# Problem Solving Select Many Practice Test 1

**Question 1**

**Which of the following is a correct representation of the set X = {1, 1/4, 1/9, 1/16, 1/25,...} in the set builder form? Indicate all correct options.[n**

^{2}= n*n sqrt(n) = square root of n]

- X = {x: x is an element of N}
- X = {n: 1/n
^{2}is an element of N} - X = {1/n
^{2}: n is an element of N} - X = {1/x
^{2}: x is a natural number} - X = {1/sqrt(x): x is a natural number}

**Correct Answer: C and D **

**Explanation:**

For X = {x: x is an element of N}, X is the set of natural numbers and hence option (A) is not correct.

For X = {n: 1/n^{2} is an element of N}, X = {-1, 1, 1/sqrt(2), 1/sqrt(3),...} and hence option (B) is not correct.

For X = {1/n^{2}: n is an element of N}, X = {1, 1/4, 1/9, 1/16, 1/25,...} and hence option (C) is correct.

For X = {1/x^{2}: x is a natural number}, X = {1, 1/4, 1/9, 1/16, 1/25,...} and hence option (D) is correct.

For X = {1/sqrt(x): x is a natural number}, X = {-1, 1, 1/sqrt(2), 1/sqrt(3),...} and hence option (E) is not correct.

**Question 2**

**Which of the following statements is true for two points A(6,3) and B(-4,5)? Indicate all correct options.**

- They lie in the same quadrant
- The mid point of AB is (1, 4)
- The point P(0, 21/5) divides AB in the ratio 3:2
- The point P(0, 21/5) divides AB in the ratio 2:3
- The two points are in different quadrants

**Correct Answer: B, C and E **

**Explanation:**

A lies in the first quadrant and B lies in the second quadrant.

Option A is false and E is true.

The mid point of AB is given by ((6-4)/2, (3+5)/2)

= (2/2, 8/2)

= (1,4)

Option B is true.

The co-ordinates of a point that divides AB in ratio 3:2 are given by

((3*-4+2*6)/(3+2), (3*5+2*3)/(3+2))

= ((-12+12)/5, (15+6)/5)

= (0, 21/5)

This is the same point as P. Hence, option C is true and D is false.

**Question 3**

**How many diagonals are there in a polygon with 9 sides? Indicate all correct options.**

- C(9,2)
- P(9,2)
- 36
- 72
- C(9,4)

**Correct Answer: A and C**

**Explanation:**

Number of diagonals = C(9,2) = 9!/[(9-2)!2!]

= 9!/(7!2!) = 9*8/2 = 9*4 = 36

Options (A) and (C) are correct.

**Question 4**

**The following table shows the percentage expenditure on different items in constructing a flat. If the cost of the flat is Rs.5,40,000, then
which of the following statements is true? Indicate all correct options.
**

Item | Labour | Timber | Cement | Bricks | Steel |
---|---|---|---|---|---|

Percentage Expenditure | 250/9% | 25% | 125/6% | 125/9% | 25/2% |

- The amount spent on timber is twice that spent on steel
- Labour is the costliest commodity
- Steel is the cheapest commodity
- The amount spent on steel was Rs. 11250
- The amount spent on timber was Rs. 67500

**Correct Answer: A, B and C**

**Explanation:**

It is clear from the table that the amount spent on timber is twice the amount spent on steelsince the percentage expenditure on timber is 25% and the percentage expenditure on steel is 25/2%.

Option A is true.

250/9% = 27.78%

125/6% = 20.83%

125/9% = 13.89%

25/2% = 12.5%

Hence, the maximum amount was spent on Labour. Option B is true.

Steel is the cheapest commodity. Option C is true.

The amount spent on steel = 25/2*540000/100 = Rs.67500

The amount spent on timber = 25*540000/100 = Rs. 135000

Options D and E are false.

**Question 5**

**A bag contains 5 red balls and 5 blue balls. In how many ways can 5 balls be drawn at random such that there are at least two balls of each colour?
Indicate all correct options.**

- 200
- 2C(5,2)*C(5,3)
- 100
- C(5,2)*C(5,3)
- C(10,5)

**Correct Answer: A and B**

**Explanation:**

We either select 2 red and 3 blue balls or 3 red and 2 blue balls.

