CUET UG Mathematics Booster Test 2 - Properties of Indefinite Integrals -

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CUET UG Mathematics Booster Test 2 - Properties of Indefinite Integrals

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

CUET UG Mathematics Booster Test 2 - Properties of Indefinite Integrals

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QUESTION 1 OF 20

Let
F(x)=d/dx (∫₀ˣ t sin t dt).
By the Fundamental Theorem of Calculus (inverse operations), evaluate the numerical output of F(π/2).

Answer Review

1 Let
F(x)=d/dx (∫₀ˣ t sin t dt).
By the Fundamental Theorem of Calculus (inverse operations), evaluate the numerical output of F(π/2).

A 0

B π/2

C 1

D π

💡 Short Explanation

Apply the Fundamental Theorem of Calculus. F(x)=x sin x. At x=π/2, sin(π/2)=1.

✅ Correct Answer Explanation

The Fundamental Theorem of Calculus states that d/dx[∫₀ˣ f(t)dt]=f(x). Thus F(x)=x sin x. Substituting x=π/2 gives F(π/2)=(π/2)×1=π/2. Hence Option B is correct. Options A, C, and D do not satisfy the direct substitution result obtained from the theorem.

❓ Why Other Options Are Incorrect

Option A → F(π/2) is not zero because sin(π/2)=1.
Option C → Ignores the factor π/2 from x.
Option D → Incorrectly doubles the actual value.

🎯 Strategy

Used: Substitution

Application: First apply the Fundamental Theorem, then substitute x=π/2.

Final Logic: F(x)=x sin x ⇒ F(π/2)=π/2.

🧠 Memory Trick

"Upper limit becomes the function."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Fundamental Theorem of Calculus

2 Which of the following is INCORRECT regarding the difference between
∫d/dx[f(x)] dx and d/dx [∫f(x) dx]?

A The first results in f(x)+C.

B The second results exactly in f(x).

C Both operations yield identical functions without any constants.

D Integration recovers the function up to an arbitrary constant.

💡 Short Explanation

Integration introduces a constant. Differentiation of an integral does not. Therefore both results are not identical.

✅ Correct Answer Explanation

∫f'(x)dx=f(x)+C, whereas d/dx[∫f(x)dx]=f(x). The first expression contains an arbitrary constant while the second does not. Therefore Option C is incorrect. Options A, B, and D correctly describe the inverse relationship between differentiation and integration.

❓ Why Other Options Are Incorrect

Option A → Correct statement of anti-differentiation.
Option B → Correct Fundamental Theorem result.
Option D → Integration always determines a family of functions differing by constants.

🎯 Strategy

Used: Odd One Out

Application: Compare the presence or absence of the integration constant.

Final Logic: Only Option C ignores the arbitrary constant.

🧠 Memory Trick

"Integral adds C; derivative removes C."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration as Inverse of Differentiation

3 Given
d/dx[sin²x]=sin2x and d/dx[−cos²x]=sin2x,
which of the following are correct?
1. ∫sin2x dx can be written as sin²x+C₁.
2. ∫sin2x dx can be written as −cos²x+C₂.
3. sin²x+cos²x depends strictly on the variable x.

A 1 and 2 only

B 2 and 3 only

C 1 and 3 only

D 1, 2, and 3

💡 Short Explanation

Both functions differentiate to sin2x. Anti-derivatives differ by constants. sin²x+cos²x=1.

✅ Correct Answer Explanation

Since both sin²x and −cos²x have derivative sin2x, statements 1 and 2 are correct. Statement 3 is false because sin²x+cos²x=1, a constant independent of x. Therefore only statements 1 and 2 are valid, making Option A correct.

❓ Why Other Options Are Incorrect

Option B → Statement 3 is false.
Option C → Statement 2 is also true and cannot be omitted.
Option D → Includes incorrect statement 3.

🎯 Strategy

Used: Elimination

Application: Verify each statement individually using derivatives and identities.

