Quadratic Equation-PART-II- (E)-03-Assignment

227. If the difference of the roots of the equation be 1, then (a) (b) (c) (d) 228. If the roots of the equations and differ by the same quantity, then is equal to (a) 4 (b) 1 (c) 0 (d) – 4 229. If the roots of are two consecutive integers, then (a) 1 (b) 2 (c) 3 (d) 4 230. If ,  are the roots of and then (a) (b) (c) (d) None of these 231. If ,  be the roots of such that and then the number of integral solutions of  is (a) 5 (b) 6 (c) 2 (d) 3 232. If X denotes the set of real numbers p for which the equation has its roots greater than p then X is equal to (a) (b) (c) Null set (d) (– , 0) 233. If one root of the quadratic equation is equal to the nth power of the other root, then the value of (a) b (b) – b (c) (d) 234. If one root of the equation is square of the other, the (a) (b) (c) (d) 235. For the equation if one of the root is square of the other, then p is equal to (a) (b) 1 (c) 3 (d) 236. If one root of equation is double of the other, then (a) (b) (c) (d) 237. The value of k for which one of the roots of is double of one of the roots of is (a) 1 (b) – 2 (c) 2 (d) None of these 238. The function has one double root (a) (b) (c) (d) 239. If are the roots of the equation , then (a) (b) (c) (d) 240. If the roots of are and root of are , then is equal to (a) (b) (c) (d) 241. If the product of roots of the equation is 7, then its roots will real w (a) (b) (c) (d) None of these 242. If a and b are rational and b is not a perfect square then the quadratic equation with rational coefficients whose one root is is (a) (b) (c) (d) None of these 243. If is a root of , where a, b are real, then (a) (b) (c) (d) None of these 244. If be the roots of the equation then the value of is (a) – 3 (b) (c) (d) None of these 245. If the roots of are in A.P. then their common difference is (a) (b) (c) (d) 246. The roots of the equation are in (a) A.P. (b) G.P. (c) H.P. (d) None of these 247. If 3 and are two roots of a cubic equation with rational coefficients, then the equation is (a) (b) (c) (d) None of these 248. What is the sum of the squares of roo (a) 5 (b) 7 (c) 9 (d) 10 249. If and , then and are the roots of (a) (b) (c) (d) None of these 250. For what value of the sum of the squares of the roots of is minimum (a) 3/2 (b) 1 (c) 1/2 (d) 11/4 251. The value of for which the sum of the cubes of the roots of , assumes the least value is (a) 3 (b) 4 (c) 5 (d) None of these 252. Let be the roots of . The value of for which is minimum, is (a) 0 (b) 1 (c) 2 (d) 3 253. If the sum of squares of the roots of the equation is least, then the value of a is (a) 0 (b) 2 (c) – 1 (d) 1 254. If are roots of and are roots of , then p is equal to (a) (b) (c) (d) 255. If are roots of the equation and are roots of the equation , then p equals (a) (b) 1 (c) (d) 2 256. If are real and are the roots of the equation , then (a) (b) (c) (d) 257. The H.M. of the roots of the equation is (a) 1 (b) 2 (c) 3 (d) None of these 258. If are the roots of the equation , then the value of and are (a) and (b) and (c) and (d) and 259. If p and q are the roots of , then (a) (b) (c) or 0 (d) or o 260. If roots of the equation are in opposite sign, then (a) (b) (c) (d) 261. Which of the following equation has 1 and –2 as the roots (a) (b) (c) (d) 262. If the roots of the equation are in the ratio m : n then (a) (b) (c) (d) 263. If the roots of the equation are in the ratio then is equal to (a) (b) (c) (d) 264. If the roots of the equation are in the ratio , then (a) (b) (c) (d) None of these 265. If the ratio of the roots of the equation be , then (a) (b) (c) (d) None of these 266. The two roots of an equation are in the ratio . The roots will (a) 6, 4, –1 (b) 6, 4, 1 (c) –6, 4, 1 (d) –6, –4, 1 267. The condition that one root of the equation is three times the other is (a) (b) (c) (d) 268. If the roots of