VKR Classes TIME BOUND TESTS -7 Target JEE ADVANCED For Class XI VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
Single Choice Question : PRACTICE TEST-. The smallest integer greater than log + log, is (A) (B) (C) (D). If x = cos + cos cos ( + ) and y = sin sin cos, then (x y) equals (A) 0 (B) (C) (D). If, are the roots of the quadratic equation x log ( + log log log ) x ( ) = 0, then the value of + + is equal to (A) (B) 5 (C) 7 (D). Let f(x) = ax bx + c (a, b, c R, a 0) where a, b, c are in A.P. Then the equation f(x) = 0 has (A) no real solution (B) two unequal real roots (C) sum of roots always negative (D) product of roots always positive 5. Let the vertices of an equilateral ABC are (, ), (, ) and (a, b). Then consider the following four statements. I. a + b must be equal to 6 II. a + b must be equal to zero III. a + b can be equal to IV. length of its median is 6 6. Let f() = (A) cosec + sec. The least value of f() for all permissible value of, is 8 6 (B) 8 6 (C) 5 6 (D) 7. Let A(, ) and B(, ) be vertices of a ABC. If the centroid of ABC moves on the line x + y =, then the locus of the vertex C is (A) x + y = 9 (B) x y = 7 (C) x + y = 5 (D) x y = 8. The value of the expression sin x cos x + sin x cos 9x + sin 9x cos 7x equals (A) (tan 9x tan x) (B) (tan 9x tan x) (C) (tan 7x tan x) (D) (tan 7x tan x) 9. If the angles subtended by the sides of a triangle at orthocentre and incentre are equal, then the triangle is (A) Scalene (B) Isosceles but not equilateral (C) Equilateral (D) Obtuse angled 0. Let there exist a unique point P inside a ABC such that PAB = PBC = PCA =. If PA = x, PB = y, PC = z, = area of ABC and a, b, c are the sides opposite to tb angle A, B, c respectively. then tan is equal to (A) a b c (B) a b c (C) a b c (D) a b c VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
. Let,, are the roots of the cubic equation a 0 x + ax + a x + a = 0 (a 0 0). Then the value of ( ) + ( ) + ( ) equals 8(a a (A) a 0 0 a ) 8(a a (B) a 0 0 a ) 8(a 0 aa ) (C) a 0 8(a a 0a (D) a 0 ) 5...upton terms. If 7 0...upton terms = 0 7 log0 and n = log 0 x + log 0 x / + log 0 x / + log 0 x /8 +... +, then x equal to (A) 0 (B) 0 5 (C) 0 6 (D) 0 7 x. The value of n n ( ) n n 5 equals (A) 5 (B) 5 (C) 6 5 (D) 6 5. In a ABC with usual notations, if r =, r = 7 and R =, then the ABC is (A) equilateral (B) acute angled which is not equilateral (C) obtuse angled (D) right angled One or more than one correct : 5. sin sin sin sin (A) can not equals three for atleast one value of R (B) is zero for some value of R (C) lies in [, ] (D) lies in [, ] Subjective : 6. Let the lengths of the altitudes drawn from the vertices of a ABC to the opposite sides are, and. If the area of ABC is, then find the value of 5 p 7. If the expression cos + cos + cos + cos + cos has the value equal to in their q lowest has integral roots. 8. Find the sum of all positive integral value(s) of n, n [, 00] for which the quadratic equation x x n = 0 has integral roots. 9. If (x x ) + (y y ) = a (x x ) + (y y ) = b and (x x ) + (y y ) = c. then x x x y y y = (a + b + c) (b + c a) (c + a b) (a + b c). Find the value of. 0. If sin cos 7 = 0, then find the number of values of in [0, ] 7 VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
