CLASS- VIII
Cube and Cube root
1) Multiple Choice Questions
A) Cube of a negative number is
a) negative b) positive c) d) negative or positive d) none
B) The unit digit of cube of 476 is
a) 4 b) 6 c) 8 d) 2
C) Cube of (-8) is
a) -512 b) 512 c) -64 d) 64
D) Cube of (-3/7) is
a) 27/343 b) -27/343 c) 9/49 d) -9/49
E) Cube root of -1331 is
a) 11 b) 21 c) -11 d) -21
F) Cube root of 2744 is
a) 16 b) 18 c) -14 d) 14
G) The smallest number by which 192 should be multiplied to make it a perfect cube is
a) 9 b) 6 c) 3 d) 2
H) The smallest number by which 686 should be divided to make it a perfect cube is
a) 1 b) 2 c) 3 d) 4
I) The volume of a cube is 729 m³. Length of its side is
a) 3m b) 6m c) 9m d) 27m
CLASS- X
1) If the height of a vertical pole is equal to the length of its shadow on the ground, the angle of elevation of the sun is
a) 0 b) 30 c) 45 d) 60
2) If the height of a vertical pole is √3 times the length of its shadow on the ground then angle of elevation in the sun at that time is
a) 30 b) 45 c) 60 d) 75
3) If the length of the shadow of a Tower is √3 times its height then the angel of elevation of the sun is
a) 45 b) 30 c) 60 d) 90
4) If a pole 12 m high casts a shadow of 4√3m long on the ground then the sun's elevation is
a) 60 b) 45 c) 30 d) 90
5) The shadow of a 5 meter long stick is 2 m long. At the same time, the length of the shadow of a 12.5 m high tree is
a) 3m b) 3.5m c) 4.5m d) 5m
6) A ladder makes an angle of 60° with the ground when placed against a wall. if the foot of the ladder is 2m away from the wall, the length of the ladder is
a) 4/√3m b) 4√3m c) 2√2m d) 4 m
7) A ladder 15m long makes an angle of 60° with the wall. Find the height of the point, where the ladder touches is the wall.
a) 15√3m b) 15√3/2 m c) 15/2 m d) 15m
8) Fom a point on the ground, 30m away from the foot of a tower, the angle of elevation of the top of the tower is 30°. The height of the tower is
a) 30m b) 10√3m c) 10m d) 30√3m
9) The angle of depression of a car parked on the road from the top of a 150m high tower is 30°. The distance of the car from the tower is
a) 50√3m b) 150√3m c) 150√2m d) 75m
10) A kite is flying at a height of 30m from the ground. The length of string from the kite to the ground 60m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is
a) 45° b) 30° c) 60° d) 90°
11) From the top of a cliff 20m high, the angle of the elevation of the top of a tower is found to be equal to the angles of depression of the foot of the tower. The height of the tower is
a) 20m b) 40m c) 60m d) 80m
12) If a 1.5m tall girl stands at a distance of 3m from a lamp post and casts a shadow of lengthh 4.5m on the ground then the height of the lamp post is
a) 1.5m b) 2m c) 2.5m d) 2.8m
13) The length of the shadow of a tower standing on level ground is found to be 2x metres longer when the sun's elevation is 30° than when it was 45°. The height of the tower is
a) (2√3x)m b) (3√2x)m c) (√3 - 1)x m d) (√3 + 1)x m
14) The length of a vertical rod and its shadow are in the ratio 1: √3. The angle of elevation of the sun is
A) 30° b) 45° c) 60° d) 90°
15) A pole casts a shadow of length 2√3m of the ground when the sun's elevation is 60°. The height of the pole is
a) 4√3 b) 6m c) 12m d) 3m
16) In the given figure,
a tower AB is 20m high and BC, its shadow on the ground is 20√3m long. The sun's attitude is
a) 30° b) 45° c) 60° d) none
17) The tops of two towers of heights and standing on a level ground subtend angles 30° and 60° respectively at the centre of the line joining their feet. Then x: y is
a) 1: 2 b) 2: 1 c) 1:3 d) 3:1
18) The angle of elevation of the top of the tower from a point on the ground 30m away from the foot of the towel is 30°, The height of the tower is
a) 10√3m b) 10√3m c) 20m d) 10√2m
19) The string of a kite is 100m long and it makes an angle 60° with the horizontal. If there is no slack in the string the height of the kite from the ground is
a) 50√3m b) 100√3m c) 50√2m d) 100m
20) If the angles of elevation of the top of a tower from two points at distances a and b from the base and in the same straight line with it are complementary then height of the tower is
a) √(a/b) b) √(ab) c) √(a+ b) d) √(a - b).