the equation are such that , then the value of is (a) (b) c (c) (d) 269. For the equation , if the product of the roots is zero, then the sum of the roots is (a) 0 (b) (c) (d) 270. If the sum of two of the roots of is zero, then (a) (b) r (c) 2r (d) 271. If the roots of the equation are equal in magnitude but opposite in sign, then the product of the roots will be (a) (b) (c) (d) 272. The value of m for which the equation has two roots equal in magnitute but opposite in sign, is (a) 1/2 (b) 2/3 (c) 3/4 (d) 4/5 273. If , then is equal to (a) (b) (c) (d) 274. If are the roots of the equation and are the roots of for some constant, then (a) (b) (c) (d) None of these 275. In a triangle . If and are the roots of the equation , then (a) (b) (c) (d) 276. The product of all real roots of the equation is (a) (b) 6 (c) 9 (d) 36 277. If the sum of the roots of the equation is equal to the sum of the squares of their reciprocals then are in (a) A.P. (b) G.P. (c) H.P. (d) None of these 278. The roots of the equation are and the roots of the equation are r, s. If are is A.P., then [] (a) (b) (c) (d) 279. If the roots of the equation and are in the same ratio, then (a) (b) (c) (d) 280. If one root of the equation is , then values of and are (a) –4, 1 (b) 4, –1 (c) (d) 281. If is a root of the equation , then (a) –2 (b) –1 (c) 1 (d) 2 282. If are the roots of and are the roots of , then (a) (b) (c) (d) None of these 283. If be the roots of and be the roots of , then the value of is (a) (b) (c) (d) 284. If and are the roots of the equation and , then which of the following is true (a) (b) (c) (d) 285. If roots of an equation are , then the value of will be (a) n (b) (c) (d) 286. If and are the roots of , then the value of is (a) (b) (c) (d) None of these 287. If are the roots of equation and be those of equation and vector is parallel to , then (a) (b) (c) (d) None of these 288. If the roots of are and and those of are and such that , then (a) (b) (c) (d) None of these 289. If the sum of the roots of the equation is equal to their product, then the value of q is equal to (a) (b) (c) 3 (d) –6 290. If , then the value of is (a) 0 (b) (c) (d) None of these 291. If are the roots of the equation , then is equal to (a) (b) (c) (d) None of these 292. If are the roots of the equation , then (a) (b) (c) (d) 293. If are the roots of and , then is (a) (b) 0 (c) (d) None of these 294. Let be the roots of the equation and let for . Then the value of the determinant is (a) (b) (c) (d) 295. If are roots of the equation , then is less then (a) 2 (b) –2 (c) 18 (d) None of these 296. If are roots of the equation , then is greater then (a) 0 (b) 1 (c) 2 (d) None of these 297. If are the roots of the equation , then the value of is (a) 5 (b) 9 (c) 11 (d) 13 298. If and are the roots of the equation , then the value of is (a) 2 (b) 3 (c) 0 (d) 1 299. If A, G, H be respectively, the A.M., G.M. and H.M. of three positive number a, b, c then the equation whose roots are these number is given by (a) (b) (c) (d) 300. Let and then A and B are roots of the equation (a) (b) (c) (d) None of these 301. If are the roots of the equation , then the quadratic equation whose roots are and i (a) (b) (c) (d) None of these [Where and ] 302. Let A, G and H are the A.M., G.M. and H.M. respectively of two unequal positive integers. Then the equation has (a) Both roots as fractions (b) At least one root which is a negative fraction (c) Exactly one positive root (d) At least one root which is an integer 303. Let , where , have the roots such that then (a) (b) (c) (d) None of these 304. The cubic equation whose roots are the A.M., G.M. and H.M. of the roots of is (a) (b) (c) (d) None of these 305. If are the roots of and also of and if are the roots of , then n is (a) An odd integer (b) An even integer (c) Any integer (d) None of these 306. If has real