PRACTICE TEST- Paragraph for question nos. to Consider a ABC whose sides BC, CA and AB are represented by the straight lines x y + 5 = 0, x + y + = 0 and 8x y 0 = 0 respectively.. The area of ABC equals (A) (B) 5 (C) 7 (D). If AD be the median of the ABC then the equation of the straight line passing through (, ) and parallel to AD is (A) x y = 0 (B) x y 0 = 0 (C) x + y + 5 = 0 (D) x + y = 0. The orthocentre of the ABC is (A) (, ) (B), (C) (, ) (D), Assertion & Reason :. Statement- : If a + b + c > 0 and a < 0 < b < c, then both roots of the quadratic equation a(x b) (x c) + b (x c) (x a) + c (x a) (x b) = 0 are real and unequal. because Statement- : If both roots of the quadratic equation px + qx + r = 0 are of opposite sign then product of roots is negative and sum of roots is positive. (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true. 5. Let 0 <,, < Statement- : If tan, tan, tan are the roots of the cubic equation x 6x + kx 8 = 0, then tan=tan = tan. because Statement- : If a + b + c = abc and a, b, c are positive numbers then a = b = c. (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true. 9 R 6. Statement- : In any ABC, maximum value of r + r + r =. because Statement- : In any ABC, R r. (All symbols have their usual meaning in a triangle) (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true. VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
7. Statement- : If the sides of the ABC are along the lines L, L and L then there is only one point in the plane of ABC which is equidistant from the lines lines L, L and L. because Statement- : Incentre of the ABC is equidistant from the lines L, L and L. (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true. 8. Let l, l, l be the lengths of the internal bisectors of angles A, B, C respectively of a ABC. Statement- : because A cos l + B cos l + C cos a Statement- : l = bc b c, l = ca l = a b c b c c a, l = ab a b (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true. One or morethan one correct : 9. Let P(x) = ax + bx + c, Q(x) = ax + cx + b and R(x) = ax + bcx + b + c abc where a, b, c R and a 0. The equation R(x) = 0 will have non real roots if (A) P(x) = 0 has distinct real roots and Q(x) = 0 has non-real roots (B) P(x) = 0 has non-real roots Q(x) = 0 has distinct real roots (C) Both P(x) = 0 and Q(x) = 0 has non-real roots (D) Both P(x) = 0 and Q(x) = 0 have distinct real roots 0. Let a, b, c be unequal real numbers. If a, b, c are in G.P. and a + b + c = bx, athen x can not be equal to (A) (B) 0 (C) (D). If + log 5 (x + ) log 5 (ax + x + a), x R then a can be equal to (A) (B) 5/ (C) (D) /. The expression ( tan + cot ) ( cot + tan ) cot is (A) independent of, (B) independent of (C) dependent on (D) dependent on,. In a AEX, T is the mid point of XE, and P is the mid point of ET. If the APE is equilateral of side length equal to unity then which of the following alternative(s) is/are correct? (A) AX = (B) EAT = 90º (C) cos XAE = (D) AT = VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 5
Match the Column :. Column-I Column-II (A) If sum of the solution of the equation cot x + cosec x + sec x = tan x (P) k in [0, ] is, then the value of k is greater than (B) If x =... (0 digits), y =... (0 digits) (Q) x y and z =... (0 digits), then equals (R) (C) Possible integral values of a for which a 6 sin x 5 a 0, x R, is (S) 5 5. Let P be an interior point of ABC. Match the correct entries for the ratios of the Area of PBC : Area of PCA : Area of PAB depending on the position of the point P w.r.t. ABC. Marks will be given only if all the entries of the column-i are correctly matched. Column-I Column-II (A) If P is centroidd (G) (P) tan A : tan B : tan C (B) If P is incentre (I) (Q) sin A : sin B : sin C (C) If P is orthocentre (H) (C) sin A : sin B : sin C (D) If P is circumcentre (S) (S) : : (T) cos A : cos B : cos C VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 6
