CSIR ASO/SO (Numerical Ability): Memory Based Test: (Held on: 5th Feb 2024) (Previous Year Paper)

Given:

Numbers = 12, 18, 21 and 30.

Calculation:

the least number which when doubled will be exactly divisible by 12, 18, 21 and 30.

LCM of (12, 18, 21, 30) = 1260.

∴ Required answer will be half of LCM = 1260/2 = 630.

Given:

Successive discounts of 15% and 35%

Calculation:

⇒ Equivalent discount = 1 - (1 - 15/100) × (1 - 35/100) = 0.4475 or 44.75%

Therefore, successive discounts of 15% and 35% is equivalent to a single discount of 0.4475 or 44.75%.

Shortcut Trick

Area of a rhombus = 2 × (Product of its half-diagonals).

Given half-diagonals are 75 cm and 90 cm.

Area = 2 × 75 × 90

Area = 150 × 90 = 13500 cm².

∴ The correct answer is 13500 cm².

Alternate Method

Given:

Half of first diagonal (d₁/2) = 75 cm

Half of second diagonal (d₂/2) = 90 cm

Formula Used:

Area of Rhombus = (1/2) × d₁ × d₂

⇒ First diagonal (d₁) = 2 × 75 = 150 cm

⇒ Second diagonal (d₂) = 2 × 90 = 180 cm

⇒ Area = (1/2) × 150 × 180

⇒ Area = 150 × 90

⇒ Area = 13500 cm²

∴ The correct answer is 13500 cm².

Additional Information

Diagonal Property

The diagonals of a rhombus bisect each other at right angles (90°).

Side Length Formula

The side (a) of a rhombus can be calculated using half-diagonals: a = √((d₁/2)² + (d₂/2)²).

Perimeter of Rhombus

The perimeter is 4 × side length, as all four sides of a rhombus are equal.

Given:

15.5% of 850 + 24.8% of 650 = ?

Calculation:

Let the unknown number be x.

15.5% of 850 + 24.8% of 650 = x

⇒ (\dfrac{15.5}{100}) × 850 + (\dfrac{24.8}{100}) × 650 = x

⇒ 131.75 + 161.2 = x

⇒ x = 292.95 ≈ 293

∴ 15.5% of 850 + 24.8% of 650 ≈ 293

Shortcut Trick

The fastest way to solve this is by testing the options to find a common ratio.

Let's check Option 2, which gives the value 4.

Subtracting 4 from 20, 24, and 29 gives the numbers 16, 20, and 25.

Three numbers are in continued proportion if the ratio of the first to second equals the second to third.

Checking the ratios: 16 ÷ 20 = 4/5 and 20 ÷ 25 = 4/5. The condition is perfectly satisfied.

∴ The correct answer is 4.

Alternate Method

Given:

Three numbers: 20, 24, and 29

Formula Used:

If three numbers a, b, and c are in continued proportion, then a/b = b/c or b² = a × c.

Calculations:

Let the number subtracted from each value be x.

⇒ The new numbers become (20 − x), (24 − x), and (29 − x).

⇒ According to the continued proportion rule: (20 − x) ÷ (24 − x) = (24 − x) ÷ (29 − x)

⇒ Cross-multiplying, we get:

⇒ (20 − x) × (29 − x) = (24 − x)²

⇒ 580 − 20x − 29x + x² = 576 − 48x + x²

⇒ 580 − 49x = 576 − 48x

⇒ 580 − 576 = 49x − 48x

⇒ x = 4

∴ The correct answer is 4.

Additional Information

Proportion

Four numbers a, b, c, and d are said to be in proportion if a ÷ b = c ÷ d, denoted as a:b :: c:d.

Continued Proportion

Three numbers a, b, and c are in continued proportion if a:b = b:c, meaning b² = a × c. Here, b is called the mean proportional between a and c.

Componendo and Dividendo Rule

For any valid proportion a/b = c/d, the rule states that (a + b) ÷ (a − b) = (c + d) ÷ (c − d). This property is highly useful for rapidly simplifying complex fractional equations without cross-multiplying.