solutions then (a) (b) (c) (d) None of these 307. If the ratio of the roots of is equal to the ratio of the roots of then are in (a) A.P. (b) G.P. (c) H.P. (d) None of these 308. P, q, r and s are integers. If the A.M. of the roots of and G.M. of the roots of are equal then (a) q is an odd integer (b) r is an even integer (c) p is an even integer (d) s is an odd integer 309. If the roots of differ by unity then the negative value of k is (a) –3 (b) (c) (d) None of these 310. The harmonic mean of the roots of the equation is (a) 2 (b) 4 (c) 6 (d) 8 311. If are the roots of then the equation in y has the roots (a) (b) (c) (d) 312. If the roots of change by the same quantity then the expression in a, b, c that does not change is (a) (b) (c) (d) None of these 313. If are the roots of then the product of the roots of the quadratic equation whose roots are and is (a) (b) (c) (d) None of these 314. The quadratic equation whose roots are the A.M. and H.M. of the roots of the equation is (a) (b) (c) (d) None of these 315. If , then the roots of the cubic equation are (a) (b) (c) (d) 316. Let a, b, c be real numbers and . If is a root of , is a root of , and then the equation has a root that always satisfies (a) (b) (c) (d) 317. If for all , then belongs to the interval (a) (b) (c) (d) None of these 318. The least integral value of k for which for all , is (a) 5 (b) 4 (c) 3 (d) None of these 319. The set of possible values of for which has roots whose sum and product are both less then 1 is (a) (b) (c) (d) 320. The set of the possible values of x such that is always positive is (a) (b) (c) (d) None of these 321. If all real value of x obtained from the equation are nonpositive then (a) (b) (c) (d) None of these 322. If does not have two distinct real roots , then the least value of is (a) 4 (b) –1 (c) 1 (d) –2 323. If then the minimum value of ab is (a) 12 (b) 24 (c) (d) None of these 324. The number of values of k for which is a perfect square is (a) 1 (b) 2 (c) 0 (d) None of these 325. If has equal integral roots then (a) b and c are integers (b) b and c are even integers (c) b is an even integer and c is a perfect square of a positive integer (d) None of these 326. Let A, G and H be the A.M., G.M. and H.M. of two positive number a and b. The quadratic equation whose roots are A and H is (a) (b) (c) (d) None of these 327. If , then the value of lies in the interval (a) (b) (c) (d) 328. If has no real roots and are real such that , then (a) (b) (c) (d) All of these 329. The quadratic equation (a) Cannot have a real root if (b) Can have a rational root if is a perfect square (c) Cannot have an integral root if where (d) None of these 330. A quadratic equation whose roots are and , where are the roots of , is (a) (b) (c) (d) 331. If a, b are the real roots of and c, d are the real roots of , then is divisible by (a) (b) (c) (d) 332. If and is satisfied for at least one real x then the greatest value of is (a) (b) (c) (d) 333. can be resolved into linear rational factors. Then (a) (b) (c) (d) None of these 334. If are the roots of the equation then equation has a root (a) (b) (c) (d) None f these 335. If are the roots of and are the roots of , then (a) A.M. of G.M. of (b) G.M. of A.M. of (c) are in A.P. (d) are in G.P. 336. If the roots of the equation are imaginary and the sum of the roots is equal to their product then a is (a) –2 (b) 4 (c) 2 (d) None of these 337. If equations and have one root common and , then (a) (b) (c) (d) 338. If equations and have one non-zero root common, then is equal to (a) 2 (b) –1 (c) 1 (d) 3 339. If and have a common root, then is equal to (a) 10 (b) 20 (c) 30 (d) 40 340. If two equations and have a common root, then the value of is (a) (b) (c) (d) 341. If the roots of and are the same, then (a) (b) (c) (d) 342. If one root of the equation is reciprocal of the other then k has