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Quadratic Equation & Expression, , EXERCISE-1, Q.1, , If the roots of the equation 3x 2  2px  4  0 are equal then p is equal to, [1] 12, , Q.2, , Q.7, , [2] real and equal, , [3] imaginary, , [4] irrational, , [2] 1, 2, , [3] 2, , [4] 1, 2, , [2] a  2,b  3, , [3] a  5,b  3, , [4] a  5,b  3, , [1] AM of the roots of second, , [2] GM of the roots of second, , [3] square root of the GM of the roots of second, , [4] None of these, , If ,  are roots of the equation ax 2  bx  c  0 , then  a  b  a  b  is equal to, [2] bc, , [3] ca, , [4] a+c, , If ,  are roots of the equation ax 2  bx  c  0 , then      is equal to, 2, , b2  3ac, a2, , [1] x 2  qx  p  0, , [2], , b2  3ac, a2, , 2, , b2  2ac, a2, q q, If , are roots of the equation x 2  px  q  0 then the equation whose roots are , will be,  , [1], , Q.9, , [4] equal, , Let x 2  2ax  b2  0 and x 2  2bx  a2  0 be two equations. Then the AM of the roots of the first equation is, , [1] ab, Q.8, , [3] irrational, , If the roots of the equation 2x 2  3x  5  0 are reciprocals of the roots of the equation ax 2  bx  2  0 , then, [1] a  2,b  3, , Q.6, , [2] rational, , If the roots of the equation 6x 2  7x  k  0 are rational then k is equal to, [1] 1, , Q.5, , [4] 0, , The roots of the equation a2 x2   a  b  x  b2  0 are, [1] real and different, , Q.4, , [3] 2 3, , The roots of the equation x 2  x  2  x  1 are, [1] imaginary, , Q.3, , [2] 2 3, , [3], , [2] x 2  px  q  0, , b2  2ac, a2, , [3] x 2  px  q  0, , [4], , [4] qx 2  px  q  0, , Q.10 If a  0,b  0 then the roots of the equation a  bx  x 2  0 are, [1] both positive, , [2] both negative, , [3] of opposite sign and numerically greater root is positive, [4] of opposite sign and numerically greater root is negative, Q.11 The quadratic equation with one root, [1] x 2  x  1  0, , , , , , 1, 1  3 is, 2, , [2] x 2  x  1  0, , [3] x 2  x  1  0, , [4] x 2  x  1  0, , Q.12 The equation whose roots are the squares of the roots of the equation ax 2  bx  c  0 is, [1] a2 x 2  b2 x  c 2  0, , 2 2, 2, 2, [2] a x   b  4ac  x  c  0, , 2 2, 2, 2, [3] a x   b  2ac  x  c  0, , 2 2, 2, 2, [4] a x   b  ac  x  c  0, , Q.13 The roots of the equation ax 2  bx  c  0 will be imaginary if, [1] a  0,b  0,c  0, , [2] a  0,b  0,c  0, , [3] a  0,b  0,c  0, , [4] a  0,b  0,c  0, , Mathematics by Goyal sir........9829652673, , 1

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Quadratic Equation & Expression, , 2, 2,  1, Q.14 The equation x , has, x 1, x 1, , [1] no root, , [2] one root, , [3] two equal roots, , [4] infinite roots, , [3] h2  ab, , [4] h2  4ab, , Q.15 If ax  2hxy  by has real and unequal factors then, 2, , 2, , [1] h2  ab, , [2] h2  ab, , Q.16 If sum of the roots of the equation ax 2  bx  c  0 is equal to the sum of their squares, then, [1] 2ac  ab  c, , [2] 2ac  b  a  b , , [3] a2  b2  c 2, , [4] a2  b2  a  b, , Q.17 For what value of a the difference of the roots of the equation 2x 2   a  1 x   a  1  0 is equal to their product?, [1] 0, , [3] 1, , [2] 1, , [4] 2, , Q.18 The roots of the equation  p  2  x 2  2  p  2  x  2  0 are not real when, [1] p  1,2, , [2] p   2,3, , [3] p   2,4 , , [4] p  3,4, , Q.19 If roots of the equation 2x 2  3  k  2  x  4  k  15x are negative of each other, then k equals, [1] 4, , [2] 2, , [3] 0, , Q.20 If the product of the roots of the equation x  3kx  2e, equals, 2, , [1] 1, , [2] 2, , 2logk, , [4] 7, ,  1  0 is 7, then the roots of the equation are real if k, , [3] 2, , [4] 2, , Q.21 If sin , cos are the roots of the equation ax2 + bx + c = 0, than, [1] a2 + b2 – 2ac = 0, , [2] a2 – b2 + 2ac = 0, , [3] (a + c)2 = b2 + c2, , [4] both b and c, , [3] (– 4, 4), , [4] None of these, , Q.22 The real roots of the equation x2 + 5 | x | + 4 = 0 are, [1] (– 1, – 4), , [2] (1, 4), , Q.23 If the roots of the equation ax2 + bx + c = 0 are reciprocal