PRACTICE TEST- Single Choice Question :. If sin x + sin x + sin x =, then the value of cos 6 x cos x + 8 cos x equals (A) (B) (C) (D) 5. Number of straight lines equidistant from three non collinear point in the plane of the points equals (A) 0 (B) (C) (D). A variable straight line passes through ;the point of intersection of the lines x + y = and x y = and meets the co-ordinate axes in A and B. The locus of the middle point of AB is (A) 5xy = y + x (B) 0xy = x + y (C) 0xy = x + y (D) 5xy = x + y. The points ( a, b), (0, 0), (a, b) and (a, ab), (ab 0) (A) are collinear (B) are the vertices of a parallelogram (C) are the vertices of a rectangle (D) lie on a circle 5. Let, be the roots of ax + bx + c = 0 (a 0) and, be the roots of px + qx + r = 0 (p 0), and D,D be the respective discriminants of these equations. If,,, are in A.P., then D : D equals a (A) p a (B) b b (C) q c (D) r 6. If the points (, ) and (, ) lies on the same side of the straight line x 5y = a, then (A) < a < 7 (B) a = (C) a = 7 (D) a < or a > 7 Paragraph for question nos. 7 to 9 In ABC as shown, XX = d ; XX = d ; XX = d and X is the centre of the circumscribed circle around the ABC. a, b and c as usual are sides BC, CA and AB respectively. a b c 7. If abc X d = d d d dd d, then the value of is equal to d d B (A) (B) (C) (D) 8 X 8. If R is the radius of the circumcircle of the ABC and a(d + d ) + b (d + d ) + c (d + d ) = kr(a + b +c) then the value of k is (A) (B) / (C) / (D) 9. Let h a, h b and h c are the altitudes of the ABC from the angular points A, B and C respectively If (a + b + c ) = t (h a d + h b d + h c d ) then t equals (A) (B) (C) (D) One or more than one correct : 0. The equation of the straight line which passes through the point of intersection of the lines x y + = 0 and 5x + y = 0 and cuts off equal intercept on coordinate axes, is (A) x y + 5 = 0 (B) x y = 0 (C) x + y + 5 = 0 (D) x + y = 0 A X C. In ABC, if cos A + cos B = sin C, then which of the following hold(s) good? (A) cot A cot B = (B) cot A cot B = (C) a, c, b are in A.P. (D) a, b, c are in G.P. VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 7
. Two equal sides of an isosceles triangle are given by the equations 7x y + = 0 and x + y = 0 and its third side passes through the point (, 0). The equation of the third side can be (A) x + y + 9 = 0 (B) x y = (C) x y = (D) x + y + 7 = 0 Match the Column. Column-I Column-II (A) The value of the n n, is equal to n0 (P) (B) Number of value(s) of k for which atleast one root of the equation (A) (k k k + ) x + (k k + 6) x + (k 7k + 0) = 0 (C) In ABC, if cos A cos B + sin A sin B sin C =, (R) 6 sin A sin B sin then the value of + + sin B sin C sin C, is equal to (S) 8 A. Column-I Column-II Subjective : (A) If the value of (P) (tan 8º) (sin 6º) (cos 5º) (tan 7º) (tan 08º) (cos 6º) (sin º) (tan 6º) (cos 80º) (Q) is k sin 8º, then k has the value equal to 5 (B) If sin x cos x + cos x sin x =, then the value of sin x is (R) 8 (C) For all permissible values of x, the maximum value of the (S) f(x) = 5sin x cosx, is tan x 5 5. If the solution set of the inequality log x x (a < b < c < d). cd > is (a, b) (c, d) then the value of where ab 6. The ratios of the lengths of the sides BC and AC of ABC to the radius of circumscribed circle are equal to and respectively. If the ratio of the lengths of the bisectors of the interior angles B and C 5 8 is ( ) where,, N. Then find the value of ( + + ). 7. Tangents parallel to the three sides of ABC are drawn to its incircle. If x, y, z be the lengths of the parts of the tangents within the triangle (with respect to the sides a, b, c) then find the value of a x + b y + c z. VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 8