Given

Mark-up rate = 44%

Discount = 50%

Formula used:

M.P = (100 + m)/100 × C.P

where, m = mark-up rate, M.P = marked price and C.P = cost price

S.P = (100 - d)/100 × M.P

where, d = discount, M.P = marked price and S.P = selling price

Loss% = (C.P - S.P)/C.P × 100

Calculation

Let the C.P. be 100.

M.P = (100 + 44)/100 × 100

M.P = 144/100 × 100

M.P = 144

Now, the S.P will be:

S.P = (100 - 50)/100 × 144

S.P = 50/100 × 144

S.P = 72

Loss% = (100 - 72)/100 × 100

Loss% = 28

∴ The correct answer is option 2.

Given:

(1 + (1 - 2)^-1 + 2 - (1 - 2)^-2)^{^2}

Calculations:

(1 + (1 - 2)^-1 + 2 - (1 - 2)^-2)²

(1 - 2) is -1, so (1 - 2)^-1 is -1 and (1 - 2)^-2 is 1

⇒ (1 - 1 + 2 - 1)²

⇒ (1)² = 1

∴ The value of the given expression is 1.

Given:

2P = 3Q = 5R

P + Q + R = 6200

Concept or Formula:

Use the given equations to express P, Q, and R in terms of a common variable.

Substitute these expressions into the second given equation to solve for the common variable.

Then calculate the values of P and R using the common variable.

The difference between P and R can be found by subtracting the smaller value from the larger one.

Solution:

Let 2P = 3Q = 5R = k (some constant)

⇒ P = k/2, Q = k/3, and R = k/5

Substitute P, Q, R in P + Q + R = 6200

⇒ k/2 + k/3 + k/5 = 6200

Solve the above equation for k,

⇒ k (1/2 + 1/3 + 1/5) = 6200

⇒ k (15/30 + 10/30 + 6/30) = 6200

⇒ k × 31/30 = 6200

⇒ k = 6200 × 30/31

⇒ k = 6000

Substitute k = 6000 in P = k/2 and R = k/5

⇒ P = 6000/2 = 3000 and R = 6000/5 = 1200

Find the difference between P and R

⇒ P - R = 3000 - 1200 = 1800

Therefore, the difference between P and R is 1800.

Shortcut Trick

Radius (r) = 3.5 m / 2 = 1.75 m = 175 cm.

Inner Surface Area of dome = 2 × π × r².

Area = 2 × (22/7) × 175 × 175 = 1,92,500 cm².

Total Cost = Area × Rate = 1,92,500 × 0.20 = ₹38,500.

∴ The correct answer is ₹38,500.

Alternate Method

Given:

Diameter of the hemispherical dome (d) = 3.5 m

Cost of painting = 20 paise per cm²

Formula Used:

Inner surface area of a hemisphere = 2πr²

Total Cost = Surface Area × Rate

⇒ Radius (r) = d / 2 = 3.5 / 2 = 1.75 m

⇒ Radius in cm = 1.75 × 100 = 175 cm

⇒ Inner Surface Area = 2 × (22/7) × 175 × 175

⇒ Inner Surface Area = 2 × 22 × 25 × 175 = 1,92,500 cm²

⇒ Rate of painting = 20 paise = 20/100 = ₹0.20 per cm²

⇒ Total Cost = 1,92,500 × 0.20

⇒ Total Cost = ₹38,500

∴ The correct answer is ₹38,500.

Additional Information

Surface Area of a Sphere

The total surface area of a complete sphere is given by 4πr², which is exactly double that of the curved surface area of a hemisphere.

Total Surface Area of a Hemisphere

If the base of the hemisphere is also included (closed hemisphere), the total surface area is 3πr² (Curved area + Base area).

Volume of a Hemisphere

The volume of a hemisphere is half the volume of a sphere, calculated as V = (2/3)πr³.

Cost Calculation Rule

Always ensure that the units of dimensions (cm, m) and the rate (per cm², per m²) are consistent before performing the final multiplication.

