Test: Elementary Mathematics - 1 - CDS MCQ

The last digit of a perfect square number cannot be 2, 3, or 7.
Therefore, choices 1, 3 and 4 cannot be a perfect square.
Now, let us check choice 2 viz. 11664.
11664 = 108² and is thus a perfect square number.
Thus, choice 2 is a perfect square number and choices 1, 3 and 4 are not.
Hence, answer option 4 is correct.

For to be an odd number we need to removal all 2's from 12!
Maximum power of 2 in 12! is
Where [x] is greatest integer function.
So, N = 10
Alternate Solution:

If we divide the 12! with 2¹⁰ the result will be odd number. So the value of N will be 10.

Statement I is true, but Statement II is false.
This is because there can be two or more readings with the same frequencies.

sin x
= sin x
= sin x
=
√2

According to the data given in the question, l || m and AC is a transversal.
∴ ∠DAC = ∠ACB [Alternate interior angles]
∠ACB = 30°
Also, q is a transversal.
∴ ∠ABC + ∠BCD = 180° [Co-interior angles]
70° + ∠BCD = 180°
∠BCD = 110°
Now, ∠ACB + ∠ACD = ∠BCD
30° + ∠ACD = 110°
∠ACD = 110° - 30° = 80°
Hence, 80° is the correct answer.

Consider statement 1:
y² = sec²x + cosec²x

Hence, statement 1 is true.

Now, let us consider statement 2.

Now, cotx =
Or, cot²(x/2) - 2cotx cot(x/2) - 1 = 0
Or, cot(x/2) = (2cotx + (4cot²x + 4)^1/2)/2 (Taking only positive sign as 0° < x < 45°)
Now, the minimum value of cot(x/2) for the given range of x will be when x = 45°.
So, plugging in x as 45°, we get
cot(x/2) = y > (2 + (4 + 4)^1/2)/2
Or, y > (2 + 2(2)^1/2)/2 or y > 2
Hence, statement 2 is correct as well.
Let us now consider statement 3.
LHS of statement 3 =
=
=
=
= cosx + sinx = a say
Now, (cosx + sinx)² = a²
Or, cos²x + sin²x + 2sinxcosx = a²
Or, 1 + sin²x = a²
Now, as sin2x < 1, (as x < 45°), we have a² < 2
Thus, a < 2 as well
Hence, the third statement is false.
Thus, only 2 statements are true.
Hence, answer option 3 is correct.

We know that the hour hand completes a rotation in 12 hours, while the minute hand completes a rotation in 60 minutes.
Thus,
Angle traced by the hour hand in 12 hr = 2π
Now,
Angle traced by the hour hand in 4 hr 36 min, i.e. (4 + (36/60)) hr
Or,
(23/5) hr =
Also,
Angle traced by the minute hand in 60 min = 2π
The angle traced by the minute hand in 36 min = (2π/60) x 36 = 6π/5 
Hence, the required angle between the two hands =
This lies between 2π/5 and 3π/5.

Let α and β be the roots of the equation ax² + bx + c = 0.
So, α + β = -b/a; α​​​​​​​β = c/a
Let 1/α and 1/β be the roots of equation px² + qx + r = 0.
So,

Squaring both sides, we get

On solving, we get b²p² = q²c² ... (1)
Now, from αβ = c/a​​​ and 1/αβ = r/p, we get (c/a) = (p/r) ⇒ ap = cr  ...(2)
Dividing equation (1) by (2), we get: acq² = b²pr

Women visiting park A = (30/100) x 1600 = 480
Women visiting park F = (20/100) x 1500 = 300
Total (A and F) = 480 + 300 = 780

Required percentage = (3/5 × 16% + 3/4 × 21% + 3/8 × 8%) × 100 = x 100 = (567/20) x 100 = 28.35%.

QS/QV = 1/3 ⇒ QS = K, QV = 3K (say)

SV = QV – QS = 2K

Similarly, PR = 4r

PT = 3r

Let the remaining angle be a° as shown in the above figure.
x + y + a = 360°
120° + 100° + a = 360°
a = 140°
z + a = 180°
z + 140° = 180°
z = 40°
Hence, 40° is the correct answer.

tan 45° = h/BC
⇒ h = BC
tan 60° = 300/BC
√3 = 300/h
h = (300/√​​​​​​​3) × (√​​​​​​​​​​​​​​3/√​​​​​​​3) = 100√​​​​​​​3 m

Speed of A = 6 km/hr
Speed of A in m/s = (5/3) m/s
Time taken by A in 100 m race = 60 seconds
Let speed of B = x m/s
Time taken by B in (100 − 8) m race = (92/x) seconds
A.T.Q.
(92/x) - 60 = 9