PRACTICE TEST- Single Choice Question :. Two circles of radii r and r are both touching the coordinates axes and intersecting each other orthogonally. The value of r r (where r > r ) equals (A) + (B) + (C) (D) + 5. Consider the family of lines (x y 6) + (x + y + ) = 0 and (x + y ) + (x y ) = 0. If the lines of these families are at right to each other then the locus of their point of intersection is (A) x + y x + y = 0 (B) x + y + x + y = 0 (C) x + y x + y = 0 (D) x + y x + y + = 0. The feet of the perpendicular from the origin on a variable chord of the circle x + y x y = 0 is N. If the variable chord makes and angle of 90º at the origin, then the locus of N has the equation (A) x + y x y = 0 (B) x + y + x + y = 0 (C) x + y x y = 0 (D) x + y + x y = 0 Paragraph for question nos. to 6 Consider XYZ whose sides x, y and z opposite to angular points X, Y and Z are in geometric progression.. If r be the common ratio of G.P. then (A) 5 5 < r < (B) 5 < r < 5 (C) 5 5 < r < (D) 5 < r < 5 sin Y 5. The integral values of is sin X (A) prime only (B) even (C) composite (D) odd 6. The maximum value of (A) irrational number (C) integer sin Z sin Y is (B) rational number but not integer (D) not defined Paragraph for question nos. to 6 Consider two circles S : x + y 9 = 0 and S : x + y 0x + 9 = 0 7. Length of the common chord of S = 0 and S = 0 is (A) 5 6 (B) 5 (C) 5 (D) 5 8. If is the angle between the two common tangents of S = 0 and S = 0, then cos equals (A) 5 (B) 5 (C) 5 (D) 5 9. Distance of a common tangent of S = 0 and S = 0 from point (9, 0) is 6 (A) 5 (B) 5 (C) 5 (D) 6 5 VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 9
Assertion & Reason : 0. Statement- : There lies exactly points on the curve 8x + y + 6xy =, which form an euilateral triangle because Statement- : The locus of all point P(x, y) satisfying 8x + y + 6xy = consists of union of a straight line and a point not on the line. (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true.. Let ABC be a triangle with centroid G and incentre I. Statement- : If GI is parallel to the side CA, then a, b, c are in A.P. because Statement- : In a triangle, incentre from the angular point A divides the angle bisector in the ratio of a:(b + c) reckoning from the vertex. (A) Statement- is true, Statement- is true and Statement- is correct explanation for Statement- (B) Statement- is true, Statement- is true and Statement- is NOT correct explanation for Statement (C) Statement- is true, Statement- is false (D) Statement- is false, Statement- is true. One or more than one correct :. In ABC, D is a point on BC such that DB =, DA = and DC =. If the circumcentre of the ADB is congruent to the circumcircle of the ADC then which of the following is/are correct? (A) Aangle B > 5º but angle C < 5º (B) both the angles B and C are greater than 5º (C) area of the triangle is 08 sq. units (D) measure of angle A equal to tan 7. Let f n () n n0 n sin ( n ). Then which of the following alternative(s) is/are correct? (A) f = (B) f 8 = (C) f = (D) f 5 () = 0. If one vertex of a equilateral triangle of side lies at the origin and other lies on the line x y = 0, then the coordinates of the third vertex are (A) (0, ) (B) (, ) (C) (, 0) (D) (, ) VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 0