to each other, then, [1] a + c = 0, , [2] b = 0, , [3] a – c = 0, , [4] None of these, , Q.24 The value of p > 0, for which both the equations x2 + px + 64 = 0 & x2 – 8x + p = 0 have real roots is, [1] 8, , [2] 16, , [3] 32, , [4] 64, , Q.25 If p, q, r are real and p  q then the roots of the equation (p – q)x2 – 5 (p + q)x – 2 (p – q) = 0, are, [1] real and equal, , [2] complex, , [3] real and unequal, , [4] none of these, , Q.26 I f t h e r o o t s o f t h e e q u a t io n x 2  2cx  ab  0 b e r e a l a n d u n e q u a l , t h e n t h e r o o t s o f, , x 2  2(a  b) x  (a2  b2 )  2c 2  0 will be, [1] Real & unequal, , [2] Real & equal, , [3] Imaginary, , [4] None of these, , [3] ± 2, ± i, , [4] none of these, , Q.27 The roots of the equation x4 – 8x2 – 9 = 0 are [1] ± 3, ± 1, , [2] ± 3, ± i, , Q.28 The number of roots of the quadratic equation 8sec2  – 6 sec + 1 = 0 is [1] Infinite, , [2] 1, , [3] 2, , [4] 0, , Mathematics by Goyal sir........9829652673, , 2

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Quadratic Equation & Expression, 2, , Q.29 If the roots of x – 4x – log2 a = 0 are real then [1] a , , 1, 4, , [2] a , , 1, 8, , [3] a , , 1, 16, , [4] none of these, , Q.30 If the sum of the roots of the equation ax2 + 4x + c = 0 is half of their difference, then the value of ac is [1] 4, [2] 8, [3] 12, [4] – 12, Q.31 If , , be the roots of the equation p(x2 + n2) + pnx + qn2x2 = 0 then the value of p(2 + 2) + p + q22 is, [1]  + , [2] 0, [3] p + q, [4]  +  + p + q, Q.32 If x  k is a common factor of the expressions x 2  px  q and x 2  lx  m , then k is equal to, [1], , pq, lm, , [2], , pl, qm, , [3], , qm, pl, , [4], , qm, pl, , 2, 2, Q.33 If both the roots of the equations k  6x 2  3   rx  2x 2  1  0 and 6k  2x  1  px  4x  2  0 are common,, , then 2r  p is equal to, [2] 1, , [1] 1, , [3] 2, , [4] 0, , Q.34 If x  11x  a and x  14x  2a have a common factor then a is equal to, [1] 24, [2] 1, [3] 2, 2, , 2, , [4] 12, , Q.35 If one root of the equation a  b  c  x 2  b  c  a  x  c  a  b   0 is 1 then, its other root is, a b  c , [1] b c  a, , , , c a  b, [2] a b  c, , , , b c  a, [3] a b  c, , , , (d) None of these, , 2, 2, 2, 2, Q.36 The equation  a  a  2  x   a  4  x  a  3a  2  0 will have more than two solutions if a equals, , [1] 2, , [3] 2, , [2] 1, , [4] not possible, , Q.37 The value of a for which the equation 3x  2  a  1 x   a  3a  2   0 will have roots of opposite sign, lies in, 2, , 2, , 2, , 3 , [4]  ,2 , 2 , 2, Q.38 If , , ,  are in GP where ,  are roots of the equation ax  2bx  c  0 and ,  are roots of the equation, [1]  ,1, , [2]  ,0 , , [3] 1,2 , , px 2  2qx  r  0 , then, , [1], , ac pr, , b2 q2, , [2], , ac pr, , b, q, , [3], , ab pq,  2, c2, r, , Q.39 If , are roots of the equation ax 2  3x  2  0  a  0  , then, [1] 0, , [2] 1, , [3] 2, ,  2 2, is greater than, , , , [4] None of these, , Q.40 If the absolute difference between two roots of the equation x 2  px  3  0 is, [1] 3,4, , [2] 4, , [4] None of these, , p , then p equals, , [3] 3, , [4] None of these, , [3] p = – 6, q = – 7, , [4] p = – 6, q = 25, , Q.41 If 3 + 4i is a root of the equation x2 + px + q = 0, then, [1] p = 6, q = 25, , [2] p = 6, q = 1, , Q.42 If   , then the equation whose roots are & is, , , [1] x2 + 5x – 3 = 0, , , , [2] 3x2 + 12x + 3 = 0, , [3] 3x2 – 19x + 3 = 0, , Q.43 If and are the roots of ax2 + bx + c. Then the value of, [1] 2/a, , [2] – 2/a, , [4] None of these, , , , , is, , a  b a   b, , *[3] a/2, , [4] None of these, , Mathematics by Goyal sir........9829652673, , 3

