NUMBER SYSTEM
Part - 1
1) Add (2√3+ 5 √5- 7 √7) and (3√5 - √3+ √7).
2) Multiply:
a) 7√6 by 5√24.
b) ³√32 by ³√250.
c) ³√7 by 2.
3) Divide:
a) ³√18 by ³√9.
b) ³√128 by ⁵√64.
4) Simplify: 2 ³√40 + 3 ³√625 - 4 ³√320.
5) Simplify:
a) (√2+ 1)(√2+ 3).
b) (3√5 - 2√3)(3√5 + 2√3).
c) (2√5+ 2√3)²
Part- 2
1) Rationalize the denominator :
a) 1/(√5+ 1)
b) 1/(√5-1)
c) 1/{√3+ √5)
d) 1/(√3- √5).
e) 2/√5
f) 1/√18
g) (√3+ √5)/√2
h) √7/(√5+1)
Part- 3
1) Express each one of the following with a rational denominator:
a) √7/(√5 +1).
b) 7√3/(√10+ √3).
c) 10/(7- 2√3)
d) 5/(4√3 - 3√2)
Part- 4
1) Rationalize the denominator and simplify:
a) (6- 4√2)/(6+ 4√2).
b) (√7- √6)/(√7+ √6).
c) (7√3 - 5√2)/(√48 + √18).
d) 1/(√5+ √6 - √11).
e) 3/(√3 - √2 + √5)
Part- 5
1) If a and b are rational numbers, find the values of a and b in each of the following equalities:
a) (5+ 2√3)/(7+ 4√3)= a + b √3
b) (√2+ √3)/(3√2- 2√3)= a - b √6.
c) (7 +√5)/(7 - √5) - (7 - √5)/(7 + √5)= a + 7√5 b.
2) Simplify:
a) 2/(2√5 - √3) + 3/(2√5 + √3)
Part- 6
1) Simplify:
a) 2√6/(√2 + √3) + 6√2/(√6 + √3).
b) (4 +√5)/(4 - √5) + (4 - √5)/(4 + √5).
c) 3√2/(√3+ √6) + 4√3/(√2+ √6) + √6/(√2+ √3).
d) 7√3/(√10+ √3) - 2√5/(√6 + √5) - 3√2/(√15 + 3√2).
Part- 7
1) Find the values of each of the following correct to three places of decimals, it being given that
(√2= 1.414, √3= 1.732, √5 = 2.236, √6= 2.449)
a) √78 + (1/2) √48 - √192.
b) 3/√2.
c) 2/(√3 -1)
2) If a= 9- 4√5, find the value of √a - 1/√a.
3) If x= 3+ 2√2, find the value of x²+ 1/x².
Part- 8
1) If x= 1/(2- √3), find the value of x³- 2x²- 7x +5.
2) If x= (√3+1)/(√3 -1) and y= (√3-1)/(√3 +1), find the value of x²- xy+ y².
3) If x= [{√(p+ 2q) + √(p - 2q)}/{√(p+ 2q) - √(p - 2q)}]. Then show that qx²- px + q=0.
Part- 9
IDENTITY, EXPANSION ETC(1)
1) Simplify:
a) (2x + 3y)(2x +3y).
b) (3- 2x)(3- 2x).
c) (3x - 1/x)²
d) (x -1/10)(x + 1/10).
e) (x + 4/3)(x + 4/3).
f) (z²+2)(z²-3).
2) Find the product of each of the following:
a) (x - 3/2)(x + 3/2)(x² + 9/4).
b) (x + 1 + y)(x - 1 - y).
c) (2x²+ 5x +1)(2x²+ 5x -1).
d) (x²+1+ y)(x²+1- y).
e) (x³+ y²/2 -3)(x³+ y²/2 + 3).
f) (2x - 3 + x - y)(2x - 3 - x + y).
3) Evaluate:
a) (104)².
b) (499)²
c) (100 x 100 - 83 x 83)/17.
d) 1.92 x 2.08.
e) 103 x 96.
f) 95 x 97.
4) Simplify using identities:
a) 225 x 225 + 2 x 225 x 75 + 75 x 75.
b) 2.2 x 2.2 - 2 x 2.2 x 0.2 + 0.2 x 0.2.
c) (3.7 x 3.7 - 2.9 x 2.9)/0.8.
d) 5.1 x 5.1 - 0.1 x 0.1.
5) Find the value of:
a) If x+ 1/x = 2, find the value of x²+ 1/x².
b) If x- 1/x = 4, find the value of x²+ 1/x².
c) If x+ 1/x = √7, find the value of x²+ 1/x² and x⁴+ 1/x⁴.
d) If x² + 1/x² = 38, find the value of x+ 1/x and x - 1/x.
e) If 2x - √7 y = 10 and xy= - √7, find the value of 4x²+ 7y².
f) If 4x²+ 9y²= 69 and xy= 1, find the value of 2x + 3y.
MISCELLANEOUS
1) Find the following products using identities:
a) (x +2)(x + 2).
b) (2y -5)(2y -5).
c) (2+ 3x)(2- 3x).
d) (x +9)(x - 12).
2) Using identities, find the product of each of the following:
a) (x -1)(x +1)(x²+1)(x⁴+1).
b) (x - 1/x)(x + 1/x)(x²+ 1/x²)(x⁴+ 1/x⁴).
c) 3ab(a - 9b)(a + 9b).
d) (x²+ x - 1)(x²- x +1).
3) Evaluate
a) 108 x 108.
b) (0.98)².
c) 105 x 98.
4) Simplify each of the following using appropriate identities:
a) 212 x 212 - 2 x 12 x 212 + 12 x 12.
b) (1.85 x 0.85+ 2 x 0.85 x 0.15 + 0.15 x 0.15.
c) (231 x 231 - 180 x 180)/411.
5) Find the value:
a) If x+ 1/x= 3, find the value of x²+ 1/x².
b) If x² + 1/x² = 83, find the value of x - 1/x.
c) If 4x²+ 9y²= 136 and xy= 10, find 2x + 3y.
IDENTITY, EXPANSION ETC(2)
1) Write the following in expanded form:
a) (2x + 5y +1)².
b) (-2x + 3y - 2z)².
c) (xy+ yz + zx)²
d) (a/4 - b/2 +1)².
2) Simplify: (a + b + c)²+ (a - b + c)²+ (a + b - c)².
3) If x+ y + z = 7 and xy+ yz + zx= 6, find x²+ y²+ z².
4) If x² + y² + z² = 40 and xy+ yz + zx= 30, find x + y + z.
5) Write the following cubes in expanded form:
a) (3x + 4y)³.
b) (2x + 1/3x)³
c) (5x - 3y)³.
d) (1/3x - 2/5y)³.
e) (4x + 3y/4)³.
f) (x²- 3y²/2)³.
6) Simplify:
a) (3x + 4y)³+ (3x - 4y)³.
b) (x/3 + y/5)³- (x/3 - y/5)³.
7) Evaluate with the help of identity:
a) (23)³.
b) (102)³.
c) (995)³.
8) Find the value type questions:
a) If x + 1/x = 7, find the value of x³ + 1/x³.
b) If x - 1/x = 5, find the value of x³ - 1/x³.
c) If (3x + 2y)= 10 and xy= 2, find the value of 27x³+ 8y³.
d) If x² + 1/x² = 23, find the value of x³ + 1/x³.
e) If x¹ + 1/x² = 18, find the value of x³ - 1/x³.
f) If x + y = 3 and xy= 2, find the value of x³+ y³
g) If x - y = 5 and xy= -6, find the value of x³- y³.
9) Find the following products:
a) (5x - 3y)(25x²+ 15xy + 9y²).
b) (2xy + 3z)(4x²y²- 6xyz+ 9z²).
10) Evaluate:
a) (25)²+ (5)³
b) (1100)³- (100)³.
11) Simplify: (27 x 27 x 27 - 7 x 7 x 7)/(27 x27 + 27 x 7 + 7 x 7).
12) Find the following products:
a) (p - 3q + 2r)(p²+ 9q²+ 4r²+ 3pq + 6qr - 2pr).
b) (2x - y -1)(4x²+ y²+ 1+ 2xy - y + 2x).
c) (√2x + 2√2 y + z)(2x²+ 8y²+ z²- 4xy - 2√2 yz - √2 xz).
13) Evaluate with the help of Identity:
a) (28)³+ (-15)³+ (-13)³.
b) (0,1)³+ (0.2)³ - (0.3)³.
c) (1/4)³+ (1/3)³ - (7/12)³.
14) Is (8/15)³ - (1/3)³ - (1/5)³= 8/75? How will you justify your answer without actually calculating the cubes ?
15) Find the value of:
a) If x + y +z = 8 and xy + yz + zx = 20, find the value of x³+ y³+ z³.
b) If x + y +z = 15 and x² + y² + z² = 33, find the value of x³+ y³+ z³- 3xyz.
16) Find the value of x³+ y³+ 15xy - 125, where x + y = 5.
17) If a+ b + c= 6, find the value of (2- a)³+ (2- b)³ + (2- c)³ -3(2- a)(2- b)(2- c).
18) Find the value of (x - a)³ + (x - b)³+ (x - c)³ - 3(x - a)(x - b)(x - c), if a+ b + c = 3x.
Miscellaneous - 2
1) Find the following products by identities:
a) (3x +5)(9x²- 15x +25).
b) (7y - 5z)(49y²+ 35yz + 25z²)
c) (0.4x + 0.5y)(0.16x²- 0.2xy + 025y²)
d) x(y +9x)(y²- 9xy + 81x²)
2) Evaluate:
a) (104)³+ (96)³
b) 2³- (1/6)³
3) Simplify: (136 x 136 x 136+ 64 x 64 x 64)/(136x 136 - 136 x 64 + 64 x 64).
4) If x+ y= 8 and xy= 6, find the value of x³+ y³.