the value (a) (b) (c) 1 (d) None of these 343. If the product of the roots of the equation is 8 then is (a) (b) (c) 3 (d) None of these 344. If the absolute value of the difference of roots of the equation exceeds then (a) or (b) (c) (d) 345. If are roots of and are the roots of , then is equal to (a) (b) (c) (d) 346. If the equation and have a common root, then (a) 0 (b) –1 (c) 0, –1 (d) 2, –1 347. If a root of the equations and is common, then its value will be (where and ) (a) (b) (c) or (d) None of these 348. If and have a common root and , then (a) 1 (b) 2 (c) 3 (d) None of these 349. If the equation and , have a common root, then (a) 0 (b) 1 (c) 2 (d) –1 350. If every pair from among the equation , and has a common root, then the product of three common roots is (a) (b) (c) (d) None of these 351. If the equation and have a common root, then the sum and product of their other roots are respectively (a) r, pq (b) –r, pq (c) pq, r (d) –pq, r 352. The value of ‘a’ for which the equations and have a common root is (a) 2 (b) –2 (c) 0 (d) None of these 353. If the equations and have a negative common root then the value of is (a) 0 (b) 2 (c) 1 (d) None of these 354. If and , have a common root then (a) (b) (c) (d) 355. If is a root of the equation then the other root is (a) (b) (c) (d) None of these 356. The common roots of the equations and are (where is a nonreal cube root of unity) (a) (b) (c) –1 (d) 357. If a, b, c are rational and no two of them are equal then the equations and (a) Have rational roots (b) Will be such at least one has rational roots (c) Have exactly one root common (d) Have at least one root common 358. If the equations and have two common roots, then (a) (b) (c) (d) None of these 359. The equations and have 2 roots in common. Then must be equal to (a) 1 (b) –1 (c) 0 (d) None of these 360. If a, b, c are in G.P. then the equations and have a common root if are in (a) A.P. (b) G.P. (c) H.P. (d) None of these 361. If the equations , have a common root then (a) a is real (b) (c) (d) The other root is also common 362. If are three quadratic equations of which each pair has exactly one root common then the number of solutions of the triplet is (a) 2 (b) 1 (c) 9 (d) 27 363. If x, y, z are three consecutive terms of a G.P., where and the common ratio is r, then the inequality holds for (a) (b) (c) (d) 364. If x is real, then the value of will not be less the (a) 4 (b) 6 (c) 7 (d) 8 365. If x be real, the least value of is (a) 1 (b) 2 (c) 3 (d) 10 366. The smallest value of in the interval is (a) (b) 5 (c) –15 (d) –20 367. If , then equals (a) 2 (b) –2 (c) 0 (d) 1 368. If x be real, then the minimum value of is (a) –1 (b) 0 (c) 1 (d) 2 369. If x be real, then the maximum value of will be equal to (a) 5 (b) 6 (c) 1 (d) 2 370. The expression has the same sign as of ‘a’ of (a) (b) (c) (d) have the same sign as a. 371. The value of is positive if (a) (b) (c) (d) 372. The values of ‘a’ for which is positive for any x are (a) (b) (c) (d) or 373. If x is real, then the maximum and minimum values of the expression will be (a) 2, 1 (b) (c) (d) None of these 374. If x is real, then the value of does not lie between (a) –9 and –5 (b) –5 and 9 (c) 0 and 9 (d) 5 and 9 375. The adjoining figure shows the graph of . Then (a) (b) (c) (d) a and b are of opposite signs 376. If is a common factor of and , then (a) (b) or (c) or (d) or 377. and will have a common factor, if (a) 24 (b) 0, 24 (c) 3, 24 (d) 0, 3 378. If is a factor of , then (a) (b) (c) (d) None of these 379. If is a factor of , then p is equal to (a) – 4 (b) 4 (c) –1 (d) 1 380. If is a factor of the expression , then (a) (b) (c) (d) None of these 381. The condition that may be divisible by a factor of the form is (a) (b) (c) (d) 382. If x be real then will take all real values when (a) (b) (c) (d) Always 383. Let , then all real values of x