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Quadratic Equation & Expression, , Q.44 A quadratic equation whose roots are, , a, a  ab, , , will be, , [1] bx 2  2a ax  a2  0, , [2] bx 2  2a ax  a2  0, , [3] ax 2  2a ax  b2  0, , [4] None of these, , Q.45 Roots of ax2 + b = 0 are real & distinct if, [1] ab > 0, , [2] ab < 0, , [3] a, b > 0, , [4] a, b < 0, , Q.46 Both roots of the equation (x – b) (x – c) + (x – a) (x – c) + (x – a) (x – b) = 0 are always, [1] positive, , [2] negative, , [3] real, , [4] none of these, , Q.47 If the equation x2 – m (2x – 8) – 15 = 0 has equal roots then m =, [1] 3, –5, Q.48 If the equation, , [1], , [2] –3, 5, , [3] 3, 5, , [4] –3, –5, , x 2  bx m  1, , has roots equal in magnitued but opposite in sign, then m =, ax  c, m 1, , a b, ab, , [2], , ab, a b, , [3], , ba, b a, , [4] None of these, , Q.49 If difference of roots of the equation x2 – px + q = 0 is 1, then p2 + 4q2 equals [1] 2q + 3, [2] (1 – 2q)2, [3] (1 + 2q)2, [4] 2q – 3, 2, Q.50 If one of the roots of equation x(x + 2) = 4 – (1 – ax ) tends to , then a will tend to [1] 0, [2] 1, [3] – 1, [4] 2, Q.51 If  and  are roots of x2 – 2x + 3 = 0, then the equation whose roots are, ,  1,  1, and, will be  1,  1, , [1] 3x2 – 2x – 1 = 0, [2] 3x2 + 2x + 1 = 0, [3] 3x2 – 2x + 1 = 0, [4] x2 – 3x + 1 = 0, 2, 2, Q.52 If f(x) = 4x + 3x – 7 and  is a common root of the equation x – 3x + 2 = 0 and x2 + 2x – 3 = 0 then the value of, f() is [1] 3, [2] 2, [3] 1, [4] 0, Q.53 For what values of p, the roots of the equation 12 (p + 2)x2 – 12(2p – 1) x – 38 p – 11 = 0 are imaginary[1] p = R–, ,  1 , [2] p  (–, – 1)   – ,  ,  2 , , 1, , [3] p   – 1 – , 2, , , [4] p = – 1, , Q.54 If x – 2 is a common factor of x2 + ax + b and x2 + cx + d, then [1] d – b = 2 (c – a), [2] b – d = (c – a), [3] 4 + 2c + b = 0, [4] b – d = 2 (c – a), Q.55 If the quadratic equations 3x2 + ax + 1 = 0 and 2x2 + bx + 1 = 0 have a common roots, then the value of the, expression 5ab – 2a2 – 3b2 is [1] 0, [2] 1, [3] – 1, [4] none of these, 2, 2, 2, 2, Q.56 If roots of the equation x (1 + m ) + 2 mc x + c – a = 0 are equal, then value of c is [1] a (1  m 2 ), , [2] a (1  m 2 ), , [3] m (1  a 2 ), , [4] m (1  a 2 ), , Q.57 The roots of the equation (a + b – 2c) x2 – (2a – b + c)x + (a – 2b + c) = 0 are, [1] a + b + c and a – b + c [2], , 1, and a – 2b + c, 2, , [3] a – 2b + c and, , 1, [4] none of these, abc, , Mathematics by Goyal sir........9829652673, , 4

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Quadratic Equation & Expression, , Q.58 If x  R and a>0 then the minimum value of ax 2  bx  c is, [1], , b2  4ac, 4a, , [2], , 4ac  b2, 4a, , b2  4ac, 2a, , [3] b2  4ac, , [4], , [3] x  2, , [4] never, , Q.59 If x be real, then 3x 2  14x  11  0 when, [1] x  , , 3, 2, , [2] x  , , 3, 4, , Q.60 For all real values of x, the maximum value of the expression, [1] 1, , [2] 45, , x, is, x 2  5x  9, , [3] 90, , Q.61 If x be real then the maximum and minimum value of the expression, [2] 7,, , [1] 2,1, , Q.62 If x be real then the value of, [1] 5 and 9, , 1, 7, , [3] 5,, , [1] 0 and 8, , x2  3x  4, are, x 2  3x  4, , 1, 5, , [4] None of these, , x 2  34x  71, will not lie between, x 2  2x  7, , [3] 9 and 5, , [2] 5 and 9, , Q.63 If x be real then the value of, , [4] None of these, , [4] 0 and 9, , x 2  2x  1, will not lie between, x 1, [2] –8 and 8, [3] –8 and 0, , [4] None of these, , Q.64 The expression a2 x 2  bx  1 will be positive for all x  R if, [2] b2  4a2, [3] 4b2  a2, [4] 4b2  a2, [1] b2  4a2, Q.65 Ramesh and Mahesh solve a quadratic equation. Ramesh reads its constant term wrongly and finds its roots as, 8 and 2 where as Mahesh reads the coefficientof x wrongly and finds its roots as 11 and  1. The correct roots of, the equation are, [2]  11, 1, , [1] 11, 1, , [3] 11,  1, , [4] None of these, , Q.66 If & are the roots of the equation ax + bx + c = 0, then (1 +  ) (1 +  ) =, , , 2, , [1] 0, , [2] positive, 2, , [3] negative, , [4] none of these, , 2, , Q.67 If the two equations x – cx + d = 0 & x – ax + b = 0 have one common root and the second has equal roots, then, 2(b + d) =, [1] 0, , [2] a + c, , [3] ac, , [4] – ac, , Q.68 If x2 + 6x – 27 > 0, –x2 + 3x + 4 > 0, then x lies in the interval, [1] (3, 4), , [2] [3, 4], , [3] (–, 3]  [4, ), , [4] , , Q.69 If the equations ax2 + bx + c = 0 and x3 + 3x2 + 3x + 2 = 0 have two common roots then, [1] a = 2b = c, , [2] a = b = c, , [3] b2 = 4ac, , [4] None, , [3] x  2, , [4] x  3, , Q.70 If x  2  2  2  ..... then, [1] x  1, , [2] 1  x  2, , Q.71 The number of the roots of the equation x  1  x  2  3 is, [1] 1, , [2] 2, , [3] 3, , [4] 4, , Mathematics by Goyal sir........9829652673, , 5