Calculation:

Total production of wheat by 6 states = 360° = 3600

⇒ 1° = 10

Production of state P = 110° = 110 × 10 = 1100

Production of state Q = 70° = 70 × 10 = 700

Average production by state P & Q = (1100 + 700)/2 = 900

Production of state R = 80° = 80 × 10 = 800

Production of state T = 20° = 20 × 10 = 200

Average production by state R & T = (800 + 200)/2 = 500

Difference between the average production of P and Q and R & T = 900 - 500 = 400

∴ The correct answer is 400.

Given

Time (T1) = 7 years

Time (T2) = ?

Formula used

S.I = PRT/100

Where SI = simple interest

P = principal

R = interest rate (in percentage)

T = time duration (in years)

Solution

Let the principal = a

Amount = 2a

S.I = 2a - a = a

a = (a × r × 7)/100

r = 100/7%

Now since the amount becomes 8 times,

S.I = 8a - a = 7a

7a = (a × 100/7 × T₂)/100

T₂ = 49 years

The correct option is 2.

Formula used:

(a +b)² = a² + b² + 2ab

Calculation:

(\sqrt{4-\sqrt{7}+\sqrt{19+2 \sqrt{84}}} )

⇒ (\sqrt{4-\sqrt{7}+\sqrt{12+7+2 \sqrt{84}}} )

⇒ (\sqrt{4-\sqrt{7}+\sqrt{(\sqrt12)^2+(\sqrt7)^2+2 \sqrt{12} \sqrt{7}}} )

⇒ (\sqrt{4-\sqrt{7}+\sqrt{(\sqrt12+\sqrt7)^2}} )

⇒ (\sqrt{4-\sqrt{7}+\sqrt12+\sqrt7} )

⇒ (\sqrt{4+\sqrt12} )

⇒ (\sqrt{1+3+\sqrt12} )

⇒ (\sqrt{1+3+2\sqrt3} )

⇒ (\sqrt{(1)^2+(\sqrt3)^2+2\times1\times\sqrt3} )

⇒ (\sqrt{(1+\sqrt3)^2} ) = (1+\sqrt{3} )

∴ The correct answer is option (1).

Shortcut Trick

Sum of ratio parts = 2 + 3 + 5 = 10 units.

Each variable appears twice in the sum, so 2 × (a + b + c) = 10 units.

Since a + b + c = 20, then 2 × 20 = 40 = 10 units ⇒ 1 unit = 4.

Therefore, a + b = 2 units = 2 × 4 = 8.

Value of c = (a + b + c) − (a + b) = 20 − 8 = 12.

∴ The correct answer is 12.

Alternate Method

Given: (a + b) : (b + c) : (c + a) = 2 : 3 : 5 and a + b + c = 20.

⇒ Let a + b = 2k, b + c = 3k, and c + a = 5k.

⇒ Adding the ratios: (a + b) + (b + c) + (c + a) = 2k + 3k + 5k.

⇒ 2(a + b + c) = 10k.

⇒ Given a + b + c = 20, substitute: 2(20) = 10k.

⇒ 40 = 10k ⇒ k = 4.

⇒ Now, a + b = 2k = 2 × 4 = 8.

⇒ We know c = (a + b + c) − (a + b).

⇒ c = 20 − 8 = 12.

∴ The correct answer is 12.

Additional Information

Constant of Proportionality

When terms are given in a ratio (x : y = m : n), they can be expressed as x = mk and y = nk, where k is the common multiplier.

Cyclic Ratio Sums

For expressions involving pairs like (a+b), (b+c), and (c+a), the sum of all pairs is always equal to 2(a + b + c).

Finding Individual Variables

To find a single variable like c from the total sum S = a+b+c, subtract the sum of the other two variables: c = S − (a+b).

Given:

The ratio of the radius and height of the cylinder = 3: 2

Radius = 21 cm

Formula used:

The curved surface area of the cylinder = 2πrh

Here,

r = radius

h = height

Calculation:

Let the radius of the base and height of the cylinder be 3x and 2x

So,

3x → 21

2x → 21/3 × 2 = 14

Curved surface area = 2π × 21 × 14

 

⇒ 2 × (22/7) × 21 × 14

⇒ 2 × 22 × 21 × 2

⇒ 1848

∴ The curved surface area (in cm2) of the cylinder is 1848.

