1. (c)
First, calculate the value on the left side:
7059−2350+1936=6645
Now, divide by 50:
6645/50=132.9
2. (c) 7/4+1.4142 = 7/ 5.4142 ≈ 1.2929
3. (d) 786 × 964 = 757704
4. (d)
First, calculate each term:
999 + 588 = 1587, 999 − 588 = 411
Now:
15872+4112 = 2510169 + 168921 = 2679090
Now divide by 999 × 588 = 587412
2679090/ 587412≈4.56≈4
5. (b)
First, calculate each term:
238 + 131 = 369, 238 − 131 = 107
Now:
3692 + 1072 = 136161 + 11449 = 147610
And:
2382 + 1312 = 56644 + 17161 = 73805
Now divide:
147610/ 73805 = 2
6. (d) First, calculate:
999 × 494 = 493506
Now: 997 × 99 = 98703
Now add: 1/5 + 98703 = 98703.2
Now divide by 4: 98703.2/4 = 24675.8
7. (d)
First, calculate
x = √3 + √2 , 1/x = √3 – √2/1
Now: x +1/x = (√3 + √2) + ( √3 – √2) = 2 √3
Formula Used:
a2 + b2 + c2 – 3abc = 1/2(a+b+c) [(a-b)2 + (b-c)2 + (c-a)2
Calculation:
a + b + c – 3abc = 1/2(225 + 226 +227) [(225-226)2 +(226-227)2 + (227-225)2]
= 1/2 × 678 × (1 + 1 + 4)
= 1/2 × 678 × 6
= 2034
9. (a)
First, calculate each term:
(0.96)3 = 0.884736, (0.1)3 = 0.001
Now:
0.884736−0.001/ 0.9216 + 0.096 + 0.01 = 0.883736/ 1.0276 ≈ 0.86
10. (a)
First, calculate each term:
√0.0001 = 0.01, √1000000 = 1000
√0.125 ≈ 0.3536, √0.008≈0.08944
Now:
0.01 × 1000/ 3 × 0.3536 × 2× 0.08944 =10/ 0.633 ≈ 15.8
11. (d)
First, divide 60 by 0.5:
60 ÷ 0.5 = 12060 ÷ 0.5 = 120
Now multiply by 3.5:
3.5 × 120 = 4203.5 × 120 = 420
12. (d)
Start from the innermost parentheses: 12 − 9 = 3
Now simplify the next part: 15−3=12
Next: 18−12=6
Then: 9 − 6 = 3
Finally: 6 − 3 = 3
13. (c)
The least common denominator is 12. Convert each fraction:
3/ 23 = 128/ 12, 23/4 = 69/ 12, 11/ 2 = 66/ 12
Now add the fractions:
128/ 12 + 69/ 12 + 66/ 12 = 263/ 12
This is equivalent to:
21 × (11/ 12) = 711/12
Convert the mixed number to an improper fraction:
Now multiply:
Add 13/ 8 to 4/ 9 using a common denominator of 72:
Now add:
First, multiply 10.8 by 5.5:
Now multiply by 84:
16. (a)
Step 1: Rewrite the expression using powers of 4
We know that:
– 4=41
– 64=43
Thus, we can rewrite 64 as 43.
The expression becomes:
(45)5×(43)8÷(43) = (43)?