Final Logic: Statements 1 and 2 true; statement 3 false.

🧠 Memory Trick

"Same derivative ⇒ same family."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Family of Anti-derivatives

4 Two curves y=F(x) and y=G(x) are both anti-derivatives of f(x)=eˣ. If the definite area under F(x) from 0 to 1 is A₁, and it is known that G(x)=F(x)+3, what is the exact area under G(x) from 0 to 1?

A A₁

B A₁+1

C A₁+e

D A₁+3

💡 Short Explanation

G(x)=F(x)+3. Area increases by area of constant strip. Width is 1 unit.

✅ Correct Answer Explanation

The area under G(x) is ∫₀¹ [F(x)+3]dx =∫₀¹F(x)dx + ∫₀¹3dx =A₁+3(1−0) =A₁+3. Hence Option D is correct. The constant difference between anti-derivatives adds a rectangular area of height 3 and width 1.

❓ Why Other Options Are Incorrect

Option A → Ignores the added constant.
Option B → Uses an incorrect rectangle area.
Option C → e does not arise from the calculation.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Interpret the added constant as a rectangle under the curve.

Final Logic: Extra area = 3×1 = 3.

🧠 Memory Trick

"Constant shift adds rectangle area."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Definite Integral Properties

5 Assertion (A):
∫(f(x)·g(x)) dx = ∫f(x) dx · ∫g(x) dx.
Reason (R): The integral of a quotient is simply the quotient of their integrals.

A Both false

B A true, R false

C Both true, R explains A

D A false, R true

💡 Short Explanation

Integral of a product is not product of integrals. Integral of a quotient is not quotient of integrals. Both statements violate linearity rules.

✅ Correct Answer Explanation

Integration is linear over sums and constant multiples only. In general, ∫fg dx ≠ (∫f dx)(∫g dx) and ∫(f/g)dx ≠ (∫f dx)/(∫g dx). Therefore both the assertion and reason are false. NCERT integration rules never permit product or quotient separation.

❓ Why Other Options Are Incorrect

Option B → Assertion is false.
Option C → Both statements are not true.
Option D → Reason is also false.

🎯 Strategy

Used: Extreme Word Filter

Application: Words like "simply" and universal equalities often indicate incorrect statements.

Final Logic: Integration preserves sums, not products or quotients.

🧠 Memory Trick

"Sums split, products don't."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Linearity of Integration

6 Match the complex linear additions with their integrated forms:

List I	List II
1. ∫(sec x tan x+1/x) dx	a. 3ˣ/ln3 + 4ˣ/ln4 + C
2. ∫(eˣ−xᵉ) dx	b. sin⁻¹x − tan⁻¹x + C
3. ∫(3ˣ+4ˣ) dx	c. sec x + ln\|x\| + C
4. ∫(1/√(1−x²)−1/(1+x²)) dx	d. eˣ − x^(e+1)/(e+1) + C

A 1-a, 2-b, 3-c, 4-d

B 1-b, 2-d, 3-a, 4-c

C 1-c, 2-d, 3-a, 4-b

D 1-d, 2-c, 3-b, 4-a

💡 Short Explanation

Apply standard integrals separately. Use linearity property. Match each result directly.

✅ Correct Answer Explanation

1→c because ∫sec x tan x dx=sec x and ∫1/x dx=ln|x|. 2→d by polynomial and exponential integration. 3→a using ∫aˣdx=aˣ/ln a. 4→b because ∫1/√(1−x²)dx=sin⁻¹x and ∫1/(1+x²)dx=tan⁻¹x.

❓ Why Other Options Are Incorrect

Option A → Multiple mismatches in standard integrals.
Option B → First and fourth pairings are incorrect.
Option D → None of the listed pairings match correctly.

🎯 Strategy

Used: Option Grouping

Application: Evaluate each integral independently before matching.

Final Logic: Standard integral formulas uniquely determine the mapping.

