Q. 1
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
How much is $\frac{1}{4} of \frac{x}{4}$?
💡
Step-by-Step Explanation & Concept Rationale
$\frac{4}{4}×\frac{x}{4}=\frac{x}{16}$.
Q. 2
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
What is $\frac{1}{5}$ of $\frac{x}{5}$?
💡
Step-by-Step Explanation & Concept Rationale
$\left(\frac{1}{5}\right) \times \left(\frac{x}{5}\right) = \frac{x}{25}$
Q. 3
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Value of $64×128×512×2^{-3}$ is?
💡
Step-by-Step Explanation & Concept Rationale
To solve this, express each number as a power of base 2:
\(64 = 2^6\)
\(128 = 2^7\)
\(512 = 2^9\)
Now, substitute these back into the expression:
\(2^{6}\times 2^{7}\times 2^{9}\times 2^{-3}\)
When multiplying terms with identical bases, you add the exponents together:
\(2^{6+7+9+(-3)}\)
\(2^{22-3}=\mathbf{2}^{\mathbf{19}}\)
\(64 = 2^6\)
\(128 = 2^7\)
\(512 = 2^9\)
Now, substitute these back into the expression:
\(2^{6}\times 2^{7}\times 2^{9}\times 2^{-3}\)
When multiplying terms with identical bases, you add the exponents together:
\(2^{6+7+9+(-3)}\)
\(2^{22-3}=\mathbf{2}^{\mathbf{19}}\)
Q. 4
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Geometric mean of $a^{2n}×b^{2n}$?
💡
Step-by-Step Explanation & Concept Rationale
The Geometric Mean (GM) of two quantities \(x\) and \(y\) is found by taking the square root of their product:
\(\text{GM}=\sqrt{x\times y}\)
Substitute the given expressions:
Here, \(x = a^{2n}\) and \(y = b^{2n}\).
\(\text{GM}=\sqrt{a^{2n}\times b^{2n}}\)
Combine the bases under the exponent:
\(\text{GM}=\sqrt{(ab)^{2n}}\)
Simplify the square root:
Taking the square root of an exponential expression is the same as dividing its exponent by 2:
\(\text{GM}=\left((ab)^{2n}\right)^{\frac{1}{2}}=(ab)^{n}=\mathbf{a}^{\mathbf{n}}\mathbf{b}^{\mathbf{n}}\)
\(\text{GM}=\sqrt{x\times y}\)
Substitute the given expressions:
Here, \(x = a^{2n}\) and \(y = b^{2n}\).
\(\text{GM}=\sqrt{a^{2n}\times b^{2n}}\)
Combine the bases under the exponent:
\(\text{GM}=\sqrt{(ab)^{2n}}\)
Simplify the square root:
Taking the square root of an exponential expression is the same as dividing its exponent by 2:
\(\text{GM}=\left((ab)^{2n}\right)^{\frac{1}{2}}=(ab)^{n}=\mathbf{a}^{\mathbf{n}}\mathbf{b}^{\mathbf{n}}\)
Q. 5
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Value of $(2/7)^{−2}$?
💡
Step-by-Step Explanation & Concept Rationale
Apply the negative exponent rule:
A negative exponent tells you to invert the fraction (find its reciprocal) and make the exponent positive:
\(\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^{n}\)
\(\left(\frac{2}{7}\right)^{-2}=\left(\frac{7}{2}\right)^{2}\)
A negative exponent tells you to invert the fraction (find its reciprocal) and make the exponent positive:
\(\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^{n}\)
\(\left(\frac{2}{7}\right)^{-2}=\left(\frac{7}{2}\right)^{2}\)
Q. 6
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Expansion of $(X+3)^{19}$ will contain __ terms?
💡
Step-by-Step Explanation & Concept Rationale
In binomial expansion (a+b)n, the number of terms is always n+1.
According to the Binomial Theorem, when you expand a binomial expression raised to a positive integer power \(n\) (in the form \((a+b)^n\)), the total number of terms in the expanded form is always equal to \(n + 1\).