5) If x- y= 10/9 and xy= 5/3, find the value of x³- y³.
6) If x+ y + z= 0, then prove x³+ y³+ z³= 3xyz.
7) Find the following products by the method of identities:
a) (x - 2y - 4z)(x²+ 4y²+ 16z²+ 2xy - 8yz + 4zx)
b) (-3x + y - z)(9x²+ y²+ z²+ 3xy + yz - 3zx).
8) Without actually calculating the cubes, evaluate each of the following:
a) 55³- 25³- 30²
b) 9.8³- 11.3³ + 1.5³.
9) If x+ y + z = 1, and xy + yz + zx= -1 and xyz= -1, find the value of x³+ y³+ z³- 3xyz.
FACTORISATION
Type- 1
1) Factorise the following:
a) 3a²+ 6ab.
b) 5xy - 25x³y².
c) 46x²+ 2xy + 10y²
d) 7x³y - 21x²y²+ 35y².
e) 8(3x + 2y)² - 16(3x + 2y).
f) 2x(x²+ y²) - 4y(x²+ y²).
g) p²(q - r)+ q(r - q).
h) 30a(b - c) - 25(c - b).
i) a²(a²+ b²- c²) - b²(c²- a²- b²)
j) (a+ b)(x + y) + (2a + 3b)(x +y)+ (3a + 4b)(x +y).
k) 1+ x²y²+ x²+ y²
l) 4a³- 8a²+ 3a -6.
m) x³- x²y - xy + y².
n) x³- x²+ x -1.
o) a²xy + abx²+ b²xy + aby²
p) a²x²+ (ax²+1)x + a.
q) x³+ xy(1- 3x) - 3y².
r) abc²+ ab - ac - abc + xy + bxy.
s) x³- x²- ax + x + a - 1.
t) x²+ 1/x² + 2 - 5x - 5/x.
u) ax²/b +(a/b + c/d)x + c/d
v) 2p(a - b)+ 3q(5a - 5b) + 4r(2b - 2a).
Miscellaneous
1) Factorise
a) x²- xy + y - x.
b) 2xy - 3ab + 2bx- 3ay.
c) a²b² -(ab²-5)b - 5a.
d) ax - bx + cy + by - cx - ay.
e) x(2a + b)²+ 4ay + 2by + 8a + 4b.
f) x²+ 1/x² - 2 - 3x + 3/x.
Type- 2
1) Factorise
a) ) x²- 4y².
b) 25x²- 36y².
c) 100- 9x².
d) 3x²- 4
e) 1- (x - y)².
f) 4(x + y)² -1.
g) 16 - 9(x + y)².
h) 4(2x -3)² -9(y +1)²
i) x²- 1/25.
j) 3 - 12(a - b)²
k) 162x⁴ - 50.
l) 16x⁴ - 625.
m) 81 - 256x⁴
n) x⁹y - xy⁹
o) x - y - x²+ y².
p) 4(x + y)² -1.
q) 4a²- 4b²+ 4a +1.
r) 4a²- 25b²+ 30b -9.
s) 25x²- 10x +1 - 36z².
t) 9a²- 25b² - 36c² + 16d²+ 2(12ad + 30bc).
u) x⁴+ 7x²+ 16.
v) (1- 4x²)(1- 4y²)+ 16xy.
w) a⁸+ a⁴b⁴ + b⁸.
Miscellaneous
1) Factorise the following:
a) 81x²- 25y².
b) (2a + 3b)² - 9c².
c) 16x² - 1/x².
d) 3xy - 243xy⁵.
e) x⁸ - y⁸.
f) 9 - 2xy - (x²+ y²).
g) (1- a²)(1- b²) + 4ab.
h) a⁴+ a²+ 1.
i) 9a⁴+ 4b⁴ - 13a²b².
MIDDLE TERM FACTOR
1) Factorise the following:
a) x²+ 19x +88.
b) x²+ 14x + 45.
c) x²+ 2x - 3.
d) x²+ 9x - 36.
e) 12x² - 7x + 1.
f) x² - x -132.
g) x² -11 x - 42.
h) x²- 11x + 18.
i) x² - 24x + 108.
j) x² - 2x - 15.
k) 6x²+ 19x + 10.
l) 12x² - 25x + 12.
m) 4x²- 17x -21.
n) 10x²+ 3x - 4.
o) 4x² - 25x + 21.
p) 3x²- 10x +8.
q) (1/2)x²+ 3x +4.
r) x²/5 + 2x - 15.
s) 9x² - 2x - 1/3.
t) √3 x²+ 5x + 2√3.
u) 4√3x²+ 10x + 2√3.
v) 4√5x²+ 17x - 3√5.
w) 5√3 x² - 32x - 7√3.
y) 3(x+5)² - 2(x +5) - 8.
z) (a² - 2a) - 23(a²- 2a) + 120.
a) 12(x² + 7x) - 8(x² + 7x)(2x -1) - 15(2x -1)².
b) 8(x +1)² - 2(x +1)(y+ 2) - 15(y +2)².
c) 4(x - y)² - 12(x +y)(x + y) + 9(x + y)².
d) x⁴+ 19x²- 150.
e) x⁴+ 3x²- 28.
f) (x²- 4x)(x²- 4x -1) - 20.
g) (5x - 1/x)² + 4(5x - 1/x)+ 4.
h) The area of a rectangle is given by the algebraic expression y²+ 5y - 24. Find the possible expressions for the dimensions of the rectangle.
Miscellaneous
Factorise the following:
a) x²+ 14x +48.
b) x² - 30x + 216.
c) x² - 32x - 105.
d) 6x²+ 5x - 6.
e) x²+ x/6 - 1/6.
f) 2x² - xx + 1/8.
g) 6√3x² - 7x - √3.
h) 16√5 x² - 50x + 5√5.
i) 12(x -2)² - 25(x -2)(y +1)+ 12(y +1)².
j) 3(6x²+ 5x)² - 10(6x²+ 5x) - 8.
k) x⁴+ 3x²- 28.
l) 4x⁴ - 5x² +1.
FACTORISE
a) x³ + 6x²y + 12xy²+ 8y³.
b) 8x³ - 36x²y + 54xy² - 27y³.
c) 64x³/27 + 4x²y + 9xy²/4 + 27y³/64.
d) x³/8 + x²y/4 + xy²/6 + y³/27.
e) 8x³/27 + 28x²/3 + 97x - 343.
f) p⁶ - 27q⁶/8 - 9p⁴q²/2 + 27p²q⁴/4.
Miscellaneous
Factorise
a) 27x³ + 108x²y + 144xy²+ 64y³.
b) x³ - 27x²y - 9xy²+ 27y³.
c) x³ + 3x²/2 + 3x/4 + 1/8.
d) x³ - x²y + xy²/3 - y³/27.
FACTORISATION
1) Factorise:
a) x³+ 64.
b) 8a³+ b³.
c) 64x³+ 343.
d) 512x³+ 1/729b³.
e) 125x³+ 1/216.
f) 32x³+ 108y³.
g) 54x⁶y + 2x³y⁴.
h) 3x⁵y³+ 24x².
i) 1+ 125x³.
j) 0.343+ 8y³.
k) 8x³+ 0.125.
l) 125x⁶+ y⁶.
m) x⁷y + xy⁷.
n) 54x³y - 128y⁴.
o) x³- 125.
p) 1331 - 343y³.
q) x³/8 - 216y³.
r) a³ 2√2b³.
s) 250x³ - 16y³.
t) 8x³ - (2x - y)³.
u) 5a + 20b + a³+ 64 b³.
v) 8x³ - 27b³ - 4ax + 6bx.
w) 2x - 3y - 8x³+ 27y³.
x) x⁶ - 26x³ - 27.
y) (3x + 4)³+ (7- 3x)³.
z) (2x+ 1)³ - (x +1)³.
a) (a/3 + b/5)³- (a/3 - b/5)³.
b) x³ - 3x²+ 3x +7.
Miscellaneous
Factorise
a) a³+ 8.
b) 27x³+ 125y³.
c) x⁶/125 + 125/x⁶
d) 8x³y³ + 27z³.
e) 2x⁹+ 54y⁸.
f) x³+ y³+ x + y.
g) x⁶ + y⁶.
h) 8x³ - 343y³.
i) x³ - 27y³/8.
j) 5√5a³ - 2√2y³.
k) 3a⁷b - 24a⁴b⁴.
l) a³ - 8b³ + 2ax - 4bx.
m) x⁶- y⁶.
n) x¹² - y¹².
o) (x + 2y)³+ (2x + y)³.
p) (2x + 3y)³- (2x - 3y)³.
q) x³ - 1/x³ - 2x + 2/x.
FACTORISATION
1) Factorise:
a) 8x³+ 27y³+ 64z³ - 72xyz.
b) 8x³- 27y³+ z³ + 18xyz.
c) 27a³+ 125b³ - c³ + 45abc.
d) x³- 27y³ -1- 9xy.
e) -27x³+ y³- z³ - 9xyz.
f) x³/8 - 64y³+ 27z³ +18xyz.
g) 2√2x³+ 3√3y³+ √5(5- 3√6xy).
h) 8x³ - y³ -1 - 6xy.
i) - a⁶ + 8b⁶ + c⁶ + 6a²b²c².
j) 27x³ - 8y⁶+ 125z³ + 90xy²z.
k) 11x³ - 54y⁶ - 2z³ - 36xy²z.
l) x⁶ - 1/x⁶ - 14.
m) (x- 3y)³+ (3y- 7z)³+ (7z - x)³.
n) (5a - 6b)³+ (7c - 5a)³+ (6b - 7c)³.
o) (x + y - 2z)³+ (y + z - 2x)³+ (z + x - 2y)³.
p) {(9x²- 4y²)³ + (4y²- 25z²)³ + (25z²- 9x²)³}/{(3x - 2y)³+ (2y - 5z)³+ (5z - 3x)³}.