Match the column : 5. Column-I Column-II (A) If a, b and c are distinct real numbers, such that the quadratic expressions (P) Q (x) = ax + bx + c, Q (x) = bx + cx + a and Q (x) = cx + ax + b are always non negative, then the possible integer in the range of the (Q) (B) (C) expression y = a b c, is ab bc ca 8 The sum + + + 8... is equal to (R) n If S n = n n n n n, then S 0 is less than (S) (D) The shortest distance from the point M( 7, ) (T) 5 to the circle x + y 0x y 5 = 0, is Subjective : 6. In an isosceles ABC, if the altitudes intersect on the inscribed circle then find the secant of the vertical angle A. 7. Real number x, y satisfies x + y =. If the maximum and minimum value of the expression z = are M and m respectively, then find the value (M + 6m). y 7 x 8. A cricket player played n (n > ) matches during his career and made a total of n n (n n 9)(.6 5. ) runs. If T 5 r represent the runs made by the player in r th match such that T = 6 and T r = T r + 6 r, r n then find n. VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
PRACTICE TEST- 5 Single Choice Question :. Number of distinct point(s) with integer coordinates (both x, y integer) which can lie on a circle with centre (, ), is (A) 0 (B) (C) (D) more than. Consider line L : y = 0 and circle S : x + y y + = 0. The area enclosed by S = 0, L = 0 and the lines which touches the circle and passing through orign, is (A) (B) (C) (D) +. Let L = 0 be in a line in parametric form given by x = x + r cos and y = y + r sin where r and have the usual meaning. The line L = 0 has been rotated through and in clockwise and anticlockwise both directions about points on the line situated at a distance units from the fixed point P(x, y ). If =5º, then area of the figure so formed is (A) sq. units (B) 8 sq. units (C) sq. units (D) No closed figure is formed. Consider a family of circles which are passing through M(, ) and are tangent to x-axis. If (h, k) is the centre of circle then (A) k (B) k (C) k (D) 0 < k < 5. Number of integral values of k for which the chord of the circle x + y = 5 passing through P(8, k) gets bisected at P(8, k) and has integral slope is (A) 8 (B) 6 (C) (D) 6. A variable line L = 0 is drawn through O(0, 0) to meet the lines L : x + y = 0 and L : x + y + =0 at points M and N respectively. A point P is taken on L = 0 such that OP is (A) x + y = 5 (B) (x + y) = 5 (C) x + y = 5 = OM (D) (x y) = 5 +. Locus of P ON Paragraph for question nos. 7 to 9 : Consider a circle S = 0 of radius unit and it touches the X-axis at point A. The centre C of this circle lies in the first quadrant. The tangent from O (where O is origin) touches the circle at point T. A point P is taken on this tangent such that OAP is right angled at A and perimeter of OAP is 8. 7. The area of OAP equals (A) 5 (B) 8 (C) 7 (D) 8. Area of the triangle by the pair of tangents from the origin and the corresponding chord of contact, is (A) 5 6 (B) 5 (C) 5 8 (D) VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
9. If the circles S = 0 and x + y k y + = 0 intersect orthogonally, then k is (A) or (BN) or (C) or (D) or Paragraph for question nos. 0 to : A circle S = 0 with centre C touches the line y = x at a point P such that OP = (where O is origin). The circle S = 0 contains the point N( 0, ) in its interior and length of its chord on the line x + y = 0 is 6. 