Shortcut Trick

Marked Price (MP) = Rs. 10000

Price after 1st discount (15%) = 10000 × (85 ÷ 100) = Rs. 8500

Final Selling Price (SP) = Rs. 7650

Second discount amount = 8500 − 7650 = Rs. 850

x% = (850 ÷ 8500) × 100 = 10%

∴ The correct answer is 10 percent.

Alternate Method

Given: MP = Rs. 10000, SP = Rs. 7650, d₁ = 15%, d₂ = x%

Formula: SP = MP × (100 − d₁) ÷ 100 × (100 − d₂) ÷ 100

⇒ 7650 = 10000 × (85 ÷ 100) × (100 − x) ÷ 100

⇒ 7650 = 85 × (100 − x)

⇒ 100 − x = 7650 ÷ 85

⇒ 100 − x = 90

⇒ x = 10

∴ The correct answer is 10 percent.

Additional Information

Successive Discounts

When two discounts a% and b% are given, the single equivalent discount is (a + b − ab ÷ 100)%.

Marked Price & Selling Price

Selling Price is the final amount paid by the customer after all discounts are subtracted from the Marked Price.

Given:

Marked price of a chair is 10 percent more than its cost price

The selling price of the chair is Rs. 300

The discount is Rs. 140

Formula used:

Loss = Cost price - Selling price

Calculation:

Let cost price of the chair is X

The marked price of the chair is = 110 percent of the cost price = 1.1X

⇒ 1.1X - 140 = 300

⇒ 1.1X = 440

⇒ X = 400

The cost price is = 400

The selling price is = 300

The loss percentage = ({400\ -\ 300\over 400}\times 100\ =\ 25 %)

∴ The required result will be 25%.

Given Expression:

((6-3)\div[(9-6)\div{(6-4)\div\left(2+\frac{8}{13}\right)}] )

Concept used:

Calculation:

((6-3)\div[(9-6)\div{(6-4)\div\left(2+\frac{8}{13}\right)}] )

= 3 ÷ [(3) ÷ {(2) ÷ ((\frac{26+8}{13})) }]

= 3 ÷ [(3) ÷ {2 ÷ (\frac{34}{13}))}]

= 3 ÷ [3 ÷ {2 × (\frac{13}{34})}]

= 3 ÷ [3 ÷ {(\frac{13}{17})}]

= 3 ÷ [3 × (\frac{17}{13})]

= 3 × [(\frac{13}{3 × 17})]

= (\frac{13}{17})

Thus, the required value is (\frac{13}{17}).

Hence, the correct answer is option 3).

Given data:

Two numbers: 0.75 and 3

Concept: The mean proportional (geometric mean) of two numbers a and b is √(ab).

⇒ Mean proportional = √(0.75 ×  3) = 1.5

Therefore, the mean proportional of 0.75 and 3 is 1.5.

Shortcut Trick

In Compound Interest, if a sum becomes 'n' times in 'T' years, it becomes 'n^k' times in 'k × T' years.

Here, the sum doubles (2 times) in 10 years (n = 2, T = 10).

To become four times (4 = 2²), the power k = 2.

Time required = k × T = 2 × 10 = 20 years.

∴ The correct answer is 20 years.

Alternate Method

Given:

Initial Sum = P

Amount after 10 years = 2P

Target Amount = 4P

Formula Used:

A = P(1 + r/100)^t

Calculations:

⇒ 2P = P(1 + r/100)¹⁰

⇒ 2 = (1 + r/100)¹⁰ ---- (i)

⇒ Let the required time be 't' years for the sum to become 4P.

⇒ 4P = P(1 + r/100)^t

⇒ 4 = (1 + r/100)^t

⇒ (2)² = (1 + r/100)^t

⇒ Substituting value of 2 from (i):

⇒ [(1 + r/100)¹⁰]² = (1 + r/100)^t

⇒ (1 + r/100)²⁰ = (1 + r/100)^t

⇒ By comparing powers, t = 20 years.

∴ The correct answer is 20 years.

Additional Information

Effective Rate of Interest

When interest is compounded more than once a year, the effective annual rate is (1 + r/n)ⁿ − 1, where n is the frequency of compounding.

