Quadratic Expressions
AIMSTUTORIAL
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Question 1 [2 May 2025, Shift-I]
\( \alpha, \beta \) are the roots of the equation \( \sin^2 x + b \sin x + c = 0 \). If \( \alpha + \beta = \frac{\pi}{2} \), then \( b^2 – 1 = \)
Teacher’s Explanation
Strategy: Use sum and product of roots. Convert \( \beta = \pi/2 – \alpha \) to relate sine and cosine.
Step 1: Relations
\( \sin\alpha + \sin\beta = -b \) and \( \sin\alpha \sin\beta = c \).
Step 2: Substitution
\( \sin\beta = \sin(\pi/2 – \alpha) = \cos\alpha \). So \( \sin\alpha + \cos\alpha = -b \).
Step 3: Square
\( (\sin\alpha+\cos\alpha)^2 = b^2 \implies 1 + 2\sin\alpha\cos\alpha = b^2 \). Since \( \sin\alpha\cos\alpha = c \), \( 1+2c = b^2 \).
Answer: (b)
Question 2 [2 May 2025, Shift-I]
The number of integral values of ‘\(a\)’ for which \( ax^2 + ax + 5 = 0 \) cannot have real roots is
Teacher’s Explanation
Strategy: No real roots condition is \( D < 0 \).
Step 1: Discriminant
\( D = a^2 – 20a < 0 \implies a(a-20) < 0 \).
Step 2: Count Integers
Range is \( (0, 20) \). Integers are \( 1, 2, …, 19 \). Count = 19.
Answer: (c)
Question 3 [2 May 2025, Shift-II]
If \( x^2 – 3ax + a^2 – 2a – k = 0 \) has different real roots for every rational number \( a \), then \( k \) lies in
Teacher’s Explanation
Strategy: \( D_x > 0 \) leads to a quadratic inequality in \( a \). For this to hold for all \( a \), the discriminant of the \( a \)-quadratic must be negative.
Step 1: x-Discriminant
\( D_x = 9a^2 – 4(a^2 – 2a – k) = 5a^2 + 8a + 4k > 0 \).
Step 2: a-Discriminant
\( D_a = 64 – 80k < 0 \implies k > 64/80 = 4/5 \).
Answer: (a)
Question 4 [3 May 2025, Shift-I]
If \( \alpha \) is a root of \( x^2 – x + 1 = 0 \), sum of 12 terms of \( (\alpha^n + 1/\alpha^n)^3 \).
Teacher’s Explanation
Strategy: Roots are \( -\omega, -\omega^2 \). Cycle repeats every 3 terms.
Step 1: Calculate Cycle
Terms are \( 1, -1, -8 \). Sum = -8.
Step 2: Total Sum
12 terms = 4 cycles. Total = \( 4 \times (-8) = -32 \).
Answer: (a)
Question 5 [3 May 2025, Shift-I]
If \( x^2 + px + 2 = 0 \) and \( x^2 + x + 2p = 0 \) have a common root, sum of roots of \( x^2 + 2px + 8 = 0 \) is
Teacher’s Explanation
Strategy: Subtract equations to find common root \( \alpha = 2 \), then find \( p \).
Step 1: Find p
Substitute \( \alpha=2 \): \( 4+2p+2=0 \implies p=-3 \).
Step 2: Solve Final Eq
\( x^2 – 6x + 8 = 0 \). Sum of roots = 6.
Answer: (c)
Question 6 [3 May 2025, Shift-I]
If both roots of \( x^2 – 5ax + 6a = 0 \) exceed 1, range of ‘\( a \)’ is
Teacher’s Explanation
Strategy: Conditions: \( D \ge 0, f(1) > 0, -b/2a > 1 \).
Step 1: Intersection
Intersection of \( a \in (-\infty, 0] \cup [24/25, \infty) \), \( a > -1 \), and \( a > 0.4 \) gives \( [24/25, \infty) \).
Answer: (b)
Question 7 [3 May 2025, Shift-II]
If \( \tan\theta, \cot\theta \) are roots of \( ax^2 + bx + c = 0 \), then
Teacher’s Explanation
Strategy: Product of roots is 1, so \( a=c \). Sum gives relation.
Step 1: Relation
\( \tan\theta + \cot\theta = \frac{2}{\sin 2\theta} = -b/a \). So \( \sin 2\theta = -2a/b = -2c/b \).
Answer: (b)
Question 8 [3 May 2025, Shift-II]
Sum of roots of \( ||2x – 3| – 4| = 2 \) is
Answer: (c)
Question 9 [4 May 2025, Shift-I]
Relation for \( k \) if new roots are \( \alpha+1/\alpha \) type.
Answer: (a)
Question 10 [4 May 2025, Shift-I]
\( f(x) \ge 0, f(-3)=0, f(0)=18 \). Find \( f(3) \).
Answer: (b)
Question 11 [9 May 2024, Shift-I]
\( k \) in \( (-0.5, 0) \) for equal roots.
Answer: (a)
Question 12 [9 May 2024, Shift-I]
Common root eq. Find root of \( ax^2 – 4x – 2a = 0 \).
Answer: (d)
Question 13 [9 May 2024, Shift-I]
Root \( \alpha \) in \( (-0.5, 0) \) for rational eq.
Answer: (b)
Question 14 [9 May 2024, Shift-II]
Value of \( \frac{28(M-\alpha)}{5(m+\beta)} \).
Answer: (b)
Question 15 [10 May 2024, Shift-I]
\( \frac{2}{\sqrt{3}}|\alpha^{2024} – \beta^{2024}| \).
Answer: (b)
Question 16 [10 May 2024, Shift-I]
Ratio from \( 12x^{1/3} – 25x^{1/6} + 12 = 0 \).