2) Prove: (x + y)³+ (y + z)³+ (z + x)³ - 3(x + y)(y + z)(z + x)= 2(x³+ y³+ z³ - 3xyz).
3) If 3x + y + z= 0, show that 27x³+ y³+ z³ = 9xyz.
4) volume of a cuboid is given by the algebraic expression x³+ 2x² - x -2.
5) Find the possible expressions for the dimensions of the cuboid.
6) The volume of a cube is given by the algebraic expression x³- 9x² + 27x -27..
Find the possible expression for the side of the cube.
Miscellaneous
1) Factorize:
a) 27x³+ y³+ z³ - 9xyz.
b) 64x³ - 8y³+ 120xy + 125.
c) 2√2x³+ 8y³- 27z³ + 18√2 xyz.
d) x³+ y³+ 1 + 3xy.
e) x³+ 1/x³ - 14.
3) Verify: x³+ y³+ z³ - 3xyz = (1/2) (x + y + z){(x - y)² + (y - z)²+ (z - x)²}
4) Factorise:
a) (a - b)³+ (b - c)³+ (c - a)³.
b) (3x - 5y)³+ (5y - 9z)³+(9z - 3x)³.
c) (x/2 - 3y)³+(3y - √3 z)³+ (3z - x/2)³.
5) Prove: (a²- b²)³+ (b²- c²)³+ (c²- a²)³= 3(a + b)(b + c)(c + a)(a - b)(b - c)(c - a).
6) What are the possible expressions for the dimensions of the cuboid, whose volume is 12ky²+ 8ky - 20k ?
MULTIPLE CHOICE QUESTIONS
1) Which one of the following is a polynomial?
a) x²/3 - 2/x² b) x³+ √4x³/√x c) √(3y)+ 5 d) (x²-1)/(x²+1).
2) The coefficient of x² in (2x²- 5)(4+ 3x²) is
a) 2 b) 3 c) 8 d) -7
3) √2 is a polynomial of degree
a) 2 b) 0 c) 1 d) 1/2
4) Degree of polynomial (x³-2)(x²+11) is
a) 0 b) 5 c) 3 d) 2
5) Degree of zero polynomial is
a) 0 b) any natural number c) 1 d) not defined
6) Standard form of the polynomial 1/x⁻³ + x/8 + 6x⁵+ √3/5 is
a) x³ + x/8 + 6x⁵+ √3/5
b) 6x⁵+ x³ + x/8 + √3/5
c) 6x⁵+ √3/5+ x/8 + x³
d) x³ + 6x⁵+ √3/5 + x/8
7) x²+ 5x - 1/2 is a
a) Quadratic polynomial in x
b) binomial
c) monomial
d) cubic polynomial in x.
8) The value of p(1/2) for p(z)= z⁴- z²+ z is
a) 7/16 b) 5/16 c) 3/16 d) 1/16
9) If p(x)= 2x²- 3x +5, then the value of {p(0)+ p(1)}/p(-1).
a) 1/10 b) 4/11 c) 9/10 d) 4/5
10) A polynomial of degree 5 in x has atmost
a) 5 terms b) 10 terms c) 6 terms d) 4 terms
11) Zero of the polynomial p(x), where p(x)= ax +1, a≠ 0 is
a) 1 b) -a c) 0 d) -1/a
12) Zeroes of the polynomial p(x)= (x +2)(x +5) are
a) 2,5 b) -2,-5 c) 1/2,1/5 d) -1/2,-1/5
13) Zeroes of the polynomial p(x)= x(x -1)(x -2) are
a) 0,-1,2 b) 0,-1, -2c) 0,1,2 d) 0,1,2
14) Which of the following is a zero of the polynomial x³+ 3x²- 3x -1 ?
a) -1 b) -2 c) 1 d) 2
15) The number to be added to the polynomial x²- 5x +4, so that 3 becomes its zero, is
a) 4 b) -4 c) -2 d) 2
16) The number to be substracted from the polynomial x²- 16x +30, so that 15 becomes its zero, is
a) 15 b) 16 c) 30 d) 0
17) A polynomial whose zeroes are √2 and -√2 is
a) x²+2 b) x - 2 c) x² - 2 d) x +2
18) If x= 2 is a zero polynomial x²- 2k +2, then the value of k is
a) 1 b) 2 c) 3 d) 4
19) The value of k for which the polynomial x³+ 3x²- 3x + k has -3 as its zero, is
a) -9 b) -3 c) 9 d) 12
20) The remainder when p(x)= x³+1 is divided by x +1, is
a) -6 b) 0 c) 1 d) 6
21) The remainder when x ⁵¹+ 51 is divided by x +1, is
a) 51 b) 50 c) -1 d) 0
22) The remainder when x²+ 2x +1 is divided by x +1, is
a) 4 b) 0 c) 1 d) -2
23) The remainder when f(x)= x³+ 4x²- 3x + 1 is divided by x -2, is
a) 16 b) 12 c) 17 d) 19
24) If x +1 is a factor of the polynomial 2x²+ Kx, then the value of k is
a) -2 b) -3 c) 4 d) 2
25) If x +a is a factor of the polynomial x⁴- a²x²+ 3x - 6a, then the value of a is
a) 0 b) 1 c) -1 d) 2
26) x +1 is a factor of the polynomial
a) x³+ x²- x +1
b) x³+ x² + x +1
c) x⁴+ x³+ x² +1
d) x⁴+ 3x³+ 3x² + x +1
27) The common factor in x²-1, x⁴-1 and (x -1)² is
a) x -1 b) x +1 c) x² -1 d) x² + 1
28)
QUADRILATERAL
1) In the given figure,
ABCD is a parallelogram in which angle A=74°. Calculate angle B, C, D.
2) in the given figure
ABCD is a parallelogram in which angle DAB=70°, angle DBC=50°. Compute ang CDB, ADB.
ABCD is a parallelogram in which angle DAB=70°, angle DBC=50°. Compute ang CDB, ADB.
3) In the given figure,
ABCD is a parallelogram in which Angle DAP= 20°, angle BAP= 40° and angle ABP= 80°. Find angle APD, BPC.
4) ABCD is a parallelogram. If AB= 2AD and P is the midpoint of AB, then find angle CPD.
5) if an angle of a parallelogram is two-thirds its adjacent angle, find the angles of the parallelogram.
6) Find each of a parallelogram if two consecutive angles are in the ratio 1:5.
7) Find the measures of an angle of a parallelogram if one angle is 30° less than twice the smallest angle.
8) In the given figure,
ABCD is a parallelogram with perimeter 40cm. Find the value of x and y.
9) In the following figures , find the value of x, y and z.
10) In the given figure,
ABCD is a parallelogram. P and Q are the made points of BC and AD respectively. Prove that APCQ is a parallelogram.
11) ABCD is a parallelogram and points P and Q are the points on the sides AD and BC respectively, such that AP= 1/4 AD and CQ= 1/4 BC. Prove that BPDQ is a parallelogram.
12) ABCD is a parallelogram.
BM bisects angle ABC and DN bisects angle ADC. Prove that BNDM is a parallelogram and BM= DN
Continue.....
1) 106, 74,106 2) 60,50 3) 40,80 4) 90 5) 72,108,72,108 6) 30,150,30,150 7) 70,110,70,110 8) 4,5 9)
MID-POINT THEOREM
Multiple Choice Questions (Quadrilateral & Mid-point)
1) Three angles of a quadrilateral are 60°, 86° and 110°. The fourth angle is
a) 104° b) 124° c) 94° d) 84°
2) The value of x in the given figure
is
a) 10 b) 20 c) 30 d) 40
3) In a quadrilateral , three angles are in the 3: 3 :1 and the fourth angle is 80°, then the other angle are
a) 100°, 100°, 80° b) 120°, 120°, 40° c) 100°,110°,70° d) 110°,110°,60
4) In a quadrilateral ABCD, AB|| DC and AD= BC= 5.5cm, and one of the angles is 80°, then the other angles are
a) 90°, 90°, 100° b) 120°, 80°, 80° c) 80°,100°,100° d) 110°,85°,85°
5) The sides of a quadrilateral are extended in order to form exterior angles . The sum of these exterior angles is
a) 360° b) 270° c) 90° d) 180°
6) Which of the following is not true for a parallelogram ?
a) opposite sides are equal.
b) opposite angles are equal
c) opposite angles are always bisected by the diagonals.
d) diagonals bisect each other.
7) in a quadrilateral ABCD, if AB= BC and CD= DA, then quadrilateral ABCD is a
a) trapezium b) rhombus c) kite d) parallelogram
8) Given a quadrilateral ABCD such that angle C=90° and diagonals AC and BD bisect each other at right angles, then the quadrilateral is a
a) trapezium b) kite c) rectangle d) square
9) P and Q are the mid-points of the sides AB and AC of ∆ ABC and O is any point on side BC . O is joined to A. If S and R are the midpoints of OB and OC respectively, then PQRS is
a) a square b) a rectangle c) a Rhombus d) a parallelogram
10) if bisectors of angle P and Q of a quadrilateral PQRS intersect each other at A, of angle Q and R at B, of angle R and S at C and of angle S and P at D, then ABCD is a
a) rectangle b) rhombus c) parallelogram
d) quadrilaterals whose opposite angles are supplementary.