Calculation
Given exponent (\(n\)) = \(19\)
Total terms = \(19 + 1 = \mathbf{20}\)
According to the Binomial Theorem, when you expand a binomial expression raised to a positive integer power \(n\) (in the form \((a+b)^n\)), the total number of terms in the expanded form is always equal to \(n + 1\).
Calculation
Given exponent (\(n\)) = \(19\)
Total terms = \(19 + 1 = \mathbf{20}\)
Q. 7
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Find the value of: 1/(√5 + √3) + 1/(√5 − √3)
💡
Step-by-Step Explanation & Concept Rationale
1/(√5+√3) + 1/(√5−√3) = (√5−√3 + √5+√3)/(5−3) = 2√5 / 2 = √5.
Q. 8
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Value of $\frac{5^{3.3.3}}{5^{3+3+3}}$?
💡
Step-by-Step Explanation & Concept Rationale
$\frac{5^{27}}{5^9}=5^{(27−9)}=5^{18}$.
Q. 9
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
The value of $\frac{5.293\times5.293-3.633\times3.633}{8.926}$
💡
Step-by-Step Explanation & Concept Rationale
This expression follows the algebraic identity \(a^2 - b^2 = (a - b)(a + b)\).
Let \(a = 5.293\)
Let \(b = 3.633\)
Find \((a + b)\): \(5.293 + 3.633 = 8.926\)
Find \((a - b)\): \(5.293 - 3.633 = 1.660\)
Now, substitute these back into the original fraction:
\(\frac{(5.293+3.633)\times (5.293-3.633)}{8.926}\)
\(\frac{8.926\times 1.660}{8.926}=1.66\)
Let \(a = 5.293\)
Let \(b = 3.633\)
Find \((a + b)\): \(5.293 + 3.633 = 8.926\)
Find \((a - b)\): \(5.293 - 3.633 = 1.660\)
Now, substitute these back into the original fraction:
\(\frac{(5.293+3.633)\times (5.293-3.633)}{8.926}\)
\(\frac{8.926\times 1.660}{8.926}=1.66\)
Q. 10
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Value of $\frac{3.8x3.8x3.8+1.2x1.2x1.2}{3.8×3.8+1.44-3.8x1.2}$
💡
Step-by-Step Explanation & Concept Rationale
This expression matches the identity \(\frac{a^3 + b^3}{a^2 + b^2 - ab} = a + b\).
Let a = 3.8
Let b = 1.2
Notice that:
a³ = 3.8 × 3.8 × 3.8
b³ = 1.2 × 1.2 × 1.2
b² = 1.2 × 1.2 = 1.44
ab = 3.8 × 1.2
Since the denominator simplifies to a² + b² - ab, the entire expression cancels out to leave just:
\(a+b=3.8+1.2=5\)
Let a = 3.8
Let b = 1.2
Notice that:
a³ = 3.8 × 3.8 × 3.8
b³ = 1.2 × 1.2 × 1.2
b² = 1.2 × 1.2 = 1.44
ab = 3.8 × 1.2
Since the denominator simplifies to a² + b² - ab, the entire expression cancels out to leave just:
\(a+b=3.8+1.2=5\)
Q. 11
Quantitative Aptitude Test
Difficulty: Medium
(1 Mark)
The sum of five successive number is 100. The product of the first and the last number is
💡
Step-by-Step Explanation & Concept Rationale
Let the numbers be: \(x\), \(x+1\), \(x+2\), \(x+3\), and \(x+4\)
Add them up: \(5x + 10 = 100\)
Solve for \(x\): \(5x = 90 \implies x = 18\)
The five numbers are 18, 19, 20, 21, and 22.
Product of the first and last number: \(18 \times 22 = \mathbf{396}\)
Add them up: \(5x + 10 = 100\)
Solve for \(x\): \(5x = 90 \implies x = 18\)
The five numbers are 18, 19, 20, 21, and 22.