Rule of 72

A quick way to estimate how long it takes to double money at CI: Time ≈ 72 ÷ Interest Rate.

SI vs CI for Doubling

In Simple Interest, if a sum doubles in T years, it becomes n times in (n−1) × T years. For 4 times, it would take (4−1) × 10 = 30 years.

Given:

4374562 × 64 = ? × 7777

Calculation:

4374562 × 64 = ? × 7777

Let the unknown number be x.

⇒ 4374562 × 64 = x × 7777

⇒ x = (\dfrac{4374562 \times64}{7777}) 

⇒ x = 35999.995 ≈ 36000

∴ Option (3) is the correct answer.

Shortcut Trick

LCM of 32 and 40 = 160.

According to the given condition: X² − 9 = 160.

Adding 9 to both sides: X² = 169.

Taking square root: X = 13.

∴ The correct answer is 13.

Alternate Method

Given:

Numbers = 32 and 40.

Condition: LCM(32, 40) = X² − 9.

Step-by-step Calculation:

⇒ Prime factorization of 32 = 2 × 2 × 2 × 2 × 2 = 2⁵.

⇒ Prime factorization of 40 = 2 × 2 × 2 × 5 = 2³ × 5.

⇒ LCM (32, 40) = Highest power of all factors = 2⁵ × 5 = 32 × 5 = 160.

⇒ According to the problem: 160 = X² − 9.

⇒ X² = 160 + 9.

⇒ X² = 169.

⇒ X = √169.

⇒ X = 13 (Taking positive value as required).

∴ The correct answer is 13.

Additional Information

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest positive integer that is perfectly divisible by each of the given numbers.

Perfect Squares

A perfect square is an integer that is the square of an integer; for example, 169 is a perfect square because it equals 13².

Relationship between LCM and HCF

For any two positive integers a and b, the relationship is: LCM(a, b) × HCF(a, b) = a × b.

Given: 

The total price of all these 6 articles is 5040.

Concept Used:

All slices of the pie chart add up to make the whole equal 100 percent or 360 degrees.

Calculation:

For statement I:

The total price of A1 = 32°/360° ×  5040 = 448

The total price of A2 = 46°/360° × 5040 = 644

The total price of A4 = 50°/360° × 5040 = 700

The total price of A6 = 70°/360° × 5040 = 980

The total price of A1, A2, A4 and A6 = 448 + 644 + 700 + 980

= 2772

So, Statement I is correct.

For statement II:

The total price of A3 = 132°/360° × 5040 = 1848

The total price of A5 = 30°/360° × 5040 = 420

The total price of A4 = 50°/360° × 5040 = 700

The total price of A6 = 70°/360° × 5040 = 980

The total price of A3 and A5 together = 1848 + 420

= 2268

The total price of A4 and A6 together = 980 + 700

= 1680

The difference between the total price of A3 and A5 together and the total price of A4 and A6 together

= 2268 - 1680

= 588

So, Statement II is not correct.

Only statement II is not correct.

Given:

675.456 + 12.492 × 55.671 = ?

Calculation :

Let the unknown number be x.

⇒ 675.456 + 12.492 × 55.671 = x 

⇒ 675.5 + 12.5 × 55.5 = x         (55.671 ≈ 55.5)

⇒ 675.5 + 693.75 = x  

⇒x = 1369.25 ≈ 1371

∴ Option (4) is the correct answer.

Calculation:

The angle formed by sectors A, B and C together is  (13 + 24 +18) = 55%

The central angle formed by sectors E, F and G together is (7 + 4 + 15) = 26%

The difference is 55 - 26 = 29%

As per the question,

Value of 100% is 360° 

So, value of 29% will be 29 × 3.6 = 104.4° 

∴ The correct option is 3

Calculation:

Area of square room

= (10)2 = 100 sq m = 100 × (100)2 sq cm

= 100 × 100 × 100 sq cm

Now, area of tile

= (50)2 = 50 × 50 sq cm

∴ Number of tiles needed

(= \frac{{Area;of;square;room}}{{Area;of;tile}})

(= \frac{{100 \times 100 \times 100}}{{50 \times 50}} = 400)