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TEST - 1 COURSE NAME : VIKAAS (JA) & VIPUL (JB)

COURSE CODE : CLINIC CLASSES

TARGET : JEE (IITs) 2014 Date : 14-02-2013

Duration : 1 Hour

Max. Marks : 80

Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.

Mathematics - Quadratic Equation INSTRUCTIONS 2.

No additional sheets will be provided for rough work.

3.

Blank paper, clipboard, log tables, slide rules, calculators, cellular phones, pagers and electronic gadgets in any form are not allowed.

4.

The answer sheet, a machine-gradable Objective Response Sheet (ORS), is provided separately.

5.

Do not Tamper / mutilate the ORS or this booklet.

6.

Do not break the seals of the question-paper booklet before instructed to do so by the invigilators.

7.

Write your Name, Roll No. and Sign in the space provide on the back page of this booklet.

B. Filling the bottom-half of the ORS : Use only Black ball point pen only for filling the ORS. Do not use Gel / Ink / Felt pen as it might smudge the ORS. 8.

Write your Roll no. in the boxes given at the top left corner of your ORS with black ball point pen. Also, darken the corresponding bubbles with Black ball point pen only. Also fill your roll no on the back side of your ORS in the space provided (if the ORS is both side printed).

9.

Fill your Paper Code as mentioned on the Test Paper and darken the corresponding bubble with Black ball point pen.

10. If student does not fill his/her roll no. and paper code correctly and properly, then his/her marks will not be displayed and 5 marks will be deducted (paper wise) from the total. 11. Since it is not possible to erase and correct pen filled bubble, you are advised to be extremely careful while darken the bubble corresponding to your answer. 12. Neither try to erase / rub / scratch the option nor make the Cross (X) mark on the option once filled. Do not scribble, smudge, cut, tear, or wrinkle the ORS. Do not put any stray marks or whitener anywhere on the ORS. 13. If there is any discrepancy between the written data and the bubbled data in your ORS, the bubbled data will be taken as final. C. Question paper format and Marking scheme : 14. The question paper consists of 1 parts. The part consists of Two Sections. 15. For each question in Section–I, you will be awarded 3 marks if you darken the bubble(s) corresponding to the correct choice for the answer and zero mark if no bubbled is darkened. In case of bubbling of incorrect answer, minus one (–1) mark will be awarded. 16. For each question in Section–II, you will be awarded 5 marks if you darken the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In case of bubbling of incorrect answer, minus one (–1) mark will be awarded.

DO NOT BREAK THE SEALS WITHOUT BEING INSTRUCTED TO DO SO BY THE INVIGILATOR

A. General : 1. This Question Paper contains 22 questions.

MATHEMATICS

MATHEMATICS SECTION - I

Straight Objective Type This section contains 15 questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct. 1.

If a, b, c R and the quadratic equation ax2 + bx + c = 0 has no real roots, then (A) (a + b + c) c > 0 (B) c < 0 (C) a + b + c > 0 (D) a + b + c < 0

2.

If , , , are the roots of the equation x3 + 3x –1 = 0, the equation whose roots are 2, 2, 2 is (A) x3 + 6x2 + 9x + 1 = 0 (B) x3 + 6x2 + 9x – 1 = 0 3 2 (C) x + 6x – 9x + 1 = 0 (D) (x2)3 + 3x2 – 1 = 0

3.

The number of integral values of ‘m’ less than 50 , so that the roots of the quadratic equation mx 2 + (2m – 1) x + (m – 2) = 0 are rational, are (A) 6 (B) 7 (C) 8 (D) 5

4.

The set of values of ‘p’ so that both the roots of the equation (p – 5)x 2 – 2px + (p – 4) = 0 are positive, one is less than 2 and other is lying between 2 & 3, is 49 , 24 (A) 4

5.

6.

(B) (5, )

49 49 (C) (–, 4) U , (D) 5 , 4 4

The least integral value of ‘a’ such that (a – 2) x 2 + 8x + a + 4 > 0, x R is (A) 3 (B) 5 (C) 4 (D) 6 4 |x – 2| . 5 x – 1 | = – x 4 .3|x – 2| . 5 x – 1 is The number of integral value(s) of x satisfying the equation | x .3 (A) 2 (B) 3 (C) 1 (D) infinite

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-1

MATHEMATICS 2

2

7.