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Quadratic Equation & Expression, 2, , Q.72 The set of values of K for which both the roots of the equation 4x – 20Kx + (25K2 + 15K – 66) = 0, are less than, 2, is given by, [1] (2, ), , [3] (– ), , [2] (4/5, 2), , Q.73 If X be the set of real number x such that, [1] , , 2x  1, 3, , 2 x  3x 2  x, , [2] (– 3/2, – 1/4), , [1] x > 4, , is positive, then X contains, , [3] (– 1/4, 1/2), , Q.74 What is the value of x which satisfy the inequality, [2] x > 3, , [4] None of these, , *[4] (1/2, 3), , x  2 2x  3, , x  2 4x  1, [3] x > – 2, , [4] None of these, , [3] – 2, – 3, , [4] 2, – 3, , [3] x – 1, , [4] – 1 x  1, , 2, Q.75 7log7 ( x  4 x  5)  x  1 , x may have values, , [1] 2, 3, , [2] 7, , Q.76 The set of values for which x3 + 1  x2 + x, is, [1] x 0, , [2] x  0, , Q.77 The values of a which make the expression x2 –ax + 1 – 2a2 always positive for real values of x, are [1] –, , 2, 2, a, 3, 3, , [2] –, , 2, 2, a, 3, 3, , [3] –, , 2,  a1, 3, , Q.78 The imaginary roots of the eqaution (x2 + 2)2 + 8x2 = 6x (x2 + 2) are [1] 1 ± i, [2] 2 ± i, [3] – 1 ± i, , [4] 0  a , , 2, 3, , [4] none of these, , Q.79 If x = – 5 + 2  4 then the value of the expression x4 + 9x3 + 35 x2 – x + 4 is equal to [1] 160, , [2] – 160, , [3] 120, , [4] – 120, , Q.80 The diagram shows the graph of y = ax2 + bx + c. Then -, , y, , x, , x, (x2,0), , [1] a > 0, , [2] b2 – 4ac < 0, , (x1,0), [3] c > 0, , [4] none of these, , *****, Mathematics by Goyal sir........9829652673, , 6

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Quadratic Equation & Expression, , EXERCISE-2,  x  m   4mn, 2  x  n, 2, , Q.1, , If x be real then the value of the expression, [1] lie between m and m+n, [3] greater than m+2n, , Q.2, , Q.3, , , , x 2  2x 1, , 2, , , , 2 3, [3] 0,1, , are, [4] 0, 2, , [3] 1, , [4] 2, , If 0  a  b  c and the roots , of the equation ax  bx  c  0 are imaginary, then, [2]   1, , [3]   1, , [4] None of these, , The number of quadratic equation which are unchanged by squaring their roots is, [2] 4, , [3] 6, , If the roots of the equation ax2 + bx + c = 0 are of the form, [2] b2 + 4ac, , If x is real and K , , ( x 2  x  1), ( x 2  x  1), , [1] 1/3  K  3, Q.8, , , , 2, , [1] b2 – 4ac, Q.7, , , ,  2 3, , [2] 1, , [1] 2, Q.6, , x2  2x 1, , If equations ax 2  2cx  b  0 and ax 2  2bx  c  0  b  c  have a common root, then a  4b  4c is equal to, , [1]   , Q.5, , , , [2] 1, 2, , [1] 0, Q.4, , [2] lie between 2m and 2n, [4] greater than m+n, , The roots of the equation 2  3, [1] 1, 0, , will not, , [4] None of these, , K 1, K2, and, then (a + b + c)2 =, K, K 1, , [3] b2 – 2ac, , [4] b2 + 2ac, , [3] K , , [4] None of these, , then, , [2] K  3, 2, , 2, , For the equation x  ( K  1) x  (K  K  8)  0 if one root is greater then 2 and other is less than 2 then K lies, between, [1] – 2 & 3, , Q.9, , [4] None of these, , [2] x – 2 or x  4, , [3] x  0 or x 4, , [4] None of these, , If ,  be the roots x 2  px  q  0 and ,  be the roots of x 2  px  r  0 then, [1] 1, , [2] q, , [3] r, 2, , Q.11, , [3] 2 & – 3, , If x  1  x  2  x  3  6 , then, [1] 0 x , , Q.10, , [2] 2 & – 2, ,         , ,        , [4] q + r, , 2, , , A quadratic equation whose roots are   and    , where ,,  are the roots of x 3  27  0 , is, , , [1] x 2  x  1  0, [2] x 2  3x  9  0, [3] x 2  x  1  0, [4] x 2  3x  9  0, , Q.12 If a > 0, b > 0, c > 0, then both the roots of the equation ax2 + bx + c = 0, [1] Are real and negative, [2] Have negative real parts, [3] are rational numbers, [4] none of these, Q.13 If a, b, c are positive real numbers, then the number of real roots of the equations ax2 + b|x| + c = 0 is [1] 0, [2] 1, [3] 2, [4] none of these, Q.14 If the roots of x2 + x + a = 0 exceed a, then [1] 2 < a < 3, [2] a > 3, [3] – 3 < a < 3, [4] a < – 2, , Mathematics by Goyal sir........9829652673, , 7