Answer: (d)
Question 17 [10 May 2024, Shift-I]
Sum of squares of roots.
Answer: (a)
Question 18 [10 May 2024, Shift-II]
Solution set \( 3^x + 3^{1-x} – 4 < 0 \).
Answer: (d)
Question 19 [10 May 2024, Shift-II]
Common solution set.
Answer: (b)
Question 20 [11 May 2024, Shift-I]
Sum of cubic terms (roots of unity).
Answer: (d)
Question 21 [11 May 2024, Shift-I]
\( a – 1/b \) value.
Answer: (a)
Question 22 [11 May 2024, Shift-I]
Inequality \( \sqrt{…} > 1-x \).
Answer: (c)
Question 23 [12 May 2023, Shift-I]
Ratio value for common factor \( x-2 \).
Answer: (b)
Question 24 [12 May 2023, Shift-II]
Intersection of solution sets.
Answer: (d)
Question 25 [12 May 2023, Shift-II]
Common root value (integer ratio condition).
Answer: (c)
Question 26 [12 May 2023, Shift-II]
\( \alpha^{15} + \beta^{15} \).
Answer: (b)
Question 27 [12 May 2023, Shift-II]
Reciprocal equation factor \( P \).
Answer: (b)
Question 28 [13 May 2023, Shift-I]
Positive root calculation.
Answer: (d)
Question 29 [13 May 2023, Shift-II]
Common root.
Answer: (b)
Question 30 [14 May 2023, Shift-I]
Set of \( p \) for positive quad.
Answer: (d)
Question 31 [14 May 2023, Shift-I]
Eq with sin/cos roots.
Answer: (c)
Question 32 [14 May 2023, Shift-I]
Relation \( a, c \) for trig roots.
Answer: (b)
Question 33 [14 May 2023, Shift-II]
Sum of values.
Answer: (d)
Question 34 [18 July 2022, Shift-II]
Sign of expression.
Answer: (d)
Question 35 [18 July 2022, Shift-I]
Zeroes of \( p(p(x)) \).
Answer: (d)
Question 36 [18 July 2022, Shift-I]
Set of real values.
Answer: (c)
Question 37 [19 July 2022, Shift-I]
\( \alpha^6+\beta^6 \).
Answer: (d)
Question 38 [19 July 2022, Shift-I]
Match List.
Answer: (c)
Question 39 [20 July 2022, Shift-II]
Statements I & II.
Answer: (b)
Question 40 [20 July 2022, Shift-II]
Relation \( \beta = ? \).
Answer: (a)
Question 41 [20 July 2022, Shift-I]
\( b+c = ? \).
Answer: (b)
Question 42 [4 Aug 2021, Shift-II]
Smallest neg integer.
Answer: (b)
Question 43 [4 Aug 2021, Shift-II]
Nature of roots.
Answer: (b)
Question 44 [5 Aug 2021, Shift-II]
Product.
Answer: (c)
Question 45 [5 Aug 2021, Shift-II]
\( k \) condition.
Answer: (c)
Question 46 [5 Aug 2021, Shift-II]
Assertion Reason.
Answer: (d)
Question 47 [5 Aug 2021, Shift-II]
Sum roots f+g.
Answer: (c)
Question 48 [3 Aug 2021, Shift-I]
\( \alpha + \beta \).
Answer: (b) (Based on text) or (a) (Based on key)
Question 49 [6 Aug 2021, Shift-II]
\( a^2 \).
Answer: (c)
Question 50 [6 Aug 2021, Shift-II]
\( M/a \).
Answer: (a)
Question 51 [9 Sep 2020, Shift-I]
Assertion Reason.
Answer: (d)
Question 52 [9 Sep 2020, Shift-II]
Interval for a.
Answer: (c)
Question 53 [10 Sep 2020, Shift-I]
Interval r.
Answer: (d)
Question 54 [10 Sep 2020, Shift-II]
Min value.
Answer: (d)
Question 55 [11 Sep 2020, Shift-II]
Max value.
Answer: (d)
Question 56 [14 Sep 2020, Shift-I]
Number of integral values.
Answer: (b)
Question 57 [14 Sep 2020, Shift-II]
Value.
Answer: (c)
Question 58 [14 Sep 2020, Shift-II]
Max, Min.
Answer: (a)
Question 59 [14 Sep 2020, Shift-II]
Set.
Answer: (c)
Question 60 [3 May 2019, Shift-I]
Set of a.
Answer: (c)
Question 61 [3 May 2019, Shift-II]
Solution set.
Answer: (a)
Question 62 [4 May 2019, Shift-I]
Excluded interval.
Answer: (d)
Question 63 [4 May 2019, Shift-I]
Inequality intersection.
Answer: (a)
Question 64 [4 May 2019, Shift-II]
Min value a.
Answer: (d)
Question 65 [6 May 2019, Shift-I]
Intercepts relation.
Answer: (b)
TEACHER BOX: Complete Answer Key (Q1-65)
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| (b) | (c) | (a) | (a) | (d) | (b) | (b) | (c) | (a) | (b) |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| (a) | (d) | (b) | (b) | (b) | (d) | (a) | (d) | (b) | (d) |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| (a) | (b) | (c) | (b) | (c) | (b) | (d) | (d) | (b) | (d) |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| (d) | (b) | (d) | (d) | (d) | (a) | (d) | (a) | (b) | (a) |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| (b) | (b) | (d) | (c) | (c) | (d) | (c) | (a) | (b) | (a) |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| (d) | (c) | (d) | (d) | (d) | (b) | (c) | (a) | (c) | (c) |
| 61 | 62 | 63 | 64 | 65 | |||||
| (a) | (d) | (a) | (d) | (b) |