11) ABCD is a Rhombus in angle BCD= 100°, then (x + y) equals
a) 40 b) 69 c) 80 d) 70
12) ABCD is a parallelogram. If angle A= (3x -20)° and angle C= (x +40)°, then the value of x is
a) 30 b) 40 c) 50 c) 60
13) D and E are the midpoints of the side AB and AC respectively of ABC DE is produced to F. To prove that DA is equal and parallel to FC, we need an additional information, which is
a) angle DAE= angle EFC b) AE= EF c) DE= EF d) angle ADE= angle ECF
14) ABCD is a parallelogram. If its diagonals are equal, then the measure of angle ABC is
a) 60 b) 90 c) 75 d) 120
15) Diagonals AC and BD of a parallelogram ABCD interesect each other at O. If OA= 5cm and OD= 4cm, then the length of AC and BD respectively are
a) 5cm, 4cm b) 10cm, 8cm c) 2.5 cm, 2cm d) 15cm, 12cm
16) If the angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60°, then the angles of the parallelogram are
a)105°, 75°, 105°, 75° b) 115°, 65°, 115°, 65° c) 120°,60°, 120°,60° d) 110°,70°, 110°,70°
17) In a parallelogram, if angle A=60°, then angle D is equals to
a) 110° b) 140° c) 120° d) 130°
18) One angle of a quadrilateral is 114° and the remaining three angles are equal. Then , the measure of each of the three equal angles is
a) 82° b) 84° c) 86° d) 92°
19) Given a trapezium PQRS such that PQ= 12cm, RS= 5cm, PQ|| SR, PS= QR= 8cm. If angle R= 130, then angle P is
a) 130 b) 50 c) 150 d) 120
20) In a parallelogram ABCD, AB= 3cm and the diagonals AC and BD are 5.8 cm and 4.2 respectively. If the diagnols AC and BD interesect at O, then the perimeter of ∆ AOB is
a) 10cm b) 8.8cm d) 7.28cm
21) If an angle of a parallelogram is four-fifths of its adjacent angle, then the angles of the parallelograms are
a) 70°, 110°, 70° , 110°
b) 80°, 100°, 80°, 110
c) 72°,118°,72°, 108°
d) 60°,120°,60°, 120°
22) Four angles of a quadrilateral are (2x +20)°, (3x -30)°, (x +10)° and (2x)°. Value of x is
a) 40 b) 45 c) 50 d) 55
23) ABCD is a rectangle where BC= (4x -5)cm and AD= (2x +3)cm. Then, BC is
a) 11cm b) 12cm c) 10cm d) 15cm
24) In Rhombus PQRS , PQ= 3x, QR=2(x +3)cm. Each side of the Rhombus is
a) 17 b) 19cm c) 18cm d) 28cm
25) ABCD is a rhombus in which altitude from D to side AB bisect AB. Then the angles of the rhombus
a) 100°, 80°, 100°, 80°
b) 110°, 70°, 110°, 70°
c) 120°,60°,120°,60°
d) 130°,50°,130°, 50°
26) P is the midpoint of side BC of a parallelogram ABCD such that angle BAP= Angle DAP. If AD= 10cm, then length of CD is
a) 10cm b) 5cm c) 6cm d) 8cm
27) ABCD is a parallelogram. P and Q are respectively the mid-points of AB and CD . PQ and diagonal AC intersect at M. If AM= 3cm, then the length of the diagonals AC is
a) 3 cm b) 4.5cm c) 6 cm d) 7.5cm
28) The diagonals AC and BD of a parallelogram ABCD intersect each other at point O. If angle BOA= 68° and angle CAD= 25°, then angle DBC is equals to
a) 40 b) 43 c) 68 d) 25
29, in a parallelogram PQRS, PQ= 9cm and PS= 5 cm. The bisector of angle P meets SR in A. PA and QR produced meet at B. Then, the length of RB is
a) 5cm b) 4cm c) 9cm d) 6cm
30) M is the mid point of side CD of a parallelogram ABCD. A line through C parallel to MA intersect AB at P and DA produced at R. If DA = 3.5cm, then the length of DR is
a) 3.5cm b) 5cm c) 7cm d) 10.5cm
31) ABCD is a trapezium in which AB|| DC. M and N are the midpoints of AD and BC respectively. If AB=12 cm, MN= 14cm, then the length of CD is
a) 16cm b) 14 cm c) 12cm d) 10cm
32) PQRS is a parallelogram. A and B are respectively the midpoints of the sides PQ and SR. AS and BQ meet the diagonal PR of length 12cm at C and D respectively. Then , the length of CD is
a) 6cm b) 3cm c) 4cm d) 5cm
33) The side AB of the parallelogram ABCD is produced to X and the bisectors of angle CNX meets DA produced and DC produced at E and F respectively. If DE= 10cm, then the length of DF is
a) 5cm b) 10cm c) 7.5cm d) 15cm
34) If the Diagonals of a rhombus are 18cm and 24cm respectively, then its side is equals to
a) 16cm b) 15cm c) 20 cm d) 17cm
35) In ∆ ABC , angrA, B & C are 30°, 40°, 110°. Then, the angles of the triangle formed by joining the midpoint of the sides of the triangle are
a) a) 70°, 70°, 40° b) 60°, 40°, 80° c) 30°,40°,110° d) 60°,70°,50°
1a 2b 3b 4c 5a 6c 7c 8d 9d 10d 11c 12a 13c 14b 15b 16c 17c 18a 19b 20d 21b 22b 23a 24c 25c 26b 27c 28b 29b 30c 31a 32c 33b 34b 35cq
SHORT ANSWER QUESTIONS
1) In the adjoining figure,
AB|| QP|| CD, Q is the midpoint of AC. If AB = 4cm, CD= 6cm, find the measure of PQ.
2) In the given figure,
ABC is an equilateral triangle. O is a point inside ∆ ABC. P, Q and R are the midpoints of AO, BO and CO respectively. Find the measures of x, y and z.
3) In the given figure,
D and E are midpoints of the sides AB and AC respectively of ∆ ABC. Q is any point on BC. AQ interesects DE at P. If AP= 3cm, find the measure of AQ.
4) In the given figure,
the straight lines l, m and n, are parallel to each other and G is the midpoint of CD. Calculate :
a) BG, if AD= 7cm
b) CF, if GE= 2.5cm
c) AB, if AC= 9cm
d) ED, if FE= 4cm.
5) ABCD is a quadrilateral,
P,Q,R and S are the midpoints of AB, BC CD and DA respectively. If AC= 6cm, BD=8.6cm, calculate PQ, QR, SR & PR.
6) Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that OA: OC = 2:3. Is ABCD a parallelogram ? justify your answer.
7) ABCD and AEFG are two parallelograms.
If angle C= 58°, determine angle F.
If angle C= 58°, determine angle F.
8) Angles of a quadrilaterals are in the ratio 3:4:4:7. Find all the angles of the quadrilateral.
9) In the given figure,
ABCD is a parallelogram. AX and CY are respectively the bisectors of opposite angles A and C. If angle DCB = 80°, find the measures of angle DAX.
10) ABC is a triangle and through A, B and C lines are drawn parallel to BC, CA, AB respectively intersecting at P, Q, R. if the perimeter of triangle PQR is 48cm, then find the perimeter of ∆ ABC.
1) 5cm 2) 20,25,20 3) 6cm 4) 3.5,5,4.5,4 5) 3.5, 5, 4.5,4 6) no 7) 58° 8) 60,80,80,140 9) 40 10) 24cm
VALUE - BASED QUESTIONS
1) The sports teacher and students class IX were in the process of preparing a drill for the sports day. An equilateral ∆ ABC was marked on the ground for some students to stand along its sides AB, BC, CA. The teacher wanted another smaller equilateral triangle PQR drawn within ∆ ABC along whose sides the remaining students could stand. The students suggested taking points P,Q, R as midpoints of AB, AC & BC and then joining PQ, QR, & PR.
i) Where the students right ? justify your answer.
ii) Write two values exhibited by the students.
2) ABCD is a rhombus
in which angle ABC= 110°. Ram's house is located at point A and Anil's house is situated at point B. There is an old age home at point O.
i) If Ram and Anil go along OA and BO respectively to reach there every Sunday to help out the senior citizens living there, then which of them has to cover shorter distance to reach there ?
ii) What values of shown by Ram and Anil ?
1)i) Yes , the students were correct
PQ= 1/2 BC, QR= 1/2 AB and PR= 1/2 AC, by mid-point theorem
ii) Problem-solving and helpfulness.
2) i) Anil has to cover a shorter distance than Ram.
As angle AOB= 90, angle BAO= 35, angle ABO= 55, BO< AO)
ii) emphaty, concern for senior citizens, helpfulness , responsibility and carying for others.
UNIT TEST
1) ABCD Is a parallelogram, if angle A=65°, then angle B+ angle D is equa to
a) 180 b) 115 c) 155 d) 230
2) In a quadrilateral, the diagonals are equal, then it cannot be a
a) square b) parallelogram c) rhombus d) rectangle
3) Diagonals of the parallelogram ABCD intersect at O. If angle BOC= 90° and angle BDC= 50°, then angle OAB is
a) 40 b) 50 c) 10 d) 90
4) A transversals cuts two parallel lines at A and B. The two interior angles at A are bisected and so are the two interior angles at B. The four bisectors form a quadrilateral ABCD which is
a) trapezium b) rhombus c) kite d) rectangle
5) In the given figure, ABCD is a rhombus. AO = 4cm and DO= 3cm. Then, the perimeter of the rhombus is
a) 18cm b) 20cm c) 21cm d) 22cm
Short Answer type -I Questions
6) In a parallelogram ABCD, determine the sum of angles C and D.
7) ABCD is a parallelogram. The bisectors of angle A and angle D intersect at O. Find the measure of angle AOD.
8) Prove that the straight line joining the midpoints of the opposite sides of a parallelogram are parallel to the other pair of parallel sides.
9) In the given figure, find the perimeter of BDEF.
10) In the adjoining figure, ABC and DBC are two triangles on the same side of BC. The midpoints of AB, AC, DC and DB are E, F, G and H respectively. Find the measure of x and y .
11) In the given figure, AB= 6cm, AC= 8cm, BC= 10cm, AD = DF= 1.5cm and AE= EG= 2cm. Find the measures of FG and DE.