21) On the level ground , the angle of elevation of a tower is 30°. on the moving 20m nearer, the angle of elevation is 60°. The height of the tower is 10
a) 10m b) 10√3m c) 15m d) 20m
22) In a rectangle, the angle between a diagonal and a side is 30° and the length of this diagonal is 8cm. The area of the rectangle is
a) 16 cm² b) 16/√3 cm² c) 16√3cm² d) 8√3 cm²
23) From the top of a hill, the angles of depression of two consecutive kilometres stones due east are found to be 30° and 45°. The height of the hill is
a) (1/2)(√3 -1) km
b) (1/2)(√3 +1) km
c) (√3 -1) km
d)(√3 +1) km
24) If the elevation of the sun changes from 30° to 60° then the difference between the lengths of shadows of a pole 15m high, is
a) 7.5m b) 15m c) 10√3m d) 5√3m
25) An observer 1.5m tall is 28.5m away from a tower and the angle of elevation of the top of the tower from the eye of the observer is 45°. The height of the tower is
a) 27m b) 30m c) 28.5 m d) none
1c 2c 3b 4a 5d 6d 7c 8b 9b 10b 11a 12c 13d 14a 15b 16a 17c 18b 19a 20b 21b 22c 23b 24c 25b
CLASS- XII
1) The area(in square unit) of the region bounded by the curve x²= 4y, the line x= 2 and x-axis is -
a) 1 b) 2/3 c) 4/3 d) 8/3
2) Let(a sec θ, b tanθ) and Q(a secα, b tanα) where θ + α=π/2, be two on the hyperbola x²/a²- y²/b²= 1. If (h,k) be the point of intersection of the normals at P and Q, then the value of k is
a) (a²+ b²)/a
b) - (a²+ b²)/a
c) (a²+ b²)/b
d) -(a²+ b²)/b
3) The equation of the tangent to the curve (1+ x²)y = 2- x where it crosses the x-axis is
a) x+ 5y = 2 b) x- 5y = 2 c) 5x - y = 2 d) 5x + y = 2
4) The area (in square unit) bounded by the parabola y²= 4ax and x²= 4ay is
a) 64a²/3 b) 32a²/3 c) 16a²/3 d) 7a²/3
5) Equations of the tangent and the normal drawn at the point (6,0) on the ellipse x²/36 + y²/9 = 1 respectively are
a) x= 6, y=0
b) x+ y= 6, y - x +6=0
c) x= 0, y=3
d) x= - 6, y=0
6) The area (in square unit) of the figure by the curve y = cosx and y = sinx and the ordinates x= 0, x=π/4 is
a) √2 +1 b) √2 - 1 c) 1/√2 d) (√2 -1)/√2
7) The straight line x + y = a will be a tangent to the ellipse x²/9+ y²/16= 1 if the value of a is
a) 8 b) ±10 c) ±5 d) ±6
8) The equation of the tangent to the parabola y²= 8x which is perpendicular to the line x - 3y +8= 0 is
a) 3x + y + 2 = 0
b) 3x - y - 1 = 0
c) 9x - 3y + 2 = 0
d) 9x + 3y + 2 = 0
9) The area (in square unit) bounded by the Parabola y²= 8x and its latus rectum is
a) 16/3 b) 25/3 c) 16√2/3 d) 32/3
10) If the curve y²= 4x and xy= k cut orthogonally, then the value of k² will be
a) 16 b) 32 c) 36 d) 8
11) The area (in square unit) bounded by the curve -3y²= x -9 and the lines x= 0, y= 0 and y= 1 is
a) 8/3 b) 3/8 c) 83
12) If the slope of the normal to the curve x³= 8a²y at P is (-2/3), then the coordinates of P are
a) (2a,a) b) (a,a) c) (2a, -a) d) none
13) If a> 2b > 0, then the positive value of m for which the line y= mx - b √(1+ m²) is a common tangent to the circles x²+ y²= b² and (x - a)²+ y²= b² is
a) 2b/√(a²- 4b²)
b) √(a²- 4b²)/2b
c) 2b/(a- 2b)
d) b/(a- 2b)
14) The area in square unit of the region bounded by the lines y= |x -1| and y= 3- |x| is
a) 6 b) 2 c) 4 d) 3
15) The minimum value of f(x)= x²+ 250/x is
a) 55 b) 25 c) 50 d) 75
16) If f(x)=kx³- 9x²+ 9x +3 is increasing function then
a) k< 3 b) k ≤ 3 c) k> 3 d) k is indeterminate
17) If f(x)= 1/(4x²+ 2x +1), then its maximum value is
a) 2/3 b) 4/3 c) 3/4 d) 1
18) If f(x)= 1/(x +1) - log(1+ x), x> 0, then f(x) is
a) a decreasing function
b) an increasing function
c) neither increasing nor decreasing
d) increasing when x> 1.