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Quadratic Equation & Expression, , Q.15 If x2/3 + x1/3 – 2 = 0 then x [1] – 2, 1, [2] – 8, – 2, [3] – 8, 1, [4] none of these, Q.16 If a < b < c < d. Then the roots of the eqaution (x – a)(x – c)– 3(x – b) (x – d) = 0, [1] real, [2] unreal, [3] rational, [4] irrational, 2x 2  4 x  1, , Q.17 If x is the real, the value of the expression, [1] any number, , x 2  4x  2, , is -, , [2] only positive number [3] only negative, , [4] only 1, , Q.18 If P(x) = ax2 + bx + c and Q(x) = – ax2 + dx + c where ac  0, then P(x). Q(x) = 0, has at least [1] Four real roots, [2] Two real roots, [3] Four imageinary roots, [4] none of these, Q.19 If the product of the roots of the equation x2 – 3kx + 2esin k –1 = 0 is 7 then its roots will be real if [1] |k|  22, , 7/9, , [2] |k|  2 7 / 9, , [3] |k|  2 7 / 9, , [4] never, , Q.20 If x > 1, then the minimum value of the expression 2 log10 x – logx (0.01) is [1] 2, [2] 4, [3] 1, , [4] none of these, , Q.21 If both roots of the equation x2 – (m + 1) x + (m + 4) = 0 are negative, then m equal [1] – 7 < m < – 5, [2] – 4 < m  – 3, [3] 2 < m < 5, [4] none of these, Q.22 If tanA and tanB are the roots of x2 – px + q = 0, then the value of sin2(A + B) is [1], , p2, 2, , p q, , 2, , [2], , p2, 2, , p  (1  q), , [3], , 2, , q2, 2, , p  (1  q), , [4], , 2, , p2, (p  q)2, , Q.23 The expression x2 + 2(a + b + c) x + 3(bc + ca + ab) will be a perfect square if, [1] a + b + c = 0, [2] ac + bc + ab = 0, [3] a = b = c, [4] none of these, 1/ 3, ,  2 , , Q.24 If ,  are roots of the equations 8x2 – 3x + 27 = 0, then the value of , , , , , [1], , 1, 3, , [2], , 7, 2, , [3] 4, , Q.25 The number of real solution of the equation x [(log 3 x ), [1] 0, [2] 1, , 2, , ( 9 / 3 ) log 3 x  5 ], ,  2 ,  ,  ,  , [4], , 1/ 3, , is -, , 1, 4, , = 3 3 is -, , [3] 2, , [4] 3, , Q.26 If equations ax + by = 1 and cx2 + dy2 = 1 has only one solution, then [1], , a2 b2, , 1, c, d, , [2], , c 2 d2, , 1, a, b, , [3], , a2 c 2, , 1, b, d, , [4] none of these, , Q.27 If the roots of the given equation (cosp – 1)x2 + (cos p) x + sin p = 0 are real, then [1] p  (–, 0), ,   , [2] p   , ,  2 2, , [3] p  (0, ), , [4] p  (0, 2), , *****, Mathematics by Goyal sir........9829652673, , 8

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Quadratic Equation & Expression, , EXERCISE-3, Q.1, , 2, , If expression e {(sin, , x  sin4 x  sin6 x  ..... ) n2}, , satisfies the equation x2 – 9x + 8 = 0, find the value of, , cos x, , ,0  x , cos x  sin x, 2, , –, , [1], , Q.2, , [IIT-91], 1, , [2], , 1 3, , 1, 1 3, , [3], , 1, 1 2, , [4] none of these, , Let a, b, c be real. If ax2 + bx + c = 0 has two real roots , , and under  < – 1 and  > 1, then 1 +, less then -, , [IIT Sc.-94], , [1] 1, , Q.3, , b, c, +, is, a, a, , [2] 2, , [3] 0, , [4] 4, , Let p, q  {1, 2, 3, 4}. The number of equations of the form px2 + qx + 1 = 0 having real roots is [IIT Sc.-94], [1] 15, , Q.4, , r, 7  4 3, p, , The equation, [1] no solution, , Q.6, , [2], , p, 7  4 3, r, , [3] for all values of p, r, , [4] for no value of p, r, , ( x  1)  ( x  1)  ( 4 x  1) has -, , [2] one solution, , [IITSc.