for which y takes real values, are (a) or (b) or (c) or (d) None of these 384. The graph of the curve is (a) Between the lines and (b) Between the lines and (c) Strictly below the line (d) None of these 385. If is a factor of the expression then (a) (b) (c) (d) None of these 386. If and are factors of the expression , then (a) (b) (c) (d) None of these 387. Given that, for all real x, the expression lies between and 3. The values between which the expression lies are (a) and 3 (b) –2 and 0 (c) –1 and 1 (d) 0 and 2 388. If x, y, z are real and distinct, then is always (a) Non-negative (b) Non-positive (c) Zero (d) None of these 389. If x + y and are two factors of the expression , then the third factor is (a) (b) (c) (d) None of these 390. If then the smallest possible value of is (a) 10 (b) 30 (c) 20 (d) None of these 391. If be the number of solutions of equation , where denote the integral part of x and m be the greatest value of on the interval ,then (a) (b) (c) (d) 392. If , then for (a) Only one value of x (b) No value of x (c) Only two values of x (d) Infinitely many values of x 393. If and and are odd numbers, then for any integer x (a) is odd or even according as x is odd or even (b) is even or odd according as x is odd or even (c) is even for all integral values of x (d) is odd for all integral values of x 394. If then for the expression (a) The least value (b) The greatest value (c) The least value (d) The greatest value 395. The value of ‘a’ for which is positive for any x are (a) (b) (c) (d) or 396. Let be a quadratic expression which is positive for all real values of x, then for all real is (a) > 0 (b) (c) (d) 397. The constant term of the quadratic expression as is (a) –1 (b) 0 (c) 1 (d) None of these 398. Let and let be the minimum value of . As b varies, the range of is (a) (b) (c) (d) 399. If be a polynomial satisfying the identity , then is given by (a) (b) (c) (d) 2x – 3 400. Let , then (a) y may be equal to (b) y may be equal to 3 (c) Set of possible value of y is (d) Set of possible values of y is 401. If , and equation of lines AB and CD be and respectively, then for all real x, point (a) Lies in the acute angle between lines AB and CD (b) Lies in the obtuse angle between lines AB and CD (c) Cannot be in the acute angle between lines AB and CD (d) Cannot lie in the obtuse angle between lines AB and CD 402. If a, b, c are real numbers such that , then the quadratic equation has (a) At least one root in [0, 1] (b) At least one root in [1, 2] (c) At least one root in [–1, 0] (d) None of these 403. The number of values of k for which the equation has two real and distinct roots lying in the interval (0, 1), are (a) 0 (b) 2 (c) 3 (d) Infinitely many 404. The value of k for which the equation has both real, distinct and negative is (a) 0 (b) 2 (c) 3 (d) –4 405. Let a, b, c be real number . If is a root of is a root of and then the equation has a root which always satisfies (a) (b) (c) (d) 406. Let a, b, c be non-zero real numbers such that , then the quadratic equation has (a) No root in (0, 2) (b) At least one root in (0, 1) (c) A double root in (0, 2) (d) Two imaginary roots 407. For the equation (a) Roots are rational (b) If one root is then the other is (c) Roots are irrational (d) If one root is then the other is 408. The values of a for which both roots of the equation lie between 0 and 1 are given by (a) (b) (c) (d) None of these 409. If p, q be non-zero real numbers and in [0, 2] and then equation has (a) Two imaginary roots (b) No root in (0, 2) (c) One root in (0, 1) and other in (1, 2) (d) One root in and other in 410. If and , then the number of real roots of the system of equation (in three unknowns ) , , is (a) 0 (b) 1 (c) 2 (d) 3 411. If , equation has (a) Both roots in (b) Both roots in (c) One root in and other in (d) One root in and other in 412. For equation to