Question

SECTION – A
Questions 1 to 6 carry 1 mark each.
1. ABC and BDE are two equilateral triangles such that 2
3
BD BC  . Find the ratio of the areas of
triangles ABC and BDE.
2. Find the coordinates of the point on y-axis which is nearest to the point (–2, 5).
3. If sinA =
1
2
, find the value of
A
A
2
1 tan
2sec

.
4. Find the values of k for quadratic equation 2x
2
+ kx + 3 = 0, so that they have two equal roots.
5. Express 7429 as a product of its prime factors.
6. Which term of the AP: 3, 8, 13, 18 ,…. Is 78?
SECTION – B
Questions 6 to 12 carry 2 marks each.
7. A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the
probability that the number on the drawn card is (i) a square number (ii) a multiple of 7
8. A box contains 12 balls of which some are red in colour. If 6 more red balls are put in the box
and a ball is drawn at random, the probability of drawing a red ball doubles than what it was
before. Find the number of red balls in the bag.
9. Using Euclid’s division algorithm, find the HCF of 2160 and 3520.
10. Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which y = mx + 3.
11. Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by
(–1, 6).
12. The sum of first n terms of an AP is given by Sn = 2n2
+ 3n . Find the sixteenth term of the AP.

Answer 1

1. ABC and BDE are two equilateral triangle such that BD =2/3BC.

find the ratio of the area of triangle ABC and BDEIn triangle ABC,

Since it is an equilateral traingle.

          AB =BC = AC

Also in triangle BDE,
 
               BD = DE = BE

Also it is given that,

                     BD = 2/3 BC

Also we know area of equilateral triangle 

     area = (root3/4) * a^2
      
in triangle ABC

          Area                          A1 = root3/4 * BC^2

Also, in triangle BDE,

                    Area                 A2 = root3/4 * BD^2

Ratio of the areas 

                          R = A1/A2

                          R = BC^2/BD^2

      Putting the value of BD
            
R = BC^2/ (2/3BC)^2

            R = 9/4

Thus the ratio of the areas of triangle to area of triangle BDE

R = 9:4
_____________________________________________________________

2. Find the coordinates of the point on y-axis which is nearest to the

point (–2, 5).

It has to have coordinates x = 0  [on y-axis]
y = 5  [along perpendicular line from y-axis to point] The point is (0,5).
____________________________________________________________________

4. Find the values of k for quadratic equation 2x²+ kx + 3 = 0, so that

they have two equal roots.

equal roots ⇒ b²-4ac=0
         
              (k)²-4(2)(3)=0

                        k²-24=0

                               k=+√24,-√24
_____________________________________________________________

5. Express 7429 as a product of its prime factors.

7429=(17 × 19 × 23)
_____________________________________________________________

6. Which term of the AP: 3, 8, 13, 18 ,…. Is 78?

Then a = 3      d = 5     l = 78

l=a+(n-1)d

78=3+(n-1)5

75/5=n-1      ⇒ n=16th term
_____________________________________________________________

7. A box contains cards numbered 11 to 123. A card is drawn at random

from the box. Find the probability that the number on the drawn card is

(i) a square number (ii) a multiple of 7

i)P(Square no)=6/113
 
(16,25,36,49,64,81,100,121)

ii)P(multiple of 7)=16/113

(14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119)
_____________________________________________________________

8. A box contains 12 balls of which some are red in colour. If 6 more red

balls are put in the box and a ball is drawn at random, the probability of

drawing a red ball doubles than what it was before. Find the number of

red balls in the bag.

Let the total number of red balls initially be x

Total number of balls in the box = 12

P(getting a red ball) = x/12

Now, 6 red balls are put in the box, then the total number of balls 

= 12 + 6 

= 18 balls

Then total number of red balls = (x + 6)

Now, 

P(getting a red ball) = (x + 6)/18

According to the question.

2(x/12) = (x + 6)/18

2x/12 = (x + 6)/18

Cross multiplying

⇒ 36x = 12x + 72

⇒ 36x - 12x = 72

⇒ 24x = 72

x = 3

So, initially the number of red balls in the box were 3 
_____________________________________________________________

9. Using Euclid’s division algorithm, find the HCF of 2160 and 3520

a=bq+r

3520=2160×1+1360

2160=1360×1+800

1360=800×1+560

800=560×1+240

560=240×2+80

240=80×3+0

Therefore the hcf of 2160 and 3520 is 80
_____________________________________________________________

10. Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’

for which y = mx + 3.

2x + 3y = 11                                                                                            .......(i)

2x - 4y = -24                                                                                            ......(ii)

Now, from (ii), we get, 

2x-4y=-24

2x=-24+4y

x=(-24+4y)/2

x=-24/2+4y/2

x=-12+2y                                                                                               ........(iii)

Put this value of x in (i), we get, 

 2(-12+2y) + 3y = 11

-24+4y+3y=11

7y=11+24

7y=35

y=35/7

y=5

now put this value of y in (iii)

x=-12+2y

x=-12+2(5)

x=-12+10

x=-2

y=mx+3                                      {y=5,x=-2)

5=m(-2)+3

5=-2m+3

5-3=-2m

2=-2m

2/2=-m

1=-m

m=-1
_____________________________________________________________

11. Find the ratio in which the line segment joining the points (– 3, 10)

and (6, – 8) is divided by (–1, 6).

Ratio=2:7
_____________________________________________________________

12. The sum of first n terms of an AP is given by Sn = 2n²+ 3n . Find the sixteenth term of the AP.
an=Sn-Sn-1

Sn=2n²+3n

Sn-1=2(n-1)²+3(n-1)

       =2n²-4n+2+3n-3

       =2n²-n-1

Then an=2n²+3n-2n²+n+1

         an=4n+1
             
        a16=4(16)+1
 
              =64+1  ⇒ 65
_____________________________________________________________

:) Hope this helps!!!!!!!!!!!!!

Answer 2

I hope it is helpful for you.
Please mark as a Brainliest answer