Let p, q be roots of the equation x – 4x + A = 0 and r and s be the roots of the equation x – 20x + B = 0. If p < q < r < s are in A.P., then (A, B) is (A) (0, – 96) (B) (96, 0) (C) (0, 96) (D) (–96, 0)

8.

Number of real ordered pair (x, y) satisfying x2 + 1 = y and y2 + 1 = x is (A) 0 (B) 1 (C) 2

(D) 4

9.

Number of integral values of a for which (a + 2) x2 + 2(a + 1) x + a = 0 will have integer roots is/are (A) 3 (B) 2 (C) 5 (D) 4

10.

If p & q are distinct reals , then 2 {(x p) (x q) + (p x) (p q) + (q is satisfied by : (A) no value of ‘ x ‘ (C) exactly two values of ‘ x ‘

11.

(B) exactly one value of ‘ x ‘ (D) infinite values of ‘ x ‘

If one root of the quadratic equation 2x 2 – 2kx + k – 4 = 0 is smaller than 1 & other is greater than 2, then complete set of values of k is (A) (–2, )

12.

x) (q p)} = (p q)2 + (x p)2 + (x q)2

4 (B) , 3

4 (C) 2 , 3

(D) (–, –2)

If , are the roots of the equation x2 – 2x + 3 = 0 then the equation whose roots are 3 – 3 2 + 5 – 2 and 3 – 2 + + 5 is (A) x2 + 3x – 2 = 0 (B) x2 – 3x + 2 = 0 (C) x2 + 3x + 2 = 0 (D) x2 – 3x – 2 = 0

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-2

MATHEMATICS

13.

14.

2

2

Sum of real solutions of the equation 5 2x – 2.5 x x 1 – 3.5 2 x 3 = 0 is (A) 1 (B) 2 (C) 3

(D) 4

Let the values of 'a' for which the roots of the equation (x – a) (x – a – 1) = 0 lie between the roots of the

equation (x + a) (x + a2 – 2) = 0 be a (–, p) q , 1 5 , then the value of q – p is 2 (A) 1 15.

(B) 2

(C) 3

If the quadratic equation ax 2 + bx + a2 + b2 + c 2 ab bc has imaginary roots then : (A) 2 (a b) + (a b)2 + (b c)2 + (c a)2 > 0 (B) 2 (a b) + (a b)2 + (b c)2 + (c a)2 < 0 (C) 2 (a b) + (a b)2 + (b c)2 + (c a)2 = 0 (D) none of these

(D) 4

ca = 0, where a , b , c are distinct reals,

SECTION - II

Multiple Correct Answer Type This section contains 7 questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct. 16.

The equation x2 – 6x + 8 + (x2 – 4x + 3) = 0, R – { –1} has (A) real and unequal roots for all (B) real roots for < 0 (C) real roots for > 0 (D) real and unequal roots for = 0

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-3

MATHEMATICS

17.

2

2

3

3

3

If a + b + c = 1, a + b + c = 2, a + b + c = 3, then 1 1 (A) ab + bc + ca = – (B) abc = 6 2 (C) ab + bc + ca =

18.

2

Let f(x) =

1 2

(D) abc =

1 9

(x – a)(x – b) . Which of the following are TRUE ? (x – c)

(A) Range of f(x) is R, for b c a (B) Range of f(x) is R, for a c b (C) Range of f(x) is proper subset of R, for a b c (D) Range of f(x) is proper subset of R, for c b a 19.

If x 2 – ax – 3 = 0 and x 2 + ax – 15 = 0 have a common root, then a = (A) 2 (B) 3 (C) – 3

(D) – 2

20.

If the quadratic equation ax 2 + 2bx – 4 = 0, where a, b, c R, does not have two real & distinct roots, then (A) a – b > 1 (B) 2b – a –4 (C) 2b – a –4 (D) a + b 1

21.

Let (a – 1) (x2 +

22.

3 x + 1)2 – (a + 1) (x4 – x2 + 1) 0 x R, then which of the following is/are correct ?

1 4 , (A) a – 3 3

(B) Largest possible value of a is

(C) Number of possible integral values of a is 3

(D) Sum of all possible integral values of a is '0'

3

If R, equation ax 2 + (b – )x + (a – b – ) = 0, a 0, a,b R has real roots, then which of the following may be true ? (A) a = b (B) b < a < 0 (C) b > a > 0 (D) a > b > 0

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-4

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0

TEST - 1 COURSE NAME : VIKAAS (JA) & VIPUL (JB)

COURSE CODE : CLINIC CLASSES

TARGET : JEE (IITs) 2014 Date : 14-02-2013

Duration : 1 Hour

Max. Marks : 80

Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.