Product of the first and last number: \(18 \times 22 = \mathbf{396}\)
Q. 12
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Solve:
\(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\text{ is equal to}\)
\(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\text{ is equal to}\)
💡
Step-by-Step Explanation & Concept Rationale
Find a common denominator to combine the two fractions:
\(\frac{(\sqrt{7}+\sqrt{5})^{2}+(\sqrt{7}-\sqrt{5})^{2}}{(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5})}\)
Expand the numerator:
\((\sqrt{7}+\sqrt{5})^{2}=7+5+2\sqrt{35}=12+2\sqrt{35}\)
\((\sqrt{7}-\sqrt{5})^{2}=7+5-2\sqrt{35}=12-2\sqrt{35}\)
\(\text{Total Numerator}=(12+2\sqrt{35})+(12-2\sqrt{35})=24\)
Simplify the denominator using the difference of squares identity
\((a-b)(a+b) = a^2 - b^2\):\((\sqrt{7})^{2}-(\sqrt{5})^{2}=7-5=2\)
Divide:
\(\frac{24}{2}=12\)
\(\frac{(\sqrt{7}+\sqrt{5})^{2}+(\sqrt{7}-\sqrt{5})^{2}}{(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5})}\)
Expand the numerator:
\((\sqrt{7}+\sqrt{5})^{2}=7+5+2\sqrt{35}=12+2\sqrt{35}\)
\((\sqrt{7}-\sqrt{5})^{2}=7+5-2\sqrt{35}=12-2\sqrt{35}\)
\(\text{Total Numerator}=(12+2\sqrt{35})+(12-2\sqrt{35})=24\)
Simplify the denominator using the difference of squares identity
\((a-b)(a+b) = a^2 - b^2\):\((\sqrt{7})^{2}-(\sqrt{5})^{2}=7-5=2\)
Divide:
\(\frac{24}{2}=12\)
Q. 13
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Solve:
\(\sqrt{121}+?=1569+74\)
\(\sqrt{121}+?=1569+74\)
💡
Step-by-Step Explanation & Concept Rationale
11+x=1643⟹x=1632.
Q. 14
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Solve:
\(\frac{4}{3}-\frac{?}{12}=1\)
\(\frac{4}{3}-\frac{?}{12}=1\)
💡
Step-by-Step Explanation & Concept Rationale
16/12−x/12=12/12
⟹16−x=12
⟹x=4.
⟹16−x=12
⟹x=4.
Q. 15
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
208÷13=2×?
💡
Step-by-Step Explanation & Concept Rationale
16=2x⟹x=8.
Q. 16
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
If y−7=12, then y+19= ——?
💡
Step-by-Step Explanation & Concept Rationale
y=12+7=19. Therefore, y+19=19+19=38.
Q. 17
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Тo a certain number 8 is added. The sum is multiplied by 3, then the product is divided by 2 and 7 is subtracted from the quotient. The remainder left is 50. The number is:
💡
Step-by-Step Explanation & Concept Rationale
We can solve this problem by working backward from the final remainder of 50:
Undo the subtraction:
Before subtracting 7, the number was:
\(50+7=57\)
Undo the division:
Before dividing by 2, the product was:
\(57\times 2=114\)
Undo the multiplication:
Before multiplying by 3, the sum was:
\(114\div 3=38\)
Undo the addition:
Before adding 8, the original number was:
\(38-8=\mathbf{30}\)
Undo the subtraction:
Before subtracting 7, the number was:
\(50+7=57\)
Undo the division:
Before dividing by 2, the product was:
\(57\times 2=114\)
Undo the multiplication:
Before multiplying by 3, the sum was:
\(114\div 3=38\)
Undo the addition:
Before adding 8, the original number was:
\(38-8=\mathbf{30}\)
Q. 18
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Minimum value of $x^{3}+10x+7$?