12) In the given figure, D is the mid point of AB and PC= (1/2) AP = 3cm. If AD= DB= 4cm and DE|| BP, Find the measure of AC.
13) In the given figure, AB|| DC|| EG. If E is the midpoint of AD. AB= 6cm and CD= 10cm, find the measure of EG.
14) in the adjacent figure, ABCD is a parallelogram. E is the midpoint of CD and through D, a line is drawn parallel to EB to meet CB produced at G. If EB= 5cm, find the measure of DG.
15) In the given figure, ABCD and PQRB are rectangles where Q is the mid point of BD . If QR= 5cm, find the measure of AB.
16) In ∆ ABC, E and F are the midpoints of AC and AB respectively. The altitude AP to BC intersects FE at Q. If QP= 5cm, find the ratio AQ: AP.
17) ABCD is a parallelogram, E and F are midpoints of the sides AB and CD respectively. PQ is any straight line that meets AD, EF & BC in points P, O and Q respectively. If PO= 6cm, find the lengths of PQ.
Short Answer type II Questions
18) ABC is an isosceles triangle in which AB= AC= 10cm, D, E and F are the midpoints of BC, CA and AB respectively. AD and FE intersect at O. Name the type of quadrilateral formed by AFDE. If FE= 6cm, find the length of AD.
19) P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP. Also, prove that AC bisects PQ.
20) Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
21) In the given figure, side AC of ∆ ABC is produced to E, so that CE= (1/2) AC. D is the midpoint of BC and ED produced meets AB at F. RC and DP are drawn parallel to BA. Prove that FD= (1/3) FE.
22) D, E and F are the midpoints of the sides AB, BC and CA respectively of an isosceles ABC . Prove that ∆ EDF is also isosceles.
OR
in the given figure, ABCD is a parallelogram. E is the midpoint of CD and through D a line is drawn parallel to EB to meet CB produce at G and it cuts AB at F. Prove that
i) AD= (1/2) GC
ii) DG= 2EB .
Long Answer Questions
23) Given a ∆ ABC. Parallel lines are drawn through A, B, C respectively to sides BC, CA and AB forming ∆ PQR . Prove that BC= (1/2) QR.
24) PQRS is a parallelogram. PO and QO are respectively the angle bisectors of angles P and Q. Line LOM is drawn parallel to PQ. Prove that
i) PL= QM
ii) LO= OM
OR
Prove that the line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides and equal to half their difference.
1d 2c 3a 4d 5b 6) 180 7) 90 9) 14cm 10) 40,30 11) 5cm, 2.5cm 12) 9cm 13) 8cm 14) 10cm 15) 10cm 16) 1:2 17) 12cm 18) rhombus, 8cm
CUBOID AND PRODUCTS CUBE
1) Find surface area of a cube whose edge is 6m. 216m²
2) Find the edge of a cube whose surface area is 432m². 6√2 m
3) The perimeter of each face of a cube is 32cm. Find lateral surface area. 256cm²
4) Find the lateral surface area and total surface area of a cuboid whose length, breadth and height are 20cm, 10cm and 40 cm respectively. 2400,2800
5) The length, breadth and height of a cuboid are in the ratio 4: 2:1 and its total surface area is 1372m². Find the dimension of the cuboid. 28,14,7
6) A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30cm long, 25cm wide and 25cm high.
a) What is the area of the glass ?
b) How much tape is needed for all the 12 edges? 4250,320
7) How many bricks of dimensions 22.5cm x 10cm x 7.5cm can be painted, if the paint is sufficient for the surface area 2812.50 m²? 30000
8) Three cubes of each side 4cm are joined end to end. Find the attached surface area of the day resulting cuboid. 224 cm²
9) An open box is made of wood 3cm thick. Its external length, breadth and height are 1.48m, 1.16m, and 8.3 dm. Find the cost of painting the inner surface at the rate of Rs 50 per square metre. Rs279.70
10) A room is 8m long, 5m broad and 4m high . There are 2 doors each 3m x 1.2 m and two windows each 1.5m x 1.2m. Find the cost of
a) distempering the walls at Rs 12.50 per square m
b) carpenting the floor at Rs 50 per m. Rs1165, Rs 2000
CYLINDER
1) The diameter of the base of a right circular cylinder is 10cm and its height is 21 cm. Find the cost of painting the curved surface at the rate of Rs1.50 per cm². 990
2) The diameter of the roller, 1m 40cm long, is 80 cm. if it takes 600 complete revolutions to level a playground, find the cost of levelling the ground at 75 paise per square metre. Rs1584
3) A cylinder is 12cm high and the circumference of its base is 44cm. Find the curved surface area and total surface area. 528cm², 836cm²
4) A rectangular sheet of 88cm x 45cm is the rolled along its length to form a cylinder. Find the curved surface area of the cylinder. 3960cm²
5) The area of the curved surface of a right circular cylinder with radius of the base as 10cm is 880cm². Determine its height. 14cm
6) The total surface area of a right circular cylinder is 165π cm². if the radius of its base is 5 cm, find its height. 11.5 cm
7) It times costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10m deep. if the cost of painting is 20 per m², find radius of the base. 1.75m
8) Find the ratio of the curved surface area of a cylinder to its total surface area, given that its height and radius are 13.5cm and 7.5cm respectively. 9:14
9) The cost of polishing the total surface area of a closed cylindrical tank at the rate of 20 paise per dm² is Rs 154. if the height is one and a half times the radius of the base, determine the radius and height of the closed cylindrical tank. 10.5 dm
10) A metal pipe is 77cm long.
The inner diameter of the cross section is 4cm, the outer diameter being 4.4 cm, Find its
a) inner curved surface area
b) outer curved surface area of
c) total surface area. 968cm², 1064.8cm² 2038.08cm²
CONE
CUBOID AND A CUBE
1) Find the surface area of a cube whose
a) edge is 8cm. 384cm²
b) perimeter of one face is 20cm. 150cm²
2) Find the edge and lateral surface area of a cube whose total surface area is 864 cm². 12cm, 576 cm²
3) If the lateral surface area of a cube is 324cm², find its total surface area. 486 cm²
4) If the difference between the total surface area and the lateral surface area of a cubical object is 392cm², find its total surface area. 1176 cm²
5) An edge of a cube is increased by 10%. Find the percentage by which the surface area of the cube has increased. 21%
6) The length of diagonal of a cube is 7√3cm. Find its surface area. 294 cm²
7) Find the lateral surface area and the total surface area of a cuboid whose dimensions are as follows:
a) length= 40cm, breadth= 30cm, height= 25cm. 3500cm², 5900cm²
b) length= 4cm, breadth= 1.7cm, height= 2.3cm. 26.22cm², 39.82cm²
c) length= 25cm, breadth= 14cm, height= 6.5cm. 520cm²,1248 cm²
CYLINDER
1) Find the curved surface area and total surface area of a cylinder whose radii and height are given:
a) radius = 7cm, height= 20cm. 880,1188
b) radius = 10.5cm, height= 14cm. 924,1617
c) radius = 3.5cm, height= 6cm. 132,209
d) radius = 10.5cm, height= 16cm. 1056,1749
2) The diameter of a cylindrical garden roller is 1.4m and its length is 2m. How much area will it cover in 5 revolutions ? 44m²
3) Find the cost of polishing the curved surface of a right circular cylinder, whose diameter is 7cm and height is 12mcm, at the rate of 50 paise per cm². Rs132
4) The diameter of a road roller is 70cm and its length is 1m 50cm. It takes 750 complete revolution to level the surface. Find the cost of leveling the surface of the rate of Rs 75 per 100 m². Rs 1856.25
5) A road roller is cylindrical in shape . Its circular end has a diameter 250cm and its width is 1m40cm. Find the least number of revolutions that the roller must make in order to level a playground 110mx 25m. 250
6) A roller 5m in length and 7m in diameter when rolled on a road was found to cover an area 5500m². How many revolations did it make ? 50
7) A circular tunnel of diameter 2m and length 1.4 km is drug out. Find the cost of plastering it at the rate of Rs 150 per 100 m². Rs 13200
8) A factory manufacturers 120000 pencils daily. The pencils are cylindrical in shape , each of length 25cm and circumference 1.5 cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at Rs 0.05 per dm². Rs2250
9) A cylinder of circumference 10cm and height 22cm rolls without sliding for 11/2 seconds at the rate of 5 complete rounds per second. Find the area covered by the cylinder in 11/2 seconds. 6050cm²
10) The figure shows the frame of a lampshade.