Mathematics - Quadratic Equation INSTRUCTIONS 2.

No additional sheets will be provided for rough work.

3.

Blank paper, clipboard, log tables, slide rules, calculators, cellular phones, pagers and electronic gadgets in any form are not allowed.

4.

The answer sheet, a machine-gradable Objective Response Sheet (ORS), is provided separately.

5.

Do not Tamper / mutilate the ORS or this booklet.

6.

Do not break the seals of the question-paper booklet before instructed to do so by the invigilators.

7.

Write your Name, Roll No. and Sign in the space provide on the back page of this booklet.

B. Filling the bottom-half of the ORS : Use only Black ball point pen only for filling the ORS. Do not use Gel / Ink / Felt pen as it might smudge the ORS. 8.

Write your Roll no. in the boxes given at the top left corner of your ORS with black ball point pen. Also, darken the corresponding bubbles with Black ball point pen only. Also fill your roll no on the back side of your ORS in the space provided (if the ORS is both side printed).

9.

Fill your Paper Code as mentioned on the Test Paper and darken the corresponding bubble with Black ball point pen.

10. If student does not fill his/her roll no. and paper code correctly and properly, then his/her marks will not be displayed and 5 marks will be deducted (paper wise) from the total. 11. Since it is not possible to erase and correct pen filled bubble, you are advised to be extremely careful while darken the bubble corresponding to your answer. 12. Neither try to erase / rub / scratch the option nor make the Cross (X) mark on the option once filled. Do not scribble, smudge, cut, tear, or wrinkle the ORS. Do not put any stray marks or whitener anywhere on the ORS. 13. If there is any discrepancy between the written data and the bubbled data in your ORS, the bubbled data will be taken as final. C. Question paper format and Marking scheme : 14. The question paper consists of 1 parts. The part consists of Two Sections. 15. For each question in Section–I, you will be awarded 3 marks if you darken the bubble(s) corresponding to the correct choice for the answer and zero mark if no bubbled is darkened. In case of bubbling of incorrect answer, minus one (–1) mark will be awarded. 16. For each question in Section–II, you will be awarded 5 marks if you darken the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In case of bubbling of incorrect answer, minus one (–1) mark will be awarded.

DO NOT BREAK THE SEALS WITHOUT BEING INSTRUCTED TO DO SO BY THE INVIGILATOR

A. General : 1. This Question Paper contains 22 questions.

MATHEMATICS

MATHEMATICS SECTION - I

Straight Objective Type This section contains 15 questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct. 1.

If a, b, c R and the quadratic equation ax2 + bx + c = 0 has no real roots, then (A) (a + b + c) c > 0 (B) c < 0 (C) a + b + c > 0 (D) a + b + c < 0

2.

If , , , are the roots of the equation x3 + 3x –1 = 0, the equation whose roots are 2, 2, 2 is (A) x3 + 6x2 + 9x + 1 = 0 (B) x3 + 6x2 + 9x – 1 = 0 3 2 (C) x + 6x – 9x + 1 = 0 (D) (x2)3 + 3x2 – 1 = 0

3.

The number of integral values of ‘m’ less than 50 , so that the roots of the quadratic equation mx 2 + (2m – 1) x + (m – 2) = 0 are rational, are (A) 6 (B) 7 (C) 8 (D) 5

4.

The set of values of ‘p’ so that both the roots of the equation (p – 5)x 2 – 2px + (p – 4) = 0 are positive, one is less than 2 and other is lying between 2 & 3, is 49 , 24 (A) 4

5.

6.

(B) (5, )

49 49 (C) (–, 4) U , (D) 5 , 4 4

The least integral value of ‘a’ such that (a – 2) x 2 + 8x + a + 4 > 0, x R is (A) 3 (B) 5 (C) 4 (D) 6 4 |x – 2| . 5 x – 1 | = – x 4 .3|x – 2| . 5 x – 1 is The number of integral value(s) of x satisfying the equation | x .3 (A) 2 (B) 3 (C) 1 (D) infinite

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-1

MATHEMATICS 2

2

7.

