D-01 / DC-01

TYPICAL QUESTIONS & ANSWERS

PART – I

OBJECTIVE TYPES QUESTIONS

Each Question carries 2 marks.

Choose correct or the best alternative in the following:

Q.1 The co-ordinates of the middle points of the sides of a triangle are (4, 2) (3, 3) and (2, 2). Then the co-ordinates of the centroid are

(A) . (B) (3, 3).

(C) (4, 3). (D) (4, 7).

Ans: A

Coordinate of the Centroid is

Coordinate of the centroid =

Q.2 If x, 2x +2, 3x + 3 are first three terms of a G.P. then its 4^th term is

(A) 27. (B) -27.

(C) 13.5. (D) -13.5.

Ans: D

If x = -1 then three terms are

-1, 0, 0

If x = -4 then the first three terms are

-4, -6, -9

Therefore common ratio is

4^th terms = -13.5

Q.3 The angle made by any diameter of a circle at any point on the circumference is

(A) 90° (B) 180°

(C) 45° (D) 60°

Ans: D

Q.4 If then the value of r is

(A) 6. (B) 5.

(C) 4. (D) 7.

Ans:A

Or,

Q.5 is equal to

(A) . (B) .

(C) . (D) .

Ans: A

Q.6 If then is equal to

(A) 4896. (B) 816.

(C) 1632. (D) 408.

Ans: B

Q.7 is

(A) . (B)5.

(C) Not defined. (D) .

Ans: A

Q.8 is equal to

(A) . (B) .

(C) . (D) .

Ans: D

Q.9 The equation of the straight line which makes equal intercepts on the axes and passes through the point (1, 2) is

(A) x + y = 3 (B) x + 2y = 5

(C) x – y = 1 (D) 2x + y = 4

Ans: A

Straight line having equal intercepts on axes is x + y = a. If it passes through (1, 2), then a = 3. Hence required straight line x + y = 3.

Q.10 Area of the triangle whose vertices are (a, b) (a, a + b), (-a, -a + b) is

(A) a²b² (B) a² + b²

(C) a² (D) b²

Ans: C

Area of reqd.

= a²

Q.11

(A) 1 (B)

(C) (D) Zero

Ans: B

Q.12 The point on the curve y² = 4x at which the tangent to the curve is parallel to y = x is

(A) (0, 0) (B) (2,

(C) (4, 4) (D) (1, 2)

Ans: D

Here . If tangent is parallel to y = x,

Q.13 is equal to

(A) tan x – cot x (B) tan x + cot x

(C) sec x + cosec x (D) sec x - cosec x

Ans: C

Q.14 Sin³x dx is equal to

(A) (B)

(C) (D)

Ans: A

(By formula)

Q.15 Solution of differential equation is

(A) e^x + e^y = const (B) e^x – e^y = const

(C) e^x . e^y = const (D) e^x / e^y = const

Ans:

Q.16 Period of Sin (2x + 3) is

(A) 2π (B)

(C) π (D)

Ans: C

= = ……….Hence period is π.

Q.17 The value of Sin 105⁰ + Cos 105⁰ is

(A) (B)

(C) (D)

Ans: D

Q.18 If pth, (2p)^th and (3p)^th terms of a G.P. are x, y, z respectively, then x, y, z are in

(A) A.P. (B) H.P.

(C) G.P. (D) None of these

Ans: C

If a, ar, ar², ar³, ……….. be the G.P. then T_p = x = ar^p-1, T_2p = y ar^2p-1, T_3p = Z = ar^3p-1. Evidently y² = xz. Hence x, y, z are in G.P.

Q.19 Sum of the series is equal to

(A) 348551 (B) -1000

(C) 5151 (D) None of the above

Ans: C

Q.20 The value of tan is

(A) (B)

(C) (D)

Ans: A

Q.21 In a triangle ABC, let a = BC, b = CA and c = AB. If , then

(A) (B)

(C) (D) None of the above

Ans: C

Q.22 The circles and cut orthogonally if the

value of p is

(A) 3 (B) -2

(C) -3 (D) 1

Ans: A

Q.23 The eccentricity of the ellipse is

(A) 3 (B)

(C) 5 (D)

Ans: D

Q.24 The derivative of – cos (log x) is

(A) sin (log x) (B)

(C) – sin (log x) (D)

Ans: B

Q.25 The value of the is

(A) 0 (B) 1

(C) e (D) Does not exist

Ans: B

Q.26 The integral is equal to

(A) (B)

(C) 0 (D) 1

Ans: D

Q.27 The area under the curve between x = 0 and x = 1 is

(A) 1 (B)

(C) (D)

Ans: C

Q.28 The solution of is

(A) (B)

(C) (D)

Ans: A

Q.29 If one root of the equation is of the other root, then K is

(A) 2 (B) 8

(C) 10 (D) 12

Ans: D.