-95], , [IIT-97 cancelled], [3] two solution, , [4] more than two solutions, , [2] 2  a  8, , [3] 2  a  8, , [RPET-97], , [4] 2  a  8, , GM of the roots of equation x 2  18x  9  0 is, [1] 6, , Q.8, , [4] 8, , If the roots of the equation x 2  8x  a2  6a  0 are real then the value of a will be, [1] 2  a  8, , Q.7, , [3] 7, , If p, q r are positive and are in AP, then roots of the equation px2 + qx + r = 0 are real if -, , [1], , Q.5, , [2] 9, , [2], , 3, , [RPET-97], [3] 3, , [4], , 3, , If one root of the equation x 2  px  q  0 and x 2  p' x  q2  0  p  p' and q  q'  is common, then the root is, [RPET-97], [1], , Q.9, , q  q', p  p', , [2], , pq' p 'q, q  q', , [3], , q  q', pq' p' q, or, p' p, q  q', , [4], , q  q', pq  p 'q', or, p' p, q  q', , If one root of the equation x2 – 30x + p = 0 is square of the other then p is equal to, [1] only 125, , [2] 125, – 216, , [3] 125, 215, , [REE 98], , [4] only 216, , Mathematics by Goyal sir........9829652673, , 9

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Quadratic Equation & Expression, , Q.10, , If x be real then the least value of x2 – 6x + 10 is, [1] 1, , Q.11, , [2] 2, , [2] a > 0, , [4] a = b = 0, [Aligarh 98], , [3] – 2, 2, 4, , [2] 33x2 – 4x + 1 = 0, , [4] – 2, 2, 3, , 1, 1, and, is, 2  3, 2  3, , [3] 33x2 – 4x – 1 = 0, , [PET 98], , [4] 33x2 + 4x + 1 = 0, , 1, 1, 1, ,  are negatives of each other, then r =, x p xq r, [2] p – q, , [3], , pq, 2, , [4], , [PET 98], , pq, 2, , [2] 17x2 – 20x + 3 = 0, , [3] 17x2 + 20x + 3 = 0, , [2] 174/5, , [CET 98], , [4] None of these, , If are the roots of the equation 5x2 – 20x + 12 = 0, the , , [BITS 98], , [3] 35, , [4] None of these, , x2  x  1, cannot take any value between, x 1, [1] 1 and 3, , Q.19, , [Aligarh 98], , The equation whose roots are reciprocal of the roots of the equation 3x2 – 20x + 17 = 0, is, , [1] 176/5, Q.18, , [3] a = b, c = 0, , [2] 1, 2, 4, , If the roots of the equation, , [1] 3x2 + 20x – 17 = 0, Q.17, , [4] – 4 < a < 0, , If are roots of x2 – 5x – 3 = 0, then the equation with roots, , [1] p + q, Q.16, , [Aligarh 98], , The roots of the equation | x2 – x – 6 | = x + 2 are, , [1] 33x2 + 4x – 1 = 0, Q.15, , [3] a < 8, , [2] 4b2 – ac = 0, , [1] 0, 1, 2, Q.14, , [4] 10, , If roots of the equations ax2 + 2bx + c = 0 and bx2 – 2 ac x + b = 0 are real, then, [1] ac = b2, , Q.13, , [3] 3, , If roots of the equation 2x2 – (a2 + 8a + 1) x + a2 – 4a = 0 are of opposite sign, then, [1] 0 < a < 4, , Q.12, , [CEE 98], , [BITS 98], , [2] –1 and 1, , [3] –1 and 3, , [4] –3 and 1, , The number of values of x in the interval 0,5 satisfying the equation 3 sin2 x  7sin x  2  0 is, [IIT-98], [1] 0, , Q.20, , [4] 10, , [2] 2  a  3, , [3] 3  a  4, , , , , , , , [IIT-99], , [4] a  4, , , , The harmonic mean of the roots of the equation 5  2 x 2  4  5 x  8  2 5  0 is, [1] 2, , Q.22, , [3] 6, , If the roots of the equation x 2  2ax  a2  a  3  0 are real and less than 3, then, [1] a  2, , Q.21, , [2] 5, , [2] 4, , [3] 6, , [IIT-99], , [4] 8, , If , , be the roots of the equation ax + bx + c =0 and  + k,  + k be the roots of the equation, 2, , Ax2 + Bx + C = 0 then, , [1], , a, A, , b 2  4ac, B 2  4 AC, , [2], , A, a, , -, , [RPET-99, IIT(Mains)2000], , a, [3]  , A, , 2, , A, [4]  , a, , 2, , Mathematics by Goyal sir........9829652673, , 10

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Quadratic Equation & Expression, , Q.23, , If are the roots of the equation x2 – 3x + 1 = 0, then the equation with roots, , 1, 1, , ,, will be,  2  2, [RPET 99], , [1] x2 – x – 1 = 0, Q.24, , [2] x2 + x – 1 = 0, , [4] None of these, , Which of the following equation has 1 and – 2 the roots ?, [1] x2 – x – 2 = 0, , Q.25, , [3] x2 + x + 2 = 0, , [2] x2 + x – 2 = 0, , [3] x2 – x + 2 = 0, , [SCRA 99], [4] x2 + x + 2 = 0, , For the equation 3x 2  px  3  0,p  0 , if one of the roots is square of the other, then p is equal to, [IIT-2000], [1], , Q.26, , 1, 3, , [2] 1, , Q.28, , [2]   0    , , [3]     0, , 2, 3, , [IIT-2000], , [4]   0    , , If b  a , then the equation  x  a  x  b   1  0 , has, , [IIT-2000], , [1] both roots in a,b, , [2] both roots in  ,a , , [3] both roots in  b, , , [4] one