Total number of ways = C(5,2)*C(5,3) + C(5,3)*C(5,2)

= 2[5!/(2!3!) * 5!/(2!3!)]

= 2[(5*4)/2*(5*4)/2]

= 2(100) = 200

Options (A) and (B) are correct.

**Question 6**

**Which of the following is an element of the set A = {x: x is a positive integer and x is a divisor of 12}? Indicate all correct options. **

- 2
- 1
- 0
- 9
- 24

**Correct Answer: A and B**

**Explanation:**

The set A = {x: x is a positive integer and x is a divisor of 12}

= {1, 2, 3, 4, 6, 12}

1 and 2 are elements of A.

**Question 7**

**A bag contains 5 red balls, 8 white balls and 7 green balls. Which of the following is true? Indicate all correct options.**

- The probability of drawing a blue ball is 7/25
- The probability of drawing a red ball is 1/4
- The probability of drawing a ball which is not green is 13/20
- The probability of drawing two red balls is 1/4
- The probabilty of drawing three green balls is 3/7

**Correct Answer: B and C **

**Explanation:**

The probability of drawing a blue ball is 0 since there are no blue balls in the bag.

Option A is false.

The probability of drawing a red ball = 5/(5+8+7)= 5/20 = 1/4|

Option B is true.

The probability of drawing a ball which is not green = (5+8)/(5+8+7)= 13/20

Option C is true.

The probability of drawing two red balls is = C(5,2)/C(20,2)

= 5!/(2!3!)/20!/(18!2!)= (5*4)/2/(20*19)/2= 20/(20*19) = 1/19

Option D is false.

The probability of drawing three green balls is = C(7,3)/C(20,3)

= 7!/(4!3!)/20!/(17!3!)

=(7*6*5)/(3*2)/(20*19*18)/(3*2)= 35/(20*19*3)= 7/(4*19*3)= 7/228

Option E is false.

**Question 8**

**The given table shows the weight of the students of a class in kilograms. Which of the given statements is true? Indicate all correct options.
Weight (in kg): 67 70 72 73 75
Number of students: 4 3 2 2 1
**

Weight (in kg): | 67 | 70 | 72 | 73 | 75 |
---|---|---|---|---|---|

Number of students: | 4 | 3 | 2 | 2 | 1 |

- There are in all 12 students
- The average weight is 70 kgs
- Most of the students weigh less than 72 kgs
- The weight of the students is proportional to their heights
- The average weight is 70.25 kgs

**Correct Answer: A, C and E**

**Explanation:**

Total students in the class = 4+3+2+2+1 = 12

Option A is true.

Average weight = (67*4+70*3+72*2+73*2+75*1)/12

= (268 + 210 + 144 + 146 + 75)/12

= 843/12 = 70.25 kgs

Option B is false and E is true.

7 students weigh less than 72 kgs. Option C is true.

Since we do not know the heights of the students, we cannot say that option D is true.

**Question 9**

**A cubical block of iron of 88 cm edge is recast into small spheres of 4 cm diameter. Which of the following statemetns is true? Indicate all correct options.**

- 2541 spheres can be made
- number of spheres = volume of iron cube/volume of sphere
- number of spheres = surface area of iron cube/surface area of sphere
- 20328 spheres can be made
- 5082 spheres can be made

**Correct Answer: B and D**

**Explanation:**

Volume of a block of iron = 88*88*88

= 681472

Volume of sphere = (4/3)*pi*r^{3}, where r is the radius of the sphere

= (4/3)*22/7*2^{3}

Number of spheres = volume of cube/volume of sphere

= [88*88*88]/[(4/3)*22/7*2^{3}]

= (88*88*88*3*7)/(4*22*8)

= 20328

Options B and D are true.