Step 2: Simplify the left side
Using the power of a power property (am)n=am⋅n, we can simplify:
(45)5=45⋅5=425
(43)8=43⋅8=424
Now, substituting these back into the expression gives:
425×424÷43
Step 3: Combine the powers
Using the property am×an=am+n:
425+24÷43=449÷43
Step 4: Simplify the division
Using the property am÷an=am−n:
449−3=446
Step 5: Rewrite in terms of 64
Now we need to express 446 in terms of 64:
446 =(43)? since 64=43
Step 6: Set the exponents equal
We know:
446=(43)?⟹446=43x
Thus, we can set the exponents equal to each other:
46=3x
Step 7: Solve for x
To find x, divide both sides by 3:
x=46/3 = 15.33
17. (a)
First, subtract: 8648 − 7652 = 996
Now solve for : 996 ÷ 40 = 24.9
18. (a)
First, subtract: 683.46 − 227.39 = 456.07
Now subtract: 456.07 − 341.85 = 114.22
19. (d)
First, multiply 104/ 14 × 52/ 19:
(104/ 14) × (52/ 19) =5408/ 266 = 1352/ 67
Now add 13/ 63:
(13/ 63) + (1352/67) = 18/ 171
⇒ 4 + 12 = (?)1/2
= √169 + 7 – 27 + 107
= √256
= √16
= 4
(70 ÷ 100)? + 3 = (0.49)4 x (0.343)4 ÷ (0.2401)4
As we know that
70 ÷ 100 = 0.7
0.49 = 0.7 2
0.343 = 0.7 3
0.2401 = 0.7 4
Put these value in given question , we will get
⇒ (0.7)? + 3 = (0.72)4 x ( 0.73 )4 ÷ ( 0.74 )4
⇒ (0.7) ? + 3 = 0.7 2 x 4 x 0.7 3 x 4 ÷ 0.7 4 x 4
⇒ (0.7) ? + 3 = 0.7 8 x 0.7 12 ÷ 0.7 16
Apply the law of Fractional Exponents and Laws of Exponents
(am)(an) = am+n
am÷an=am-n
Or
am/an=am – n
we will get,
⇒ (0.7)? + 3 = (0.7) 8 + 12 – 16
⇒ (0.7)? + 3 = (0.7) 20 – 16
⇒ (0.7)? + 3 = (0.7) 4
On comparing the exponents both sides, we get
? + 3 = 4
∴ ? = 4 – 3 = 1
= 5 – [3/4 + {5/2 – (1/2 + 1/42)}]/2
= 5 – [3/4 + {5/2 – (21 + 1/42)}]/2
= 5 – [3/4 + {5/2 – 22/42)}]/2
= 5 – [3/4 + {105 – 22/42)}]/2
= 5 – [3/4 + 83/42]/2 = 5 – [63 + 166/84]/2
= 5 – 229/84/2 = 420 – 229/84/2 = 191/84/2
= 191/84 x 2 = 191/168 = 123/168
25. (a)
Numerator: 0.53 = 0.125, 0.23 = 0.008, 0.33 = 0.0270
3 × 0.5 × 0.3 × 0.2 = 0.09
So, the numerator is: 0.125 + 0.008 + 0.027 − 0.09 = 0.07
Denominator:
0.52 = 0.25, 0.22 = 0.04, 0.32 = 0.09
0.3 × 0.2 = 0.1,
So, the denominator is: 0.25 + 0.04 + 0.09 − 0.1 − 0.06 − 0.15 = 0.07
Thus, the fraction is: 0.07/ 0.07=1
26. (c)
This is a perfect cube identity:
a3+b3+c3−3abc = (a+b+c) (a2+b2+c2−ab−bc−ac)
Thus, the answer simplifies to:
a+b+c=2.247 +1.730 + 1.023 =5
27. (c)
This is a sum of cubes formula:
a3+b3/ a2−ab+b2 = a+b