Let p, q be roots of the equation x – 4x + A = 0 and r and s be the roots of the equation x – 20x + B = 0. If p < q < r < s are in A.P., then (A, B) is (A) (0, – 96) (B) (96, 0) (C) (0, 96) (D) (–96, 0)

8.

Number of real ordered pair (x, y) satisfying x2 + 1 = y and y2 + 1 = x is (A) 0 (B) 1 (C) 2

(D) 4

9.

Number of integral values of a for which (a + 2) x2 + 2(a + 1) x + a = 0 will have integer roots is/are (A) 3 (B) 2 (C) 5 (D) 4

10.

If p & q are distinct reals , then 2 {(x p) (x q) + (p x) (p q) + (q is satisfied by : (A) no value of ‘ x ‘ (C) exactly two values of ‘ x ‘

11.

(B) exactly one value of ‘ x ‘ (D) infinite values of ‘ x ‘

If one root of the quadratic equation 2x 2 – 2kx + k – 4 = 0 is smaller than 1 & other is greater than 2, then complete set of values of k is (A) (–2, )

12.

x) (q p)} = (p q)2 + (x p)2 + (x q)2

4 (B) , 3

4 (C) 2 , 3

(D) (–, –2)

If , are the roots of the equation x2 – 2x + 3 = 0 then the equation whose roots are 3 – 3 2 + 5 – 2 and 3 – 2 + + 5 is (A) x2 + 3x – 2 = 0 (B) x2 – 3x + 2 = 0 (C) x2 + 3x + 2 = 0 (D) x2 – 3x – 2 = 0

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-2

MATHEMATICS

13.

14.

2

2

Sum of real solutions of the equation 5 2x – 2.5 x x 1 – 3.5 2 x 3 = 0 is (A) 1 (B) 2 (C) 3

(D) 4

Let the values of 'a' for which the roots of the equation (x – a) (x – a – 1) = 0 lie between the roots of the

equation (x + a) (x + a2 – 2) = 0 be a (–, p) q , 1 5 , then the value of q – p is 2 (A) 1 15.

(B) 2

(C) 3

If the quadratic equation ax 2 + bx + a2 + b2 + c 2 ab bc has imaginary roots then : (A) 2 (a b) + (a b)2 + (b c)2 + (c a)2 > 0 (B) 2 (a b) + (a b)2 + (b c)2 + (c a)2 < 0 (C) 2 (a b) + (a b)2 + (b c)2 + (c a)2 = 0 (D) none of these

(D) 4

ca = 0, where a , b , c are distinct reals,

SECTION - II

Multiple Correct Answer Type This section contains 7 questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct. 16.

The equation x2 – 6x + 8 + (x2 – 4x + 3) = 0, R – { –1} has (A) real and unequal roots for all (B) real roots for < 0 (C) real roots for > 0 (D) real and unequal roots for = 0

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-3

MATHEMATICS

17.

2

2

3

3

3

If a + b + c = 1, a + b + c = 2, a + b + c = 3, then 1 1 (A) ab + bc + ca = – (B) abc = 6 2 (C) ab + bc + ca =

18.

2

Let f(x) =

1 2

(D) abc =

1 9

(x – a)(x – b) . Which of the following are TRUE ? (x – c)

(A) Range of f(x) is R, for b c a (B) Range of f(x) is R, for a c b (C) Range of f(x) is proper subset of R, for a b c (D) Range of f(x) is proper subset of R, for c b a 19.

If x 2 – ax – 3 = 0 and x 2 + ax – 15 = 0 have a common root, then a = (A) 2 (B) 3 (C) – 3

(D) – 2

20.

If the quadratic equation ax 2 + 2bx – 4 = 0, where a, b, c R, does not have two real & distinct roots, then (A) a – b > 1 (B) 2b – a –4 (C) 2b – a –4 (D) a + b 1

21.

Let (a – 1) (x2 +

22.

3 x + 1)2 – (a + 1) (x4 – x2 + 1) 0 x R, then which of the following is/are correct ?

1 4 , (A) a – 3 3

(B) Largest possible value of a is

(C) Number of possible integral values of a is 3

(D) Sum of all possible integral values of a is '0'

3

If R, equation ax 2 + (b – )x + (a – b – ) = 0, a 0, a,b R has real roots, then which of the following may be true ? (A) a = b (B) b < a < 0 (C) b > a > 0 (D) a > b > 0

Space for Rough Work

RESONANCE

JA&JBTEST1140213C0-4

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