🧠 Memory Trick

"Sec-Tan→Sec, 1/x→Log."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Standard Integrals and Linearity

7 If the area under f(x) represents physical work done, what does the geometrical interpretation of
∫5f(x) dx
represent compared to ∫f(x) dx?

A Horizontal stretch by 5

B Vertical scaling by a factor of 5, representing 5 times the work

C A linear shift of 5 units upwards

D No change in area

💡 Short Explanation

Constant multiple scales function values. Area scales by same factor. Work becomes five times larger.

✅ Correct Answer Explanation

The constant multiple rule gives ∫5f(x)dx = 5∫f(x)dx. Geometrically, every ordinate becomes five times larger, producing vertical scaling. Since area under the curve scales proportionally, the interpreted work becomes five times the original value.

❓ Why Other Options Are Incorrect

Option A → Scaling is vertical, not horizontal.
Option C → No vertical translation occurs.
Option D → Area changes by a factor of five.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Compare scaling effects on area and physical quantities.

Final Logic: Multiplying function values by 5 multiplies area by 5.

🧠 Memory Trick

"Constant outside = area multiplied."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Constant Multiple Rule

8 Evaluate the scalar multiple applied to this vector derivative integral:
∫(2 d/dx(sin x) î −4 d/dx(eˣ) ĵ) dx.

A 2cos x î−4eˣ ĵ+C⃗

B 2sin x î−4eˣ ĵ+C⃗

C 2sin x î+4eˣ ĵ+C⃗

D −2cos x î−4eˣ ĵ+C⃗

💡 Short Explanation

Integrate derivatives directly. Integration reverses differentiation. Preserve scalar multiples.

✅ Correct Answer Explanation

Since ∫ d/dx(sin x) dx = sin x and ∫ d/dx(eˣ) dx = eˣ, the vector integral becomes 2sin x î−4eˣ ĵ+C⃗. Therefore Option B correctly applies inverse operations component-wise.

❓ Why Other Options Are Incorrect

Option A → Uses derivative instead of anti-derivative.
Option C → Incorrect sign in the ĵ component.
Option D → Incorrect trigonometric function and sign.

🎯 Strategy

Used: Elimination

Application: Replace each derivative by its original function.

Final Logic: Integration undoes differentiation.

🧠 Memory Trick

"Integral cancels derivative."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Inverse Property

9 A firm's marginal cost data is given by
MC(x)=3x²−6x+9.
The total cost function is
C(x)=∫MC(x) dx.
If the fixed cost (when x=0) is C(0)=15, find the full expression for C(x).

A 3x³−3x²+9x+15

B x³−6x²+9x+15

C x³−3x²+9x

D x³−3x²+9x+15

💡 Short Explanation

Integrate marginal cost. Determine constant from C(0)=15. Obtain total cost function.

✅ Correct Answer Explanation

Integrating: C(x)=x³−3x²+9x+C. Using C(0)=15 gives C=15. Therefore C(x)=x³−3x²+9x+15. Option D correctly satisfies both the derivative condition and the initial cost condition.

❓ Why Other Options Are Incorrect

Option A → Derivative becomes 9x²−6x+9.
Option B → Incorrect coefficient of x² term.
Option C → Missing fixed-cost constant.

🎯 Strategy

Used: Substitution

Application: Integrate first, then use C(0)=15.

Final Logic: Initial condition fixes the arbitrary constant.

🧠 Memory Trick

"Marginal → Integrate → Add fixed cost."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Determination of Constant

10 Arrange the process to solve
∫(x³+5x²+4x+1)/x² dx:
1. Integrate each isolated term to get x²/2+5x+4ln|x|−1/x+C.
2. Divide each term in the numerator by x².
3. Apply general linearity to separate into four separate integrals.
4. Write the expanded integrand as x+5+4/x+1/x².

A 1, 2, 3, 4

B 2, 3, 4, 1

C 2, 4, 3, 1

D 4, 2, 3, 1

💡 Short Explanation

Simplify the rational expression first. Separate integrals next. Integrate term-wise.