root in  ,a  and other in  b, , , If are the roots of the quadratic equation 6x2 – 6x + 1 = 0, then, , 1, 1, (a  b  c 2  d 3)  (a  b  c 2  d 3) , 2, 2, b c d, 12d  6c  4b  a, [1], [2] 12a + 6b + 4c + 9d, [3] a   , 2 3 4, 12, Q.29, , [4], , If  and       , are the roots of the equation x 2  bx  c  0 , where c  0  b , then, [1] 0    , , Q.27, , [3] 3, , [PET 2000], [4] None of these, , Let ,  be the roots of x 2  x  p  0 and ,  be the roots of x 2  4x  q  0 . If , , ,  are in G.P. then the, integral values of p and q respectively, are, [1] 2, 32, , Q.30, , [2] 2,3, , [2] 1, , [2] 1, , [IIT-2001], [3] 2, , [4] 0, , [3] 2, , [2] 2, 3, , [3] 2, – 6, , [PET 2001], [4] 6, 2, , 2, , If roots of x – 12x + 39x – 28 = 0 are in A. P. then their common difference is, [1] , , [2] , , [3] , , [PET 2001], , [4] 0, –1, , If difference of the roots of equation x2 – px + 8 = 0 is 2 then ‘P’ equals, [1] 6, , Q.33, , [4] 6, 32, , If p, q are the roots of equation x2 + px + q = 0 then value of p must be equal to, [1] 0, 1, , Q.32, , [3] 6,3, , The number of solutions of log4  x  1  log2  x  3 , [1] 3, , Q.31, , [IIT-2001], , [PET 2001], [4] None of these, , Mathematics by Goyal sir........9829652673, , 11

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Quadratic Equation & Expression, , Q.34, , The set of all real numbers x for which x 2  x  2  x  0 , is, [1]  , 2    2,  , , Q.35, , , , [3]  , 1  1,  , , [4], , , , 2,, , , , [3] 2 10, , [PET 2002], [4] None of these, , If (x – a) (x – b) + (x – c) (x – a) + (x – c) (x – b) = 0 then it's roots are, [2] Complex, , [3] Imaginary, , [PET 2002], [4] Rational, , If one root of quadratic equation is 2  5 then find the quadratic equation, [1] x2 – 4x – 1 = 0, , Q.38, , 2, , , [2] 3 10, , [1] Real, Q.37, ,  , , If roots of the equation 12x2 + mx + 5 = 0 are in 3 : 2 them m =, [1] 5 10, , Q.36, , , , [2] ,  2 , , [IIT-2002], , [2] x2 + 4x + 1 = 0, , [3] x2 + 4x – 1 = 0, , [PET 2003], [4] x2 – 4x + 1 = 0, , If and are the roots of equation ax2 + bx + c = 0 then find the roots of equaiton cx2 + bx + a = 0, [PET 2003], , 1 1, ,,  , , [1], Q.39, , 1, 1, ,,  , , [3], , 1, 1, ,,  , , [4] , , 1 1, ,,  , , If are the roots of equaiton 4 x 2  13 x  7  0 , then find the value , [1], , Q.40, , [2] , , 5 5, 4, , [2], , 5, 6, , [3], , 5, 6, , [PET 03], [4] 2 5, , The value of a for which one roots of the quadratic equation (a2 – 5a + 3) x2 + (3a – 1) x + 2 = 0 is twice as large, as the other is, , [AIEEE-03], , [1] – 2/3, Q.41, , Q.45, , [AIEEE-03, PET-2000], , [1] geometric progression, , [2] harmonic progression, , [3] arithmetic-geometric progression, , [4] arithmetic progression, , The number of real solutions of the equation x2 – 3| x | + 2 = 0, is, [2] 1, , [3] 3, , [AIEEE-03], [4] 2, , The real number x when added to its inverse gives the minimum value of the sum at x equal to [AIEEE-03], [1] 1, , Q.44, , [4] 2/3, , a b, c, , and are in, c a, b, , [1] 4, Q.43, , [3] – 1/3, , If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the square of their, reciprocals, then, , Q.42, , [2] 1/3, , [2] –1, , [3] –2, , [4] 2, , If one root of the quadratic equation x 2  px  q  0 be square of the other then, [1] p3  q  3p  q  q  0, , [2] p3  q  3p  q  q  0, , [3] p3  q  3p  q  q  0, , [4] p3  q  3p  q  q  0, , [IIT-2004], , Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the, quadratic equation, [1] x2 + 18x – 16 = 0, , [AIEEE-04], [2] x2 – 18x + 16 = 0, , [3] x2 + 18x + 16 = 0, , [4] x2 – 18x – 16 = 0, , Mathematics by Goyal sir........9829652673, , 12

