The problem requires us to analyze a given statement and determine which of the two conclusions logically follow from it. We must assume the statement to be absolutely true and not introduce any outside information or personal opinions.
A) Only conclusion II follows. This is because the statement directly implies that real-world, hands-on experiences (playing in a garden) are superior in benefit to virtual, screen-based experiences (watching videos on devices) for small children.
To find the ratio of powers, we will use the formula for power in terms of current and resistance, which is \(P = I^2 R\). We are given the ratio of currents and the ratio of resistances for two circuits. We will express the power for each circuit and then find their ratio.
The ratio of resistances is given as \(R_1 : R_2 = 2 : 1\). This can be written as:
\[ \frac{R_1}{R_2} = \frac{2}{1} \]For the second circuit, the power \(P_2\) is:
\[ P_2 = I_2^2 R_2 \]This can be rearranged as:
\[ \frac{P_1}{P_2} = \left(\frac{I_1}{I_2}\right)^2 \times \left(\frac{R_1}{R_2}\right) \]Thus, the ratio of their powers is \(1 : 8\).
C) 1 : 8
If sec θ + tan θ = p, then find 4 sec θ
\(\frac{2(1-p^2)}{p}\)
\(\frac{2(1+p^2)}{p}\)
\(\frac{1+p^2}{2p}\)
\(\frac{1+p^2}{p}\)
The problem provides an equation involving trigonometric functions secant and tangent. We need to find the value of \(4 \sec \theta\) in terms of \(p\). The key is to use the fundamental trigonometric identity relating secant and tangent: \(\sec^2 \theta - \tan^2 \theta = 1\).
We are given the equation:
\[ \sec \theta + \tan \theta = p \quad \ldots (1) \]Recall the trigonometric identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]Factor the left side using the difference of squares formula \((a^2 - b^2 = (a-b)(a+b))\):
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]Substitute the given value from equation (1) into this factored identity:
\[ (\sec \theta - \tan \theta)(p) = 1 \]Solve for \((\sec \theta - \tan \theta)\):
\[ \sec \theta - \tan \theta = \frac{1}{p} \quad \ldots (2) \]Now we have a system of two linear equations in terms of \(\sec \theta\) and \(\tan \theta\):
Equation (1): \( \sec \theta + \tan \theta = p \)
Equation (2): \( \sec \theta - \tan \theta = \frac{1}{p} \)
Add equation (1) and equation (2) to eliminate \(\tan \theta\):
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = p + \frac{1}{p} \] \[ 2 \sec \theta = p + \frac{1}{p} \]Combine the terms on the right side:
\[ 2 \sec \theta = \frac{p^2 + 1}{p} \]The question asks for \(4 \sec \theta\). Multiply both sides by 2:
\[ 2 \times (2 \sec \theta) = 2 \times \left( \frac{p^2 + 1}{p} \right) \] \[ 4 \sec \theta = \frac{2(p^2 + 1)}{p} \]B) The derived expression for \(4 \sec \theta\) is \( \frac{2(p^2 + 1)}{p} \), which matches option B.
Factorise: y3 − xy2 − x2y + x3
(y − x)2(y + x)
(y − x)2(y - x)
(y − x)(y2 + x2)
(y + x)2(y − x)
To factorise the given expression, we will use the method of grouping terms. We will group terms that have common factors and then factor out those common factors. This process will help us identify a common binomial factor, leading to the fully factorised form.
Write down the given expression:
\[ y^3 - xy^2 - x^2y + x^3 \]Group the terms: Group the first two terms and the last two terms. It's often helpful to keep the signs with the terms.
\[ (y^3 - xy^2) + (-x^2y + x^3) \]Factor out common factors from each group:
So the expression becomes:
\[ y^2(y - x) - x^2(y - x) \]Factor out the common binomial factor: Now, \((y - x)\) is a common factor in both terms.
\[ (y - x)(y^2 - x^2) \]Apply the difference of squares formula: Recall that \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = y\) and \(b = x\).
\[ y^2 - x^2 = (y - x)(y + x) \]Substitute this back into the expression:
\[ (y - x)(y - x)(y + x) \]Simplify the expression: Combine the \((y - x)\) terms.
\[ (y - x)^2 (y + x) \]A) \((y - x)^2 (y + x)\) is the correct factorised form, as derived in the step-by-step solution.
The problem involves decoding a pattern in a given code language. We need to identify the rule applied to transform the letters of the original words into their coded forms. Once the rule is found, we apply it to the new word 'MINES' to find its coded form.
Correct Option: C) LGKAN
The problem presents a series of number pairs and asks us to find the missing number in the third pair based on the relationship observed in the first and last pairs. We need to identify a consistent mathematical operation or pattern that transforms the first number of each pair into the second number.
Correct Option: B) 64
This is a linear seating arrangement puzzle. We need to deduce the positions of the seven friends based on the given clues. We will place the friends one by one, starting with the most definitive clues, and then use relative positions to complete the arrangement.
A) Ben. As per the final arrangement (L D B F H P T), Ben is immediately to the left of Flint.
To solve this problem, we need to track Vimal's movements step-by-step, noting the distance and direction for each segment. We can visualize this as movement on a 2D coordinate plane or simply keep track of net displacement in North-South and East-West directions. Finally, we determine the displacement from Point B to Point A.
So, Point A is at \((0,0)\) and Point B is at \((13,0)\).
To reach Point A from Point B, Vimal needs to move from \((13,0)\) to \((0,0)\).
The change in position required is:
\[ \text{Change in X-coordinate} = 0 - 13 = -13 \]\[ \text{Change in Y-coordinate} = 0 - 0 = 0 \]A negative change in the X-coordinate means moving towards the West. The Y-coordinate change is 0, meaning no North-South movement is required.
Therefore, Vimal must drive 13 km towards the West to reach Point A from Point B.
B) 13 km towards West. This matches our calculated displacement from Point B to Point A.
(x+2y−3z)2
is equal to:
x2 + 4y2 + 9z2 + 4xy − 12yz − 6xz
x2 + 4y2 + 9z2 + 4xy − 12yz + 6xz
x2 + 4y2 + 9z2+ 4xy + 12yz − 6xz
x2 + 4y2 + 9z2 − 4xy − 12yz − 6xz
The problem requires us to expand the given algebraic expression \((x+2y-3z)^2\). This can be solved using the algebraic identity for the square of a trinomial: \((a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\).
Correct Option: A) \(x^2 + 4y^2 + 9z^2 + 4xy - 12yz - 6xz\)
This matches our derived expansion.
The problem presents a number series, and we need to find the next number in the sequence. We will analyze the differences between consecutive terms to identify the pattern.
Therefore, the next number in the series is 136.
B) 136 — The series follows a pattern where each subsequent number is obtained by subtracting 8 from the previous number. Applying this rule to 144 gives \(144 - 8 = 136\).