Q.30 The centroid of the triangle formed by the straight lines is

(A) (0, 0) (B) (1, 0)

(C) (0, 1) (D) (1, 1)

Ans: C.

The points of intersection of the

Given straight lines are A(3, 0), B(0, 3), C(-3, 0)

Q.31 The distance between the parallel lines 3x + 4y + 5 = 0 and 3x + 4y + 15 = 0 is

(A) 1 (B) 2

(C) 3 (D) 5

Ans: B.

The distance of origin from the line 3x + 4y +5 = 0 is

The distance of origin from the line 3x + 4y +15 = 0 is

\ distance between parallel lines =

Q.32 , where m n is equal to

(A) m (B) n

(C) m – n (D) m + n

Ans: C.

Q.33 If then is equal to

(A) 2 sin 4x (B) 4 sin 2x

(C) sin 4x (D) 2 sin 2x

Ans: A.

Q.34 is equal to

(A) (B) log

(C) tan x + sec x (D) tan x – sec x

Ans: D.

Q.35 is equal to

(A) (B)

(C) 1 (D) 0

Ans: B.

Q.36 The solution of the differential equation is

(A) (B)

(C) (D)

Ans: A.

Q.37 The value of is equal to

(A) 1 (B)

(C) (D) zero

Ans: C.

Q.38 The value of is

(A) (B)

(C) (D)

Ans: D.

Q.39 If , then n is equal to

(A) 8 (B) 12

(C ) 16 (D) 20

Ans: D

(n - 8)(n - 9)(n - 10)(n - 11) = 12.11.10.9 = (20 - 8)(20 - 9)(20 - 10)(20 - 11)

Q.40 is equal to

(A) 0 (B) 1

(C) 2 (D) 3

Ans: B

= 1 x 1 = 1

Q.41 If the point P(x, y) is equidistant from the points and , then

(A) bx = ay (B) ax = by

(C) x = y (D) x + y = 0

Ans: A

Q.42 The area of the triangle formed by the lines y = a + x, y = a – x, y = 0, where a > 0, is

(A) 1 (B) a

(C) (D) zero

Ans: C

y = a + x

y = a – x

Solving, we get y = a, x = 0

Therefore vertex is (0, a) another vertices are (-a, 0), (a, 0).

Area of the triangle is

Q.43 If is equal to

(A) 1 (B) 2

(C) (D)

Ans: A

thus

Q.44 is equal to

(A) sec x + cosec x (B)

(C) (D)

Ans: C

Q.45 The area bounded by the parabola and its latus rectum is

(A) (B)

(C) (D)

Ans: D

The parabola is symmetrical about the x – axis

Area =

Q.46 The solution of differential equation is

(A) (B)

(C) (D)

Ans: B

Q.47 Value of is

(A) (B)

(C) (D)

Ans: C

Q.48 Value of is

(A) 0 (B) 1

(C) (D)

Ans: A

Q.49 The number of terms in the sequence are

(A) 8 (B) 9

(C) 10 (D) 6

Ans: B

C.R = 2

Q.50 First three terms in the expansion of are

(A) (B)

(C) (D)

Ans: D

Q.51 Value of is

(A) (B)

(C) (D)

Ans: A

Q.52 If , then the value of cos 2A is

(A) (B)

(C) (D)

Ans: D

Q.53 The value of ‘x’ such that PQ = QR, where P, Q and R are (6, -1), (1, 3) and (x, 8) respectively is given by

(A) 5, –3 (B) 3, 5

(C) 2, 5 (D) 2, 3

Ans: A

Q.54 Slope of the line passing through the points & is

(A) (B)

(C) (D)

Ans: A

Q.55 is equal to

(A) (B)

(C) (D)

Ans: C

Q.56 If then is equal to

(A) (B)

(C) (D)

Ans: A

Q.57 is equal to

(A) (B)

(C) (D)

Ans: C

Q.58 Order and degree of the differential equation is given by

(A) 3, 2 (B) 2, 3

(C) 1, 3 (D) 3, 1

Ans: D

Order – 3 [Power of higher directive]

Degree – 1

Q.59 Which term of the series 37+32+27+22+.............. is –103?

(A) 24^th (B) 30^th

(C) 15^th (D) 29^th

Ans: D

a = 37, d = -5

n = 29

Q.60 How many terms are there in the expansion of

(A) 4 (B) 6

(C) 16 (D) 10

Ans: C

The given expansion is

No. of terms in the expansion is = 15 + 1 = 16

(ONE more than the power of given expansion)

Q.61 If and , find the value of cot

(A) (B)

(C) (D)

Ans: B

, and

Q.62 Expansion of is equal to

(A) (B)

(C) (D)

Ans: A

Q.63 For what value of k do the points & lie on a straight line.

(A) 3 (B) 4

(C) 0 (D) 1

Ans: C

The points say A(-1, 4), B(-3, 8), C(-k + 1, 3k) lies on straight line if area of ABC = 0

Q.64 Mid point of the line joining (3, 5) and is given by

(A) (B) (1, 2)

(C) (2, 3) (D) (2, 1)

Ans: A

The midpoint of the line joining (3, 5) and (-7, -3) is

mid point.