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Quadratic Equation & Expression, , Q.46, , 2, , If (1 – p) is a root of quadratic equation x + px + (1 – p) = 0 then its roots are, [AIEEE-04], [1] 0, – 1, , Q.47, , [2] – 1, 1, , [3] 0, 1, , [4] – 1, 2, , If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value, of ‘q’ is, , Q.48, , [AIEEE-04], , [1] 3, , [2] 12, , In a triangle PQR, R , , , . If tan, 2, , [3] 49/4, , P,   and tan, 2, , [4] 4, , Q,   are the roots of ax2 + bx + c = 0, a  0 then, 2, [AIEEE-05], , [1] b = a + c, Q.49, , [2] b = c, , [3] c = a + b, , [4] a = b + c, 2, , If value of a for which the sum of the squares of the roots of the equation x - (a - 2)x - a - 1 = 0 assume the least, value is, [1] 2, , Q.50, , [2] 3, , [3] 0, , [4] 1, , If the roots of the equation x2 - bx + c = 0 be two consecutive integers, then b2 - 4c equals, [1] 1, , Q.51, , [AIEEE-05, PET-2005], , [2] 2, , [3] 3, , [AIEEE-05], , [4] –2, , If both the roots of the quadratic equation x2 - 2kx + k2 + k - 5 = 0 are less than 5, then k lies in the interval, [AIEEE-05], [1] [4, 5], , Q.52, , [3] (6, ), , [2] (-, 4), , [4] (5, 6), , In the quadratic equation ax2 + bx + c = 0, if  = b2 – 4ac and  + , 2 + 2, 3 + 3 are in G.P. where ,  are, the roots of ax2 + bx + c = 0 then, [1]  0, , Q.53, , [3] c = 0, , [4]  = 0, , [2] 0, , [3] 1, , [4] 2, , All the values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0 are greater then – 2 but less than, 4, lie in the interval, [AIEEE-2006], [1] m > 3, , Q.55, , [2] b = 0, , If the roots of the quadratic equation x2 + px + q = 0 are tan30º and tan15º, respectively then the value of, 2 + q – p is, [AIEEE-2006], [1] 3, , Q.54, , [IIT 2005], , [2] – 1 < m < 3, , [3] 1 < m < 4, , [4] – 2 < m < 0, , The number of values of x in the interval [0, 3] satisfying the equation 2 sin2x + 5 sinx– 3 = 0 is [AIEEE-2006], [1] 6, , Q.56, , [2] 1, , If x is real, the maximum value of, , [1] 41, , [2] 1, , [3] 2, 3 x 2  9 x  17, 3x 2  9x  7, , [4] 4, , is -, , [AIEEE-2006], , [3], , 17, 7, , [4], , 1, 4, , Mathematics by Goyal sir........9829652673, , 13

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Quadratic Equation & Expression, , Q.57, , 2, , If difference of the roots of the equation x + ax + 1 = 0 is less than, , 5 , then the set of possible values of a is:, [AIEEE-2007], , [1] (–), Q.58, , [3] (–), , [4] (3), , If roots of the equations bx2 + cx + a = 0 be imaginary, then for all real values of x, the value of the experession, 3b2x2 + 6bcx + 2c2 is :, [AIEEE-2009], [1] < 4ab, , Q.59, , [2] (–), , [2] > – 4ab, , [3] < – 4ab, , [4] > 4ab, , Let p and q be real numbers such that p  0, p3  q and p3  –q. If  and  are non-zero complex numbers, satisfying  +  = – p and 3 + 3 = q, then a quadratic equation having  and  as its roots is :, [IIT-JEE -2010], , Q.60, , [1] (p3 + q)x2 – (p3 + 2q)x + (p3 + q) = 0, , [2] (p3 + q)x2 – (p3 – 2q)x + (p3 + q) = 0, , [3] (p3 – q)x2 – (5p3 – 2q)x + (p3 – q) = 0, , [4] (p3 – q)x2 – (5p3 + 2q)x + (p3 – q) = 0, , The real number k for which the equation, 2 x 3  3x  k  0 has two distinct real roots in [0, 1] :, [JEE Mains – 2013], , Q.61, , [1] Lies between –1 and 0, , [2] Does not exist, , [3] Lies between 1 and 2, , [4] Lies between 2 and 3, , If the equations x 2  2 x  3  0 and ax 2  bx  c  0, a, b, c  R , have a common root, then a : b : c is :, [JEE Mains – 2013], [1] 1 : 3 : 2, , Q.62, , [2] 3 : 1 : 2, , [3] 1 : 2 : 3, , [4] 3 : 2 : 1, , If a  R and the equation, , [JEE Mains – 2014], ,  3( x  [x ]) 2  2( x  [x ])  a 2  0, , (where [x] denotes the greatest integer  x ) has no integral solution, then all possible values of a lie in the, interval :, (A) (1, 0)  (0, 1), Q.63, , (D) (,  2)  (2, ), 1 1, Let  and  be the roots of equation px 2  qx  r  0, p  0 . If p, q, r are in A.P. and   4 , then the value,  , (B) (1, 2), , (C) (–2, –1), , of |    | is :, (A), , 61, 9, , [JEE Mains – 2014], (B), , 2 17, 9, , (C), , 34, 9, , (D), , 2 13, 9, , Mathematics by Goyal sir........9829652673, , 14

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Quadratic Equation & Expression, , EXERCISE # 1, , EXERCISE # 2, , EXERCISE # 3, , Mathematics by Goyal sir........9829652673, , 15