19) Let α, β be the roots of x²+ (3- λ)x - λ=0, then the value of λ for which α²+ β² is minimum, is
a) 0 b) 1 c) 3 d) 2
20) The function f(x)=2x³- 3x² -12x +4 has
a) no maxima and minima
b) one maximum and one minimum
c) two maxima
d) two minima
21) The height of the cylinder of maximum volume that can be inscribed in a sphere of radius a, is
a) 3a/2 b) √2 a/3 c) 2a/√3 d) a/√3
22) Maximum value of (logx)/x in [0, ∞) is
a) (log2)/2 b) 0 c) 1/e d) e
23) Let the function f: R--R be defined by f(x)=2x + cosx; then f(x)
a) has maximum value at x=0
b) has minimum value at x=π
c) is a decreasing function
d) is an increasing function
24) The maximum distance from the origin of a point on the curve x= a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a, b> 0, is
a) a- b b) a+ b c) √(a²+ b²) d) √(a²- b²)
25) The velocity v of a particle moving along a straight line is given by a+ bv²= x², where x is the distance of the particle from the origin. Then the acceleration of the particle is
a) x/b b) bx c) x/a d) b/x
26) If the slope of the tangent at (x,y) to a curve passing through the point (2,1) is (x²+ y²)/2xy , then the equation of the curve is
a) 2(x²- y²)= 3x
b) 2(x²- y²)= 3y
c) x(x²- y²)= 6
d) 2px(x² + y²)= 6
27) A particle moves uniform acceleration f along a straight line. If v be it's velocity at time t and s be the distance described during the interval, then
a) s= 2vt - ft²
b) s= vt - ft²/2
c) s= vt/2 - ft²
d) s= vt - ft²/2
28) A particle moving in a straight line traverses a distance x in time t, if t= x²/2 + x, then the retardation of the particle is
a) equal to its velocity
b) constant
c) is equal to the cube of its velocity
d) equal to the square of its velocity
29) If y= 3x²+ 2 and if x changes from 10 to 10.1, then the approximate change in y will be
a) 8 b) 6 c) 5 d) 4
30) The rate of change of surface area of a sphere of radius r when the radius is increasing at the rate of 2 cm/sec is proportional to
a) 1/r² b) r² c) r d) 1/r
1b 2d 3a 4c 5a 6b 7c 8d 9d 10b 11c 12a 13a 14c 15d 16c 17b 18a 19d 20b 21c 22c 23d 24b 25a 26a 27d 28c 29b 30c
ALGEBRA/COORDINATE GEOMETRY
5) A problem in Mathematics is given to three students A, B and C and their respective probability of solving the problem is 1/2, 1/3, and 1/4. Then the probability that the problem is solved, is
a) 3/4 b) 1/2 c) 2/3 d) 7/8.
6) The probability that a leap year will have 53 Tuesday or Saturday is
a) 2/7 b) 3/7 c) 4/7 d) 1/7.
8) The value of the determinant
1+ a 1 1
1 1+ b 1
1 1 1+ c is
a) 1+ ab+ bc+ ca+ abc
b) abc (1+ 1/a + 1/b + 1/c)
c) 4abc d) abc (1/a+ 1/b + 1/c).
9) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A + I.
11) If P(A)= 2/3, P(B)= 1/2 and P(AUB) = 5/6, then the events A and B are
a) mutually exclusive
b) independent as well as mutually exclusive
c) independent d) none.
12) The roots of the determinants equation are
x. 3. 7
2 x -2 = 0
7 8 x
a) -2, -7, 5 b) -2, -5, 7 c) 2, 5, -7 d) 2, 5, 7.
16) The multiplicative inverse of matrix
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
17) The probability that atleast one of the events A and B occurs is 3/5. If A and B occur simultaneously with probability 1/5, then the value of P(A')+ P(B') is
a) 2/5 b) 4/5 c) 6/5 d) 7/5.
20) If the system of equations x+ 2y + 3z= 1, 2x + ky + 5z = 1; 3x + 4y + 7z = 1 has no solutions, then
a) k= -1 b) k= 1 c) k= 3 d) k= 2.
21) The probability that the same number appears on throwing three dice simultaneously is
a) 1/6 b) 1/36 c) 5/36 d) none.
22) A is a square matrix such that A³= I; then Inverse of A is equals to
a) A² b) A c) A³ d) none.
23) If the determinant
1 a a²- bc
D= 1 b b²- ca
1 c c²- ab
Thed D is
a) 0 b) independent of a c) independent of b d) independent of x.
28) If y x 0
0 y x = 0
x. 0 y
and x≠ 0, then the which one of the following is correct for determinant
a) x is one of the cube roots of 1
b) y is one of the cube roots of 1
c) y/x is one of the cube roots of 1
d) y/x is one of the cube roots of (-1).
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