Quadratic Equations PYQs Set 11 (PDF)

Set 11 of 11 · Questions 301-316 of 316

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Quadratic Equations · Set 11 (Q301-316)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
301. $In △ ABC, the value of ∠ A is obtained from the equation 3 cos A + 2 = 0 . The quadratic equation, whose roots are sin A and tan A, is$ (1) $3 x^{2} + 5 x - 5 = 0$ (2) $6 x^{2} - 5 x - 5 = 0$ (3) $6 x^{2} + 5 x - 5 = 0$ (4) $6 x^{2} + 5 x + 5 = 0$ 302. $If 5, 5 r, 5 r^{2} are the lengths of the sides of a triangle, then r cannot be equal to:$ (1) $\frac{3}{4}$ (2) $\frac{5}{4}$ (3) $\frac{7}{4}$ (4) $\frac{3}{2}$ 303. $Let α and β be the roots of x^{2} + 3 x - 16 = 0, and γ and δ be the roots of x^{2} + 3 x - 1 = 0 . If P_{n} = α^{n} + β^{n} and Q_{n} = γ^{n} + o^{n}, then \frac{P _{25} + 3 P _{24}}{2 P _{23}} + \frac{Q _{25} - Q _{23}}{Q _{24}} is equal to$ (1) $4$ (2) $3$ (3) $5$ (4) $7$ 304. $The number of real roots of the equation e^{4 x} + 2 e^{3 x} - e^{x} - 6 = 0 is:$ (1) $2$ (2) $4$ (3) $1$ (4) $0$ 305. $Let α and β be the roots of the equation x^{2} - x - 1 = 0 If p_{k} = (α)^{k} + (β)^{k}, k \geq 1, then which of the following statements is not true?$ (1) $P_{5} = P_{2} \times P_{3}$ (2) $P_{1} + P_{2} + P_{3} + P_{4} + P_{5} = 25$ (3) $P_{3} = P_{5} - P_{4}$ (4) $P_{5} = 11$ 306. $Let x be a real number. Match the following: List-I (A) The minimum value of 2 x^{2} + 4 x + 5 (B) The maximum value of \frac{x ^{2} + 4 x + 1}{x ^{2} + x + 1} (C) If 1 \leq \frac{3 x ^{2} - 5 x + 6}{x ^{2} + 1} \leq 2, \forall x \in [a, b] then b = (D) If 1 \leq \frac{3 x ^{2} - 5 x + 6}{x ^{2} + 1} \leq 2, \forall x \in [a, b] then a = List-II (I) - 1 (II) 1 (III) 2 (IV) 3 (V) 4 The correct answer is$ (1) $A - IV, B - III, C - II, D - V$ (2) $A - IV, B - V, C - II, D - III$ (3) $A - IV, B - III, C - V, D - II$ (4) $A - III, B - V, C - IV, D - I$ 307. $Let α be a root of the equation (a - c) x^{2} + (b - a) x + (c - b) = 0 where a, b, c are distinct real numbers such that the matrix α^{2} 1 a α 1 b 11 c is singular. then the value of \frac{( a - c ) ^{2}}{( b - a ) ( c - b )} + \frac{( b - a ) ^{2}}{( a - c ) ( c - b )} + \frac{( c - b ) ^{2}}{( a - c ) ( b - a )} is$ (1) $6$ (2) $3$ (3) $9$ (4) $12$ 308. $Let P (x) = x^{2} + b x + c be a quadratic polynomial with real coefficients such that \int_{0}^{1} P (x) d x = 1 and P (x) leaves remainder 5 when it is divided by (x - 2) . Then the value of 9 (b + c) is equal to:$ (1) $9$ (2) $15$ (3) $7$ (4) $11$ 309. $The number of integral values of m for which the quadratic expression (1 + 2 m) x^{2} - 2 (1 + 3 m) x + 4 (1 + m), x \in R, is always positive, is:$ (1) $3$ (2) $8$ (3) $7$ (4) $6$ 310. $Let for a \neq = a_{1} \neq = 0, f (x) = a x^{2} + b x + c, g (x) = a_{1} x^{2} + b_{1} x + c_{1} and p (x) = f (x) - g (x) . If p (x) = 0 only for x = - 1 and p (- 2) = 2, then the value of p (2) is:$ (1) $3$ (2) $9$ (3) $6$ (4) $18$ 311. $For the equation x^{4} + x^{3} - 4 x^{2} + x + 1 = 0 the ratio of the sum of the squares of all the roots to the product of the distinct roots is$ (1) $1 : 4$ (2) $3 : 5$ (3) $9 : 1$ (4) $4 : 3$ 312. $If α_{1}, β_{1}, γ_{1}, δ_{1} are the roots of the equation ax^{4} + bx^{3} + cx^{2} + dx + e = 0 and α_{2}, β_{2}, γ_{2}, δ_{2} are the roots of the equation ex^{4} + dx^{3} + cx^{2} + bx + a = 0 such that 0 < α_{1} < β_{1} < γ_{1} < δ_{1}, 0 < α_{2} < β_{2} < γ_{2} < δ_{2}, α_{1} - δ_{2} = 2 = β_{1} - γ_{2}; γ_{1} - β_{2} = δ_{1} - α_{2} = 4 then a + b + c + d + e =$ (1) $10$ (2) $12$ (3) $6$ (4) $8$ 313. $The number of elements in the set {x \in R : (∣ x ∣ - 3) ∣ x + 4∣ = 6} is equal to$ (1) $3$ (2) $2$ (3) $4$ (4) $1$ 314. $Let S = {x \in [- 6, 3] - (- 2, 2) : \frac{∣ x + 3∣ - 1}{∣ x ∣ - 2} \geq 0} and T = {x \in Z : x^{2} - 7∣ x ∣ + 9 \leq 0} . Then the number of elements in S \cap T is$ (1) $7$ (2) $5$ (3) $4$ (4) $3$ 315. $If the set of all a \in R - 1, for which the roots of the equation (1 - a) x^{2} + 2 (a - 3) x + 9 = 0 are positive is (- \infty, - α] \cup [β, γ), then 2 α + β + γ is equal to_____.$ 316. $Let m be the minimum possible value of log_{3} (3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}) where y_{1}, y_{2}, y_{3} are real numbers for which y_{1} + y_{2} + y_{3} = 9 . Let M be the maximum possible value of (log_{3} x_{1} + log_{3} x_{2} + log_{3} x_{3}), where x_{1}, x_{2}, x_{3} are positive real numbers for which x_{1} + x_{2} + x_{3} = 9 . Then the value of log_{2} (m^{3}) + log_{3} (M^{2}) is____. (JEE Advanced 2019)$