Substitute, a=725 and b=371
725 + 371 = 1096
28. (b)
29. (c)
√289 – √625 ÷ √25
= √17 x 17 – √25 x 25 ÷ √5 x 5
= 17 – 25 ÷ 5 = 17 – 5 = 12
30. (c)
We know that, (a + b)2 = a2 + b2 + 2ab
= 234 + 2 x 108 = 450
(a – b)2 = a2 + b2 – 2ab
= 234 – 2 x 108 = 18
∴ (a + b)2/(a – b)2 = 450/18 = 25
⇒ [(a + b)/(a – b)]2 = 25
∴ (a + b)/(a – b) = √25 = 5
31. (b)
a2 + 1 = a
⇒ a + 1/a = 1
On squaring both sides, we get
a2 + 1/a2 + 2 = 1
On cubing both sides, we get
(a2 + 1/a2)3 = (-1)3
⇒ a6 + 1/a6 + 3a2 x 1/a2 (a2 + 1/a2) = -1
⇒ a6 + 1/a6 + 3 x (-1) = -1
Now, a6 + 1/a6 + 1 = 3
As, a12 + a6 + 1 can be written as a6 + 1/a6 + 1
∴ a12 + a6 + 1 = 3
32. (b)
Given, 0.764 x P = 1.236 x Q
⇒ Q / P = 1.236 / 0.764
Now, (Q – P) / (Q + P) = [(Q/P) – 1] / [(Q/P) + 1]
⇒ [(1.236/0.764) – 1] / [(1.236/0.764) + 1]
= (1.236 – 0.764) / (1.236 + 0.764)
= 0.472 / 2.000
= 0.236
33. (c)
∵ a + (1/b) = 1
ab + 1 = b ……(i)
Also, b + (1/c) = 1
⇒ b = 1 – (1/c) ….(ii)
From Eqs. (i) and (ii), we get
ab + 1 = 1 – (1/c)
⇒ ab = -1/c
⇒ abc = -1
34. (a)
Notice that from the given condition a+b+c=0a + b + c = 0, we can use substitutions for the terms involving sums. For example:
- a+b=−c
- b+c=−a
- c+a=−b
Using these in the first term:
(a+b)/ c+(b+c)/ a+ (c+a)/ b =(−c/ c) + (−a/ a) + (−b/ b)
This simplifies to:
−1+ (−1) + (−1) = −3
Next, simplify the second part:
a/ (b+c) +b/ (c+a) +c/ (a+b)
Again using the fact that a+b+c=0, we can substitute:
- b+c=−a
- c+a=−b
- a+b=−c
This becomes:
a/ (−a) +b/(−b) + c/(−c) =−1−1−1=−3
Now multiply the results from both parts:
(−3) × (−3) = 9
35. (b)
Since a * b = a + b + a/b
∴ 12 * 4 = 12 + 4 + 12/4
= 12 + 4 + 3 = 19
36. (a)
37. (d)
x + (a/x) = 1
⇒ x2 + a = x….(1)
⇒ x2 – x = -a….(2)
Now, (x2 +x+a)/(x3 –x2 ) = 2x/[x(x2 -x)]
= 2/(-a) [From (1) and (2)]
= -2/a
38. (a)
Given equation is,
√28 – 6√3 = √3a + b
By expanding equation,
√ 1 + 27 – 6√3 = √3a + b
⇒ √(1)2 + (3√3)2 – 2 x 3√3 = ⇒ √3a + b
⇒ √(1 – 3√3)2 = √3a + b
⇒ 1 – 3√3 = √3a + b
On comparing the left and right side equation, we get,
a = -3, b = 1
∴ a + b = -3 + 1 = – 2
39. (b)
GIVEN:
x + y = 1
Formula used:
( x + y)3 = x3 + y3 + 3xy ( x + y)
Calculation:
Here, we have
x + y = 1
On cubing both sides we get
(x + y)3 = 13
⇒ x3 + y3 + 3xy ( x +y) = 1
⇒ x3 + y3 + 3xy × 1 = 1
⇒ x3 + 3xy + y3
= 1
40. (a)
Given that,
a = (√2 – 1)1/3
By cubing the both side of the given equation.