have exactly one root in (1, 3), the set of values of k is (a) (– 4, 0) (b) (1, 3) (c) (0, 4) (d) None of these 413. Let , then has no value in (a) (b) (c) (d) None of these 414. If , then equation has (a) At least one root in (0, 1) (b) At least one root in (0, 2) (c) At least one root in (–1, 1) (d) None of these 415. If has two distinct real roots in (0, 1), where , then (a) (b) (c) (d) 416. If , then the solution of , is given by (a) (b) or (c) (d) or 417. The solution of is (a) (b) (c) (d) None of these 418. For all , if , then m lies in the interval (a) (b) (c) (d) 419. If is a factor of , then (a) (b) (c) (d) None of these 420. If is factor of then the other factor is (a) (b) (c) (d) None of these 421. The set of values of x which satisfy and , is (a) (2, 3) (b) (c) (d) (1, 3) 422. The solution of the equation is given by (a) (b) (c) (d) 423. The complete solution of the inequation is (a) or (b) (c) (d) 424. If x is real and satisfies , then (a) (b) (c) (d) 425. If then the inequality has the solution represented (a) (b) (c) (d) 426. If x satisfies , then (a) (b) or (c) (d) None of these 427. The number of positive integral solutions of is (a) 4 (b) 3 (c) 2 (d) 1 428. If , then the solution set for x is (a) (b) {2} (c) (d) [0, 2] 429. The inequality is valid when x lies in (a) (3, 4) (b) (1, 2) (c) (–1, 2) (d) (–4, 3) 430. The graph of the function is strictly above the x-axis, then ‘a’ must satisfy the inequality (a) (b) (c) (d) None of these 431. If x is a real number such that are three consecutive terms of an A.P. then the next two consecutive term of the A.P. are (a) 14, 6 (b) –2, –10 (c) 14, 22 (d) None of these 432. If x, y are rational numbers such that , then (a) x and y connot be determined (b) (c) (d) None of these 433. If the greatest interger less than or equal to x, and (x)= the least integer greatest than or equal to x and then x belongs to (a) [3, 4] (b) (c) (d) 434. The set of real values of x satisfying and is (a) [2, 4] (b) (c) (d) None of these 435. The set of real values of x satisfying is (a) (b) (c) (d) None of these 436. If (the set of integers) such that then the number of possible values of x is (a) 3 (b) 4 (c) 6 (d) None of these 437. If x is an interger satisfying and then the number of possible values of x is (a) 3 (b) 4 (c) 2 (d) Infinite 438. The solution set of the ineuation is (a) (b) (c) (d) 439. If then the solution set is (a) R (b) (c) (d) None of these 440. The solution set of , is (a) (b) (c) (d) None of these 441. The equation can have real solutions for x if a belongs to (a) (b) (c) (d) None of these 442. If has two rational factors, then the value of m will be (a) (b) (c) (d) 6, 2 443. If has a common root, then the value of h is equal to (a) 1 (b) 2 (c) 3 (d) 4 444. Minimum value of is (a) 4 (b) 9 (c) 16 (d) 25 445. Let . If when divided by leaves 5 as remainder, and is divisible by then (a) (b) (c) (d) None of these 446. is divisible by if (a) n is any integer (b) n is an odd positive integer (c) n is an even positive integer (d) n is a rational number 447. The number of solution of the equation is (a) One (b) Two (c) Three (d) Zero 448. The line cuts the curve whose equation is at (a) Three real points (b) One real point (c) At least one real point (d) No real point 449. Let R=the set of real numbers,  = the set of integers, N= the set of natural numbers. If S be the solution set of the equation , where the least integer greater then or equal to x and the greatest integer less than or equal to x, then (a) (b) (c) (d) None of these 450. The number of real roots or is equal to (a) 0 (b) 2 (c) 4 (d) 6 451. The number of positive real roots of is (a) 3 (b) 2 (c) 1 (d) 0 452. The number of negative real roots of is (a) 3 (b) 2 (c) 1 (d) 0 453. The number of complex roots of the equation is (a) 3 (b) 2 (c) 1 (d) 0 454. is a factor of if (a) (b) (c) (d)