Elementary Algebra problems
- 1. If the cost of 2 apples, 3 oranges and a pear is Rs.62 and the cost of 5 apples, 6 oranges and 4 pears is Rs. 120, then find the cost of 3 apples, 3 oranges and 3 pears?
Options- A. Rs. 39
- B. Rs. 46
- C. Rs. 58
- D. Rs. 24 Discuss
Correct Answer: Rs. 58
Explanation:
Let the cost of each apple,orange nd pear be Rs.x,y and z, respectively.Then,
2x + 3y + z = 62 ...(i)
5x + 6y + 4z = 20 ...(ii)
On subtracting Eq.(i)from Eq.(ii), we get
3x + 3y + 3z = 58
So, the cost of 3 apple,3 oranges and 3 pears is Rs. 58.
- 2. Two candidates attempt to solved a quadratic equation of from x 2 + px + q = 0. One starts with a wrong value of p and finds the roots to be 2 and 6. Other starts with a wrong value of q and find the root to be 2 and -9. Find the correct roots and equation?
Options- A. -3, -2 and x x2 + 9x + 18
- B. 3, 4 and x2 + 8x + 15
- C. -3, -4 and x2 + 7x + 12
- D. 2, 6 and x2+ 7x + 12 Discuss
Correct Answer: -3, -4 and x2 + 7x + 12
Explanation:
When p is wrong i.e., -b/a = (? + ?) is wrong but c/a =(??) is correct.
Then ?? = c/a = 2 x 6 = 12 ....(i)
Again, when q is wrong i.e.,c/a = (?.? ) is wrong but -b/a = ?+? is correct.
Then, -b/a = ?+?= 2 + (-9) = -7
So, the required correct quadratic equations is
x2 -(? + ?)x + ?.?=0
? x2 - (-7)x + 12 = 0
? x2 + 7x + 12 = 0
and correct roots this equations are -3, -4.
- 3. If a + b + 1 = 0, then the value of (a 3 + b 3 + 1 ? 3ab) is
Options- A. 3
- B. 0
- C. ?1
- D. 1 Discuss
Correct Answer: 0
Explanation:
As we know from the formula,
A3 + B3 + C3 = ( A + B + C) (A2 + B2 + C2 ? AB ? BC ? CA) + 3ABC
If A + B + C = 0 then,
A3 + B3 + C3 = 0 x (A2 + B2 + B2 ? AB ? BC ? CA) + 3ABC
A3 + B3 + C3 = 0 + 3ABC
A3 + B3 + C3 - 3ABC= 0
As we know that from given question,
A = a, B = b and C = 1, Put these value in above equation, we will get
a3 + b3 + 13 - 3 x a x b x 1 = 0
a3 + b3 + 1 - 3ab = 0
- 4.
If a + 1 = 1, then the value of a3 is : a
Options- A. ? 2
- B. 2
- C. ? 1
- D. 4 Discuss
Correct Answer: ? 1
Explanation:
- 5. If ax + by = 6, bx ? ay = 2 and x 2 + y 2 = 4, then the value of ( a 2 + b 2) would be :
Options- A. 10
- B. 2
- C. 4
- D. 5 Discuss
Correct Answer: 10
Explanation:
As we know that given in question,
ax + by = 6 ........................ (i)
bx ? ay = 2 ........................ (ii)
On squaring the both equation and adding,
a2 x2 + b2 y2 + 2abxy + b2 x2 + a2 y2 ? 2abxy = 36 + 4
? x2 (a2 + b2) + y2 (a2 + b2) = 40
? (a2 + b2) (x2 + y2) = 40
As per given question, x2 + y2 = 40 , Put this value in above equation, we will get
? (a2 + b2) × 4 = 40
? a2 + b2 = 10
- 6. In xy?plane, P and Q are two points having co? ordinates (2, 0) and (5, 4) respectively. Then the numerical value of the area of the circle with radius PQ is
Options- A. 16 ?
- B. 32 ?
- C. 14 ?
- D. 25 ? Discuss
Correct Answer: 25 ?
Explanation:
According to question ,
PQ = ?( 5 - 2 )2 + ( 4 - 0 )2
PQ = ?9 + 16 = 5
? Area of circle = ?r2
Area of circle = 25 ? sq. units
- 7. In the xy?coordinate system, if ( a, b) and ( a + 3, b + k) are two points on the line defined by the equation x = 3y ? 7, then k =?
Options- A. 7 3
- B. 1
- C. 9
- D. 3 Discuss
Correct Answer: 1
Explanation:
Given in the question,
Points (a, b) and [(a + 3), (b + k)] will satisfy the equation x ? 3y + 7 = 0.
Points (a, b) in the equation x ? 3y + 7 = 0, We will get
? a ? 3b + 7 = 0 ...................... (i)
Points (a + 3), (b + k) in the equation x ? 3y + 7 = 0, We will get
a + 3 ? 3 (b + k) + 7 = 0
? a + 3 ? 3b ? 3 k + 7 = 0 (Rearrange the equation)
? a ? 3b + 7 + 3 ? 3 k = 0 (a ? 3b + 7 = 0 , Put the value from Equation (1) ), We will get
? 0 + 3 ? 3k = 0
? 3k = 3
? k = 1
- 8.
If a = ? 3 - ? 2 , b = ? 3 + ? 2 , then the value of ?3 + ?2 ?3 - ?2 a 2 + b 2 is : b a
Options- A. 900
- B. 970
- C. 1030
- D. 930 Discuss
Correct Answer: 970
Explanation:
- 9. Area of the trapezium formed by x-axis; y-axis and the lines 3x + 4y = 12 and 6x + 8y = 60 is:
Options- A. 37.5 sq. unit
- B. 31.5 sq. unit
- C. 48 sq. unit
- D. 36.5 sq. unit Discuss
Correct Answer: 31.5 sq. unit
Explanation:
- 10. If x = 1 + ?2 + ?3, then the value of ( 2x4 ? 8x3 ? 5x2 + 26x ? 28) is
Options- A. 6 ?6
- B. 0
- C. 3 ?6
- D. 2 ?6 Discuss
Correct Answer: 6 ?6
Explanation:
Given in the question ,
x - 1 = ?2 + ?3
On squaring both sides ,
x 2 ? 2x + 1 = 2 + 3 + 2?6
? x 2 - 2x - 4 = 2 ?6
On squaring again,
? x4 + 4x2 + 16 ? 4x3 ? 8x2 + 16x = 24
? x4 ? 4x3 ? 4x2 + 16x ? 8 = 0
? 2x4 ? 8x3 ? 8x2 + 32x ? 16 = 0
? 2x4 ? 8x3 ? 5x2 + 26x ? 28 ? 3x2 + 6x + 12 = 0
? 2x4 ? 8x3 ? 5x2 + 26x ? 28 = 3x2 ? 6x ? 12
? 2x4 ? 8x3 ? 5x2 + 26x ? 28 = 3 (x2 ? 2x ? 4)
? 2x4 ? 8x3 ? 5x2 + 26x ? 28 = 3 x 2?6 = 6 ?6
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