The diameter of the frame is 28cm and its height is 18 cm. A margin of 1cm is to be given for folding it over the top and 1cm for bottom of the frame and a margin of 1cm is to be provided for stitching its side (height). Find how much cloth is required for covering the lampshade. 1780cm²
11) A rectangular sheet of metal long 88cm long and 20cm broad is rolled along its length to form a cylinder. Find the curved surface area of the cylinder. 1760cm²
12) The area of the curved surface of a cylinder is 4400 cm² and the circumference of its base is 110 cm. Find the height of the cylinder. 40cm
13) Find the diameter of the cylinder if the area of its curved surface is 88cm² and height is 14cm. 2cm
14) Find the height of the solid cylinder if its total surface area 660 cm² and radius is 5cm. 16cm
15) The cost of painting the outer curved surface of a cylinder at Rs 1:50 per cm² is Rs 660. If the height of the cylinder is 0.1m, find the radius of the base. 7cm
16) The ratio of the curved surface of a cylinder to its surface area is 1:2. Find the ratio between the height and the radius of the cylinder. 1:1
17) The total surface area of a right circular cylinder is 231cm². Its curved surface is 2/3 of its total surface area. Determine the radius of the base and height of the cylinder. 3.5,7
18) The internal base radius of a cylindrical container 7cm. It contains water up to a height of 10cm. Find the surface area of the wetted surface of the cylinder. 440cm²
19) it is required to make a closed cylindrical tank of height to 1.3m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same ? 8.8 m²
20) Find the total surface area of a hollow cylindrical pipe open at both ends, if its length is 12cm, external diameter is 8cm and thickness is 2cm. (leave your answer in terms of π). 168π cm²
21) The total surface area of a hollow metallic cylinder open at both ends of external radius 8 cm and height 10cm is 338π cm². Taking r to be inner radius, find the thickness of the metal in the cylinder. 3cm
22) A hollow pipe is 100cm long and the difference between its inner and outer radius is 3.5 cm. What is the difference between its outer and inner lateral surface areas? 2200cm²
23) A closed cylindrical tank is to be made with metal sheet. If 12% of the metal sheet gets wested in constructing the tank, find the total area of the metal sheet used, given that height of the tank is 2.4m and its diameter is 2.8m. 38m½
24) The radii of two right circular cylinders are in the ratio 2:3. if the heights are in the ratio 5:4, Find the ratio of their curved surface areas . 5:6
25) The curved surface area of a 15cm high cylinder is 2310cm². A wire of diameter 6mm is wound around it so as to covere it completely. Find the length of the wire. 38.5m
VOLUME OF A RIGHT CIRCULAR CONE
1) Find the volume of the right circular cone with:
a) radius = 7cm, height= 12cm. 616cm³
b) radius = 21cm, height= 49cm. 18480cm³
c) diameter = 56m, height= 1.02m. 83776cm³
d) radius = 35 dm, slant height= 37 dm. 15400dm³
e) perimeter of the base = 8cm, height= 5cm. 280/33 cm³
2) Find the capacity in litres/kilolitres of a conical vessel, with
a) radius= 28cm, slant height= 35 cm. 17.248L
b) height= 48m, slant height= 50m. 9856 kL
3) The circumference of the base of a 12m high solid cone is 22m. Find the volume of the cone . 154m³
4) A conical tent has the area of its base is 154 m² and that of its curved surface as 550m². Find the volume of the tent. 1232m³
5) A cone of slant height 10m has a curved surface area 188.4m². Find its volume (π= 3.14). 301.44 m³
6) A right circular cone of height 4cm has a curved surface area 47.1cm². Find its volume (π= 3.14). 37.68 cm³
7) The floor area of a tent in the form of a right circular cone is 3168/7 m². The Canvas required for the tent is 3960/7 m². Find the air capacity of the tent. 9504/7 m³
8) A military tent is in the form of a right circular cone of vertical height 6m, the diameter of the base 7m. if 12 soldiers can sleep in it, find the average cubic metre of air space available per soldier. 77/12 m³
9) In a conical tent of base radius 7m and height 12m, how many bags of wheat can be emptied if volume of each bag is 3.5m³? 176 bags
10) A heap of wheat is in the form of a cone of diameter 9 m and height 3.5m. Find its volume. Also find how much canvas cloth is needed to cover the heap. 74.25m³, 80.614 m²(app)
LINES AND ANGLES
1) The measure of an angle which is 24° more than its complement is
a) 66° b) 57° c) 156° d) 114°
2) The measure of an angle which is 32° less than the supplement is
a) 148° b) 58° c) 74° d) 122°
3) The measure of an angle which is four times is complement is
a) 78° b) 76° c) 72° d) 74°
4) If the supplement of an angle is 4 times of its complement, then the angle is
a) 60° b) 40° c) 50° d) 70°
5) if two complementary angles are in the ratio 2 :3, then the angles are
a) 58°, 32° b) 50°,40° c) 56°, 34° d) 36°,54°
6) angle P and Q are complementary angles. if they are represented by the expressions m angle Q and m angle P= 2y + 30°, then their measures respectively are
a) 70°, 20° b) 20°,70° c) 10°,80° d) 80°,10°
7) In the given figures ,
which pairs of angles represent a linear pair ?
a) (i and (iii) b) (iii) and (iv) c) (iii) and (v) d) (i), (ii) and (v)
8) If in the given figure,
OA and OB are opposite rays, then the value of x is
a) 40 b) 44 c) 46 d) 42
9) The value of x that will make POQ a straight line is 
a) 30 b) 25 c) 35 d) 40
10) In the given figure,
if POQ is a straight line, then the value of x is
a) 20° b) 30° c) 40° d) 50°
11) In the given figure,
angle AOC=50°, then Angle AOD+ angle COB is equal to
a) 100° b) 140° c) 260° d) 130°
12) In the given figure
AB||CD . Transversal PQ intersects AB at E and CD at F. Given angle CFQ= 47°, the measure of X and y respectively are
a) 30°,150° b) 37°,143° c) 47°,133° d) 39°,141°
13) In the given figure,
l ||| m || n. If x: y= 5:4, then the measure of angle z is
a) 40° b) 50° c) 90° d) 80°.
14) In the given figure,
AB|| CD || EF and GH|| KL. The measure of angle HKL is
a) 95° b) 145° c) 130° d) 135°
15) The measure of x in the given figure is 
a) 125° b) 70° c) 105° d) 100°
16) In the given figure,
if l || m, then the value of x is
a) 18° b) 72° c) 54° d) 100°
17) In the given figure
AB|| CD and EF || BD. If angle= 65°, then the measure of angle CFE is
a) 120° b) 115° c) 65° d) 165°
18) In the given figure,
if l || m, then the measure of x is
a) 70° b) 100° c) 40° d) 30°
19) In the adjoining figure,
if l || m and n ⊥ m, then the measure of angle P is
a) 48° b) 42° c) 90° d) 38°
20) In the given figure,
if l || m then the measure of angle x is
a) 65° b) 40° c) 25° d) 90°
21) If two angles of a triangle are complementary, then it is
a) a right triangle
b) ab obtuse angled triangle
c) an acute angled triangle
d) an equilateral triangle
22) An exterior angle of a triangle is 110° and its two opposite interior angles are equal. Each of these equal angles is
a) 70° b) 55° c) 35° d) 110°
23) The angles of a triangle are in the ratio 4:5:9. The triangle is
a) an isosceles triangle
b) an obtuse angled triangle
c) an acute angled triangle
d) a right triangle
24) An exterior angle is drawn to a triangle. if this exterior angle is acute, then the triangle must be
a) an acute angled triangle
b) a right triangle
c) an obtuse angled triangle
d) an equilateral triangle
25) If the measure of each base angle of an isosceles triangle is seven times the measure of the vertex angle, then the measure of the vertex angle is
a) 84° b) 48° c) 12° d) 24°
26) if the vertex of an isosceles triangle is 80° the measure of an exterior angle to one of the base angles of this triangle is
a) 100° b) 120° c) 110° d) 130°
27) In the given figure,
if PQ || RS and angle ACS = 120°, then angle BAC is equal to
a) 53° b) 77° c) 50⁰ d) 107°
28) The value of x in the given figure
is
a) 100° b) 70° c) 110° d) 150°
29) The base of triangle ABC is produced both ways and the measures of exterior angles formed are 94° and 126°. 
Then, the measure of angle BAC is
a) 94° b) 54° c) 40° d) 44°
30) The value of x
in the given figure is
a) 65° b) 95° c) 80° d) 120°
31) If one of the angles of an isosceles triangle is 125°, then the angle between the bisectors of the other two angles is
a) 125.5° b) 152.5° c) 152° d) 125°
32) ∆ ABC is a right triangle in which angle A is a right angle. AL is drawn perpendicular to BC. If angle BAL is 35°, then the measure of angle ACB is
a) 70° b) 17.5° c) 35° d) 105°
33) ABC is an equilateral triangle and BDC is an isosceles triangle right angled at D. Angle ABD is equal to 
a) 45° b) 60° c) 105° d) 120°
34) The side BC of ∆ ABC is produced to point D.
The bisector of angle ABC and ACD meet at a point E. If angle BAC= 68°, then the measure of angle BEC is
a) 30° b) 32° c) 36° d) 34°
35) In the given figure,
AB || CD, then the value of x and y respectively are
a) 25°, 65° b) 60°,30° c) 65°, 25° d) 40°,50°
36) In the given figure,
ABCD is a quadrilateral in which angle ABC= 73°, angle C= 97° and angle D= 110°. If AE|| DC and BE || AD and AE intersect BC at F, then the measure of angle EBF is
a) 23° b) 70° c) 10° d) 27°
37) The angle between the bisectors of two acute angles of a right triangle is
a) 135° b) 120° c) 90° d) 150°
38) The measure of x in the given figure is 
a) 35° b) 25° c) 30° d) 20°
39) in the given figure,
AB || CD. Transversal EF intersects AB at P and CD at Q. Angle PRQ= x, angle RPQ= y. If angle APR= 25°, angle RQC= 30° and CQF= 65°, then the measures of angle x and y respectively are
a)55°,40° b) 50°,45° c) 60°, 35° d) 35°,60°
40) If the bisector of the bese angles of a triangle enclose an angle of 135°, then the triangle is
a) an acute angle triangle
b) an obtuse angled triangle
c) an equilateral triangle
d) a right triangle
41) In the figure given alongside,
if AB || CF and CD || FE, then the value of x is
a) 40° b) 65° c) 75° d) 105°
42) In the given figure,
if AB || CD , then the value of x is
a) 97° b) 100° c) 107° d) 45°
43) BO and CO, the bisectors of angle B and C respectively, of ∆ ABC, meet at O. If angle = 60°, then the measure of angle BOC is 
a) 100° b) 90° c) 120° d) 150°
44) If two parallel lines of cut by a transversal, then the bisectors of the interior angles on the same side of the transversal intersect each other at
a) 60° b) 90° c) 100° d) 120°
45) if two parallel lines are interesected by a transversal, then the bisectors of the interior angles form a
a) kite b) Rhombus c) rectangle d) trapezium
46) ABC is a triangle in which BE ⊥ AC and CD⊥ AB, BE and CD intersect at O.