[pi=22/7, r^{3}=r*r*r]

**Question 10**

**Which of the following statements is true for the quadratic equation 2x ^{2}+x-1=0? Indicate all correct options.[n^{2} = n*n]**

- The equation has equal roots
- The equation has real and distinct roots
- x = 1
- x = -1
- x = -1/2

**Correct Answer: B and D**

**Explanation:**

2x^{2}+x-1=0

2x^{2} +2x-x-1=0

2x(x+1)-1(x+1)=0

(2x-1)(x+1)=0

x=1/2, -1

Clearly, the equation has real and distinct roots.

Options B and D are correct.

**Question 11**

**A is the set of vowels. How many elements does the power set of A have? Indicate all correct options. [2**

^{4}= 2*2*2*2]

- 2
^{4} - 0
- 5
- 32
- 2
^{5}

**Correct Answer: D and E**

**Explanation:**

The set A is A = {a, e, i, o, u}

The set has five elements and the number of elements in the power set of A is given by 2^{5} = 32

**Question 12**

**Which of the following is true for the equation x ^{2}+px+2q=0, where D is the discriminant? Indicate all correct options.[n^{2} = n*n]**

- D = 0
- D >0 if p
^{2}= 8q - D<0
- D<0 if p
^{2}<8p - D>0

**Correct Answer: B and D**

**Explanation:**

D = p^{2} - 4*1*2q

= p^{2} - 8q

D=0 if p^{2}-8q = 0

p^{2} = 8q

Option A is wrong and B is true.

D<0 if p^{2}-8q<0

p^{2}<8q

Option C is false and D is true.

Option E is false.

**Question 13**

**Which of the following is a root of the equation x ^{2} + 3x -(a^{2}+a-2) = 0? Indicate all correct options.**

[n^{2} = n*n]

- (a+2)
- (a-1)
- -(a+2)
- (1-a)
- (2-a)

**Correct Answer: B and C**

**Explanation:**

x^{2} + 3x -(a^{2}+a-2) = 0

x^{2} + 3x - (a+2)(a-1) = 0

x^{2} +{(a+2)-(a-1)}x - (a+2)(a+1) = 0

{x^{2} +(a+2)x} - (a-1)x - (a+2)(a-1) = 0

x{x + (a+2)} - (a-1){x + (a+2)} = 0

{x + (a+2)}{x - (a-1)} = 0

x = -(a+2), (a-1)

Hence, options B and C are true.

**Question 14**

**For two rational expressions p(x) = (2x-1)/(x+1) and q(x) = (x+1)/(2x+1), which of the following is true? Indicate all correct options.**

- p(x) + q(x) = q(x) + p(x)
- p(x) - q(x) = q(x) - p(x)
- p(x) - q(x) = -{q(x) - p(x)}
- p(x)*q(x) = (2x-1)/(2x+1)
- p(x)/q(x) = {(2x-1)(2x+1)}/{(x+1)(x-1)}

**Correct Answer: A, C and D**

**Explanation:**

p(x) + q(x) = q(x) + p(x) since addition of rational expressions is commutative.

Option A is true.

Option B is false and C is true since subtraction of rational expressions is not commutative.

p(x)*q(x) = [(2x-1)/(x+1)]*[(x+1)/(2x+1)]

= (2x-1)/(2x+1)

Option D is true.

p(x)/q(x) = [(2x-1)/(x+1)]/[(x+1)/(2x+1)]

= [(2x-1)/(x+1)]*[(2x+1)/(x+1)]

= [(2x-1)(2x+1)]/[(x+1)(x+1)]

Option E is false.

**Question 15**

**If p(x) = (x+2)/(x-2) and q(x) = x/(x ^{2}-4), then which of the following represents p(x)*q(x)? Indicate all correct options.[n^{2} = n*n]**

- x/(x-2)
- x(x-2)
^{2} - x/[(x-2)(x-2)]
- x/(x-2)
^{2} - x(x+2)/(x-2)

**Correct Answer: C and D**

**Explanation:**

p(x)*q(x) = [(x+2)/(x-2)]*[x/(x^{2}-4)]

= [(x+2)*x]/[(x-2)(x+2)(x-2)]

= x/[(x-2)(x-2)]

= x/(x-2)^{2}