0. Radius of the circle S = 0, is (A) (B) 5 (C) 6 (D) 7. Coordinates of the centre of the circle S = 0, is (A) ( 9, ) (B) ( 7, ) (C) ( 6, ) (D) ( 8, ). Equation of the circle touching the line y = x at P and passing through (, ), is (A) x + y + 5x 9y + = 0 (B) x + y + 5x + 9y + = 0 (C) x + y + x y + = 0 (D) x + y + 5x + 9y + 6 = 0 Paragraph for question nos. to 5 : Consider two points A (, ) and B (, ). Let M be a point on the straight line L x + y = 0.. If M be a point on the line L = 0 such that AM + BM is minimum, then the reflection of M in the line x=y is (A) (, ) (B) (, ) (C( (, ) (D( (, ). If M be a point on the line L = 0 such that AM BM is maximum, then the distance of M from N (,) is (A) 5 (B) 7 (C) 5 (D) 0 5. If M be a point on the line L = 0 such that AM BM is minimum, then the area of AMB equals (A) (B) (C) 6 (D) 8 One or more than one correct : 6. Let P be a point on the circle S x + y 6x 8y + 6 = 0. OX is the positive side of X-axis where O is the origin. Which of the following is/are correct? 6 (A) OP is minimum when P is, 5 5 (B) OP is maximum when P is, 5 5 96 8 (C) POX is minimum when P is, (D) POX is maximum when P is (0, ) 5 5 7. Which of the following statement(s) holds good? (A) Locus of the centre of a variable circle S = 0 which cuts two given circles S = 0 and S = 0 orthogonally (with coefficient of x and y unity in S = 0, S = 0 and S = 0) is S S = 0 (B) The acute angle bisector between the lines x y + = 0 and x y + = 0 is x y = 0 (C) If the lines px + qy + r = 0, qx + ry + p = 0 and rx + py + q = 0 and concurrent p (where p, q, r are non-zero number) then qr q + pr r + =. pq (D) If is the acute angle between the line pair x 5xy + y 6x + 9y + = 0, then cos = 5 VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
8. Tangents PA and PB are drawn to the circles S x + y y = 0 from the point P(, ). Which of the following alternative(s) is/are correct? (A) The power of point P(, ) with respect to circle S = 0 is (B) The angle between tangents from P(, ) to the circle S = 0 is (C) The equation of circumcircle of PAB is x + y x 5y + = 0 (D) The area of quadrilateral PACB is 7 square units where C is the centre of circle S = 0 9. If the equation of circle touching the y-axis at (0, ) and making an intercept of 8 unit on x-axis is x + y + gx + fy + c = 0, then (g + f + c) can be (A) (B) 7 (C) (D) 0. Which of the following statements is/are incorrect? (A) Two circles always have a unique common normal (B) Raidcal axis is always perpendicular bisector to the line joining the centres of two circles (C) Raidcal axis is nearer to the centre of circle of smaller radius (D) Two circles always have a radical axis Match the Column :. Column-I Column-I (A) If the equation of the image of line pair, y = x in y-axis is (P) y x x + =, then equals (B) Area of the parallelogram formed by the straight lines (Q) x + y =, x + y = 7 ; x + y = 5 and x + y =, is (C) The radius of the circle whose two normals are represented by the (R) equation x 5xy 5x + 5y = 0 and which touches externally the circle x + y x + y = 0 will be (S) (D) Let y 8xy + 5x = 0 are two tangents from origin to a unit circle (T) 7 in first quadrant. If the length of tangent on this circle from origin is a + b, then (a + b) equals Subjective :. The equation of a line through the mid point of the sides AB and AD of rhombus ABCD, whose one diagonal is x y + 5 = 0 and one vertex is A(, ) is ax + by + c = 0. Find the absolute value of (a + b+c) where a, b, c are integers expressed in lowest form.. If the straight line joining the origin to the points of intersection of x xy + y + x y + = 0 and x + y = k are at right angles, then find the value of 5k 6k.. Let S = 0 and S = 0 be two circles intersectijng at P(6, ) and both are tangent to x-axis and line y = mx. If product of radii of the circles S = 0 and S = 0 is 5, then find the value of m. VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605