✅ Correct Answer Explanation

The correct order is: 2 → Divide by x². 4 → Obtain x+5+4/x+1/x². 3 → Separate using linearity. 1 → Integrate each term individually. Hence the proper sequence is 2, 4, 3, 1, making Option C correct.

❓ Why Other Options Are Incorrect

Option A → Starts with the final answer.
Option B → Separation occurs before complete simplification.
Option D → Expansion cannot occur before division.

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Follow the natural algebra-to-integration workflow.

Final Logic: Simplify → Expand → Separate → Integrate.

🧠 Memory Trick

"Divide → Expand → Split → Integrate."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: General Linearity and Rational Functions

11 If a continuous probability density function involves an integrand
∫2xe^(x²) dx,
by direct analytical inspection, what is the anti-derivative?

A e^(x²)+C

B 2e^(x²)+C

C e^(2x)+C

D xe^(x²)+C

💡 Short Explanation

Derivative of x² is 2x. Integrand matches chain-rule pattern. Direct inspection gives the anti-derivative.

✅ Correct Answer Explanation

Since d/dx[e^(x²)] = e^(x²)·2x, the integrand 2xe^(x²) is exactly the derivative of e^(x²). Therefore the anti-derivative is e^(x²)+C. Option A correctly reverses differentiation, while the remaining options do not differentiate back to the given integrand.

❓ Why Other Options Are Incorrect

Option B → Differentiates to 4xe^(x²), not 2xe^(x²).
Option C → Derivative is 2e^(2x), unrelated to the integrand.
Option D → Product rule differentiation produces extra terms.

🎯 Strategy

Used: Inspection Method

Application: Identify a function whose derivative exactly matches the integrand.

Final Logic: 2x is the derivative of x², giving e^(x²)+C.

🧠 Memory Trick

"Inside derivative present ⇒ exponential stays."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Inspection and Substitution Forms

12 Through reverse derivative search, what is the numerical coefficient of the x term in the integral of
∫tan²x dx?
(Hint: use the identity tan²x=sec²x−1)

A 1

B 2

C 0

D −1

💡 Short Explanation

Use tan²x=sec²x−1. Integrate term-wise. Result contains −x.

✅ Correct Answer Explanation

Using the identity tan²x = sec²x − 1, we get ∫tan²x dx = ∫sec²x dx − ∫1 dx = tan x − x + C. Hence the coefficient of x is −1. Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Coefficient is not positive one.
Option B → No factor 2 appears after integration.
Option C → x-term is present and cannot vanish.

🎯 Strategy

Used: Substitution

Application: Rewrite tan²x using a standard trigonometric identity.

Final Logic: tan x−x+C contains coefficient −1.

🧠 Memory Trick

"Tan² = Sec² − 1."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Trigonometric Integrals

13 Evaluate the general linear polynomial form:
∫(ax+b)² dx.

A (ax+b)³/3+C

B 2a(ax+b)+C

C (ax+b)³/3a+C

D (ax+b)³/a+C

💡 Short Explanation

Let u=ax+b. Then du=a dx. Apply substitution formula.

✅ Correct Answer Explanation

Using u=ax+b, du=a dx ⇒ dx=du/a. Therefore, ∫(ax+b)²dx =(1/a)∫u²du =(1/a)(u³/3)+C =(ax+b)³/(3a)+C. Hence Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Missing division by a.
Option B → Represents a derivative-type expression.
Option D → Missing factor 1/3 from integration.

🎯 Strategy

Used: Substitution

Application: Replace the linear expression with a single variable.

Final Logic: u=ax+b gives (ax+b)³/(3a)+C.