💡
Step-by-Step Explanation & Concept Rationale
To find the minimum value of the cubic function \(f(x) = x^3 + 10x + 7\), we look at its first derivative to check for critical points (peaks or valleys):
Find the derivative:
\(f^{\prime }(x)=3x^{2}+10\)
Analyze the derivative:
Since any real number squared (\(x^{2}\)) is always zero or positive, \(3x^2 \ge 0\). Adding \(10\) means that:
\(f^{\prime }(x)\ge 10\quad \text{for all real values of }x.\)
Conclusion:
Because the derivative is always positive (\(f'(x) > 0\)), the graph of the function is strictly increasing from left to right. It has no local minimum or local maximum points.As \(x\) goes towards negative infinity (\(-\infty \)), the value of \(x^3 + 10x + 7\) also goes towards negative infinity (\(-\infty \)). Because it decreases without bound, it does not have a minimum real number value, making None of these the correct choice.
Find the derivative:
\(f^{\prime }(x)=3x^{2}+10\)
Analyze the derivative:
Since any real number squared (\(x^{2}\)) is always zero or positive, \(3x^2 \ge 0\). Adding \(10\) means that:
\(f^{\prime }(x)\ge 10\quad \text{for all real values of }x.\)
Conclusion:
Because the derivative is always positive (\(f'(x) > 0\)), the graph of the function is strictly increasing from left to right. It has no local minimum or local maximum points.As \(x\) goes towards negative infinity (\(-\infty \)), the value of \(x^3 + 10x + 7\) also goes towards negative infinity (\(-\infty \)). Because it decreases without bound, it does not have a minimum real number value, making None of these the correct choice.
Q. 19
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Expression \(x^{2}−3x+5\) has a minimum value for x equal to:
💡
Step-by-Step Explanation & Concept Rationale
The expression \(f(x) = x^2 - 3x + 5\) is a quadratic equation where the coefficient of \(x^{2}\) is positive (\(a = 1\)). This means its graph is a parabola that opens upward, and its lowest point (minimum value) occurs at its vertex.
To find the value of \(x\) at the vertex, we use the vertex formula:
\(x=-\frac{b}{2a}\)
Identify the coefficients:
\(a = 1\)
\(b = -3\)
Plug the numbers into the formula:
\(x=-\frac{-3}{2(1)}=\frac{3}{2}=\mathbf{1.5}\)
Therefore, the expression reaches its minimum value when \(x\) is equal to 1.5.
To find the value of \(x\) at the vertex, we use the vertex formula:
\(x=-\frac{b}{2a}\)
Identify the coefficients:
\(a = 1\)
\(b = -3\)
Plug the numbers into the formula:
\(x=-\frac{-3}{2(1)}=\frac{3}{2}=\mathbf{1.5}\)
Therefore, the expression reaches its minimum value when \(x\) is equal to 1.5.
Q. 20
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
2/5 of students are boys. If 50 boys, how many girls?
💡
Step-by-Step Explanation & Concept Rationale
2/5x=50⟹x=125 total. Girls = 125−50=75.
Q. 21
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
HCF of $(x^4+3x^2−4) \ and \ (x^4−4x^2+3)$?
💡
Step-by-Step Explanation & Concept Rationale
Factorization:
$(x^2+4)(x^2−1)$
and
$(x^2−3)(x^2−1)$.
Common is $(x^2−1)$.
$(x^2+4)(x^2−1)$
and
$(x^2−3)(x^2−1)$.
Common is $(x^2−1)$.
Q. 22
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
The sum of two numbers is twice their difference. If one of the number is 10, the other number is
💡
Step-by-Step Explanation & Concept Rationale
x+10=2(x−10)
⟹x=30;
x+10=2(10−x)
⟹3x=10.
⟹x=30;
x+10=2(10−x)
⟹3x=10.
Q. 23
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
7/?=2/26
💡
Step-by-Step Explanation & Concept Rationale
7/x=1/13⟹x=7×13=91.
Q. 24
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
208 ÷ 13 = 2 × ?
💡
Step-by-Step Explanation & Concept Rationale
208 ÷ 13 = 16. Then 16 = 2 × ? ⇒ ? = 8.
Q. 25
Quantitative Aptitude Test
Difficulty: Hard
(1 Mark)
Divide 37 into 2 parts: 5 times one + 11 times other = 227.
💡
Step-by-Step Explanation & Concept Rationale
5x+11(37−x)=227⟹407−6x=227⟹6x=180⟹x=30. Parts 30, 7.
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