PRACTICE TEST- 6 Single Choice Question :. If (, 6) is the image of the point (, ) with respect to the line L = 0, then L s equal to (A) x y + = 0 (B) x y + = 0 (C) x y + 5 = 0 (D) 6x y + = 0. Solution set of the equation ( cos x ) + cosx cosx = is (A) n ± ; n I (B) n ; n I (C) n ; n I (D) (n + ) ; n I. What is the co-efficient of x 00 in the binomial expansion of x x (A) 009 (B) 009 (C) 00 (D) 008. The number of three digit numbers satisfying the condition that there is no repetition in digits, the number must contain 5, and is less than 800 (A) 70 (B) 80 (C) 67 (D) 68 5. In a certain tournament, a player must be defeated three to be eliminated. If 576 contestants enter th tournament, then the greatest number of games that could be played equals to (A) 75 (B) 77 (C) 78 (D) 70 6. In a triangle ABC ; line joining incentre and circumcentre is parallel to the side AC. Then the value of cos A + cos C is equal to (A) (B) (C) 7. Co-ordinates of a point P from which the lengths of tangents drawn to the circle x + y + x + 7 = 0, x + y + x + 5y + 9 = 0 and x + y + y = 0 are equal, is (A) (, ) (B) (, ) (C) (, ) (D) None of these 8. If x, y R and satisfy (x + 5) + (y ) =, then the minimum value of x + y is (A) (B) (C) (D) Comprehension (Q.9 to Q.) Let P(a, b) be a point in the first quadrant. Circles are drawn through P touching the co-ordinate axes. 9. Radius of one of the circle be (A) ( a b ) (B) ( a + b ) (B) a + b ab (D) a + b ab 0. If two of the circles are orthogonal then a and b satisfy the relation (A) a + b = ab (B) (a + b) = ab (C) a + b = ab (D) a b = ab. Equation of the common chord of the two circles is (A) x + y = a b (B) x + y = ab (C) x + y = a + b (D) x + y = ab (D) 009 VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 5
Comprehension (Q. to Q.) In a ABC ; If O is circum centre and OE is perpendicular bisector of BC. Of and OG are perpendicular bisectors of AC and AB respectively. R and r denote circumradius and inradius respectively of triangle ABC (triangle is neither right angled nor equilateral) A G 0 F B E C. Circum radius of BOE is (A) R sec (A/) R sec(a / ) (B) (C) R sec (A/) R sec(a / ) (D). Length of BE is equal to (A) R sin (A/) (B) R cos (A/) (C) R sin (A/) (D) R cos (A/). The ratio of areas of the triangle GEF and ABC is (A) Rr (B) r R Match the column : 5. If a + b + c = and (a, b, c > 0), then Column-I Column-II (A) abc (P) 6 (B) (C) (C) 8 (Q) a b c 7 R r (R) 9 a b c (D) ( a) ( b) ( c) (S) 7 (D) R r 6. Column-I Column-II (A) Number of subsets of {a, b, c, d, e, f, g} (P) 0 which contain both a and b, are (B) Number of different arrangements of the (Q) letters in CONTEST in which first two places are occupied by vowels, are (C) Number of permutation of the letter ABCDEFG (R) 65 which contain the word BAD are (D) Number of different ways in which atleast fruits (S) 0 can be selected from apples, mangoes, 5 oranges and bananas is (fruits of the same species are identical) VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 6
PRACTICE TEST- 7 Single Choice Question :. The coefficient of a b c 7 in the expansion of (bc + ca + ab) 8 is (A) 60 (B) 0 (C) 0 (D) 80. Three circles of same radius 5 intersect at a point O and each two intersect at A, B, and C respectively. Then the radius of the circle that circumscribes ABC is (A) 0 (B).5 (C) 5 (D) 7.5. Number of ways in which the letters of the word DECISIONS be arranged so that letter N is some where to the right of letter D, is 9! (A) 8! (B) 9! (C) 8 (C).8!. There are 0 stations enroute. A train has to be stopped at of them. Number of ways in which the train can be stopped if atleast two of the stopping stations are consecutive, is (A) 5 (B) 56 (C) 6 (D) 6 One or more than one correct : 5. Given that, are the roots of the equation Ax x + = 0 and, the roots of the equation Bx 6x + = 0. If,, and are in H.P. then (A) A = (B) B = 6 (C) A = (D) B = 8 6. A triangle has altitude of length 5 and 7. The length of third altitude CAN-NOT be (A) 7 (B) 8 (C) 9 (D) 0 7. The system of equations x + y =, x + y = a will have (A) four solutions if a = (B) two solutions if a = / (C) four solutions if < a < (D) All of the above 8. (a x + a y + ) 009 is a polynomial in x and y. If the sum of the co-efficients vanishes for some real value of a. Then possible of is/are (A) (B) (C) (D) Comprehension (Q.9 to Q.) Consider two externally touching circles S and S having centres at points A and B whose radii are & respectively. A tangent to circle S from point B touch the circle S at point C. D is chosen on circle S so that AC is parallel to BD and the two segments BC & AD do not intersect. Segment AD intersects the circle S at E. The line through B and E intersects the circle S at another point F. 9. The length of segment EF is (A) (B) (C) (D) 0. The area of triangle ABD is (A) (B) (C) 6 (D) 8. The length of the segment DE is (A) (B) (C) (D) VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 7
Subjective :. Circle S is centered at (0, ) with radius. Circle S is externally tangent to circles S and also tangent x to x-axis. If the locus of the centre of the variable circle S can be expressed as y = +. Find.. For 0 <, if the point ( cos, sin ) lies in the angle between the lines y = ± (x ) in which origin lies, then lies in the interval of length k. Then value of k must be.. Let ABC be isoscles and AB = AC. Points M and N are mid-points of AB and AC respectively. AB Medians MC and NB intersects at right angle. If BC find the value of (p + q). = q p where p and q are relatively prime. Then 5. Let S be the sum of all divisors of the least natural number having divisors. If be the sum of the digits of S. Then find /. 6. Let O be an octagon with vertices labelled V, V,..., V 8 consecutively. All the diagonals of the octagon are drawn except for diagonals between V and V 5, V and V 6, V and V 7 and V and V 8. Then number of all triangles, whose vertices are vertices of the octagon, and whose edges are the diagonals which have been drawn, is. find (/). 7. Lattice paths are paths consisting of one-unit setps in the positive horizontal or positive vertical directions. Let distinct lattice paths from the point (, 0) to the point (, 5) ; if at the most one diagonal step (a vertical unit and a horizontal unit at once) is allowed ; are. Let be the sum of all digits of. Then find the value of /. VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 8
ANSWER KEY PRACTICE TEST-. C. B. C. B 5. B 6. D 7. A 8. C 9. C 0. D. D. B. C. D 5. AC6. 9 7. 8. 600 9. 0. 5 PRACTICE TEST-. C. D. B. C 5. A 6. A 7. D 8. D 9. AB 0. ABCD. AB. BD. ABC. A P,Q,R ; B P ; C QR 5. A S ; B R ; C P ; D Q PRACTICE TEST-. C. D. C. A 5. A 6. D 7. C 8. A 9. D 0. AD. BC. BD. A R;B P;C Q. A R ; B P ; C S 5. 0 PRACTICE TEST-. A. C. A. A 5. D 6. D 7. D 8. A 9. B 0. A. C. BCD. CD. AD 5. A Q,R,S ; B P ; C P,Q,R,S,T ; D Q 6. 9 7. 8. 6 PRACTICE TEST-5. B. C. B. A 5. B 6. B 7. B 8. C 9. D 0. D. A. A. B. D 5. A 6. ABCD 7. AC8. AC9. AC0. AC0. ABD. A T ; B S ; C Q ; D P PRACTICE TEST-6. C. C. B. D 5. B 6. C 7. C 8. B 9. D 0. A. C. B. A. C 5. A S,Q ; B R ; C P,R ; D Q 6. A Q ; B P ; C S ; D R PRACTICE TEST-7. D. C. C. D 5. AD 6. BCD 7. AC8. ABD 9. C 0. D. C. 8.. 7 5. 5 6. 8 7. 5 VKR Classes, C-9-0, Indra Vihar, Kota. Mob. No. 9890605 9