🧠 Memory Trick

"Linear inside ⇒ divide by derivative."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Substitution Method

14 Evaluate the higher degree rational polynomial:
∫((x+1)(x−2))/√x dx.

A 2/5 x^(5/2)+2/3 x^(3/2)+4x^(1/2)+C

B 2/5 x^(5/2)+2/3 x^(3/2)−4x^(1/2)+C

C 5/2 x^(5/2)−3/2 x^(3/2)−2x^(1/2)+C

D 2/5 x^(5/2)−2/3 x^(3/2)−4x^(1/2)+C

💡 Short Explanation

Expand numerator first. Convert to fractional powers. Integrate term-wise.

✅ Correct Answer Explanation

\((x+1)(x-2)=x^{2}-x-2\) Dividing by √x: \(x^{3/2}-x^{1/2}-2x^{-1/2}\) Integrating: \(\frac{2}{5}x^{5/2}-\frac{2}{3}x^{3/2}-4x^{1/2}+C\) Therefore Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Wrong signs for the last two terms.
Option B → Middle term sign incorrect.
Option C → Incorrect coefficients from power-rule integration.

🎯 Strategy

Used: Elimination

Application: Expand, simplify exponents, then integrate.

Final Logic: Power rule produces Option D exactly.

🧠 Memory Trick

"Expand → Fractional powers → Integrate."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Algebraic Function Integration

📄 Passage

Integration by parts is formulated from the product rule of differentiation:
∫u v dx=u∫v dx−∫(u'∫v dx)dx.
It is exceptionally useful for logarithmic and inverse trigonometric functions, which do not have straightforward primitive forms. For instance, to integrate log x, we mathematically treat 1 as the second function.

Based on the passage, what is the integral of log x?

A 1/x+C

B xlog x−x+C

C xlog x+x+C

D (log x)²/2+C

💡 Short Explanation

Use integration by parts. Take u=log x and dv=dx. Apply the standard formula.

✅ Correct Answer Explanation

Using integration by parts: u=log x, dv=dx ⇒ du=dx/x, v=x. Hence, ∫log x dx =xlog x−∫1 dx =xlog x−x+C. Therefore Option B is the standard NCERT result.

❓ Why Other Options Are Incorrect

Option A → Derivative does not equal log x.
Option C → Sign error after applying integration by parts.
Option D → Derivative gives (log x)/x.

🎯 Strategy

Used: Substitution

Application: Apply the standard integration-by-parts formula.

Final Logic: xlog x−x+C differentiates to log x.

🧠 Memory Trick

"Log becomes xlog x minus x."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration by Parts

📄 Passage

When applying integration by parts on
∫xlog x dx,
what is the analytical result?

A x²/2 log x−x²/4+C

B x²/2 log x+x²/4+C

C xlog x−x²/2+C

D x²/2 log x−x/2+C

💡 Short Explanation

Use integration by parts. Let u=log x and dv=x dx. Simplify carefully.

✅ Correct Answer Explanation

Take u=log x, dv=x dx. Then du=dx/x, v=x²/2. Therefore, ∫xlog x dx =(x²/2)log x−(1/2)∫x dx =(x²/2)log x−x²/4+C. Thus Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Wrong sign in the second term.
Option C → Missing factor x²/2 before log x.
Option D → Incorrect integration of x.

🎯 Strategy

Used: Substitution

Application: Apply integration by parts systematically.

Final Logic: uv−∫vdu yields Option A.

🧠 Memory Trick

"Log first, polynomial second."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Integration by Parts

17 Evaluate
∫(sin²x)/(1+cos x) dx.

A x−sin x+C

B x+sin x+C

C cos x−x+C

D sin x−x+C

💡 Short Explanation

Simplify using identities. sin²x=(1−cos²x). Cancel common factors.

✅ Correct Answer Explanation

\(\frac{{sin⁡}^{2}x}{1+cos⁡x}=\frac{\left(1-cos⁡x)(1+cos⁡x\right)}{1+cos⁡x}=1-cos⁡x\) Therefore, \(\int (1-cos⁡x)dx=x-sin⁡x+C.\) Hence Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Sign of sin x is incorrect.
Option C → Differentiation does not return the integrand.
Option D → Signs and terms are incorrect.