If angle BAC = 75°, then the measure of angle BOC is
a) 100° b) 105° c) 75° d) 115°
47) ABC is a right angle triangle, right angled at B.
BC = BA. D is a point on AC produced and a line DEF cuts CB at E, AB at F. If angle D = 13° and angle FAE= 29°, then the measure of angle FEA is
a) 31° b) 42° c) 29° d) 16°
48) In ∆ XYZ, XY= XZ.
A straight line cuts XZ at P, YZ at Q and XY produced at R. If YQ= YR and QP = QZ, then the measure of angle PQY is
a) 100° b) 124° c) 144° d) 140°
49) ABCD is a square,
if AP= PQ and QRC= 35°, then the measure of angle PAQ is
a) 40° b) 35° c) 30° d) 25°
50) In the given figure,
if AB divides angle DAC in the ratio 1:3, then the measure of angle marked x is
a) 108° b) 100° c) 80° d) 90°
1b 2c 3c 4a 5d 6a 7d 8d 9a 10b 11c 12c 13d 14b 15a 16c 17b 18c 19a 20d 21a 22b 23d 24c 25c 26d 27b 28a 29c 30d 31b 32c 33c 34d 35b 36d 37a 38c 39a 40d 41d 42a 43c 44b 45c 46b 47c 48c 49a 50d
BOOSTER - B
Short Answer Questions
1) ABC is a right angle triangle in which angle A= 90° and AB= AC. Find the values of angle B and C.
2) In the given figure,
show that AB || EF.
3) OD is the bisector of angle AOC. OE is the bisector of angle BOC and OD ⊥ OE,
show that the point A, O and B are collinear.
4) in the given figure,
bisectors PR and QS of the alternate interior angles are parallel. Show that l || m.
5) in the given figure, 
BA || QP and BC|| QR. Show that anyABC= Angle PQR.
6) in the given figure, 
BA || PQ and BC || QR. Show that angle ABC+ angle PQR= 180°.
7) In the given figure alongside,
AB || DE. Find the measure of angle BCD.
8) in the given figure,
find x if AB || CD.
9) Three coplanar lines intersect at O forming angles as shown in the figure.
Find the values of x,y,z and y.
10) The exterior angles obtained on producing the base of a triangle both ways are 100° and 120°. Find all the angles of the triangle.
11) in the given figure,
AB|| CD and EF|| BC. angle BAC= 65°, angle DHF= 35°, find angle AGH.
12) Prove that if two parallel lines are intersected by a transversal , then bisectors of any two corresponding angles are parallel.
13) Prove that if two lines are perpendicular to the same line then these lines are parallel to each other.
14) m and n are two plane mirrors parallel to each other.
Prove that the incident ray CA is parallel to the reflected ray BD.
15) Find the value of x and terms of right angles.
1) 45, 45. 7) 40 8) 130 9) 40, 50,90,40 10) 80,60,40 11) 100 15) 3 rt angle s
UNIT TEST - A (MM 50)
Multiple choice questions (1 mark each)
1) In the given figure,
ABCD is a rectangle in which angle AOB= 100°. The value of x is
a) 40° b) 50° c) 60° d) 70°
2) In the given figure,
the value of x is
a) 55° b) 70° c) 35° d) 110°
3) In the given figure,
If AOB is a straight line, then the value of x is
a) 27° b) 126° c) 63° d) 54°
4) In the given figure,
if x: y = 1:4, then the value of x and y are respectively.
a) 36° and 144° b) 18° and 72° c) 144° and 36° d) 72° and 18°
5) In the given figure,
if l || m, then the values of x and y are respectively.
a) 60°, 120° b) 55°, 125° c) 70°,110° d) 50°,130°
6) In the given figure
if p || q and l || m, then the measure of x is
a)100° b) 90° c) 80° d) 105°
7) The measure of x in
the given figure is
a) 115° b) 85° c) 80° d) 95°
8) If AB|| CD, find the measure of x.
a) 9° b) 11° c) 22° d) 18°
Short Answer Questions (2 Marks each)
9) in the given figure,
POQ is a straight line. Find the value of x.
10) In the adjoining figure,
ABC is a straight line. Find the value of x.
11) In the given figure,
a: b: c= 2 :5:3. If AOB is a straight line, find the value of a, b and c.
12) If a ray OZ stands on the line XY such that Angle XOZ= Angle ZOY, show that angle XOZ = 90°.
13) Two lines
AB and CD intersect at O such that angle BOC+ AOD = 290°. Find all the four angles.
14) If a line is perpendicular to one of the two given parallel lines, prove that it is also perpendicular to the other line.
15) in the given figure,
the side BC of ∆ ABC is produced to a point D. If the bisectors of angle ABC and angle ACD meet at E, then show that angle BEC= (1/2) Angle BAC.
16) In the given figure,
PS bisects angle TPQ and ∆ PQR is an isosceles triangle. Prove that PS || RQ.
Short Answer Questions (3 Marks each)
17 AB|| DE and BD|| FG.
If angle FGH= 125° and angle B= 55°, find angle x and y.
18) ABCD is a trapezium.
EF is parallel to AD and BC. Find the measures of the angles marked x,y,z and p.
19) In the given figure,
AB and CD are two straight lines, intersecting each other at point O. If angle COE= 90°, find the values of x, y and z.
20) in the given figure, 
if AB || CD , angle 1= (3x +15)°, angle 3= (x + 5y)° and angle 5= (7y +2)°, find the measure of angle 6.
21) In the given figure,
AB || CD. Find the value of x.
22) In the given figure,
AE || BD, and CA|| DE. Find the measure of x and y.
Long Answer Questions (4 Marks each)
23) if two parallel lines are intersected by a transversal , then show that the quadrilateral formed by the bisectors of two pairs of interior angles is a rectangle.
24) In the given figure,
AB= AC. D is a point on AC and E on AB such that AD = ED= BC. Prove that angle AED= angle BCE.
1b 2a 3c 4a 5b 6b 7a 8d
9) 59 10) 28 11) 36,90,54 13) 145,35,145,35 17) 70,55 18) 50,60,50,70 19) 30,60,120. 20) 45 21) 46 22) 65,65
⊥ °
HERON FORMULA
BOOSTER - A
1) The area of the triangle with the base 8cm and height 10cm is
a) 80cm² b) 40cm² c) 20cm² d) 18 cm²
2) The sides of a triangle are 12cm, 16cm and 20 cm. Its area is
a)48cm² b) 120cm² c) 96 cm² d) 160 cm²
3) The area of a triangle whose sides are 3cm, 4cm and 5cm is
a) 42cm² b) 6cm² c) 84cm² d) 100cm²
4) If the perimeter of an equilateral triangle is 24m, then its area is
a) 20√3m² b) 16√3m² d) 8√3 m² d) 24√3 m²
5) If the area of the equilateral triangle is 16√3cm², then the perimeter of the triangle is
a) 12cm b) 24cm c) 48cm d) 36cm
6) the edges of a triangle board are 6cm, 8cm and 10 cm. The cost of painting it at the rate of 70 paise per cm² is
a) Rs 7 b) Rs 16.80 c) Rs 17 d) Rs 16
7) The perimeter of a Rhombus is 20cm. If one of its diagonals is 6cm, then its area is
a) 28 cm² b) 36cm² c) 24 cm² d) 20cm²
8) An isosceles right triangle has an area 8cm². The length of the hypotenuse is
a) 6cm b) √32cm c) 8cm d) 4cm
9) The area of an isosceles triangle having base 24 cm and length of one of the equal side is 20cm is
a) 480cm² b) 196cm² c) 240cm² d) 192cm²
10) The perimeter of an isosceles triangle is 32 cm. The ratio of equal side to its base is 3:2. Then area of the triangle is
a) 32√2 cm² b) 32cm² c) 16 √2cm² d) 16cm²
11) If the perimeter and base of an isosceles triangle are 11cm and 5cm respectively, then it's area is
a) 5√11cm² b) 5√11/2 cm² c) 5√11/8 cm² d) 5√11/4 cm²
12) If the difference between the semi-perimetre 's' and the sides 'a', 'b' and 'c' of ∆ ABC are 8cm, 7 cm and 6cm respectively, then ar(∆ ABC) is
a) 63cm² b) 42 cm² c) 84 cm² d) 168 cm²
13) thre sides of a triangle are 13cm, 14cm and 15cm. The length of the shortest altitude is
a) 12cm b) 11.2cm c) 12.9cm d) 11.9cm
14) The sides of a triangle are 17cm, 25cm and 26cm. The length of the altitude to the longest side correct up to two places of decimals is
a) 16.32cm b) 34.00cm c) 15.69cm d) 24.00cm
15) If the perimeter of rhombus whose diagonals measure by 12cm and 16 cm is equal to the perimeter of an isosceles triangle having the equal side and the base in the ratio 3:2, then the area of the isosceles triangle is
a) 500√2cm² b) 25√2 cm² c) 75√2 cm² d) 100√2cm²
1b 2c 3b 4b 5b 6b 7c 8b 9d 10a 11d 12c 13b 14c 15a
BOOSTER - A(1)
Short answer Questions
1) Find the area of a triangle whose base and attitude is 10cm and 7cm respectively .
2) Find the area of a triangle whose sides are 13 cm, 14 cm and 15cm.
3) Find the area of a triangle, two sides of which are 9cm and 12cm and the perimeter is 36 cm.
4) The sides of a triangle are in the ratio 3 : 5 : 7. Find its area if its perimeter is 60cm.
5) Find the area of an equilateral triangle whose perimeter is 24cm.
6) The height of an equilateral triangle is 6cm. Find the area of the triangle.(Take √3= 1.732).
7) Find the area of an isosceles triangle each of whose equal sides is 13cm and whose base is 24cm.
