{x} \\ I & II \end{aligned} d x$
Using integration by parts
$I = {[x . e^{x}]}_{0}^{1} - \int_{0}^{1} 1 . e^{x} d x$
$= (1 . e - 0) - {[e^{x}]}_{0}^{1}$
$= 1$
Let $I = \int_{0}^{1} \begin{array}{cc} x & e^{x} \\ I & II \end{array} d x$
Using integration by parts
$I = {[x . e^{x}]}_{0}^{1} - \int_{0}^{1} 1 . e^{x} d x$
$= (1 . e - 0) - {[e^{x}]}_{0}^{1}$
$= 1$
Question 6/10
1 / -0

If ∆ABC ~ ∆DEF such that AB = 1.2 cm & DE = 1.4cm. Find the ratio of areas of ∆ABC & ∆DEF.

Correct

Wrong

Skipped
A
6 : 7

Correct

Wrong

Skipped
B
16 : 81

Correct

Wrong

Skipped
C
36 : 49

Correct

Wrong

Skipped
D
4 : 9

Solutions

We know that the ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

$\begin{array}{l} ∴ \frac{area (Δ A B C)}{area (Δ D E F)} = \frac{A B^{2}}{D E^{2}} \\ \Rightarrow \frac{area (Δ A B C)}{area (Δ D E F)} = \frac{(1.2)^{2}}{(1.4)^{2}} = {(\frac{12}{14})}^{2} = \frac{36}{49} \end{array}$
Question 7/10
1 / -0

A circle is inscribed in a ∆ABC having sides 8 cm, 10cm and 12cm as shown in figure. Find AD, BE & CF.

Correct

Wrong

Skipped
A
4cm, 2cm, 3cm

Correct

Wrong

Skipped
B
7cm, 5cm, 3cm

Correct

Wrong

Skipped
C
3cm, 2cm, 1cm.

Correct

Wrong

Skipped
D
7cm, 6cm, 3cm

Solutions
Question 8/10
1 / -0

The ratio in which the line joining (2, 4, 5), (3, 5, –4) is divided by the yz plane is?

Correct

Wrong

Skipped
A
2 : 3

Correct

Wrong

Skipped
B
3 : 2

Correct

Wrong

Skipped
C
–2 : 3

Correct

Wrong

Skipped
D
4 : –3

Solutions
Question 9/10
1 / -0

The sides of a triangle are 6.5 cm, 10 cm & x cm, where x is a positive number. What is the smallest possible value of x among the following?

Correct

Wrong

Skipped
A
3.5

Correct

Wrong

Skipped
B
4

Correct

Wrong

Skipped
C
4.5

Correct

Wrong

Skipped
D
2.8

Solutions

A triangle can be formed only when sum of two sides of triangle is greater than the third side. Hence the smallest possible value of x will be 4.

i.e. 6.5 + 10 > 4

6.5 + 4 > 10

4 + 10 > 6.5
Question 10/10
1 / -0

If $α & β$ are the zeros of the polynomial $f (x) = x^{2} - 5 x + k$ such that $α - β = 1,$ Find the value of $k$ .

Correct

Wrong

Skipped
A
5

Correct

Wrong

Skipped
B
4

Correct

Wrong

Skipped
C
6

Correct

Wrong

Skipped
D
7

Solutions

Question 1/10

2,13

3,112

97,127

32,112

Question 2/10

6

7

5

8

Question 3/10

cosx2+c

2cosx+c

-cosx2+c

-2cosx+c

Question 4/10

Mode = 3 Mean – 2 Median

Mode = 2 Median – 3 Mean

Mode = 3 Median –2 Mean

Mode = 3 median + 2 Mean

Question 5/10

1

-1

0

E

Question 6/10

6 : 7

16 : 81

36 : 49

4 : 9

Question 7/10

4cm, 2cm, 3cm

7cm, 5cm, 3cm

3cm, 2cm, 1cm.

7cm, 6cm, 3cm

Question 8/10

2 : 3

3 : 2

–2 : 3

4 : –3

Question 9/10

3.5

4

4.5

2.8

Question 10/10

5

4

6

7

$(2, \frac{1}{3})$

$(3, \frac{11}{2})$

$(\frac{9}{7}, \frac{12}{7})$

$(\frac{3}{2}, \frac{11}{2})$

$\frac{\cos \sqrt{x}}{2} + c$

$2 \cos \sqrt{x} + c$

$\frac{- \cos \sqrt{x}}{2} + c$

$- 2 \cos \sqrt{x} + c$