🎯 Strategy

Used: Elimination

Application: Simplify the trigonometric expression before integrating.

Final Logic: Integrand reduces to 1−cos x.

🧠 Memory Trick

"Factor, cancel, integrate."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Trigonometric Integrals

18 Evaluate
∫√(1+sin 2x) dx
inside an interval where sin x+cos x>0.

A sin x+cos x+C

B −sin x+cos x+C

C cos x−sin x+C

D sin x−cos x+C

💡 Short Explanation

Use identity for sin2x. Simplify the square root. Apply interval condition.

✅ Correct Answer Explanation

\(1+sin⁡2x=1+2sin⁡xcos⁡x=(sin⁡x+cos⁡x)^{2}\) Since \(sin⁡x+cos⁡x>0\), \(\sqrt{1+sin⁡2x}=sin⁡x+cos⁡x.\) Thus \(\int (sin⁡x+cos⁡x)dx=-cos⁡x+sin⁡x+C\) which equals Option D.

❓ Why Other Options Are Incorrect

Option A → Differentiates to cos x−sin x.
Option B → Derivative gives −cos x−sin x.
Option C → Derivative gives −sin x−cos x.

🎯 Strategy

Used: Substitution

Application: Convert the expression into a perfect square identity.

Final Logic: √(1+sin2x)=sin x+cos x.

🧠 Memory Trick

"1+sin2x = (sin+cos)²."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Trigonometric Identities in Integration

19 A particle's acceleration is
a(t)=6t.
It starts from rest at the origin (v(0)=0, s(0)=0). What is the exact position function s(t)?

A 3t²

B t³+3

C t³

D 6t³

💡 Short Explanation

Integrate acceleration to get velocity. Use v(0)=0. Integrate velocity to get position.

✅ Correct Answer Explanation

Given a(t)=6t, integrating gives v(t)=3t²+C₁. Using v(0)=0 gives C₁=0. Again integrating, s(t)=t³+C₂. Using s(0)=0 gives C₂=0. Hence s(t)=t³ and Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Represents velocity-like behavior.
Option B → Violates s(0)=0.
Option D → Differentiation twice gives 36t, not 6t.

🎯 Strategy

Used: Substitution

Application: Apply initial conditions after each integration step.

Final Logic: Double integration yields s(t)=t³.

🧠 Memory Trick

"Acceleration → Velocity → Position."

📚 Class XII – NCERT Mathematics – Chapter: Differential Equations – Topic: Initial Value Problems

20 If
∫1/(x²+a²) dx=f(x)+C,
and f(a)=π/4a, what is f(x)?

A tan⁻¹(x/a)

B 1/a tan⁻¹(x/a)

C 1/a sin⁻¹(x/a)

D a tan⁻¹(x/a)

💡 Short Explanation

Use the standard integral formula. Substitute x=a to verify. Match the given condition.

✅ Correct Answer Explanation

The standard result is \(\int \frac{dx}{x^{2}+a^{2}}=\frac{1}{a}{tan⁡}^{-1}\left(\frac{x}{a}\right)+C.\) At x=a, \(f(a)=\frac{1}{a}{tan⁡}^{-1}(1)=\frac{1}{a}⋅\frac{\pi }{4}=\frac{\pi }{4a}.\) Hence Option B satisfies the condition exactly.

❓ Why Other Options Are Incorrect

Option A → Missing factor 1/a.
Option C → Uses the wrong inverse trigonometric function.
Option D → Incorrect scaling factor.

🎯 Strategy

Used: Option Grouping

Application: Compare options with the standard NCERT formula.

Final Logic: Standard integral directly matches Option B.

🧠 Memory Trick

"x²+a² ⇒ (1/a)tan⁻¹(x/a)."

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Standard Integral Forms

CUET UG Mathematics Booster Test 2 - Properties of Indefinite Integrals

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