⇒ (92/x) = 69

⇒​​​​​​​ x = (92/69) m/s
Speed of B in km/hr:
4.8 km/hr
This is the required answer.

x = a cosθ+ b sinθ … (1)
y = a cosθ– b sinθ … (2)
Squaring x and y, we get
⇒ x² = a² cos²θ+ b² sin²θ+ 2ab cosθ sinθ
⇒ y² = a² sin²θ+ b² cos²θ– 2ab cosθ sinθ
⇒ x² + y² = a²(cos²θ + sin²θ) + b²(cos²θ + sin²θ)

x² + 2x - 24 has factors: x² + 6x - 4x - 24 = (x + 6)(x - 4).
F(x) = 3x³ - ax² - 74x + b
F(-6) = 3(-6)³ - a(-6)² - 74(-6) + b
= -648 - 36a + 444 + b
= -36a + b - 204
F(4) = 3(4)³ - a(4)² - 74(4) + b
= 192 - 16a - 296 + b
= -16a + b - 104
Put F(- 6) = 0 and F(4) = 0.
-36a + b = 204 ... (1)
-16a + b = 104 … (2)
(1) - (2) will give:
-20a = 100
⇒ a = -5
Putting this value in (2),
-16(-5) + b = 104
b = 104 - 80 = 24
Hence, a = -5, b = 24

Let the speed of the car be c and that of the bus be b
As the distance travelled by both the vehicles is the same, 6c = 8.5b.
Or, c/b = 8.5/6 = 85/60 = 17/12
Now, increasing the speed of the car by 50%, the new ratio becomes:
25.5 : 12 = 255 : 120 = 17 : 8
Hence, answer option (2) is correct.

x² + y² + 2x - 10y + 26 = 0
⇒ (x + 1)² + (y - 5)² = 0
Since x and y are both real, we get
(x + 1)² = 0
⇒ x = -1 and (y - 5)² = 0
⇒ y = 5
∴ x + y = 4

Radius, r = 5 cm
Height of the cone, (h) = 12 cm
Height of the cylinder, (H) = 12 cm

Total surface area of the solid = Curved surface area of the cone + Curved surface area of the cylinder + Curved surface area of the hemisphere
Total surface area of the solid = πrl + 2πrH + 2πr²
Total surface area of the solid = πr(l + 2H + 2r)
Now, first find the slant height of the cone.
Slant height, l =
l =
l =
l = = 13 cm
Now, substitute all the values in the formula of total surface area of the solid:
Total surface area of the solid = πr (l + 2H + 2r)
Total surface area of the solid = (22/7) × 5 [13 + 2(12) + 2(5)]
Total surface area of the solid = (110/7) [13 + 24 + 10]
Total surface area of the solid = (110/7) × 47
Total surface area of the solid = 5,170/7 
Total surface area of the solid = 738.57 cm²
∴ The total surface area of the model of pencil is 738.57 cm².
Hence, option 2 is correct.

Let us assume Rs. 100 is his salary.
He spent Rs. 20 on food, Rs. 20 on rent and 30% of the remaining Rs. 60 = Rs. 18 on shopping.
So, he can save 60 – 18 = Rs. 42.
But, in a particular month, he spent 2 × 18 = Rs. 36
So, he can save 60 – 36 = Rs. 24
Percentage decrease =

Multiplying both sides of the first equation by abc, we get
bc + ac + ab = 0 … (1)
Also, (a + b + c)² = 5²
a² + b² + c² + 2(ab + bc + ac) = 25 … (2)
Substituting (1) in (2),
a² + b² + c² = 25
We have a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ac - bc) = 5(25 - 0)
⇒ a³ + b³ + c³ - 3abc = 125

Sides of the triangle are 5, 13 and 12.
ΔABC is a pythagorean triplet

∠BDA = 90°, ∠AHD = 36°, AB = 2AD
∠DBA = 30°
So, ∠DCA = 30° (because angles by the same segment at the circumference are equal)
Now, we know that triangle ABD is a right triangle.
So, ∠DAB = 60° and ∠DBA = 30°
∠TDA = ∠DBA (Angles in alternate segment)
∠DCA = 30°
∠CDH = 6° (Exterior angle = Sum of interior opposite triangles for triangle DCH)
∠CDT = 6 + 90 + 30 = 126°
Hence, 126° is the correct answer.

As the man covers 18 km while rowing against the stream in 10 hours, his speed against the stream is 1.8 km/hr.
So, statement 1 is correct.
Let the speed of rowing of man in still water be v km/hr.
Let the speed of flow of water in the stream be u km/hr.
So, 18 = 4(v + u) ... (i)
And, 18 = 10(v - u) ... (ii)
From equations (i) and (ii), we get v = 3.15 and u = 1.35
Thus, statements 2 and 3 are correct as well.
Hence, option 4 is correct.