Q.65 is equal to

(A) (B) 2

(C) (D)

Ans: B

Q.66 If y = x sin x, then is equal to

(A) cos x + sin x (B) cos x + x sin x

(C) x cos x + sin x (D) x cos x – sin x

Ans: C

If y = x sin x

Differentiating both side w.r. to x we have

Q.67 is equal to

(A) tan x + c (B)

(C) (D)

Ans: D

Q.68 The solution of the differential equation is

(A) (x + y) = k (1 – xy) (B) y – x = kxy

(C) (D)

Ans: A

Using variable separable method

Q.69 The square root of 12 – 5i is

(A) . (B) .

(C) . (D) .

Ans: C

Let

12-5i = =

Q.70 If be the roots of then the quadratic equation whose roots are is

(A) . (B) .

(C) . (D) .

Ans: A

We know that , . Since = , equation is

Q.71 The 12^th term in the binomial expansion of is

(A) . (B) .

(C) . (D) .

Ans: B

12^th term in the expansion of

is =

Q.72 The area of the triangle formed by the coordinate axes and the line 2x + 3y=6 is

(A) 3 sq. units. (B) 6 sq. units.

(C) 9 sq. units. (D) 12 sq. units.

Ans: A

Area of triangle is base ´ height = =3 sq. unit

Q.73 The eccentricity of the ellipse if its latus rectum is equal to one half of its minor axis is

(A) . (B) .

(C) ½ . (D) .

Ans: B

Eccentricity =

Q.74 In a triangle ABC, sin A – cos B = cos C, then angle B is

(A) . (B) .

(C) . (D) .

Ans: A

Given sin A-cos B=cos C Sin A= cos B+ cos C =

Q.75 is equal to

(A) e – 1. (B) e + 1.

(C) 0. (D) 1.

Ans: D

= =e-e+1 =1

Q.76 If , then is equal to

(A) . (B)

(C) . (D) .

Ans: C

Given

\ or

Q.77 The point is equidistant from points (7,6) and (-3, 4) if

(A) (B)

(C) (D)

Ans: A

Q.78 The value of is

(A) (B)

(C) 1 (D) 0

Ans: D

= -

= = 0

Q.79 The equation of a line through point (2, -3) and parallel to y-axis is

(A) y = -3 . (B) y = 2.

(C) x = 2. (D) x = -3.

Ans: C

The equation of line parallel to y-axis and at a distance 2 is

Q.80 The length of tangent from point (5,1) to the circle is

(A) 81. (B) 29.

(C) 7. (D) 21.

Ans: C

Here S =

= 49

length of tangent =

Q.81 The differential coefficient of log tan x is

(A) 2 sec2 x. (B) 2 cosec 2 x.

(C) 2 sec³ x. (D) 2 cosec³ x.

Ans: B

Let y = log tan x

Q.82 The expression where w is a cube root of unity, equals

(A) 16. (B) 16 w.

(C) 16 w² . (D) 0.

Ans: B

= = = = 16w

= 16w

Q.83 The complex number z = x +iy which satisfies the equation lie on

(A) The x-axis.

(B) The straight line y =5.

(C) A circle passing through the origin.

(D) The y-axis.

Ans: A

-10 y + 25 = 10 y + 25

y =0

Q.84 If , then the value of xyz is

(A) 0. (B) 1.

(C) 2. (D) 3.

Ans: B

Given

=()=

Q.85 The equation whose roots are the reciprocals of the roots of the equation is

(A) . (B) .

(C) . (D) .

Ans: D

We have --------(1)

Let are roots of (1), then

Again and .

Equation is Þ

Q.86 The smallest positive integer n for which is

(A) 2 (B) 4

(C) 3 (D) 5

Ans: B

We have

n is a multiple of 4

the smallest positive value of n is 4

Q.87 If then is equal to

(A) . (B) .

(C) . (D) .

Ans: A

Given

Q.88 is equal to

(A) . (B) .

(C) . (D) .

Ans: B

Q.89 Integrating factor of the differential equation is

(A) 1 + x. (B) .

(C) . (D) .

Ans: C

I.F. = = =

Q.90 The distance between two parallel lines 3x + 4y = 5 and 6x + 8y = 35 is

(A) 1.0. (B) 1.5.

(C) 2.0. (D) 2.5.

Ans: D

Putting y=0 in we get

Thus ( lie on

The length of perpendicular from to 6x+8y =35 is

d= Hence, the distance between the given lines is 2.5

Q.91 The angle between the vectors and is

(A) . (B) .

(C) . (D) .

Ans: C

We know that :

= Þ

Q.92 The value of is

(A) 2. (B) 4.

(C) 6. (D) zero.

Ans: A

Q.93 is equal to