a3 = √2 – 1
Then 1/a3 = 1/√2 – 1
Now multiply and divide the above equation by √2 + 1
⇒ 1/a3 = ( √2 + 1 ) / ( √2 + 1 ) x 1/(√2 – 1 )
⇒ 1/a3 = ( √2 + 1 ) x 1 / ( √2 + 1 ) x (√2 – 1 ) [ Use the formula (A + B)(A – B) = A2 – B2 ]
⇒ 1/a3 = ( √2 + 1 ) / ( 2 – 1 )
⇒ 1/a3 = √2 + 1
According to the question
( a – a-1 )3 + 3 ( a – a-1 )
= (a – 1/a)3 + 3(a – 1/a) Use the formula [ ( A – B)3 = A3 – B3 + 3AB(A – B) ]
= a3 – 1/a3 – 3 a x 1/a (a -1/a) + 3(a – 1/a)
= a3 – 1/a3 – 3 (a -1/a) + 3(a – 1/a)
= a3 – 1/a3
Pu the value of a3 and 1/a3
= √2 -1 – ( √2 + 1 )
= √2 -1 – √2 – 1
= -2
41. (a)
42. (b)
Given:
p + q = 10 and pq = 5
Concept used:
a2 + b2 = (a + b)2 – 2ab
Calculation:
Here,
p + q = 10 and pq = 5
⇒ p/q + q/p = (p2 + q2)/ pq
⇒ [(p + q)2 – 2 × pq]/pq
⇒ [102 – 2 × 5]/5
⇒ (100 – 10)/5 = 90/5 = 18
43. (c)
Given in question,
x + y = 18
By using algebraic formula
∵ x2 + y2 = (x + y)2 – 2xy
Put the value of x + y and xy as per given question,
⇒ x2 + y2 = (18)2 – 2 x 72
⇒ x2 + y2 = 324 – 144
⇒ x2 + y2 = 180
44. (a)
We can utilize the identity for the sum of cubes. The sum of cubes of three variables a, b, and is:
a3+b3+c3=3abc if a+b+c = 0
In our case, we have: (m−n) + (n−r) + (r−m)
Simplifying this:
(m−n) + (n−r) + (r−m) = m − n + n − r + r − m = 0
Since the sum is zero, we can apply the identity for the sum of cubes:
(m−n)3 + (n−r)3 + (r−m)3 = 3(m−n)(n−r)(r−m)
Now substitute 3(m−n)(n−r)(r−m) into the original expression:
3(m−n)(n−r)(r−m)/ 6(m−n)(n−r)(r−m)
Cancel out (m−n)(n−r)(r−m) from the numerator and denominator:
We know that,
(a + b + c)2 = (a2 + b2 + c2) + 2(ab + bc + ca)
⇒ 196 = 96 + 2(ab + bc + ca)
⇒ 2(ab + bc + ca) = 196 – 96 = 100
∴ (ab + bc + ca) = 100/2 = 50
46. (a)
Given expression can be written as
= 1/(1 x 4) + 1/(4 x 7) + 1/(7 x 10) + 1/(10 x 13) + 1/(13 x 16)
= 1/3{ 3/(1 x 4) + 3/(4 x 7) + 3/(7 x 10) + 3/(10 x 13) + 3/(13 x 16) }
= 1/3{ (4 – 1)/(1 x 4) + (7 – 4)/(4 x 7) + (10 – 7 )/(7 x 10) + (13 – 10)/(10 x 13) + (16 – 13/(13 x 16) }
= 1/3[(1 – 1/4) + (1/4 – 1/7) + (1/7 – 1/10) + (1/10 – 1/13) + (1/13 – 1/16)]
= 1/3[1 – 1/4 + 1/4 – 1/7 + 1/7 – 1/10 + 1/10 – 1/13 + 1/13 – 1/16]
= 1/3[1 – 1/16]
= 1/3[ (16 – 1)/16]
= 1/3[ (15)/16]
= 1 x 15/ 3 x 16
= 5/16
47. (a)
Let’s simplify it step by step:
(1−1/2)=1/2, (1−1/3)=2/3, (1−1/4) =3/4, …, (1−1/m)=(m−1)/m
The entire product becomes:
1/ 2 × 2/ 3 × 3/ 4 ×4/ 5⋯(m−1)/m
Notice that most terms in the numerator and denominator cancel out, leaving: 1/m
We can simplify each term inside the parentheses: 2 – (1/3) = 5/3, 2 – (3/5) = 7/5, 2 – (5/7) = 9/7
In general, the term for nn-th factor is:
2−(2n−1)/(2n+1) = 2(2n+1)−(2n−1)/ 2n+1
= 4n+2−2n+1/ 2n+1
=2n+3/ 2n+1
Thus, the product becomes:
5/ 3 × 7/ 5 × 9/ 7⋅⋯⋅1001/ 999
Notice that most terms in the numerator and denominator cancel out, leaving: 1001/ 3
(1 + 1/2) (1 + 1/3) (1 + 1/4) …. (1 + 1/150)
= 3/2 x 4/3 x 5/4 x … 151/150
= (1/2) x 151
= 151/2 = 75.5
50. (b)