8) Find the percentage increase in the area of a triangle if each of its side is doubled .
9) A rhombus shaped sheet with perimeter 40 cm and diagonal 12cm is painted on both sides at the rate of Rs 5 per cm². Find the cost of painting.
10) Find the cost of printing the shaded area shown in the given figure, at the rate of Rs 1 per cm². (Take √3= 1.73).
1) 35cm² 2) 84cm² 3) 54cm² 4) 60√3cm² 5) 16√3cm² 6) 20.784cm² 7) 60cm² 8) 300% 9) Rs 960 10) Rs 243.93
BOOSTER (A(1)(1)
Value based questions
1) ABCD represents a plot of land owned by man.
He divides it into two parts by joining diagonal BD and donate the triangular part BAD to an orphanage .
i) If angle CBD=90°, find the area of the plot donated by him.
ii) What are the values shown by the man here ?
2) Some students started a cleanliness campaign in their school. For distribution among the fellow students, they prepared hand fans by stitching 10 equal sized triangular strips of different types of paper (as shown in the given figure).
The dimensions of equal strips are 25cm, 14cm and 25 cm. They wrote slogans for maintaining cleanliness in shaded areas.
i) Find the area used for writing the slogans.
ii) What value are depicted by the students?
3) The diagram given below shows the triangular side walls of the entrance to a library with quotes by Mahatma Gandhi written on them. The sides of each of the triangular walls are 15m, 11m and 6m respectively.
a) Find the area of each of the triangular wall.
b) What values can be inculcated by the two quotes in the visitors ?
4) A woman inherits a triangular plot of land ABC as shown in the figure.
She contributes to society by donating a triangular piece ADC out of this plot for constructing an old age.
i) Find the area of the plot with her.
ii) What values are shown by the woman?
1) i) 20.98m²(approx) ii) Empathy, concern for orphans
2) i) 840cm² ii) Social responsibility, creative thinking, leadership, cooperation and awareness about maintaining cleanliness.
3) i) 20√2m² ii) Be hardworking, focussed, determined, caring, helpful, considerate and humane
4) i) 114m² ii) Empathy, concern for old people, compassion, helpful, caring and decision making ability.
UNIT TEST - A
MM- 10
(1 Mark each)
1) A triangle ABC in which AB= AC = 4cm and angle A= 90°, has an area of
a) 4cm² b) 16cm² c) 8cm² d) 12cm²
2) The area of a triangle whose sides are 8cm,15 cm and 19 cm is
a) 91√19cm² b) 6√91 cm² c) 19√91cm² d) 8√91cm²
3) Find the area of an equilateral triangle whose perimeter 18 cm. (Take √3= 1.732). (2)
4) Find the area that needs to be added to the area ∆ ADB,
so that it become equal to the area of ∆ ABC (take √3= 1.732). (3)
5) Find the area of ∆ ABC in which AB= 36cm, BC = 48cm and AC= 60cm. Find the length of the shortest altitude. (3)
1c 2b 3) 15.588cm² 4) 19.3cm² 5) 864cm², 28.8cm
SURFACE AREAS AND VOLUMES
CUBE AND CUBOID
1) The total surface area of a cube is 96 cm². The volume of the cube is
a) 27cm³ b) 64cm³ c) 8cm³ d) 512 cm³
2) The number of cubes whose edges measure 3cm, that can be found by melting a cubic block of metal of edge 15cm is
a) 125 b) 45 c) 75 d) 135
3) The difference between the total surface area of a cube of side 4cm and its lateral surface area is
a) 16 cm² b) 20 cm² c) 32 cm² d) 24 cm²
4) The volume of a cube whose diagonal is 2√3 cm is
a) 84 cm³ b) 4 cm³ c) 8√3 cm³ d) 4√3 cm³
5) The number of planks of dimensions (5m x 25cm x 10cm) that can be placed in a pit which is 20 m long, 6m wide and 80cm deep is
a) 764 b) 840 c) 768 d) 960
6) The number of 6m cubes that can be formed from another cuboid measuring 18m x 12m x 9m is
a) 9 b) 10 c) 12 d) 15
7) The length of the longest rod that can be placed in a room 12m long, 9m broad and 8m high is
a) 15m b) 20m c) 18m d) 17m
8) The edge of a cube whose volume is equal to the volume of a cuboid of dimensions 36cm x 75cm x 80cm is
a) 48cm b) 60cm c) 36cm d) 42cm
9) A rectangular pit of dimensions 30m x 15m x 12m is dug and the Earth taken out is disposed of in a carrier which can carry a maximum load of 540m³ of earth. The least number of rounds the carrier had to make to dispose of the earth dug out is
a) 20 b) 20 c) 15 d) 12
10 A granery is in the shape of a cuboid of size 16m x 12m x 9m. If a bag of grain occupies a space of 0.48m³, then the maximum number of bags that can be stored in the granary is
a) 1800 b) 3600 c) 2400 d) 3000
11) When a cuboid of dimensions 30cm x 30cm x 42.6cm is melted and converted into cubes of edge 3cm, then the number of cubes formed is
a) 2840 b) 2130 c) 1420 d) 710
1b 2a 3c 4a 5c 6a 7d 8b 9b 10b 11c
CYLINDER
12) The volume of a right circular cylinder is 2310cm³. If the radius of its base is 7cm, then its height is
a) 7.5cm b) 22.5cm c) 15cm d) 30 cm
13) If a square paper of side 25cm is rolled to form a cylinder, then its curved surface area is
a) 625 cm² b) 500cm² c) 250cm² d) 1000 cm²
14) The curved surface area of a well of diameter 3.5m and depth 10m is
a) 135m² b) 35m² c) 70m² d) 110m²
15) The curved surface area of a cylinder whose circumference of the base is 22m and height is 3m is
a) 66m² b) 132m² c) 33m² d) 99m²
16) If the outer diameter of a pipe 21m long is 1m, then its outer curves surface area is
a) 21m² b) 63m² c) 66m² d) 42m²
17) The cost of cementing the inner curved surface area of a 14m deep well of radius 2m at the rate of Rs 2 per m² is
a) Rs 350 b) Rs 56 c) Rs 122 d) Rs 176
18) The diameter of the base of a cylinder of curved surface area 88 cm² and height 14 cm is
a) 1cm b) 2cm c) 1.5cm d) 2.5 cm
19) The total surface area of a circular cylinder of height 4cm and radius 3cm is
a) 132cm² b) 66cm² c) 198cm² d) 99 cm²
20) If the lateral surface area of a cylinder is 132cm² and its height is 7cm, then its base diameter is
a) 5cm b) 3cm c) 6cm d) 4cm
21) The circumference of the base of a right circular cylinder is 44cm. If its whole surface area of 968 cm², then the sum of its height and radius is
a) 16cm b) 18cm c) 20cm d) 22cm
22) The curved surface area of a right circular cylinder is 4400cm². if the circumference of its base is 110cm, then its height is
a) 36cm b) 38cm c) 40cm d) 42cm
23) A cylindrical piece of maximum volume has to be cut out of an iron cube of edge 4 cm. Then the maximum volume of the iron cylinder is
a) 32πcm³ b) 24πcm³ c) 16πcm³ d) 28πcm³
24) If each bag containing rice occupies 2.1m³ of space, then the number of full bags which can be emptied into a drum of radius 4.2m and height 3.5m is
a) 69 b) 46 c) 92 d) 138
25) If the radius of the base of a right circular cylinder is halved , keeping the same height, then the ratio of the volume of the reduced cylinder to the volume of the original cylinder is
a) 1:4 b) 4:1 c) 1:2 d) 2:1
26) A cylindrical vessel of radius 16cm contains water to a depth of 30cm. if a spherical ball of brass is dropped into it and the water rises by 9cm, then the radius of the ball is
a) 12cm b) 15cm c) 8cm d) 18cm
27) The radii of 2 cylinders are in the ratio 2:3 and their heights are in the ratio 5:3. The ratio of their volume is
a) 20 : 27 b) 20:37 c) 17:27 d) 10:17
12c 13a 14d 15a 16c 17a 18b 19a 20c 21d 22c 23c 24c 25a 26a 27a
SPHERE
The volume of the sphere of diameter 42 3834,30038 the surface area of a radius 3.5 e77 1504 150 4 120 the volume of a sphere is numerical equal to surface area than its diameter is 6 unit 3 unit 1 units 2 minutes a cube of side 4 contains a square touching its side find the approximate volume of the gaping 3330 the ratio of the radiator spear is volume by the ratio 64 is to 27 is 16:98 is 237 4:3 given the surface area of a spherical short 26 diameter is 12 14 16 18 it's Tiara for radius 3 is melternate the casting to right circular cone is a height to in the radius of the base of the cone is 27369 spherical balloon grows to eyes AC radius of inflated in the ratio of the volume of the metal values in the original age is to 14:1621 5:1 total sarface theories 1848 then the director of 22.6 28 24 that
CONE
The total surface area to cone of DS and slant height is the total surface area of a radiation slant height 10 374 598561 282.5 the volume of the phone is 1570 if it is 15 high then its base area is 415 413 300 14540 slant height of a cone of this radius 7 is 25 then its height is 32 24 18 36 the diameter of the base of the cone of the height 15 the volume 77 1421 10.5 aconical 10 to 20 1 high and the diameter the basis for a 10 menslips in 8 then the average number of cubic affairs page per man is 448848 chronicle pendle 240 years 100 is made clothes which is 100 wise than the length of the cloth used to make the pandal is 625 676 624 if the ratio the Reddy of the base up to con is 3:21 in the ratio of the height is 1:3 then the ratio of the volume is 221 1:33 is 21
Mixed
The curved surface area of a cylinder and a cone is equals to their base radius is same in the ratio of the slunt had the cone to the height of the cylinder is 2:31 is 21 1
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