JEE Mathematics Miscellaneous Problems MCQs Set 04

Answer: (c) A is true, but R is false.

Explanation:
1. Sum of focal distances 2a = 4 so a = 2, and ae = √3 gives b² = 4 − 3 = 1. The curve is \( \frac{x^2}4+y^2=1 \).
2. On y = 1/2: x²/4 = 3/4, so x = ±√3 and the intercept is 2√3, so the Assertion is true.
3. The curve is x²/4 + y² = 1, not x²/4 + 2y² = 5/4, so the Reason is false.

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Answer: (c) A is true, but R is false.

Explanation:
1. Let \( I=\int_0^{\pi/2}\frac{dx}{1+\tan^3x} \). Replacing x by π/2 − x gives \( I=\int_0^{\pi/2}\frac{\tan^3x}{1+\tan^3x}dx \).
2. Adding: 2I = π/2, so I = π/4: the Assertion is true.
3. The correct property is \( \int_0^af(x)dx=\int_0^af(a-x)dx \); the shift f(x + a) changes the limits, so the Reason is false.

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Answer: (d) A is false, but R is true.

Explanation:
1. By the chain rule, \( F'(x)=\cos\sqrt{x^2}\cdot2x=2x\cos x \) (for x ≥ 0), not cos x, so the Assertion is false.
2. The Reason is the fundamental theorem of calculus and is true.

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Answer: (d) A is false, but R is true.

Explanation:
1. For n = 1 the left side is 1 = 1¹, and for n = 2 it is 3 < 4. By AM-GM, \( 1\cdot3\cdots(2n-1)\le\left(\frac{1+3+\cdots+(2n-1)}n\right)^n=n^n \), so the product is never greater than nⁿ: the Assertion is false.
2. The sum of the first n odd numbers is n², so the Reason is true.

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Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Equal tangent lengths give AB + CD = BC + DA, the condition for a quadrilateral to have an incircle.
2. For sides a, a + d, a + 2d, a + 3d (in order) we get a + (a + 3d) = (a + d) + (a + 2d), so the condition holds. The Assertion is true.
3. The Reason is true and is exactly why AB + CD = BC + DA, so it explains the Assertion.

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Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. By AM > HM for unequal positive numbers: \( \frac{a+b+c}3\gt \frac3{\frac1a+\frac1b+\frac1c} \).
2. Rearranging: \( (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\gt 9 \). Both statements are true and the Reason explains the Assertion.

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Answer: (d) A is false, but R is true.

Explanation:
1. Let z = e^{iθ}. Then \( z^2+\bar z=e^{2i\theta}+e^{-i\theta}=2\cos\frac{3\theta}2\,e^{i\theta/2} \).
2. Its argument is θ/2 (or θ/2 + π), not θ, so the Assertion is false.
3. For |z| = 1, \( \bar z=\cos\theta-i\sin\theta \), so the Reason is true.

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Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. With ω a cube root of unity: \( a_0+a_3+\cdots=\frac13[2^{36}+(1+\omega)^{36}+(1+\omega^2)^{36}]=\frac13(2^{36}+1+1)=\frac23(2^{35}+1) \), so the Assertion is true.
2. The two sums in the Reason are standard (x = 1 and x = ±1) and true, but they do not give the sum of every third coefficient, so the Reason does not explain the Assertion.

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Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Roots of x² + x + 1 = 0 are non-real. Complex roots of a real-coefficient equation come in conjugate pairs, so a common root forces both roots to be common.
2. Hence the two equations are proportional: a = b = c, so the Assertion is true.
3. The Reason states exactly why both roots must be common, so it explains the Assertion.

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Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. By the sine rule a : b : c = sin A : sin B : sin C, so sin A : sin B : sin C = cos A : cos B : cos C, which gives tan A = tan B = tan C.
2. So A = B = C = 60° and the triangle is equilateral: the Assertion is true.
3. The Reason (the cosine rule) is true and is the tool that links the sides to the cosines used in the argument, so it explains the Assertion.

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Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. Reduction: \( I_n+I_{n-2}=\int_0^{\pi/4}\tan^{n-2}x\sec^2x\,dx=\frac1{n-1} \), so the Reason is true.
2. I₂ = 1 − π/4, hence \( I_4=\frac13-I_2=\frac\pi4-\frac23=\frac{3\pi-8}{12} \), so the Assertion is true, and it follows from the Reason.

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Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. f(2) = 32 − 60 + 28 − 5 = −5 and f(3) = 108 − 135 + 42 − 5 = 10 have opposite signs, so a root lies in (2, 3) by the intermediate value theorem: the Assertion is true.
2. The Reason is Rolle's theorem, which is true, but it is not what proves the Assertion.

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